psychmeta: Psychometric meta-analysis toolkit

Description

Overview of the psychmeta package.

Details

The psychmeta package provides tools for computing bare-bones and psychometric meta-analyses and for generating psychometric data for use in meta-analysis simulations. Currently, psychmeta supports bare-bones, individual-correction, and artifact-distribution methods for meta-analyzing correlations and d values. Please refer to the overview tutorial vignette for an introduction to psychmeta's functions and workflows.

Running a meta-analysis

The main functions for conducting meta-analyses in psychmeta are ma_r for correlations and ma_d for d values. These functions take meta-analytic dataframes including effect sizes and sample sizes (and, optionally, study labels, moderators, construct and measure labels, and psychometric artifact information) and return the full results of psychometric meta-analyses for all of the specified variable pairs. Examples of correctly formatted meta-analytic datasets for ma functions are data_r_roth_2015, data_r_gonzalezmule_2014, and data_r_mcdaniel_1994. Individual parts of the meta-analysis process can also be run separately; these functions are described in detail below.

Preparing a database for meta-analysis

The convert_es function can be used to convert a variety of effect sizes to either correlations or d values. Sporadic psychometric artifacts, such as artificial dichotomization or uneven splits for a truly dichotomous variable, can be individually corrected using correct_r and correct_d. These functions can also be used to compute confidence intervals for observed, converted, and corrected effect sizes. 'Wide' meta-analytic coding sheets can be reformatted to the 'long' data frames used by psychmeta with reshape_wide2long. A correlation matrix and accompanying vectors of information can be similarly reformatted using reshape_mat2dat.

Meta-analytic models

psychmeta can compute barebones meta-analyses (no corrections for psychometric artifacts), as well as models correcting for measurement error in one or both variables, univariate direct (Case II) range restriction, univariate indirect (Case IV) range restriction, bivariate direct range restriction, bivariate indirect (Case V) range restriction, and multivariate range restriction. Artifacts can be corrected individually or using artifact distributions. Artifact distribution corrections can be applied using either Schmidt and Hunter's (2015) interactive method or Taylor series approximation models. Meta-analyses can be computed using various weights, including sample size (default for correlations), inverse variance (computed using either sample or mean effect size; error based on mean effect size is the default for d values), and weight methods imported from metafor.

Preparing artifact distributions meta-analyses

For individual-corrections meta-analyses, reliability and range restriction (u) values should be supplied in the same data frame as the effect sizes and sample sizes. Missing artifact data can be imputed using either bootstrap or other imputation methods. For artifact distribution meta-analyses, artifact distributions can be created automatically by ma_r or ma_d or manually by the create_ad family of functions.

Moderator analyses

Subgroup moderator analyses are run by supplying a moderator matrix to the ma_r or ma_d families of functions. Both simple and fully hierarchical moderation can be computed. Subgroup moderator analysis results are shown by passing an ma_obj to print(). Meta-regression analyses can be run using metareg.

Reporting results and supplemental analyses

Meta-analysis results can be viewed by passing an ma object to summary. Bootstrap confidence intervals, leave one out analyses, and other sensitivity analyses are available in sensitivity. Supplemental heterogeneity statistics (e.g., Q, I^{2}) can be computed using heterogeneity. Meta-analytic results can be converted between the r and d metrics using convert_ma. Each ma_obj contains a metafor escalc object in ma$...$escalc that can be passed to metafor's functions for plotting, publication/availability bias, and other supplemental analyses. Second-order meta-analyses of correlations can be computed using ma_r_order2. Example second-order meta-analysis datasets from Schmidt and Oh (2013) are available. Tables of meta-analytic results can be written as markdown, Word, HTML, or PDF files using the metabulate function, which exports near publication-quality tables that will typically require only minor customization by the user.

Simulating psychometric meta-analyses

psychmeta can be used to run Monte Carlo simulations for different meta-analytic models. simulate_r_sample and simulate_d_sample simulate samples of correlations and d values, respectively, with measurement error and/or range restriction artifacts. simulate_r_database and simulate_d_database can be used to simulate full meta-analytic databases of sample correlations and d values, respectively, with artifacts. Example datasets fitting different meta-analytic models simulated using these functions are available (data_r_meas, data_r_uvdrr, data_r_uvirr, data_r_bvdrr, data_r_bvirr, data_r_meas_multi, and data_d_meas_multi). Additional simulation functions are also available.

An overview of our labels and abbreviations

Throughout the package documentation, we use several sets of labels and abbreviations to refer to methodological features of variables, statistics, analyses, and functions. We define sets of key labels and abbreviations below.

Abbreviations for meta-analytic methods:

• bb: Bare-bones meta-analysis.

• ic: Individual-correction meta-analysis.

• ad: Artifact-distribution meta-analysis.

Abbreviations for types of artifact distributions and artifact-distribution meta-analyses:

• int: Interactive approach.

• tsa: Taylor series approximation approach.

Notation used for variables involved in correlations:

• x or X: Scores on the observed variable designated as X by the analyst (i.e., scores containing measurement error). By convention, X typically represents a predictor variable.

• t or T: Scores on the construct associated with X (i.e., scores free from measurement error).

• y or Y: Scores on the observed variable designated as Y by the analyst (i.e., scores containing measurement error). By convention, Y typically represents a criterion variable.

• p or P: Scores on the construct associated with Y (i.e., scores free from measurement error).

Note: The use of lowercase or uppercase labels does not alter the meaning of the notation.

Notation used for variables involved in d values:

• g: Group membership status based on the observed group membership variable (i.e., statuses containing measurement/classification error).

• G: Group membership status based on the group membership construct (i.e., statuses free from measurement/classification error).

• y or Y: Scores on the observed variable being compared between groups (i.e., scores containing measurement error).

• p or P: Scores on the criterion construct being compared between groups (i.e., scores free from measurement error).

Note: There is always a distinction between the g and G labels because they differ in case. The use of lowercase or uppercase labels for y/Y or p/P does not alter the meaning of the notation.

Notation used for types of correlations:

• rxy: Observed correlation.

• rxp: Correlation corrected for measurement error in Y only.

• rty: Correlation corrected for measurement error in X only.

• rtp: True-score correlation corrected for measurement error in both X and Y.

Note: Correlations with labels that include "i" suffixes are range-restricted, and those with "a" suffixes are unrestricted or corrected for range restriction.

Notation used for types of d values:

• dgy: Observed d value.

• dgp: d value corrected for measurement error in Y only.

• dGy: d value corrected for measurement/classification error in the grouping variable only.

• dGp: True-score d value corrected for measurement/classification error in both X and the grouping variable.

Note: d values with labels that include "i" suffixes are range-restricted, and those with "a" suffixes are unrestricted or corrected for range restriction.

Types of correction methods (excluding sporadic corrections and outdated corrections implemented for posterity):

• meas: Correction for measurement error only.

• uvdrr: Correction for univariate direct range restriction (i.e., Case II). Can be applied to using range restriction information for either X or Y.

• uvirr: Correction for univariate indirect range restriction (i.e., Case IV). Can be applied to using range restriction information for either X or Y.

• bvdrr: Correction for bivariate direct range restriction. Use with caution: This correction is an approximation only and is known to have a positive bias.

• bvirr: Correction for bivariate indirect range restriction (i.e., Case V).

Note: Meta-analyses of d values that involve range-restriction corrections treat the grouping variable as "X."

Labels for types of output from psychometric meta-analyses:

• ts: True-score meta-analysis output. Represents fully corrected estimates.

• vgx: Validity generalization meta-analysis output with X treated as the predictor. Represents estimates corrected for all artifacts except measurement error in X. Artifact distributions will still account for variance in effects explained by measurement error in X.

• vgy: Validity generalization meta-analysis output with Y treated as the predictor. Represents estimates corrected for all artifacts except measurement error in Y. Artifact distributions will still account for variance in effects explained by measurement error in Y.

Author(s)

Maintainer: Jeffrey A. Dahlke jeff.dahlke.phd@gmail.com

Authors:

• Brenton M. Wiernik brenton@psychmeta.com

Other contributors:

• Wesley Gardiner (Unit tests) [contributor]

• Michael T. Brannick (Testing) [contributor]

• Jack Kostal (Code for reshape_mat2dat function) [contributor]

• Sean Potter (Testing; Code for cumulative and leave1out plots) [contributor]

• John Sakaluk (Code for funnel and forest plots) [contributor]

• Yuejia (Mandy) Teng (Testing) [contributor]

See Also

Useful links:

• Report bugs at https://github.com/psychmeta/psychmeta/issues

Adjusted sample size for a non-Cohen d value for use in a meta-analysis of Cohen's d values

Description

This function is used to convert a non-Cohen d value (e.g., Glass' \Delta) to a Cohen's d value by identifying the sample size of a Cohen's d that has the same standard error as the non-Cohen d. This function permits users to account for the influence of sporadic corrections on the sampling variance of d prior to use in a meta-analysis.

Usage

adjust_n_d(d, var_e, p = NA)

Arguments

Details

The adjusted sample size is computed as:

n_{adjusted}=\frac{d^{2}p(1-p)+2}{2p(1-p)var_{e}}

Value

A vector of adjusted sample sizes.

References

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Sage. doi:10.4135/9781483398105. Chapter 7 (Equations 7.23 and 7.23a).

Examples

adjust_n_d(d = 1, var_e = .03, p = NA)

Adjusted sample size for a non-Pearson correlation coefficient for use in a meta-analysis of Pearson correlations

Description

This function is used to compute the adjusted sample size of a non-Pearson correlation (e.g., a tetrachoric correlation) based on the correlation and its estimated error variance. This function allows users to adjust the sample size of a correlation corrected for sporadic artifacts (e.g., unequal splits of dichotomous variables, artificial dichotomization of continuous variables) prior to use in a meta-analysis.

Usage

adjust_n_r(r, var_e)

Arguments

Details

The adjusted sample size is computed as:

n_{adjusted}=\frac{(r^{2}-1)^{2}+var_{e}}{var_{e}}

Value

A vector of adjusted sample sizes.

References

Schmidt, F. L., & Hunter, J. E. (2015). *Methods of meta-analysis: Correcting error and bias in research findings* (3rd ed.). Sage. doi:10.4135/9781483398105. Equation 3.7.

Examples

adjust_n_r(r = .3, var_e = .01)

Wald-type tests for moderators in psychmeta meta-analyses

Description

This function computes Wald-type pairwise comparisons for each level of categorical moderators for an ma_psychmeta object, as well as an ombnibus one-way ANOVA test (equal variance not assumed).

Currently, samples across moderator levels are assumed to be independent.

Usage

## S3 method for class 'ma_psychmeta'
anova(
  object,
  ...,
  analyses = "all",
  moderators = NULL,
  L = NULL,
  ma_obj2 = NULL,
  ma_method = c("bb", "ic", "ad"),
  correction_type = c("ts", "vgx", "vgy"),
  conf_level = NULL
)

Arguments

Value

An object of class anova.ma_psychmeta. A tibble with a row for each construct pair in object and a column for each moderator tested. Cells lists of contrasts tested.

Note

Currently, only simple (single) categorical moderators (one-way ANOVA) are supported.

Examples

ma_obj <- ma_r(rxyi, n, construct_x = x_name, construct_y = y_name,
moderators = moderator, data = data_r_meas_multi)

anova(ma_obj)

Matrix formula to estimate the standardized mean difference associated with a weighted or unweighted composite variable

Description

This function is a wrapper for composite_r_matrix that converts d values to correlations, computes the composite correlation implied by the d values, and transforms the result back to the d metric.

Usage

composite_d_matrix(d_vec, r_mat, wt_vec, p = 0.5)

Arguments

Details

The composite d value is computed by converting the vector of d values to correlations, computing the composite correlation (see composite_r_matrix), and transforming that composite back into the d metric. See "Details" of composite_r_matrix for the composite computations.

Value

The estimated standardized mean difference associated with the composite variable.

Examples

composite_d_matrix(d_vec = c(1, 1), r_mat = matrix(c(1, .7, .7, 1), 2, 2),
                   wt_vec = c(1, 1), p = .5)

Scalar formula to estimate the standardized mean difference associated with a composite variable

Description

This function estimates the d value of a composite of X variables, given the mean d value of the individual X values and the mean correlation among those variables.

Usage

composite_d_scalar(
  mean_d,
  mean_intercor,
  k_vars,
  p = 0.5,
  partial_intercor = FALSE
)

Arguments

Details

There are two different methods available for computing such a composite, one that uses the partial intercorrelation among the X variables (i.e., the average within-group correlation) and one that uses the overall correlation among the X variables (i.e., the total or mixture correlation across groups).

If a partial correlation is provided for the interrelationships among variables, the following formula is used to estimate the composite d value:

d_{X}=\frac{\bar{d}_{x_{i}}k}{\sqrt{\bar{\rho}_{x_{i}x_{j}}k^{2}+\left(1-\bar{\rho}_{x_{i}x_{j}}\right)k}}

where d_{X} is the composite d value, \bar{d}_{x_{i}} is the mean d value, \bar{\rho}_{x_{i}x_{j}} is the mean intercorrelation among the variables in the composite, and k is the number of variables in the composite. Otherwise, the composite d value is computed by converting the mean d value to a correlation, computing the composite correlation (see composite_r_scalar for formula), and transforming that composite back into the d metric.

Value

The estimated standardized mean difference associated with the composite variable.

References

Rosenthal, R., & Rubin, D. B. (1986). Meta-analytic procedures for combining studies with multiple effect sizes. Psychological Bulletin, 99(3), 400–406.

Examples

composite_d_scalar(mean_d = 1, mean_intercor = .7, k_vars = 2, p = .5)

Matrix formula to estimate the correlation between two weighted or unweighted composite variables

Description

This function computes the weighted (or unweighted, by default) composite correlation between a set of X variables and a set of Y variables.

Usage

composite_r_matrix(
  r_mat,
  x_col,
  y_col,
  wt_vec_x = rep(1, length(x_col)),
  wt_vec_y = rep(1, length(y_col))
)

Arguments

Details

This is computed as:

\rho_{XY}\frac{\mathbf{w}_{X}^{T}\mathbf{R}_{XY}\mathbf{w}_{Y}}{\sqrt{\left(\mathbf{w}_{X}^{T}\mathbf{R}_{XX}\mathbf{w}_{X}\right)\left(\mathbf{w}_{Y}^{T}\mathbf{R}_{YY}\mathbf{w}_{Y}\right)}}

where \rho_{XY} is the composite correlation, \mathbf{w} is a vector of weights, and \mathbf{R} is a correlation matrix. The subscripts of \mathbf{w} and \mathbf{R} indicate the variables indexed within the vector or matrix.

Value

A composite correlation

References

Mulaik, S. A. (2010). Foundations of factor analysis. Boca Raton, FL: CRC Press. pp. 83–84.

Examples

composite_r_scalar(mean_rxy = .3, k_vars_x = 4, mean_intercor_x = .4)
R <- reshape_vec2mat(.4, order = 5)
R[-1,1] <- R[1,-1] <- .3
composite_r_matrix(r_mat = R, x_col = 2:5, y_col = 1)

Scalar formula to estimate the correlation between a composite and another variable or between two composite variables

Description

This function estimates the correlation between a set of X variables and a set of Y variables using a scalar formula.

Usage

composite_r_scalar(
  mean_rxy,
  k_vars_x = NULL,
  mean_intercor_x = NULL,
  k_vars_y = NULL,
  mean_intercor_y = NULL
)

Arguments

Details

The formula to estimate a correlation between one composite variable and one external variable is:

\rho_{Xy}=\frac{\bar{\rho}_{x_{i}y}}{\sqrt{\frac{1}{k_{x}}+\frac{k_{x}-1}{k_{x}}\bar{\rho}_{x_{i}x_{j}}}}

and the formula to estimate the correlation between two composite variables is:

\rho_{XY}=\frac{\bar{\rho}_{x_{i}y_{j}}}{\sqrt{\frac{1}{k_{x}}+\frac{k-1}{k_{x}}\bar{\rho}_{x_{i}x_{j}}}\sqrt{\frac{1}{k_{y}}+\frac{k_{y}-1}{k_{y}}\bar{\rho}_{y_{i}y_{j}}}}

where \bar{\rho}_{x_{i}y} and \bar{\rho}_{x_{i}y{j}} are mean correlations between the x variables and the y variable(s), \bar{\rho}_{x_{i}x_{j}} is the mean correlation among x variables, \bar{\rho}_{y_{i}y_{j}} is the mean correlation among y variables, {k}_{x} is the number of x variables, and {k}_{y} is the number of y variables.

Value

A vector of composite correlations

References

Ghiselli, E. E., Campbell, J. P., & Zedeck, S. (1981). Measurement theory for the behavioral sciences. San Francisco, CA: Freeman. p. 163-164.

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Thousand Oaks, CA: Sage. doi:10.4135/9781483398105. pp. 441 - 447.

Examples

## Composite correlation between 4 variables and an outside variable with which
## the four variables correlate .3 on average:
composite_r_scalar(mean_rxy = .3, k_vars_x = 4, mean_intercor_x = .4)

## Correlation between two composites:
composite_r_scalar(mean_rxy = .3, k_vars_x = 2, mean_intercor_x = .5,
                   k_vars_y = 2, mean_intercor_y = .2)

Matrix formula to estimate the reliability of a weighted or unweighted composite variable

Description

This function computes the reliability of a variable that is a weighted or unweighted composite of other variables.

Usage

composite_rel_matrix(rel_vec, r_mat, sd_vec, wt_vec = rep(1, length(rel_vec)))

Arguments

Details

This function treats measure-specific variance as reliable.

The Mosier composite formula is computed as:

\rho_{XX}=\frac{\mathbf{w}^{T}\left(\mathbf{r}\circ\mathbf{s}\right)+\mathbf{w}^{T}\mathbf{S}\mathbf{w}-\mathbf{w}^{T}\mathbf{s}}{\mathbf{w}^{T}\mathbf{S}\mathbf{w}}

where \rho_{XX} is a composite reliability estimate, \mathbf{r} is a vector of reliability estimates, \mathbf{w} is a vector of weights, \mathbf{S} is a covariance matrix, and \mathbf{s} is a vector of variances (i.e., the diagonal elements of \mathbf{S}).

Value

The estimated reliability of the composite variable.

References

Mosier, C. I. (1943). On the reliability of a weighted composite. Psychometrika, 8(3), 161–168. doi:10.1007/BF02288700

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Thousand Oaks, CA: Sage. doi:10.4135/9781483398105. pp. 441 - 447.

Examples

composite_rel_matrix(rel_vec = c(.8, .8),
r_mat = matrix(c(1, .4, .4, 1), 2, 2), sd_vec = c(1, 1))

Scalar formula to estimate the reliability of a composite variable

Description

This function computes the reliability of a variable that is a unit-weighted composite of other variables.

Usage

composite_rel_scalar(mean_rel, mean_intercor, k_vars)

Arguments

Details

The Mosier composite formula is computed as:

\rho_{XX}=\frac{\bar{\rho}_{x_{i}x_{i}}k+k\left(k-1\right)\bar{\rho}_{x_{i}x_{j}}}{k+k\left(k-1\right)\bar{\rho}_{x_{i}x_{j}}}

where \bar{\rho}_{x_{i}x_{i}} is the mean reliability of variables in the composite, \bar{\rho}_{x_{i}x_{j}} is the mean intercorrelation among variables in the composite, and k is the number of variables in the composite.

Value

The estimated reliability of the composite variable.

References

Mosier, C. I. (1943). On the reliability of a weighted composite. Psychometrika, 8(3), 161–168. doi:10.1007/BF02288700

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Thousand Oaks, CA: Sage. doi:10.4135/9781483398105. pp. 441 - 447.

Examples

composite_rel_scalar(mean_rel = .8, mean_intercor = .4, k_vars = 2)

Matrix formula to estimate the u ratio of a composite variable

Description

This function estimates the u ratio of a composite variable when at least one matrix of correlations (restricted or unrestricted) among the variables is available.

Usage

composite_u_matrix(
  ri_mat = NULL,
  ra_mat = NULL,
  u_vec,
  wt_vec = rep(1, length(u_vec)),
  sign_r_vec = 1
)

Arguments

Details

This is computed as:

u_{composite}=\sqrt{\frac{\left(\mathbf{w}\circ\mathbf{u}\right)^{T}\mathbf{R}_{i}\left(\mathbf{w}\circ\mathbf{u}\right)}{\mathbf{w}^{T}\mathbf{R}_{a}\mathbf{w}}}

where u_{composite} is the composite u ratio, \mathbf{u} is a vector of u ratios, \mathbf{R}_{i} is a range-restricted correlation matrix, \mathbf{R}_{a} is an unrestricted correlation matrix, and \mathbf{w} is a vector of weights.

Value

The estimated u ratio of the composite variable.

Examples

composite_u_matrix(ri_mat = matrix(c(1, .3, .3, 1), 2, 2), u_vec = c(.8, .8))

Scalar formula to estimate the u ratio of a composite variable

Description

This function provides an approximation of the u ratio of a composite variable based on the u ratios of the component variables, the mean restricted intercorrelation among those variables, and the mean unrestricted correlation among those variables. If only one of the mean intercorrelations is known, the other will be estimated using the bivariate indirect range-restriction formula. This tends to compute a conservative estimate of the u ratio associated with a composite.

Usage

composite_u_scalar(mean_ri = NULL, mean_ra = NULL, mean_u, k_vars)

Arguments

Details

This is computed as:

u_{composite}=\sqrt{\frac{\bar{\rho}_{i}\bar{u}^{2}k(k-1)+k\bar{u}^{2}}{\bar{\rho}_{a}k(k-1)+k}}

where u_{composite} is the composite u ratio, \bar{u} is the mean univariate u ratio, \bar{\rho}_{i} is the mean restricted correlation among variables, \bar{\rho}_{a} is the mean unrestricted correlation among variables, and k is the number of variables in the composite.

Value

The estimated u ratio of the composite variable.

Examples

composite_u_scalar(mean_ri = .3, mean_ra = .4, mean_u = .8, k_vars = 2)

Compute coefficient alpha

Description

Compute coefficient alpha

Usage

compute_alpha(sigma = NULL, data = NULL, standardized = FALSE, ...)

Arguments

Value

Coefficient alpha

Examples

compute_alpha(sigma = reshape_vec2mat(cov = .4, order = 10))

Comprehensive d_{Mod} calculator

Description

This is a general-purpose function to compute d_{Mod} effect sizes from raw data and to perform bootstrapping. It subsumes the functionalities of the compute_dmod_par (for parametric computations) and compute_dmod_npar (for non-parametric computations) functions and automates the generation of regression equations and descriptive statistics for computing d_{Mod} effect sizes. Please see documentation for compute_dmod_par and compute_dmod_npar for details about how the effect sizes are computed.

Usage

compute_dmod(
  data,
  group,
  predictors,
  criterion,
  referent_id,
  focal_id_vec = NULL,
  conf_level = 0.95,
  rescale_cdf = TRUE,
  parametric = TRUE,
  bootstrap = TRUE,
  boot_iter = 1000,
  stratify = FALSE,
  empirical_ci = FALSE,
  cross_validate_wts = FALSE
)

Arguments

Value

If bootstrapping is selected, the list will include:

• point_estimate: A matrix of effect sizes (d_{Mod_{Signed}}, d_{Mod_{Unsigned}}, d_{Mod_{Under}}, d_{Mod_{Over}}), proportions of under- and over-predicted criterion scores, minimum and maximum differences, and the scores associated with minimum and maximum differences. All of these values are computed using the full data set.

• bootstrap_mean: A matrix of the same statistics as the point_estimate matrix, but the values in this matrix are the means of the results from bootstrapped samples.

• bootstrap_se: A matrix of the same statistics as the point_estimate matrix, but the values in this matrix are bootstrapped standard errors (i.e., the standard deviations of the results from bootstrapped samples).

• bootstrap_CI_Lo: A matrix of the same statistics as the point_estimate matrix, but the values in this matrix are the lower confidence bounds of the results from bootstrapped samples.

• bootstrap_CI_Hi: A matrix of the same statistics as the point_estimate matrix, but the values in this matrix are the upper confidence bounds of the results from bootstrapped samples.

If no bootstrapping is performed, the output will be limited to the point_estimate matrix.

References

Nye, C. D., & Sackett, P. R. (2017). New effect sizes for tests of categorical moderation and differential prediction. Organizational Research Methods, 20(4), 639–664. doi:10.1177/1094428116644505

Examples

# Generate some hypothetical data for a referent group and three focal groups:
set.seed(10)
refDat <- MASS::mvrnorm(n = 1000, mu = c(.5, .2),
                        Sigma = matrix(c(1, .5, .5, 1), 2, 2), empirical = TRUE)
foc1Dat <- MASS::mvrnorm(n = 1000, mu = c(-.5, -.2),
                         Sigma = matrix(c(1, .5, .5, 1), 2, 2), empirical = TRUE)
foc2Dat <- MASS::mvrnorm(n = 1000, mu = c(0, 0),
                         Sigma = matrix(c(1, .3, .3, 1), 2, 2), empirical = TRUE)
foc3Dat <- MASS::mvrnorm(n = 1000, mu = c(-.5, -.2),
                         Sigma = matrix(c(1, .3, .3, 1), 2, 2), empirical = TRUE)
colnames(refDat) <- colnames(foc1Dat) <- colnames(foc2Dat) <- colnames(foc3Dat) <- c("X", "Y")
dat <- rbind(cbind(G = 1, refDat), cbind(G = 2, foc1Dat),
             cbind(G = 3, foc2Dat), cbind(G = 4, foc3Dat))

# Compute point estimates of parametric d_mod effect sizes:
compute_dmod(data = dat, group = "G", predictors = "X", criterion = "Y",
     referent_id = 1, focal_id_vec = 2:4,
     conf_level = .95, rescale_cdf = TRUE, parametric = TRUE,
     bootstrap = FALSE)

# Compute point estimates of non-parametric d_mod effect sizes:
compute_dmod(data = dat, group = "G", predictors = "X", criterion = "Y",
     referent_id = 1, focal_id_vec = 2:4,
     conf_level = .95, rescale_cdf = TRUE, parametric = FALSE,
     bootstrap = FALSE)

# Compute unstratified bootstrapped estimates of parametric d_mod effect sizes:
compute_dmod(data = dat, group = "G", predictors = "X", criterion = "Y",
     referent_id = 1, focal_id_vec = 2:4,
     conf_level = .95, rescale_cdf = TRUE, parametric = TRUE,
     boot_iter = 10, bootstrap = TRUE, stratify = FALSE, empirical_ci = FALSE)

# Compute unstratified bootstrapped estimates of non-parametric d_mod effect sizes:
compute_dmod(data = dat, group = "G", predictors = "X", criterion = "Y",
     referent_id = 1, focal_id_vec = 2:4,
     conf_level = .95, rescale_cdf = TRUE, parametric = FALSE,
     boot_iter = 10, bootstrap = TRUE, stratify = FALSE, empirical_ci = FALSE)

Function for computing non-parametric d_{Mod} effect sizes for a single focal group

Description

This function computes non-parametric d_{Mod} effect sizes from user-defined descriptive statistics and regression coefficients, using a distribution of observed scores as weights. This non-parametric function is best used when the assumption of normally distributed predictor scores is not reasonable and/or the distribution of scores observed in a sample is likely to represent the distribution of scores in the population of interest. If one has access to the full raw data set, the dMod function may be used as a wrapper to this function so that the regression equations and descriptive statistics can be computed automatically within the program.

Usage

compute_dmod_npar(
  referent_int,
  referent_slope,
  focal_int,
  focal_slope,
  focal_x,
  referent_sd_y
)

Arguments

Details

The d_{Mod_{Signed}} effect size (i.e., the average of differences in prediction over the range of predictor scores) is computed as

d_{Mod_{Signed}}=\frac{\sum_{i=1}^{m}n_{i}\left[X_{i}\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right]}{SD_{Y_{1}}\sum_{i=1}^{m}n_{i}},

where

• SD_{Y_{1}} is the referent group's criterion standard deviation;

• m is the number of unique scores in the distribution of focal-group predictor scores;

• X is the vector of unique focal-group predictor scores, indexed i=1 through m;

• X_{i} is the i^{th} unique score value;

• n is the vector of frequencies associated with the elements of X;

• n_{i} is the number of cases with a score equal to X_{i};

• b_{1_{1}} and b_{1_{2}} are the slopes of the regression of Y on X for the referent and focal groups, respectively; and

• b_{0_{1}} and b_{0_{2}} are the intercepts of the regression of Y on X for the referent and focal groups, respectively.

The d_{Mod_{Under}} and d_{Mod_{Over}} effect sizes are computed using the same equation as d_{Mod_{Signed}}, but d_{Mod_{Under}} is the weighted average of all scores in the area of underprediction (i.e., the differences in prediction with negative signs) and d_{Mod_{Over}} is the weighted average of all scores in the area of overprediction (i.e., the differences in prediction with negative signs).

The d_{Mod_{Unsigned}} effect size (i.e., the average of absolute differences in prediction over the range of predictor scores) is computed as

d_{Mod_{Unsigned}}=\frac{\sum_{i=1}^{m}n_{i}\left|X_{i}\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right|}{SD_{Y_{1}}\sum_{i=1}^{m}n_{i}}.

The d_{Min} effect size (i.e., the smallest absolute difference in prediction observed over the range of predictor scores) is computed as

d_{Min}=\frac{1}{SD_{Y_{1}}}Min\left[\left|X\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right|\right].

The d_{Max} effect size (i.e., the largest absolute difference in prediction observed over the range of predictor scores)is computed as

d_{Max}=\frac{1}{SD_{Y_{1}}}Max\left[\left|X\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right|\right].

Note: When d_{Min} and d_{Max} are computed in this package, the output will display the signs of the differences (rather than the absolute values of the differences) to aid in interpretation.

Value

A vector of effect sizes (d_{Mod_{Signed}}, d_{Mod_{Unsigned}}, d_{Mod_{Under}}, d_{Mod_{Over}}), proportions of under- and over-predicted criterion scores, minimum and maximum differences (i.e., d_{Mod_{Under}} and d_{Mod_{Over}}), and the scores associated with minimum and maximum differences.

Examples

# Generate some hypothetical data for a referent group and three focal groups:
set.seed(10)
refDat <- MASS::mvrnorm(n = 1000, mu = c(.5, .2),
                        Sigma = matrix(c(1, .5, .5, 1), 2, 2), empirical = TRUE)
foc1Dat <- MASS::mvrnorm(n = 1000, mu = c(-.5, -.2),
                         Sigma = matrix(c(1, .5, .5, 1), 2, 2), empirical = TRUE)
foc2Dat <- MASS::mvrnorm(n = 1000, mu = c(0, 0),
                         Sigma = matrix(c(1, .3, .3, 1), 2, 2), empirical = TRUE)
foc3Dat <- MASS::mvrnorm(n = 1000, mu = c(-.5, -.2),
                         Sigma = matrix(c(1, .3, .3, 1), 2, 2), empirical = TRUE)
colnames(refDat) <- colnames(foc1Dat) <- colnames(foc2Dat) <- colnames(foc3Dat) <- c("X", "Y")

# Compute a regression model for each group:
refRegMod <- lm(Y ~ X, data.frame(refDat))$coef
foc1RegMod <- lm(Y ~ X, data.frame(foc1Dat))$coef
foc2RegMod <- lm(Y ~ X, data.frame(foc2Dat))$coef
foc3RegMod <- lm(Y ~ X, data.frame(foc3Dat))$coef

# Use the subgroup regression models to compute d_mod for each referent-focal pairing:

# Focal group #1:
compute_dmod_npar(referent_int = refRegMod[1], referent_slope = refRegMod[2],
             focal_int = foc1RegMod[1], focal_slope = foc1RegMod[2],
             focal_x = foc1Dat[,"X"], referent_sd_y = 1)

# Focal group #2:
compute_dmod_npar(referent_int = refRegMod[1], referent_slope = refRegMod[2],
             focal_int = foc2RegMod[1], focal_slope = foc1RegMod[2],
             focal_x = foc2Dat[,"X"], referent_sd_y = 1)

# Focal group #3:
compute_dmod_npar(referent_int = refRegMod[1], referent_slope = refRegMod[2],
             focal_int = foc3RegMod[1], focal_slope = foc3RegMod[2],
             focal_x = foc3Dat[,"X"], referent_sd_y = 1)

Function for computing parametric d_{Mod} effect sizes for any number of focal groups

Description

This function computes d_{Mod} effect sizes from user-defined descriptive statistics and regression coefficients. If one has access to a raw data set, the dMod function may be used as a wrapper to this function so that the regression equations and descriptive statistics can be computed automatically within the program.

Usage

compute_dmod_par(
  referent_int,
  referent_slope,
  focal_int,
  focal_slope,
  focal_mean_x,
  focal_sd_x,
  referent_sd_y,
  focal_min_x,
  focal_max_x,
  focal_names = NULL,
  rescale_cdf = TRUE
)

Arguments

Details

The d_{Mod_{Signed}} effect size (i.e., the average of differences in prediction over the range of predictor scores) is computed as

d_{Mod_{Signed}}=\frac{1}{SD_{Y_{1}}}\intop f_{2}(X)\left[X\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right] dX,

where

• SD_{Y_{1}} is the referent group's criterion standard deviation;

• f_{2}(X) is the normal-density function for the distribution of focal-group predictor scores;

• b_{1_{1}} and b_{1_{0}} are the slopes of the regression of Y on X for the referent and focal groups, respectively;

• b_{0_{1}} and b_{0_{0}} are the intercepts of the regression of Y on X for the referent and focal groups, respectively; and

• the integral spans all X scores within the operational range of predictor scores for the focal group.

The d_{Mod_{Under}} and d_{Mod_{Over}} effect sizes are computed using the same equation as d_{Mod_{Signed}}, but d_{Mod_{Under}} is the weighted average of all scores in the area of underprediction (i.e., the differences in prediction with negative signs) and d_{Mod_{Over}} is the weighted average of all scores in the area of overprediction (i.e., the differences in prediction with negative signs).

The d_{Mod_{Unsigned}} effect size (i.e., the average of absolute differences in prediction over the range of predictor scores) is computed as

d_{Mod_{Unsigned}}=\frac{1}{SD_{Y_{1}}}\intop f_{2}(X)\left|X\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right|dX.

The d_{Min} effect size (i.e., the smallest absolute difference in prediction observed over the range of predictor scores) is computed as

d_{Min}=\frac{1}{SD_{Y_{1}}}Min\left[\left|X\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right|\right].

The d_{Max} effect size (i.e., the largest absolute difference in prediction observed over the range of predictor scores)is computed as

d_{Max}=\frac{1}{SD_{Y_{1}}}Max\left[\left|X\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right|\right].

Note: When d_{Min} and d_{Max} are computed in this package, the output will display the signs of the differences (rather than the absolute values of the differences) to aid in interpretation.

If d_{Mod} effect sizes are to be rescaled to compensate for a cumulative density less than 1 (see the rescale_cdf argument), the result of each effect size involving integration will be divided by the ratio of the cumulative density of the observed range of scores (i.e., the range bounded by the focal_min_x and focal_max_x arguments) to the cumulative density of scores bounded by -Inf and Inf.

Value

A matrix of effect sizes (d_{Mod_{Signed}}, d_{Mod_{Unsigned}}, d_{Mod_{Under}}, d_{Mod_{Over}}), proportions of under- and over-predicted criterion scores, minimum and maximum differences (i.e., d_{Mod_{Under}} and d_{Mod_{Over}}), and the scores associated with minimum and maximum differences. Note that if the regression lines are parallel and infinite focal_min_x and focal_max_x values were specified, the extrema will be defined using the scores 3 focal-group SDs above and below the corresponding focal-group means.

References

Nye, C. D., & Sackett, P. R. (2017). New effect sizes for tests of categorical moderation and differential prediction. Organizational Research Methods, 20(4), 639–664. doi:10.1177/1094428116644505

Examples

compute_dmod_par(referent_int = -.05, referent_slope = .5,
                 focal_int = c(.05, 0, -.05), focal_slope = c(.5, .3, .3),
                 focal_mean_x = c(-.5, 0, -.5), focal_sd_x = rep(1, 3),
                 referent_sd_y = 1,
                 focal_min_x = rep(-Inf, 3), focal_max_x = rep(Inf, 3),
                 focal_names = NULL, rescale_cdf = TRUE)

Confidence limits for noncentral chi square parameters (function and documentation from package 'MBESS' version 4.4.3) Function to determine the noncentral parameter that leads to the observed Chi.Square-value, so that a confidence interval for the population noncentral chi-square value can be formed.

Description

Confidence limits for noncentral chi square parameters (function and documentation from package 'MBESS' version 4.4.3) Function to determine the noncentral parameter that leads to the observed Chi.Square-value, so that a confidence interval for the population noncentral chi-square value can be formed.

Usage

conf.limits.nc.chisq(
  Chi.Square = NULL,
  conf.level = 0.95,
  df = NULL,
  alpha.lower = NULL,
  alpha.upper = NULL,
  tol = 1e-09,
  Jumping.Prop = 0.1
)

Arguments

Details

If the function fails (or if a function relying upon this function fails), adjust the Jumping.Prop (to a smaller value).

Value

• Lower.Limit: Value of the distribution with Lower.Limit noncentral value that has at its specified quantile Chi.Square

• Prob.Less.Lower: Proportion of cases falling below Lower.Limit

• Upper.Limit: Value of the distribution with Upper.Limit noncentral value that has at its specified quantile Chi.Square

• Prob.Greater.Upper: Proportion of cases falling above Upper.Limit

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.edu), Keke Lai (University of California–Merced)

Construct a confidence interval

Description

Function to construct a confidence interval around an effect size or mean effect size.

Usage

confidence(
  mean,
  se = NULL,
  df = NULL,
  conf_level = 0.95,
  conf_method = c("t", "norm"),
  ...
)

Arguments

Details

CI=mean_{es}\pm quantile\times SE_{es}

Value

A matrix of confidence intervals of the specified width.

Examples

confidence(mean = c(.3, .5), se = c(.15, .2), df = c(100, 200), conf_level = .95, conf_method = "t")
confidence(mean = c(.3, .5), se = c(.15, .2), conf_level = .95, conf_method = "norm")

Construct a confidence interval for correlations using Fisher's z transformation

Description

Construct a confidence interval for correlations using Fisher's z transformation

Usage

confidence_r(r, n, conf_level = 0.95)

Arguments

Value

A confidence interval of the specified width (or matrix of confidence intervals)

Examples

confidence_r(r = .3, n = 200, conf_level = .95)

Confidence interval method for objects of classes deriving from lm_mat

Description

Confidence interval method for objects of classes deriving from lm_mat Returns lower and upper bounds of confidence intervals for regression coefficients.

Arguments

Control function to curate intercorrelations to be used in automatic compositing routine

Description

Control function to curate intercorrelations to be used in automatic compositing routine

Usage

control_intercor(
  rxyi = NULL,
  n = NULL,
  sample_id = NULL,
  construct_x = NULL,
  construct_y = NULL,
  construct_names = NULL,
  facet_x = NULL,
  facet_y = NULL,
  intercor_vec = NULL,
  intercor_scalar = 0.5,
  dx = NULL,
  dy = NULL,
  p = 0.5,
  partial_intercor = FALSE,
  data = NULL,
  ...
)

Arguments

Value

A vector of intercorrelations

Examples

## Create a dataset in which constructs correlate with themselves
rxyi <- seq(.1, .5, length.out = 27)
construct_x <- rep(rep(c("X", "Y", "Z"), 3), 3)
construct_y <- c(rep("X", 9), rep("Y", 9), rep("Z", 9))
dat <- data.frame(rxyi = rxyi, 
                  construct_x = construct_x, 
                  construct_y = construct_y, 
                  stringsAsFactors = FALSE)
dat <- rbind(cbind(sample_id = "Sample 1", dat), 
             cbind(sample_id = "Sample 2", dat), 
             cbind(sample_id = "Sample 3", dat))

## Identify some constructs for which intercorrelations are not 
## represented in the data object:
construct_names = c("U", "V", "W")

## Specify some externally determined intercorrelations among measures:
intercor_vec <- c(W = .4, X = .1)

## Specify a generic scalar intercorrelation that can stand in for missing values:
intercor_scalar <- .5

control_intercor(rxyi = rxyi, sample_id = sample_id, 
                 construct_x = construct_x, construct_y = construct_y, 
                 construct_names = construct_names, 
                 intercor_vec = intercor_vec, intercor_scalar = intercor_scalar, data = dat)

Control for psychmeta meta-analyses

Description

Control for psychmeta meta-analyses

Usage

control_psychmeta(
  error_type = c("mean", "sample"),
  conf_level = 0.95,
  cred_level = 0.8,
  conf_method = c("t", "norm"),
  cred_method = c("t", "norm"),
  var_unbiased = TRUE,
  pairwise_ads = FALSE,
  moderated_ads = FALSE,
  residual_ads = TRUE,
  check_dependence = TRUE,
  collapse_method = c("composite", "average", "stop"),
  intercor = control_intercor(),
  clean_artifacts = TRUE,
  impute_artifacts = TRUE,
  impute_method = c("bootstrap_mod", "bootstrap_full", "simulate_mod", "simulate_full",
    "wt_mean_mod", "wt_mean_full", "unwt_mean_mod", "unwt_mean_full", "replace_unity",
    "stop"),
  seed = 42,
  use_all_arts = TRUE,
  estimate_pa = FALSE,
  decimals = 2,
  hs_override = FALSE,
  zero_substitute = .Machine$double.eps,
  ...
)

Arguments

Value

A list of control arguments in the package environment.

Examples

control_psychmeta()

Convert logical variable indicating whether a reliability value is an internal-consistency estimate or a multiple-administration estimate to a string variable of generic reliability types

Description

Convert logical variable indicating whether a reliability value is an internal-consistency estimate or a multiple-administration estimate to a string variable of generic reliability types

Usage

convert_consistency2reltype(consistency)

Arguments

Value

String variable of generic reliability types.

Convert effect sizes

Description

This function converts a variety of effect sizes to correlations, Cohen's d values, or common language effect sizes, and calculates sampling error variances and effective sample sizes.

Usage

convert_es(
  es,
  input_es = c("r", "d", "delta", "g", "t", "p.t", "F", "p.F", "chisq", "p.chisq", "or",
    "lor", "Fisherz", "A", "auc", "cles"),
  output_es = c("r", "d", "A", "auc", "cles"),
  n1 = NULL,
  n2 = NULL,
  df1 = NULL,
  df2 = NULL,
  sd1 = NULL,
  sd2 = NULL,
  tails = 2
)

Arguments

Value

A data frame of class es with variables:

References

Chinn, S. (2000). A simple method for converting an odds ratio to effect size for use in meta-analysis. Statistics in Medicine, 19(22), 3127–3131. doi:10.1002/1097-0258(20001130)19:22<3127::AID-SIM784>3.0.CO;2-M

Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Sage.

Ruscio, J. (2008). A probability-based measure of effect size: Robustness to base rates and other factors. Psychological Methods, 13(1), 19–30. doi:10.1037/1082-989X.13.1.19

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Sage. doi:10.4135/9781483398105

Examples

convert_es(es = 1,  input_es="d", output_es="r", n1=100)
convert_es(es = 1, input_es="d", output_es="r", n1=50, n2 = 50)
convert_es(es = .2, input_es="r", output_es="d",  n1=100, n2=150)
convert_es(es = -1.3, input_es="t", output_es="r", n1 = 100, n2 = 140)
convert_es(es = 10.3, input_es="F", output_es="d", n1 = 100, n2 = 150)
convert_es(es = 1.3, input_es="chisq", output_es="r", n1 = 100, n2 = 100)
convert_es(es = .021, input_es="p.chisq", output_es="d", n1 = 100, n2 = 100)
convert_es(es = 4.37, input_es="or", output_es="r", n1=100, n2=100)
convert_es(es = 4.37, input_es="or", output_es="d", n1=100, n2=100)
convert_es(es = 1.47, input_es="lor", output_es="r", n1=100, n2=100)
convert_es(es = 1.47, input_es="lor", output_es="d", n1=100, n2=100)

Function to convert meta-analysis of correlations to d values or vice-versa

Description

Takes a meta-analysis class object of d values or correlations (classes r_as_r, d_as_d, r_as_d, and d_as_r; second-order meta-analyses are currently not supported) as an input and uses conversion formulas and Taylor series approximations to convert effect sizes and variance estimates, respectively.

Usage

convert_ma(ma_obj, ...)

convert_meta(ma_obj, ...)

Arguments

Details

The formula used to convert correlations to d values is:

d=\frac{r\sqrt{\frac{1}{p\left(1-p\right)}}}{\sqrt{1-r^{2}}}

The formula used to convert d values to correlations is:

r=\frac{d}{\sqrt{d^{2}+\frac{1}{p\left(1-p\right)}}}

To approximate the variance of correlations from the variance of d values, the function computes:

var_{r}\approx a_{d}^{2}var_{d}

where a_{d} is the first partial derivative of the d-to-r transformation with respect to d:

a_{d}=-\frac{1}{\left[d^{2}p\left(1-p\right)-1\right]\sqrt{d^{2}+\frac{1}{p-p^{2}}}}

To approximate the variance of d values from the variance of correlations, the function computes:

var_{d}\approx a_{r}^{2}var_{r}

where a_{r} is the first partial derivative of the r-to-d transformation with respect to r:

a_{r}=\frac{\sqrt{\frac{1}{p-p^{2}}}}{\left(1-r^{2}\right)^{1.5}}

Value

A meta-analysis converted to the d value metric (if ma_obj was a meta-analysis in the correlation metric) or converted to the correlation metric (if ma_obj was a meta-analysis in the d value metric).

Correct d values for measurement error and/or range restriction

Description

This function is a wrapper for the correct_r() function to correct d values for statistical and psychometric artifacts.

Usage

correct_d(
  correction = c("meas", "uvdrr_g", "uvdrr_y", "uvirr_g", "uvirr_y", "bvdrr", "bvirr"),
  d,
  ryy = 1,
  uy = 1,
  rGg = 1,
  pi = NULL,
  pa = NULL,
  uy_observed = TRUE,
  ryy_restricted = TRUE,
  ryy_type = "alpha",
  k_items_y = NA,
  sign_rgz = 1,
  sign_ryz = 1,
  n1 = NULL,
  n2 = NA,
  conf_level = 0.95,
  correct_bias = FALSE
)

Arguments

Value

Data frame(s) of observed d values (dgyi), range-restricted d values corrected for measurement error in Y only (dgpi), range-restricted d values corrected for measurement error in the grouping variable only (dGyi), and range-restricted true-score d values (dGpi), range-corrected observed-score d values (dgya), range-corrected d values corrected for measurement error in Y only (dgpa), range-corrected d values corrected for measurement error in the grouping variable only (dGya), and range-corrected true-score d values (dGpa).

References

Alexander, R. A., Carson, K. P., Alliger, G. M., & Carr, L. (1987). Correcting doubly truncated correlations: An improved approximation for correcting the bivariate normal correlation when truncation has occurred on both variables. Educational and Psychological Measurement, 47(2), 309–315. doi:10.1177/0013164487472002

Dahlke, J. A., & Wiernik, B. M. (2020). Not restricted to selection research: Accounting for indirect range restriction in organizational research. Organizational Research Methods, 23(4), 717–749. doi:10.1177/1094428119859398

Hunter, J. E., Schmidt, F. L., & Le, H. (2006). Implications of direct and indirect range restriction for meta-analysis methods and findings. Journal of Applied Psychology, 91(3), 594–612. doi:10.1037/0021-9010.91.3.594

Le, H., Oh, I.-S., Schmidt, F. L., & Wooldridge, C. D. (2016). Correction for range restriction in meta-analysis revisited: Improvements and implications for organizational research. Personnel Psychology, 69(4), 975–1008. doi:10.1111/peps.12122

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Sage. doi:10.4135/9781483398105. pp. 43–44, 140–141.

Examples

## Correction for measurement error only
correct_d(correction = "meas", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = .7, pa = .5)
correct_d(correction = "meas", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = NULL, pa = .5, n1 = 100, n2 = 200)

## Correction for direct range restriction in the continuous variable
correct_d(correction = "uvdrr_y", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = .7, pa = .5)
correct_d(correction = "uvdrr_y", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = NULL, pa = .5, n1 = 100, n2 = 200)

## Correction for direct range restriction in the grouping variable
correct_d(correction = "uvdrr_g", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = .7, pa = .5)
correct_d(correction = "uvdrr_g", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = NULL, pa = .5, n1 = 100, n2 = 200)

## Correction for indirect range restriction in the continuous variable
correct_d(correction = "uvdrr_y", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = .7, pa = .5)
correct_d(correction = "uvdrr_y", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = NULL, pa = .5, n1 = 100, n2 = 200)

## Correction for indirect range restriction in the grouping variable
correct_d(correction = "uvirr_g", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = .7, pa = .5)
correct_d(correction = "uvirr_g", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = NULL, pa = .5, n1 = 100, n2 = 200)

## Correction for indirect range restriction in the continuous variable
correct_d(correction = "uvdrr_y", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = .7, pa = .5)
correct_d(correction = "uvdrr_y", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = NULL, pa = .5, n1 = 100, n2 = 200)

## Correction for direct range restriction in both variables
correct_d(correction = "bvdrr", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = .7, pa = .5)
correct_d(correction = "bvdrr", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = NULL, pa = .5, n1 = 100, n2 = 200)

## Correction for indirect range restriction in both variables
correct_d(correction = "bvirr", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = .7, pa = .5)
correct_d(correction = "bvirr", d = .5, ryy = .8, uy = .7,
          rGg = .9, pi = NULL, pa = .5, n1 = 100, n2 = 200)

Correct for small-sample bias in Cohen's d values

Description

Corrects a vector of Cohen's d values for small-sample bias, as Cohen's d has a slight positive bias. The bias-corrected d value is often called Hedges's g.

Usage

correct_d_bias(d, n)

Arguments

Details

The bias correction is:

g = d_{c} = d_{obs} \times J

where

J = \frac{\Gamma(\frac{n - 2}{2})}{\sqrt{\frac{n - 2}{2}} \times \Gamma(\frac{n - 3}{2})}

and d_{obs} is the observed effect size, g = d_{c} is the corrected (unbiased) estimate, n is the total sample size, and \Gamma() is the gamma function.

Historically, using the gamma function was computationally intensive, so an approximation for J was used (Borenstein et al., 2009):

J = 1 - 3 / (4 * (n - 2) - 1)

This approximation is no longer necessary with modern computers.

Value

Vector of g values (d values corrected for small-sample bias).

References

Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis. Academic Press. p. 104

Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Introduction to meta-analysis. Wiley. p. 27.

Examples

correct_d_bias(d = .3, n = 30)
correct_d_bias(d = .3, n = 300)
correct_d_bias(d = .3, n = 3000)

Correct for small-sample bias in Glass' \Delta values

Description

Correct for small-sample bias in Glass' \Delta values.

Usage

correct_glass_bias(delta, nc, ne, use_pooled_sd = rep(FALSE, length(delta)))

Arguments

Details

The bias correction is estimated as:

\Delta_{c}=\Delta_{obs}\frac{\Gamma\left(\frac{n_{control}-1}{2}\right)}{\Gamma\left(\frac{n_{control}-1}{2}\right)\Gamma\left(\frac{n_{control}-2}{2}\right)}

where \Delta is the observed effect size, \Delta_{c} is the corrected estimate of \Delta, n_{control} is the control-group sample size, and \Gamma() is the gamma function.

Value

Vector of d values corrected for small-sample bias.

References

Hedges, L. V. (1981). Distribution theory for Glass’s estimator of effect size and related estimators. Journal of Educational Statistics, 6(2), 107–128. doi:10.2307/1164588

Examples

correct_glass_bias(delta = .3, nc = 30, ne = 30)

Multivariate select/correction for covariance matrices

Description

Correct (or select upon) a covariance matrix using the Pearson-Aitken-Lawley multivariate selection theorem.

Usage

correct_matrix_mvrr(
  Sigma_i,
  Sigma_xx_a,
  x_col,
  y_col = NULL,
  standardize = FALSE,
  var_names = NULL
)

Arguments

Value

A matrix that has been manipulated by the multivariate range-restriction formula.

References

Aitken, A. C. (1934). Note on selection from a multivariate normal population. Proceedings of the Edinburgh Mathematical Society (Series 2), 4(2), 106–110.

Lawley, D. N. (1943). A note on Karl Pearson’s selection formulae. Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences, 62(1), 28–30.

Examples

Sigma_i <- reshape_vec2mat(cov = .2, var = .8, order = 4)
Sigma_xx_a <- reshape_vec2mat(cov = .5, order = 2)
correct_matrix_mvrr(Sigma_i = Sigma_i, Sigma_xx_a = Sigma_xx_a, x_col = 1:2, standardize = TRUE)

Multivariate select/correction for vectors of means

Description

Correct (or select upon) a vector of means using principles from the Pearson-Aitken-Lawley multivariate selection theorem.

Usage

correct_means_mvrr(
  Sigma,
  means_i = rep(0, ncol(Sigma)),
  means_x_a,
  x_col,
  y_col = NULL,
  var_names = NULL
)

Arguments

Value

A vector of means that has been manipulated by the multivariate range-restriction formula.

References

Aitken, A. C. (1934). Note on selection from a multivariate normal population. Proceedings of the Edinburgh Mathematical Society (Series 2), 4(2), 106–110.

Lawley, D. N. (1943). A note on Karl Pearson’s selection formulae. Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences, 62(1), 28–30.

Examples

Sigma <- diag(.5, 4)
Sigma[lower.tri(Sigma)] <- c(.2, .3, .4, .3, .4, .4)
Sigma <- Sigma + t(Sigma)
diag(Sigma) <- 1
correct_means_mvrr(Sigma = Sigma, means_i = c(.3, .3, .1, .1),
means_x_a = c(-.1, -.1), x_col = 1:2)

Correct correlations for range restriction and/or measurement error

Description

Corrects Pearson correlations (r) for range restriction and/or measurement error

Usage

correct_r(
  correction = c("meas", "uvdrr_x", "uvdrr_y", "uvirr_x", "uvirr_y", "bvdrr", "bvirr"),
  rxyi,
  ux = 1,
  uy = 1,
  rxx = 1,
  ryy = 1,
  ux_observed = TRUE,
  uy_observed = TRUE,
  rxx_restricted = TRUE,
  rxx_type = "alpha",
  k_items_x = NA,
  ryy_restricted = TRUE,
  ryy_type = "alpha",
  k_items_y = NA,
  sign_rxz = 1,
  sign_ryz = 1,
  n = NULL,
  conf_level = 0.95,
  correct_bias = FALSE,
  zero_substitute = .Machine$double.eps
)

Arguments

Details

The correction for measurement error is:

\rho_{TP}=\frac{\rho_{XY}}{\sqrt{\rho_{XX}\rho_{YY}}}

The correction for univariate direct range restriction is:

\rho_{TP_{a}}=\left[\frac{\rho_{XY_{i}}}{u_{X}\sqrt{\rho_{YY_{i}}}\sqrt{\left(\frac{1}{u_{X}^{2}}-1\right)\frac{\rho_{XY_{i}}^{2}}{\rho_{YY_{i}}}+1}}\right]/\sqrt{\rho_{XX_{a}}}

The correction for univariate indirect range restriction is:

\rho_{TP_{a}}=\frac{\rho_{XY_{i}}}{u_{T}\sqrt{\rho_{XX_{i}}\rho_{YY_{i}}}\sqrt{\left(\frac{1}{u_{T}^{2}}-1\right)\frac{\rho_{XY_{i}}^{2}}{\rho_{XX_{i}}\rho_{YY_{i}}}+1}}

The correction for bivariate direct range restriction is:

\rho_{TP_{a}}=\frac{\frac{\rho_{XY_{i}}^{2}-1}{2\rho_{XY_{i}}}u_{X}u_{Y}+\mathrm{sign}\left(\rho_{XY_{i}}\right)\sqrt{\frac{\left(1-\rho_{XY_{i}}^{2}\right)^{2}}{4\rho_{XY_{i}}}u_{X}^{2}u_{Y}^{2}+1}}{\sqrt{\rho_{XX_{a}}\rho_{YY_{a}}}}

The correction for bivariate indirect range restriction is:

\rho_{TP_{a}}=\frac{\rho_{XY_{i}}u_{X}u_{Y}+\lambda\sqrt{\left|1-u_{X}^{2}\right|\left|1-u_{Y}^{2}\right|}}{\sqrt{\rho_{XX_{a}}\rho_{YY_{a}}}}

where the \lambda value allows u_{X} and u_{Y} to fall on either side of unity so as to function as a two-stage correction for mixed patterns of range restriction and range enhancement. The \lambda value is computed as:

\lambda=\mathrm{sign}\left[\rho_{ST_{a}}\rho_{SP_{a}}\left(1-u_{X}\right)\left(1-u_{Y}\right)\right]\frac{\mathrm{sign}\left(1-u_{X}\right)\min\left(u_{X},\frac{1}{u_{X}}\right)+\mathrm{sign}\left(1-u_{Y}\right)\min\left(u_{Y},\frac{1}{u_{Y}}\right)}{\min\left(u_{X},\frac{1}{u_{X}}\right)\min\left(u_{Y},\frac{1}{u_{Y}}\right)}

Value

Data frame(s) of observed correlations (rxyi), range-restricted correlations corrected for measurement error in Y only (rxpi), range-restricted correlations corrected for measurement error in X only (rtyi), and range-restricted true-score correlations (rtpi), range-corrected observed-score correlations (rxya), range-corrected correlations corrected for measurement error in Y only (rxpa), range-corrected correlations corrected for measurement error in X only (rtya), and range-corrected true-score correlations (rtpa).

References

Alexander, R. A., Carson, K. P., Alliger, G. M., & Carr, L. (1987). Correcting doubly truncated correlations: An improved approximation for correcting the bivariate normal correlation when truncation has occurred on both variables. Educational and Psychological Measurement, 47(2), 309–315. doi:10.1177/0013164487472002

Dahlke, J. A., & Wiernik, B. M. (2020). Not restricted to selection research: Accounting for indirect range restriction in organizational research. Organizational Research Methods, 23(4), 717–749. doi:10.1177/1094428119859398

Hunter, J. E., Schmidt, F. L., & Le, H. (2006). Implications of direct and indirect range restriction for meta-analysis methods and findings. Journal of Applied Psychology, 91(3), 594–612. doi:10.1037/0021-9010.91.3.594

Le, H., Oh, I.-S., Schmidt, F. L., & Wooldridge, C. D. (2016). Correction for range restriction in meta-analysis revisited: Improvements and implications for organizational research. Personnel Psychology, 69(4), 975–1008. doi:10.1111/peps.12122

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Sage. doi:10.4135/9781483398105. pp. 43-44, 140–141.

Examples

## Correction for measurement error only
correct_r(correction = "meas", rxyi = .3, rxx = .8, ryy = .8,
     ux_observed = TRUE, uy_observed = TRUE, rxx_restricted = TRUE, ryy_restricted = TRUE)
correct_r(correction = "meas", rxyi = .3, rxx = .8, ryy = .8,
     ux_observed = TRUE, uy_observed = TRUE, rxx_restricted = TRUE, ryy_restricted = TRUE, n = 100)

## Correction for direct range restriction in X
correct_r(correction = "uvdrr_x", rxyi = .3, ux = .8, rxx = .8, ryy = .8,
     ux_observed = TRUE, uy_observed = TRUE, rxx_restricted = TRUE, ryy_restricted = TRUE)
correct_r(correction = "uvdrr_x", rxyi = .3, ux = .8, rxx = .8, ryy = .8,
     ux_observed = TRUE, uy_observed = TRUE, rxx_restricted = TRUE, ryy_restricted = TRUE, n = 100)

## Correction for indirect range restriction in X
correct_r(correction = "uvirr_x", rxyi = .3, ux = .8, rxx = .8, ryy = .8,
     ux_observed = TRUE, uy_observed = TRUE, rxx_restricted = TRUE, ryy_restricted = TRUE)
correct_r(correction = "uvirr_x", rxyi = .3, ux = .8, rxx = .8, ryy = .8,
     ux_observed = TRUE, uy_observed = TRUE, rxx_restricted = TRUE, ryy_restricted = TRUE, n = 100)

## Correction for direct range restriction in X and Y
correct_r(correction = "bvdrr", rxyi = .3, ux = .8, uy = .8, rxx = .8, ryy = .8,
     ux_observed = TRUE, uy_observed = TRUE, rxx_restricted = TRUE, ryy_restricted = TRUE)
correct_r(correction = "bvdrr", rxyi = .3, ux = .8, uy = .8, rxx = .8, ryy = .8,
     ux_observed = TRUE, uy_observed = TRUE, rxx_restricted = TRUE, ryy_restricted = TRUE, n = 100)

## Correction for indirect range restriction in X and Y
correct_r(correction = "bvirr", rxyi = .3, ux = .8, uy = .8, rxx = .8, ryy = .8,
     ux_observed = TRUE, uy_observed = TRUE, rxx_restricted = TRUE, ryy_restricted = TRUE)
correct_r(correction = "bvirr", rxyi = .3, ux = .8, uy = .8, rxx = .8, ryy = .8,
     ux_observed = TRUE, uy_observed = TRUE, rxx_restricted = TRUE, ryy_restricted = TRUE, n = 100)

Correct correlations for small-sample bias

Description

Corrects Pearson correlations (r) for small-sample bias

Usage

correct_r_bias(r, n)

Arguments

Details

r_{c}=\frac{r_{obs}}{\left(\frac{2n-2}{2n-1}\right)}

Value

Vector of correlations corrected for small-sample bias.

References

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Sage. doi:10.4135/9781483398105. pp. 140–141.

Examples

correct_r_bias(r = .3, n = 30)
correct_r_bias(r = .3, n = 300)
correct_r_bias(r = .3, n = 3000)

Correct correlations for scale coarseness

Description

Corrects correlations for scale coarseness.

Usage

correct_r_coarseness(
  r,
  kx = NULL,
  ky = NULL,
  n = NULL,
  dist_x = "norm",
  dist_y = "norm",
  bin_value_x = c("median", "mean", "index"),
  bin_value_y = c("median", "mean", "index"),
  width_x = 3,
  width_y = 3,
  lbound_x = NULL,
  ubound_x = NULL,
  lbound_y = NULL,
  ubound_y = NULL,
  index_values_x = NULL,
  index_values_y = NULL
)

Arguments

Value

Vector of correlations corrected for scale coarseness (if n is supplied, corrected error variance and adjusted sample size is also reported).

References

Aguinis, H., Pierce, C. A., & Culpepper, S. A. (2009). Scale coarseness as a methodological artifact: Correcting correlation coefficients attenuated from using coarse scales. Organizational Research Methods, 12(4), 623–652. doi:10.1177/1094428108318065

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Sage. doi:10.4135/9781483398105. pp. 287-288.

Peters, C. C., & Van Voorhis, W. R. (1940). Statistical procedures and their mathematical bases. New York, NY: Mcgraw-Hill. doi:10.1037/13596-000. pp. 393–399.

Examples

correct_r_coarseness(r = .35, kx = 5, ky = 4, n = 100)
correct_r_coarseness(r = .35, kx = 5, n = 100)
correct_r_coarseness(r = .35, kx = 5, ky = 4, n = 100, dist_x="unif", dist_y="norm")

Correct correlations for artificial dichotomization of one or both variables

Description

Correct correlations for artificial dichotomization of one or both variables.

Usage

correct_r_dich(r, px = NA, py = NA, n = NULL, ...)

Arguments

Details

r_{c}=\frac{r_{obs}}{\left[\frac{\phi\left(p_{X}\right)}{p_{X}\left(1-p_{X}\right)}\right]\left[\frac{\phi\left(p_{Y}\right)}{p_{Y}\left(1-p_{Y}\right)}\right]}

Value

Vector of correlations corrected for artificial dichotomization (if n is supplied, corrected error variance and adjusted sample size is also reported).

References

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Sage. doi:10.4135/9781483398105. pp. 43–44.

Examples

correct_r_dich(r = 0.32, px = .5, py = .5, n = 100)

Correct correlations for uneven/unrepresentative splits

Description

This correction is mathematically equivalent to correcting the correlation for direct range restriction in the split variable.

Usage

correct_r_split(r, pi, pa = 0.5, n = NULL)

Arguments

Details

r_{c}=\frac{r_{obs}}{u\sqrt{\left(\frac{1}{u^{2}}-1\right)r_{obs}^{2}+1}}

where u=\sqrt{\frac{p_{i}(1-p_{i})}{p_{a}(1-p_{a})}}, the ratio of the dichotomous variance in the sample (p_{i} is the incumbent/sample proportion in one of the two groups) to the dichotomous variance in the population (p_{a} is the applicant/population proportion in one of the two groups). This correction is identical to the correction for univariate direct range restriction, applied to a dichotomous variable.

Value

Vector of correlations corrected for unrepresentative splits (if n is supplied, corrected error variance and adjusted sample size is also reported).

References

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Sage. doi:10.4135/9781483398105. pp. 287-288.

Examples

correct_r_split(r = 0.3, pi = .9, pa = .5, n = 100)

Generate an artifact distribution object for use in artifact-distribution meta-analysis programs.

Description

This function generates artifact-distribution objects containing either interactive or Taylor series artifact distributions. Use this to create objects that can be supplied to the ma_r_ad and ma_r_ad functions to apply psychometric corrections to barebones meta-analysis objects via artifact distribution methods.

Allows consolidation of observed and estimated artifact information by cross-correcting artifact distributions and forming weighted artifact summaries.

For u ratios, error variances can be computed for independent samples (i.e., settings in which the unrestricted standard deviation comes from an external study) or dependent samples (i.e., settings in which the range-restricted standard deviation comes from a sample that represents a subset of the applicant sample that provided the unrestricted standard deviation). The former circumstance is presumed to be more common, so error variances are computed for independent samples by default.

Usage

create_ad(
  ad_type = c("tsa", "int"),
  rxxi = NULL,
  n_rxxi = NULL,
  wt_rxxi = n_rxxi,
  rxxi_type = rep("alpha", length(rxxi)),
  k_items_rxxi = rep(NA, length(rxxi)),
  rxxa = NULL,
  n_rxxa = NULL,
  wt_rxxa = n_rxxa,
  rxxa_type = rep("alpha", length(rxxa)),
  k_items_rxxa = rep(NA, length(rxxa)),
  ux = NULL,
  ni_ux = NULL,
  na_ux = NULL,
  wt_ux = ni_ux,
  dep_sds_ux_obs = rep(FALSE, length(ux)),
  ut = NULL,
  ni_ut = NULL,
  na_ut = NULL,
  wt_ut = ni_ut,
  dep_sds_ut_obs = rep(FALSE, length(ut)),
  mean_qxi = NULL,
  var_qxi = NULL,
  k_qxi = NULL,
  mean_n_qxi = NULL,
  qxi_dist_type = rep("alpha", length(mean_qxi)),
  mean_k_items_qxi = rep(NA, length(mean_qxi)),
  mean_rxxi = NULL,
  var_rxxi = NULL,
  k_rxxi = NULL,
  mean_n_rxxi = NULL,
  rxxi_dist_type = rep("alpha", length(mean_rxxi)),
  mean_k_items_rxxi = rep(NA, length(mean_rxxi)),
  mean_qxa = NULL,
  var_qxa = NULL,
  k_qxa = NULL,
  mean_n_qxa = NULL,
  qxa_dist_type = rep("alpha", length(mean_qxa)),
  mean_k_items_qxa = rep(NA, length(mean_qxa)),
  mean_rxxa = NULL,
  var_rxxa = NULL,
  k_rxxa = NULL,
  mean_n_rxxa = NULL,
  rxxa_dist_type = rep("alpha", length(mean_rxxa)),
  mean_k_items_rxxa = rep(NA, length(mean_rxxa)),
  mean_ux = NULL,
  var_ux = NULL,
  k_ux = NULL,
  mean_ni_ux = NULL,
  mean_na_ux = rep(NA, length(mean_ux)),
  dep_sds_ux_spec = rep(FALSE, length(mean_ux)),
  mean_ut = NULL,
  var_ut = NULL,
  k_ut = NULL,
  mean_ni_ut = NULL,
  mean_na_ut = rep(NA, length(mean_ut)),
  dep_sds_ut_spec = rep(FALSE, length(mean_ut)),
  estimate_rxxa = TRUE,
  estimate_rxxi = TRUE,
  estimate_ux = TRUE,
  estimate_ut = TRUE,
  var_unbiased = TRUE,
  ...
)

Arguments

Value

Artifact distribution object (matrix of artifact-distribution means and variances) for use artifact-distribution meta-analyses.

Examples

## Example computed using observed values only:
create_ad(ad_type = "tsa", rxxa = c(.9, .8), n_rxxa = c(50, 150),
              rxxi = c(.8, .7), n_rxxi = c(50, 150),
              ux = c(.9, .8), ni_ux = c(50, 150))

create_ad(ad_type = "int", rxxa = c(.9, .8), n_rxxa = c(50, 150),
              rxxi = c(.8, .7), n_rxxi = c(50, 150),
              ux = c(.9, .8), ni_ux = c(50, 150))

## Example computed using all possible input arguments (arbitrary values):
rxxa <- rxxi <- ux <- ut <- c(.7, .8)
n_rxxa <- n_rxxi <- ni_ux <- ni_ut <- c(50, 100)
na_ux <- na_ut <- c(200, 200)
mean_qxa <- mean_qxi <- mean_ux <- mean_ut <- mean_rxxi <- mean_rxxa <- c(.7, .8)
var_qxa <- var_qxi <- var_ux <- var_ut <- var_rxxi <- var_rxxa <- c(.1, .05)
k_qxa <- k_qxi <- k_ux <- k_ut <- k_rxxa <- k_rxxi <- 2
mean_n_qxa <- mean_n_qxi <- mean_ni_ux <- mean_ni_ut <- mean_n_rxxa <- mean_n_rxxi <- c(100, 100)
dep_sds_ux_obs <- dep_sds_ux_spec <- dep_sds_ut_obs <- dep_sds_ut_spec <- FALSE
mean_na_ux <- mean_na_ut <- c(200, 200)

wt_rxxa <- n_rxxa
wt_rxxi <- n_rxxi
wt_ux <- ni_ux
wt_ut <- ni_ut

estimate_rxxa <- TRUE
estimate_rxxi <- TRUE
estimate_ux <- TRUE
estimate_ut <- TRUE
var_unbiased <- TRUE

create_ad(rxxa = rxxa, n_rxxa = n_rxxa, wt_rxxa = wt_rxxa,
              mean_qxa = mean_qxa, var_qxa = var_qxa,
              k_qxa = k_qxa, mean_n_qxa = mean_n_qxa,
              mean_rxxa = mean_rxxa, var_rxxa = var_rxxa,
              k_rxxa = k_rxxa, mean_n_rxxa = mean_n_rxxa,

              rxxi = rxxi, n_rxxi = n_rxxi, wt_rxxi = wt_rxxi,
              mean_qxi = mean_qxi, var_qxi = var_qxi,
              k_qxi = k_qxi, mean_n_qxi = mean_n_qxi,
              mean_rxxi = mean_rxxi, var_rxxi = var_rxxi,
              k_rxxi = k_rxxi, mean_n_rxxi = mean_n_rxxi,

              ux = ux, ni_ux = ni_ux, na_ux = na_ux, wt_ux = wt_ux,
              dep_sds_ux_obs = dep_sds_ux_obs,
              mean_ux = mean_ux, var_ux = var_ux, k_ux =
               k_ux, mean_ni_ux = mean_ni_ux,
              mean_na_ux = mean_na_ux, dep_sds_ux_spec = dep_sds_ux_spec,

              ut = ut, ni_ut = ni_ut, na_ut = na_ut, wt_ut = wt_ut,
              dep_sds_ut_obs = dep_sds_ut_obs,
              mean_ut = mean_ut, var_ut = var_ut,
              k_ut = k_ut, mean_ni_ut = mean_ni_ut,
              mean_na_ut = mean_na_ut, dep_sds_ut_spec = dep_sds_ut_spec,

              estimate_rxxa = estimate_rxxa, estimate_rxxi = estimate_rxxi,
              estimate_ux = estimate_ux, estimate_ut = estimate_ut, var_unbiased = var_unbiased)

Generate an artifact distribution object for a dichotomous grouping variable.

Description

This function generates artifact-distribution objects containing either interactive or Taylor series artifact distributions for dichotomous group-membership variables. Use this to create objects that can be supplied to the ma_r_ad and ma_d_ad functions to apply psychometric corrections to barebones meta-analysis objects via artifact distribution methods.

Allows consolidation of observed and estimated artifact information by cross-correcting artifact distributions and forming weighted artifact summaries.

Usage

create_ad_group(
  ad_type = c("tsa", "int"),
  rGg = NULL,
  n_rGg = NULL,
  wt_rGg = n_rGg,
  pi = NULL,
  pa = NULL,
  n_pi = NULL,
  n_pa = NULL,
  wt_p = n_pi,
  mean_rGg = NULL,
  var_rGg = NULL,
  k_rGg = NULL,
  mean_n_rGg = NULL,
  var_unbiased = TRUE,
  ...
)

Arguments

Value

Artifact distribution object (matrix of artifact-distribution means and variances) for use in artifact-distribution meta-analyses.

Examples

## Example artifact distribution for a dichotomous grouping variable:
create_ad_group(rGg = c(.8, .9, .95), n_rGg = c(100, 200, 250),
                mean_rGg = .9, var_rGg = .05,
                k_rGg = 5, mean_n_rGg = 100,
                pi = c(.6, .55, .3), pa = .5, n_pi = c(100, 200, 250), n_pa = c(300, 300, 300),
                var_unbiased = TRUE)
                
create_ad_group(ad_type = "int", rGg = c(.8, .9, .95), n_rGg = c(100, 200, 250),
                mean_rGg = .9, var_rGg = .05,
                k_rGg = 5, mean_n_rGg = 100,
                pi = c(.6, .55, .3), pa = .5, n_pi = c(100, 200, 250), n_pa = c(300, 300, 300),
                var_unbiased = TRUE)

Create a tibble of artifact distributions by construct

Description

Create a tibble of artifact distributions by construct

Usage

create_ad_tibble(
  ad_type = c("tsa", "int"),
  n = NULL,
  sample_id = NULL,
  construct_x = NULL,
  facet_x = NULL,
  measure_x = NULL,
  construct_y = NULL,
  facet_y = NULL,
  measure_y = NULL,
  rxx = NULL,
  rxx_restricted = TRUE,
  rxx_type = "alpha",
  k_items_x = NA,
  ryy = NULL,
  ryy_restricted = TRUE,
  ryy_type = "alpha",
  k_items_y = NA,
  ux = NULL,
  ux_observed = TRUE,
  uy = NULL,
  uy_observed = TRUE,
  estimate_rxxa = TRUE,
  estimate_rxxi = TRUE,
  estimate_ux = TRUE,
  estimate_ut = TRUE,
  moderators = NULL,
  cat_moderators = TRUE,
  moderator_type = c("simple", "hierarchical", "none"),
  construct_order = NULL,
  supplemental_ads = NULL,
  data = NULL,
  control = control_psychmeta(),
  ...
)

create_ad_list(
  ad_type = c("tsa", "int"),
  n = NULL,
  sample_id = NULL,
  construct_x = NULL,
  facet_x = NULL,
  measure_x = NULL,
  construct_y = NULL,
  facet_y = NULL,
  measure_y = NULL,
  rxx = NULL,
  rxx_restricted = TRUE,
  rxx_type = "alpha",
  k_items_x = NA,
  ryy = NULL,
  ryy_restricted = TRUE,
  ryy_type = "alpha",
  k_items_y = NA,
  ux = NULL,
  ux_observed = TRUE,
  uy = NULL,
  uy_observed = TRUE,
  estimate_rxxa = TRUE,
  estimate_rxxi = TRUE,
  estimate_ux = TRUE,
  estimate_ut = TRUE,
  moderators = NULL,
  cat_moderators = TRUE,
  moderator_type = c("simple", "hierarchical", "none"),
  construct_order = NULL,
  supplemental_ads = NULL,
  data = NULL,
  control = control_psychmeta(),
  ...
)

Arguments

Value

A tibble of artifact distributions

Examples

## Examples to create Taylor series artifact distributions:
# Overall artifact distributions (not pairwise, not moderated)
create_ad_tibble(ad_type = "tsa",
                 n = n, rxx = rxxi, ryy = ryyi,
                 construct_x = x_name, construct_y = y_name,
                 sample_id = sample_id, moderators = moderator,
                 data = data_r_meas_multi,
                 control = control_psychmeta(pairwise_ads = FALSE,
                                             moderated_ads = FALSE))

# Overall artifact distributions by moderator combination
create_ad_tibble(ad_type = "tsa",
                 n = n, rxx = rxxi, ryy = ryyi,
                 construct_x = x_name, construct_y = y_name,
                 sample_id = sample_id, moderators = moderator,
                 data = data_r_meas_multi,
                 control = control_psychmeta(pairwise_ads = FALSE,
                                             moderated_ads = TRUE))

# Pairwise artifact distributions (not moderated)
create_ad_tibble(ad_type = "tsa",
                 n = n, rxx = rxxi, ryy = ryyi,
                 construct_x = x_name, construct_y = y_name,
                 sample_id = sample_id, moderators = moderator,
                 data = data_r_meas_multi,
                 control = control_psychmeta(pairwise_ads = TRUE,
                                               moderated_ads = FALSE))

# Pairwise artifact distributions by moderator combination
create_ad_tibble(ad_type = "tsa",
                 n = n, rxx = rxxi, ryy = ryyi,
                 construct_x = x_name, construct_y = y_name,
                 sample_id = sample_id, moderators = moderator,
                 data = data_r_meas_multi,
                 control = control_psychmeta(pairwise_ads = TRUE,
                                             moderated_ads = TRUE))

Construct a credibility interval

Description

Function to construct a credibility interval around a mean effect size.

Usage

credibility(mean, sd, k = NULL, cred_level = 0.8, cred_method = c("t", "norm"))

Arguments

Details

CR=mean_{es}\pm quantile\times SD_{es}

Value

A matrix of credibility intervals of the specified width.

Examples

credibility(mean = .3, sd = .15, cred_level = .8, cred_method = "norm")
credibility(mean = .3, sd = .15, cred_level = .8, k = 10)
credibility(mean = c(.3, .5), sd = c(.15, .2), cred_level = .8, k = 10)

Hypothetical d value dataset simulated with sampling error only

Description

Data set for use in example meta-analyses of multiple variables.

Usage

data(data_d_bb_multi)

Format

data.frame

Examples

data(data_d_bb_multi)

Hypothetical d value dataset simulated to satisfy the assumptions of the correction for measurement error only in multiple constructs

Description

Data set for use in example meta-analyses correcting for measurement error in multiple variables.

Usage

data(data_d_meas_multi)

Format

data.frame

Examples

data(data_d_meas_multi)

Hypothetical dataset simulated to satisfy the assumptions of the bivariate correction for direct range restriction

Description

Data set for use in example meta-analyses of bivariate direct range restriction. Note that the BVDRR correction is only an approximation of the appropriate range-restriction correction and tends to have a noticeable positive bias when applied in meta-analyses.

Usage

data(data_r_bvdrr)

Format

data.frame

Examples

data(data_r_bvdrr)

Hypothetical dataset simulated to satisfy the assumptions of the bivariate correction for indirect range restriction

Description

Data set for use in example meta-analyses of bivariate indirect range restriction.

Usage

data(data_r_bvirr)

Format

data.frame

Examples

data(data_r_bvirr)

Meta-analysis of OCB correlations with other constructs

Description

Data set to demonstrate corrections for univariate range restriction and measurement error using individual corrections or artifact distributions. NOTE: This is an updated version of the data set reported in the Gonzalez-Mulé, Mount, an Oh (2014) article that was obtained from the first author.

Usage

data(data_r_gonzalezmule_2014)

Format

data.frame

References

Gonzalez-Mulé, E., Mount, M. K., & Oh, I.-S. (2014). A meta-analysis of the relationship between general mental ability and nontask performance. Journal of Applied Psychology, 99(6), 1222–1243. doi:10.1037/a0037547

Examples

data(data_r_gonzalezmule_2014)

Artifact-distribution meta-analysis of the validity of interviews

Description

Data set to demonstrate corrections for univariate range restriction and criterion measurement error using artifact distributions.

Usage

data(data_r_mcdaniel_1994)

Format

data.frame

References

McDaniel, M. A., Whetzel, D. L., Schmidt, F. L., & Maurer, S. D. (1994). The validity of employment interviews: A comprehensive review and meta-analysis. Journal of Applied Psychology, 79(4), 599–616. doi:10.1037/0021-9010.79.4.599

Examples

data(data_r_mcdaniel_1994)

Bare-bones meta-analysis of parenting and childhood depression

Description

Data set to demonstrate bare-bones meta-analysis.

Usage

data(data_r_mcleod_2007)

Format

data.frame

References

McLeod, B. D., Weisz, J. R., & Wood, J. J., (2007). Examining the association between parenting and childhood depression: A meta-analysis. Clinical Psychology Review, 27(8), 986–1003. doi:10.1016/j.cpr.2007.03.001

Examples

data(data_r_mcleod_2007)

Hypothetical dataset simulated to satisfy the assumptions of the correction for measurement error only

Description

Data set for use in example meta-analyses correcting for measurement error in two variables.

Usage

data(data_r_meas)

Format

data.frame

Examples

data(data_r_meas)

Hypothetical correlation dataset simulated to satisfy the assumptions of the correction for measurement error only in multiple constructs

Description

Data set for use in example meta-analyses correcting for measurement error in multiple variables.

Usage

data(data_r_meas_multi)

Format

data.frame

Examples

data(data_r_meas_multi)

Second order meta-analysis of operational validities of big five personality measures across East Asian countries

Description

Example of a second-order meta-analysis of correlations corrected using artifact-distribution methods.

Usage

data(data_r_oh_2009)

Format

data.frame

References

Oh, I. -S. (2009). The Five-Factor Model of personality and job performance in East Asia: A cross-cultural validity generalization study. (Doctoral dissertation) Iowa City, IA: University of Iowa. https://www.proquest.com/dissertations/docview/304903943/

Schmidt, F. L., & Oh, I.-S. (2013). Methods for second order meta-analysis and illustrative applications. Organizational Behavior and Human Decision Processes, 121(2), 204–218. doi:10.1016/j.obhdp.2013.03.002

Examples

data(data_r_oh_2009)

Artifact-distribution meta-analysis of the correlation between school grades and cognitive ability

Description

Data set to demonstrate corrections for univariate range restriction and cognitive ability measurement error.

Usage

data(data_r_roth_2015)

Format

data.frame

References

Roth, B., Becker, N., Romeyke, S., Schäfer, S., Domnick, F., & Spinath, F. M. (2015). Intelligence and school grades: A meta-analysis. Intelligence, 53, 118–137. doi:10.1016/j.intell.2015.09.002

Examples

data(data_r_roth_2015)

Hypothetical dataset simulated to satisfy the assumptions of the univariate correction for direct range restriction

Description

Data set for use in example meta-analyses correcting for univariate direct range restriction.

Usage

data(data_r_uvdrr)

Format

data.frame

Examples

data(data_r_uvdrr)

Hypothetical dataset simulated to satisfy the assumptions of the univariate correction for indirect range restriction

Description

Data set for use in example meta-analyses correcting for univariate indirect range restriction.

Usage

data(data_r_uvirr)

Format

data.frame

Examples

data(data_r_uvirr)

Estimation of applicant and incumbent reliabilities and of true- and observed-score u ratios

Description

Functions to estimate the values of artifacts from other artifacts. These functions allow for reliability estimates to be corrected/attenuated for range restriction and allow u ratios to be converted between observed-score and true-score metrics. Some functions also allow for the extrapolation of an artifact from other available information.

Available functions include:

• estimate_rxxa: Estimate the applicant reliability of variable X from X's incumbent reliability value and X's observed-score or true-score u ratio.

• estimate_rxxa_u: Estimate the applicant reliability of variable X from X's observed-score and true-score u ratios.

• estimate_rxxi: Estimate the incumbent reliability of variable X from X's applicant reliability value and X's observed-score or true-score u ratio.

• estimate_rxxi_u: Estimate the incumbent reliability of variable X from X's observed-score and true-score u ratios.

• estimate_ux: Estimate the true-score u ratio for variable X from X's reliability coefficient and X's observed-score u ratio.

• estimate_uy: Estimate the observed-score u ratio for variable X from X's reliability coefficient and X's true-score u ratio.

• estimate_ryya: Estimate the applicant reliability of variable Y from Y's incumbent reliability value, Y's correlation with X, and X's u ratio.

• estimate_ryyi: Estimate the incumbent reliability of variable Y from Y's applicant reliability value, Y's correlation with X, and X's u ratio.

• estimate_uy: Estimate the observed-score u ratio for variable Y from Y's applicant and incumbent reliability coefficients.

• estimate_up: Estimate the true-score u ratio for variable Y from Y's applicant and incumbent reliability coefficients.

Usage

estimate_rxxa(
  rxxi,
  ux,
  ux_observed = TRUE,
  indirect_rr = TRUE,
  rxxi_type = "alpha"
)

estimate_rxxi(
  rxxa,
  ux,
  ux_observed = TRUE,
  indirect_rr = TRUE,
  rxxa_type = "alpha"
)

estimate_ut(ux, rxx, rxx_restricted = TRUE)

estimate_ux(ut, rxx, rxx_restricted = TRUE)

estimate_ryya(
  ryyi,
  rxyi,
  ux,
  rxx = 1,
  rxx_restricted = FALSE,
  ux_observed = TRUE,
  indirect_rr = TRUE,
  rxx_type = "alpha"
)

estimate_ryyi(
  ryya,
  rxyi,
  ux,
  rxx = 1,
  rxx_restricted = FALSE,
  ux_observed = TRUE,
  indirect_rr = TRUE,
  rxx_type = "alpha"
)

estimate_uy(ryyi, ryya, indirect_rr = TRUE, ryy_type = "alpha")

estimate_up(ryyi, ryya)

estimate_rxxa_u(ux, ut)

estimate_rxxi_u(ux, ut)

Arguments

Details

#### Formulas to estimate rxxa ####

Formulas for indirect range restriction:

\rho_{XX_{a}}=1-u_{X}^{2}\left(1-\rho_{XX_{i}}\right)

\rho_{XX_{a}}=\frac{\rho_{XX_{i}}}{\rho_{XX_{i}}+u_{T}^{2}-\rho_{XX_{i}}u_{T}^{2}}

Formula for direct range restriction:

\rho_{XX_{a}}=\frac{\rho_{XX_{i}}}{u_{X}^{2}\left[1+\rho_{XX_{i}}\left(\frac{1}{u_{X}^{2}}-1\right)\right]}

#### Formulas to estimate rxxi ####

Formulas for indirect range restriction:

\rho_{XX_{i}}=1-\frac{1-\rho_{XX_{a}}}{u_{X}^{2}}

\rho_{XX_{i}}=1-\frac{1-\rho_{XX_{a}}}{\rho_{XX_{a}}\left[u_{T}^{2}-\left(1-\frac{1}{\rho_{XX_{a}}}\right)\right]}

Formula for direct range restriction:

\rho_{XX_{i}}=\frac{\rho_{XX_{i}}u_{X}^{2}}{1+\rho_{XX_{i}}\left(u_{X}^{2}-1\right)}

#### Formulas to estimate ut ####

u_{T}=\sqrt{\frac{\rho_{XX_{i}}u_{X}^{2}}{1+\rho_{XX_{i}}u_{X}^{2}-u_{X}^{2}}}

u_{T}=\sqrt{\frac{u_{X}^{2}-\left(1-\rho_{XX_{a}}\right)}{\rho_{XX_{a}}}}

#### Formulas to estimate ux ####

u_{X}=\sqrt{\frac{u_{T}^{2}}{\rho_{XX_{i}}\left(1+\frac{u_{T}^{2}}{\rho_{XX_{i}}}-u_{T}^{2}\right)}}

u_{X}=\sqrt{\rho_{XX_{a}}\left[u_{T}^{2}-\left(1-\frac{1}{\rho_{XX_{a}}}\right)\right]}

#### Formulas to estimate ryya #### Formula for direct range restriction (i.e., when selection is based on X):

\rho_{YY_{a}}=1-\frac{1-\rho_{YY_{i}}}{1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)}

Formula for indirect range restriction (i.e., when selection is based on a variable other than X):

\rho_{YY_{a}}=1-\frac{1-\rho_{YY_{i}}}{1-\rho_{TY_{i}}^{2}\left(1-\frac{1}{u_{T}^{2}}\right)}

#### Formulas to estimate ryyi #### Formula for direct range restriction (i.e., when selection is based on X):

\rho_{YY_{i}}=1-\left(1-\rho_{YY_{a}}\right)\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]

Formula for indirect range restriction (i.e., when selection is based on a variable other than X):

\rho_{YY_{i}}=1-\left(1-\rho_{YY_{a}}\right)\left[1-\rho_{TY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]

#### Formula to estimate uy ####

u_{Y}=\sqrt{\frac{1-\rho_{YY_{a}}}{1-\rho_{YY_{i}}}}

#### Formula to estimate up ####

u_{P}=\sqrt{\frac{\frac{1-\rho_{YY_{a}}}{1-\rho_{YY_{i}}}-\left(1-\rho_{YY_{a}}\right)}{\rho_{YY_{a}}}}

Value

A vector of estimated artifact values.

References

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Sage. doi:10.4135/9781483398105 p. 127.

Le, H., & Schmidt, F. L. (2006). Correcting for indirect range restriction in meta-analysis: Testing a new meta-analytic procedure. Psychological Methods, 11(4), 416–438. doi:10.1037/1082-989X.11.4.416

Hunter, J. E., Schmidt, F. L., & Le, H. (2006). Implications of direct and indirect range restriction for meta-analysis methods and findings. Journal of Applied Psychology, 91(3), 594–612. doi:10.1037/0021-9010.91.3.594

Le, H., Oh, I.-S., Schmidt, F. L., & Wooldridge, C. D. (2016). Correction for range restriction in meta-analysis revisited: Improvements and implications for organizational research. Personnel Psychology, 69(4), 975–1008. doi:10.1111/peps.12122

Examples

estimate_rxxa(rxxi = .8, ux = .8, ux_observed = TRUE)
estimate_rxxi(rxxa = .8, ux = .8, ux_observed = TRUE)
estimate_ut(ux = .8, rxx = .8, rxx_restricted = TRUE)
estimate_ux(ut = .8, rxx = .8, rxx_restricted = TRUE)
estimate_ryya(ryyi = .8, rxyi = .3, ux = .8)
estimate_ryyi(ryya = .8, rxyi = .3, ux = .8)
estimate_uy(ryyi = c(.5, .7), ryya = c(.7, .8))
estimate_up(ryyi = c(.5, .7), ryya = c(.7, .8))
estimate_rxxa_u(ux = c(.7, .8), ut = c(.65, .75))
estimate_rxxi_u(ux = c(.7, .8), ut = c(.65, .75))

Inverse Spearman-Brown formula to estimate the amount by which a measure would have to be lengthened or shorted to achieve a desired level of reliability

Description

This function implements the inverse of the Spearman-Brown prophecy formula and answers the question: "How much would I have to increase (do decrease) the length of this measure to obtain a desired reliability level given the current reliability of the measure?" The result of the function is the multiplier by which the length of the original measure should be adjusted. The formula implemented here assumes that all items added to (or subtracted from) the measure will be parallel forms of the original items.

Usage

estimate_length_sb(rel_initial, rel_desired)

Arguments

Details

This is computed as:

k^{*}=\frac{\rho_{XX}^{*}(\rho_{XX}-1)}{(\rho_{XX}^{*}-1)\rho_{XX}}

where \rho_{XX} is the inital reliability, \rho_{XX}^{*} is the predicted reliability of a measure with a different length, and k^{*} is the number of times the measure would have to be lengthened to obtain a reliability equal to \rho_{XX}^{*}.

Value

The estimated number of times by which the number of items in the initial measure would have to be multiplied to achieve the desired reliability.

References

Ghiselli, E. E., Campbell, J. P., & Zedeck, S. (1981). Measurement theory for the behavioral sciences. San Francisco, CA: Freeman. p. 236.

Examples

## Estimated k to achieve a reliability of .8 from a measure with an initial reliability of .7
estimate_length_sb(rel_initial = .7, rel_desired = .8)

## Estimated k to achieve a reliability of .8 from a measure with an initial reliability of .9
estimate_length_sb(rel_initial = .9, rel_desired = .8)

Estimate covariance matrices and mean vectors containing product terms

Description

Estimate covariance matrices and mean vectors containing product terms

Usage

estimate_matrix_prods(sigma_mat, mu_vec, prod_list)

Arguments

Value

Augmented covariance matrix and mean vector containing product terms.

Examples

## Establish mean and covariance parameters
mu_vec <- 1:4
sigma_mat <- reshape_vec2mat(c(.1, .2, .3, .4, .5, .6), var_names = LETTERS[1:4])
names(mu_vec) <- colnames(sigma_mat)

## Define a list of variables to be used in estimating products:
prod_list <- list(c("A", "B"),
                  c("A", "C"),
                  c("A", "D"),
                  c("B", "C"),
                  c("B", "D"),
                  c("C", "D"))

## Generate data for the purposes of comparison:
set.seed(1)
dat <- data.frame(MASS::mvrnorm(100000, mu = mu_vec, Sigma = sigma_mat, empirical = TRUE))

## Create product terms in simulated data:
for(i in 1:length(prod_list))
     dat[,paste(prod_list[[i]], collapse = "_x_")] <-
     dat[,prod_list[[i]][1]] * dat[,prod_list[[i]][2]]

## Analytically estimate product variables and compare to simulated data:
estimate_matrix_prods(sigma_mat = sigma_mat, mu_vec = mu_vec, prod_list = prod_list)
round(cov(dat), 2)
round(apply(dat, 2, mean), 2)

Estimation of statistics computed from products of random, normal variables

Description

This family of functions computes univariate descriptive statistics for the products of two variables denoted as "x" and "y" (e.g., mean(x * y) or var(x * y)) and the covariance between the products of "x" and "y" and of "u" and "v" (e.g., cov(x * y, u * v) or cor(x * y, u * v)). These functions presume all variables are random normal variables.

Available functions include:

• estimate_mean_prod: Estimate the mean of the product of two variables: x * y.

• estimate_var_prod: Estimate the variance of the product of two variables: x * y.

• estimate_cov_prods: Estimate the covariance between the products of two pairs of variables: x * y and u * v.

• estimate_cor_prods: Estimate the correlation between the products of two pairs of variables: x * y and u * v.

Usage

estimate_mean_prod(mu_x, mu_y, cov_xy)

estimate_var_prod(mu_x, mu_y, var_x, var_y, cov_xy)

estimate_cov_prods(mu_x, mu_y, mu_u, mu_v, cov_xu, cov_xv, cov_yu, cov_yv)

estimate_cor_prods(
  mu_x,
  mu_y,
  mu_u,
  mu_v,
  var_x,
  var_y,
  var_u,
  var_v,
  cov_xu,
  cov_xv,
  cov_yu,
  cov_yv,
  cov_xy,
  cov_uv
)

Arguments

Value

An estimated statistic computed from the products of random, normal variables.

References

Bohrnstedt, G. W., & Goldberger, A. S. (1969). On the exact covariance of products of random variables. Journal of the American Statistical Association, 64(328), 1439. doi:10.2307/2286081

Goodman, L. A. (1960). On the exact variance of products. Journal of the American Statistical Association, 55(292), 708. doi:10.2307/2281592

Estimate descriptive statistics of square-root reliabilities

Description

Estimate descriptive statistics of square-root reliabilities from descriptive statistics of reliabilities via Taylor series approximation

Usage

estimate_q_dist(mean_rel, var_rel)

Arguments

Details

var_{q_{X}}=\frac{var_{\rho_{XX}}}{4q_{X}^{2}}

Value

The estimated mean and variance of a distribution of square-root reliability values.

Examples

estimate_q_dist(mean_rel = .8, var_rel = .15)

Estimate descriptive statistics of reliabilities

Description

Estimate descriptive statistics of reliabilities from descriptive statistics of square-root reliabilities via Taylor series approximation

Usage

estimate_rel_dist(mean_q, var_q)

Arguments

Details

var_{\rho_{XX}}=4q_{X}^{2}var_{\rho_{XX}}

Value

The estimated mean and variance of a distribution of reliability values.

Examples

estimate_rel_dist(mean_q = .9, var_q = .05)

Spearman-Brown prophecy formula to estimate the reliability of a lengthened measure

Description

This function implements the Spearman-Brown prophecy formula for estimating the reliability of a lengthened (or shortened) measure. The formula implemented here assumes that all items added to (or subtracted from) the measure will be parallel forms of the original items.

Usage

estimate_rel_sb(rel_initial, k)

Arguments

Details

This is computed as:

\rho_{XX}^{*}=\frac{k\rho_{XX}}{1+(k-1)\rho_{XX}}

where \rho_{XX} is the initial reliability, k is the multiplier by which the measure is to be lengthened (or shortened), and \rho_{XX}^{*} is the predicted reliability of a measure with a different length.

Value

The estimated reliability of the lengthened (or shortened) measure.

References

Ghiselli, E. E., Campbell, J. P., & Zedeck, S. (1981). Measurement theory for the behavioral sciences. San Francisco, CA: Freeman. p. 232.

Examples

## Double the length of a measure with an initial reliability of .7
estimate_rel_sb(rel_initial = .7, k = 2)

## Halve the length of a measure with an initial reliability of .9
estimate_rel_sb(rel_initial = .9, k = .5)

Estimate u ratios from available artifact information

Description

Uses information about standard deviations, reliability estimates, and selection ratios to estimate u ratios. Selection ratios are only used to estimate u when no other information is available, but estimates of u computed from SDs and reliabilities will be averaged to reduce error.

Usage

estimate_u(
  measure_id = NULL,
  sdi = NULL,
  sda = NULL,
  rxxi = NULL,
  rxxa = NULL,
  item_ki = NULL,
  item_ka = NULL,
  n = NULL,
  meani = NULL,
  sr = NULL,
  rxya_est = NULL,
  data = NULL
)

Arguments

Value

A vector of estimated u ratios.

Examples

sdi <- c(1.4, 1.2, 1.3, 1.4)
sda <- 2
rxxi <- c(.6, .7, .75, .8)
rxxa <- c(.9, .95, .8, .9)
item_ki <- c(12, 12, 12, NA)
item_ka <- NULL
n <- c(200, 200, 200, 200)
meani <- c(2, 1, 2, 3)
sr <- c(.5, .6, .7, .4)
rxya_est <- .5

## Estimate u from standard deviations only:
estimate_u(sdi = sdi, sda = sda)

## Estimate u from incumbent standard deviations and the
## mixture standard deviation:
estimate_u(sdi = sdi, sda = "mixture", meani = meani, n = n)

## Estimate u from reliability information:
estimate_u(rxxi = rxxi, rxxa = rxxa)

## Estimate u from both standard deviations and reliabilities:
estimate_u(sdi = sdi, sda = sda, rxxi = rxxi, rxxa = rxxa,
           item_ki = item_ki, item_ka = item_ka, n = n,
           meani = meani, sr = sr, rxya_est = rxya_est)

estimate_u(sdi = sdi, sda = "average", rxxi = rxxi, rxxa = "average",
           item_ki = item_ki, item_ka = item_ka, n = n, meani = meani)

## Estimate u from selection ratios as direct range restriction:
estimate_u(sr = sr)

## Estimate u from selection ratios as indirect range restriction:
estimate_u(sr = sr, rxya_est = rxya_est)

Taylor series approximations for the variances of estimates artifact distributions.

Description

Taylor series approximations to estimate the variances of artifacts that have been estimated from other artifacts. These functions are implemented internally in the create_ad function and related functions, but are useful as general tools for manipulating artifact distributions.

Available functions include:

• estimate_var_qxi: Estimate the variance of a qxi distribution from a qxa distribution and a distribution of u ratios.

• estimate_var_rxxi: Estimate the variance of an rxxi distribution from an rxxa distribution and a distribution of u ratios.

• estimate_var_qxa: Estimate the variance of a qxa distribution from a qxi distribution and a distribution of u ratios.

• estimate_var_rxxa: Estimate the variance of an rxxa distribution from an rxxi distribution and a distribution of u ratios.

• estimate_var_ut: Estimate the variance of a true-score u ratio distribution from an observed-score u ratio distribution and a reliability distribution.

• estimate_var_ux: Estimate the variance of an observed-score u ratio distribution from a true-score u ratio distribution and a reliability distribution.

• estimate_var_qyi: Estimate the variance of a qyi distribution from the following distributions: qya, rxyi, and ux.

• estimate_var_ryyi: Estimate the variance of an ryyi distribution from the following distributions: ryya, rxyi, and ux.

• estimate_var_qya: Estimate the variance of a qya distribution from the following distributions: qyi, rxyi, and ux.

• estimate_var_ryya: Estimate the variance of an ryya distribution from the following distributions: ryyi, rxyi, and ux.

Usage

estimate_var_qxi(
  qxa,
  var_qxa = 0,
  ux,
  var_ux = 0,
  cor_qxa_ux = 0,
  ux_observed = TRUE,
  indirect_rr = TRUE,
  qxa_type = "alpha"
)

estimate_var_qxa(
  qxi,
  var_qxi = 0,
  ux,
  var_ux = 0,
  cor_qxi_ux = 0,
  ux_observed = TRUE,
  indirect_rr = TRUE,
  qxi_type = "alpha"
)

estimate_var_rxxi(
  rxxa,
  var_rxxa = 0,
  ux,
  var_ux = 0,
  cor_rxxa_ux = 0,
  ux_observed = TRUE,
  indirect_rr = TRUE,
  rxxa_type = "alpha"
)

estimate_var_rxxa(
  rxxi,
  var_rxxi = 0,
  ux,
  var_ux = 0,
  cor_rxxi_ux = 0,
  ux_observed = TRUE,
  indirect_rr = TRUE,
  rxxi_type = "alpha"
)

estimate_var_ut(
  rxx,
  var_rxx = 0,
  ux,
  var_ux = 0,
  cor_rxx_ux = 0,
  rxx_restricted = TRUE,
  rxx_as_qx = FALSE
)

estimate_var_ux(
  rxx,
  var_rxx = 0,
  ut,
  var_ut = 0,
  cor_rxx_ut = 0,
  rxx_restricted = TRUE,
  rxx_as_qx = FALSE
)

estimate_var_ryya(
  ryyi,
  var_ryyi = 0,
  rxyi,
  var_rxyi = 0,
  ux,
  var_ux = 0,
  cor_ryyi_rxyi = 0,
  cor_ryyi_ux = 0,
  cor_rxyi_ux = 0
)

estimate_var_qya(
  qyi,
  var_qyi = 0,
  rxyi,
  var_rxyi = 0,
  ux,
  var_ux = 0,
  cor_qyi_rxyi = 0,
  cor_qyi_ux = 0,
  cor_rxyi_ux = 0
)

estimate_var_qyi(
  qya,
  var_qya = 0,
  rxyi,
  var_rxyi = 0,
  ux,
  var_ux = 0,
  cor_qya_rxyi = 0,
  cor_qya_ux = 0,
  cor_rxyi_ux = 0
)

estimate_var_ryyi(
  ryya,
  var_ryya = 0,
  rxyi,
  var_rxyi = 0,
  ux,
  var_ux = 0,
  cor_ryya_rxyi = 0,
  cor_ryya_ux = 0,
  cor_rxyi_ux = 0
)

Arguments

Details

#### Partial derivatives to estimate the variance of qxa using ux ####

Indirect range restriction:

b_{u_{X}}=\frac{(q_{X_{i}}^{2}-1)u_{X}}{\sqrt{(q_{X_{i}}^{2}-1)u_{X}^{2}+1}}

b_{q_{X_{i}}}=\frac{q_{X_{i}}^{2}u_{X}^{2}}{\sqrt{(q_{X_{i}}^{2}-1)u_{X}^{2}+1}}

Direct range restriction:

b_{u_{X}}=\frac{q_{X_{i}}^{2}(q_{X_{i}}^{2}-1)u_{X}}{\sqrt{-\frac{q_{X_{i}}^{2}}{q_{X_{i}}^{2}(u_{X}^{2}-1)-u_{X}^{2}}}(q_{X_{i}}^{2}(u_{X}^{2}-1)-u_{X}^{2})^{2}}

b_{q_{X_{i}}}=\frac{q_{X_{i}}u_{X}^{2}}{\sqrt{-\frac{q_{X_{i}}^{2}}{q_{X_{i}}^{2}(u_{X}^{2}-1)-u_{X}^{2}}}(q_{X_{i}}^{2}(u_{X}^{2}-1)-u_{X}^{2})^{2}}

#### Partial derivatives to estimate the variance of rxxa using ux ####

Indirect range restriction:

b_{u_{X}}=2\left(\rho_{XX_{i}}-1\right)u_{X}

\rho_{XX_{i}}: b_{\rho_{XX_{i}}}=u_{X}^{2}

Direct range restriction:

b_{u_{X}}=\frac{2(\rho_{XX_{i}}-1)\rho_{XX_{i}}u_{X}}{(-\rho_{XX_{i}}u_{X}^{2}+\rho_{XX_{i}}+u_{X}^{2})^{2}}

b_{\rho_{XX_{i}}}=\frac{u_{X}^{2}}{(-\rho_{XX_{i}}u_{X}^{2}+\rho_{XX_{i}}+u_{X}^{2})^{2}}

#### Partial derivatives to estimate the variance of rxxa using ut ####

b_{u_{T}}=\frac{2(\rho_{XX_{i}}-1)*\rho_{XX_{i}}u_{T}}{(-\rho_{XX_{i}}u_{T}^{2}+\rho_{XX_{i}}+u_{T}^{2})^{2}}

b_{\rho_{XX_{i}}}=\frac{u_{T}^{2}}{(-\rho_{XX_{i}}u_{T}^{2}+\rho_{XX_{i}}+u_{T}^{2})^{2}}

#### Partial derivatives to estimate the variance of qxa using ut ####

b_{u_{T}}=\frac{q_{X_{i}}^{2}(q_{X_{i}}^{2}-1)u_{T}}{\sqrt{\frac{-q_{X_{i}}^{2}}{q_{X_{i}}^{2}*(u_{T}^{2}-1)-u_{T}^{2}}}(q_{X_{i}}^{2}(u_{T}^{2}-1)-u_{T}^{2})^{2}}

b_{q_{X_{i}}}=\frac{q_{X_{i}}u_{T}^{2}}{\sqrt{\frac{q_{X_{i}}^{2}}{u_{T}^{2}-q_{X_{i}}^{2}(u_{T}^{2}-1)}}(u_{T}^{2}-q_{X_{i}}^{2}(u_{T}^{2}-1))^{2}}

#### Partial derivatives to estimate the variance of qxi using ux ####

Indirect range restriction:

b_{u_{X}}=\frac{1-qxa^{2}}{u_{X}^{3}\sqrt{\frac{q_{X_{a}}^{2}+u_{X}^{2}-1}{u_{X}^{2}}}}

b_{q_{X_{a}}}=\frac{q_{X_{a}}}{u_{X}^{2}\sqrt{\frac{q_{X_{a}-1}^{2}}{u_{X}^{2}}+1}}

Direct range restriction:

b_{u_{X}}=-\frac{q_{X_{a}}^{2}(q_{X_{a}}^{2}-1)u_{X}}{\sqrt{\frac{q_{X_{a}}^{2}u_{X}^{2}}{q_{X_{a}}^{2}(u_{X}^{2}-1)+1}}(q_{X_{a}}^{2}(u_{X}^{2}-1)+1)^{2}}

b_{q_{X_{a}}}=\frac{q_{X_{a}}u_{X}^{2}}{\sqrt{\frac{q_{X_{a}}^{2}u_{X}^{2}}{q_{X_{a}}^{2}(u_{X}^{2}-1)+1}}(q_{X_{a}}^{2}(u_{X}^{2}-1)+1)^{2}}

#### Partial derivatives to estimate the variance of rxxi using ux ####

Indirect range restriction:

b_{u_{X}}=\frac{2-2\rho_{XX_{a}}}{u_{X}^{3}}

b_{\rho_{XX_{a}}}=\frac{1}{u_{X}^{2}}

Direct range restriction:

b_{u_{X}}=-\frac{2(\rho_{XX_{a}}-1)\rho_{XX_{a}}u_{X}}{(\rho_{XX_{a}}(u_{X}^{2}-1)+1)^{2}}

b_{\rho_{XX_{a}}}=\frac{u_{X}^{2}}{(\rho_{XX_{a}}(u_{X}^{2}-1)+1)^{2}}

#### Partial derivatives to estimate the variance of rxxi using ut ####

u_{T}: b_{u_{T}}=-\frac{2(\rho_{XX_{a}}-1)\rho_{XX_{a}}u_{T}}{(\rho_{XX_{a}}(u_{T}^{2}-1)+1)^{2}}

b_{\rho_{XX_{a}}}=\frac{u_{T}^{2}}{(\rho_{XX_{a}}(u_{T}^{2}-1)+1)^{2}}

#### Partial derivatives to estimate the variance of qxi using ut ####

b_{u_{T}}=-\frac{(q_{X_{a}}-1)q_{X_{a}}^{2}(q_{X_{a}}+1)u_{T}}{\sqrt{\frac{q_{X_{a}}^{2}u_{T}^{2}}{q_{X_{a}}^{2}u_{T}^{2}-q_{X_{a}}^{2}+1}}(q_{X_{a}}^{2}u_{T}^{2}-q_{X_{a}}^{2}+1)^{2}}

b_{q_{X_{a}}}=\frac{q_{X_{a}}u_{T}^{2}}{\sqrt{\frac{q_{X_{a}}^{2}u_{T}^{2}}{q_{X_{a}}^{2}u_{T}^{2}-q_{X_{a}}^{2}+1}}(q_{X_{a}}^{2}u_{T}^{2}-q_{X_{a}}^{2}+1)^{2}}

#### Partial derivatives to estimate the variance of ut using qxi ####

b_{u_{X}}=\frac{q_{X_{i}}^{2}u_{X}}{\sqrt{\frac{q_{X_{i}}^{2}u_{X}^{2}}{(q_{X_{i}}^{2}-1)u_{X}^{2}+1}}((q_{X_{i}}^{2}-1)u_{X}^{2}+1)^{2}}

b_{q_{X_{i}}}=-\frac{u_{X}^{2}(u_{X}^{2}-1)}{\sqrt{\frac{q_{X_{i}}^{2}u_{X}^{2}}{(q_{X_{i}}^{2}-1)u_{X}^{2}+1}}((q_{X_{i}}^{2}-1)u_{X}^{2}+1)^{2}}

#### Partial derivatives to estimate the variance of ut using rxxi ####

b_{u_{X}}=\frac{\rho_{XX_{i}}u_{X}}{\sqrt{\frac{\rho_{XX_{i}}u_{X}^{2}}{(\rho_{XX_{i}}-1)u_{X}^{2}+1}}((\rho_{XX_{i}}-1)u_{X}^{2}+1)^{2}}

b_{\rho_{XX_{i}}}=-\frac{u_{X}^{2}(u_{X}^{2}-1)}{2\sqrt{\frac{\rho_{XX_{i}}u_{X}^{2}}{(\rho_{XX_{i}}-1)u_{X}^{2}+1}}((\rho_{XX_{i}}-1)u_{X}^{2}+1)^{2}}

#### Partial derivatives to estimate the variance of ut using qxa ####

b_{u_{X}}=\frac{u_{X}}{q_{X_{a}}^{2}\sqrt{\frac{q_{X_{a}}^{2}+u_{X}^{2}-1}{q_{X_{a}}^{2}}}}

b_{q_{X_{a}}}=\frac{1-u_{X}^{2}}{q_{X_{a}}^{3}\sqrt{\frac{q_{X_{a}}^{2}+u_{X}^{2}-1}{q_{X_{a}}^{2}}}}

#### Partial derivatives to estimate the variance of ut using rxxa ####

b_{u_{X}}=\frac{u_{X}}{\rho_{XX_{a}}\sqrt{\frac{\rho_{XX_{a}}+u_{X}^{2}-1}{\rho_{XX_{a}}}}}

b_{\rho_{XX_{a}}}=\frac{1-u_{X}^{2}}{2\rho_{XX_{a}}^{2}\sqrt{\frac{\rho_{XX_{a}}+u_{X}^{2}-1}{\rho_{XX_{a}}}}}

#### Partial derivatives to estimate the variance of ux using qxi ####

b_{u_{T}}=\frac{q_{X_{i}}^{2}u_{T}}{\sqrt{\frac{u_{T}^{2}}{u_{T}^{2}-q_{X_{i}}^{2}(u_{T}^{2}-1)}}(u_{T}^{2}-q_{X_{i}}^{2}(u_{T}^{2}-1))^{2}}

b_{q_{X_{i}}}=\frac{q_{X_{i}}(u_{T}^{2}-1)\left(\frac{u_{T}^{2}}{u_{T}^{2}-q_{X_{i}}^{2}(u_{T}^{2}-1)}\right)^{1.5}}{u_{T}^{2}}

#### Partial derivatives to estimate the variance of ux using rxxi ####

b_{u_{T}}=\frac{\rho_{XX_{i}}u_{T}}{\sqrt{\frac{u_{T}^{2}}{-\rho_{XX_{i}}u_{T}^{2}+\rho_{XX_{i}}+u_{T}^{2}}}(-\rho_{XX_{i}}u_{T}^{2}+\rho_{XX_{i}}+u_{T}^{2})^{2}}

b_{\rho_{XX_{i}}}=\frac{(u_{T}^{2}-1)\left(\frac{u_{T}^{2}}{-\rho_{XX_{i}}u_{T}^{2}+\rho_{XX_{i}}+u_{T}^{2}}\right)^{1.5}}{2u_{T}^{2}}

#### Partial derivatives to estimate the variance of ux using qxa ####

b_{u_{T}}=\frac{q_{X_{a}}^{2}u_{T}}{\sqrt{q_{X_{a}}^{2}(u_{T}^{2}-1)+1}}

b_{q_{X_{a}}}=\frac{q_{X_{a}}(u_{T}-1)}{\sqrt{q_{X_{a}}^{2}(u_{T}^{2}-1)+1}}

#### Partial derivatives to estimate the variance of ux using rxxa ####

b_{u_{T}}=\frac{\rho_{XX_{a}}u_{T}}{\sqrt{\rho_{XX_{a}}(u_{T}^{2}-1)+1}}

b_{\rho_{XX_{a}}}=\frac{u_{T}^{2}-1}{2\sqrt{\rho_{XX_{a}}(u_{T}^{2}-1)+1}}

#### Partial derivatives to estimate the variance of ryya ####

b_{\rho_{YY_{i}}}=\frac{1}{\rho_{XY_{i}}^{2}\left(\frac{1}{u_{X}^{2}}-1\right)+1}

b_{u_{X}}=\frac{2(\rho_{YY_{i}}-1)\rho_{XY_{i}}^{2}u_{X}}{(u_{X}^{2}-\rho_{XY_{i}}^{2}(u_{X}^{2}-1))^{2}}

b_{\rho_{XY_{i}}}=\frac{2(\rho_{YY_{i}}-1)\rho_{XY_{i}}u_{X}^{2}(u_{X}^{2}-1)}{(u_{X}^{2}-\rho_{XY_{i}}^{2}(u_{X}^{2}-1))^{2}}

#### Partial derivatives to estimate the variance of qya ####

b_{q_{Y_{i}}}=\frac{q_{Y_{i}}}{\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]\sqrt{1-\frac{1-q_{Y_{i}}^{2}}{1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)}}}

b_{u_{X}}=-\frac{(1-q_{Y_{i}}^{2})\rho_{XY_{i}}^{2}}{u_{X}^{3}\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]\sqrt{1-\frac{1-q_{Y_{i}}^{2}}{1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)}}}

b_{\rho_{XY_{i}}}=-\frac{(1-q_{Y_{i}}^{2})\rho_{XY_{i}}\left(1-\frac{1}{u_{X}^{2}}\right)}{\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]\sqrt{1-\frac{1-q_{Y_{i}}^{2}}{1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)}}}

#### Partial derivatives to estimate the variance of ryyi ####

\rho_{YY_{a}}: b_{\rho_{YY_{a}}}=\rho_{XY_{i}}^{2}\left(\frac{1}{u_{X}^{2}}-1\right)+1

b_{u_{X}}=-\frac{2(\rho_{YY_{a}}-1)\rho_{XY_{i}}^{2}}{u_{X}^{3}}

b_{\rho_{XY_{i}}}=-\frac{2(\rho_{YY_{a}}-1)\rho_{XY_{i}}(u_{X}^{2}-1)}{u_{X}^{2}}

#### Partial derivatives to estimate the variance of qyi ####

b_{q_{Y_{a}}}=\frac{q_{Y_{a}}\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]}{\sqrt{1-\left(1-q_{Y_{a}}\right)\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]}}

b_{u_{X}}=\frac{(1-q_{Y_{a}}^{2})\rho_{XY_{i}}\left(1-\frac{1}{u_{X}^{2}}\right)}{\sqrt{1-\left(1-q_{Y_{a}}\right)\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]}}

b_{\rho_{XY_{i}}}=\frac{(1-q_{Y_{a}}^{2})\rho_{XY_{i}}^{2}}{u_{X}^{3}\sqrt{1-\left(1-q_{Y_{a}}\right)\left[1-\rho_{XY_{i}}^{2}\left(1-\frac{1}{u_{X}^{2}}\right)\right]}}

Examples

estimate_var_qxi(qxa = c(.8, .85, .9, .95), var_qxa = c(.02, .03, .04, .05),
                 ux = .8, var_ux = 0,
                 ux_observed = c(TRUE, TRUE, FALSE, FALSE),
                 indirect_rr = c(TRUE, FALSE, TRUE, FALSE))
estimate_var_qxa(qxi = c(.8, .85, .9, .95), var_qxi = c(.02, .03, .04, .05),
                 ux = .8, var_ux = 0,
                 ux_observed = c(TRUE, TRUE, FALSE, FALSE),
                 indirect_rr = c(TRUE, FALSE, TRUE, FALSE))
estimate_var_rxxi(rxxa = c(.8, .85, .9, .95),
                  var_rxxa = c(.02, .03, .04, .05), ux = .8, var_ux = 0,
                 ux_observed = c(TRUE, TRUE, FALSE, FALSE),
                 indirect_rr = c(TRUE, FALSE, TRUE, FALSE))
estimate_var_rxxa(rxxi = c(.8, .85, .9, .95), var_rxxi = c(.02, .03, .04, .05),
                  ux = .8, var_ux = 0,
                 ux_observed = c(TRUE, TRUE, FALSE, FALSE),
                 indirect_rr = c(TRUE, FALSE, TRUE, FALSE))
estimate_var_ut(rxx = c(.8, .85, .9, .95), var_rxx = 0,
                ux = c(.8, .8, .9, .9), var_ux = c(.02, .03, .04, .05),
                 rxx_restricted = c(TRUE, TRUE, FALSE, FALSE),
                rxx_as_qx = c(TRUE, FALSE, TRUE, FALSE))
estimate_var_ux(rxx = c(.8, .85, .9, .95), var_rxx = 0,
                ut = c(.8, .8, .9, .9), var_ut = c(.02, .03, .04, .05),
                 rxx_restricted = c(TRUE, TRUE, FALSE, FALSE),
                rxx_as_qx = c(TRUE, FALSE, TRUE, FALSE))
estimate_var_ryya(ryyi = .9, var_ryyi = .04, rxyi = .4, var_rxyi = 0, ux = .8, var_ux = 0)
estimate_var_ryya(ryyi = .9, var_ryyi = .04, rxyi = .4, var_rxyi = 0, ux = .8, var_ux = 0)
estimate_var_qyi(qya = .9, var_qya = .04, rxyi = .4, var_rxyi = 0, ux = .8, var_ux = 0)
estimate_var_ryyi(ryya = .9, var_ryya = .04, rxyi = .4, var_rxyi = 0, ux = .8, var_ux = 0)

Non-linear estimate of variance of \rho corrected for psychometric artifacts using numeric integration

Description

Functions to estimate the variance of \rho corrected for psychometric artifacts. These functions integrate over the residual distribution of correlations from an interactive artifact-distribution meta-analysis to non-linearly estimate the variance of \rho.

Available functions include:

• estimate_var_rho_int_meas: Variance of \rho corrected for measurement error only

• estimate_var_rho_int_uvdrr: Variance of \rho corrected for univariate direct range restriction (i.e., Case II) and measurement error

• estimate_var_rho_int_bvdrr: Variance of \rho corrected for bivariate direct range restriction and measurement error

• estimate_var_rho_int_uvirr: Variance of \rho corrected for univariate indirect range restriction (i.e., Case IV) and measurement error

• estimate_var_rho_int_bvirr: Variance of \rho corrected for bivariate indirect range restriction (i.e., Case V) and measurement error

• estimate_var_rho_int_rb: Variance of \rho corrected using Raju and Burke's correction for direct range restriction and measurement error

Usage

estimate_var_rho_int_meas(mean_qx, mean_qy, var_res)

estimate_var_rho_int_uvdrr(
  mean_rxyi,
  mean_rtpa,
  mean_qxa,
  mean_qyi,
  mean_ux,
  var_res
)

estimate_var_rho_int_uvirr(
  mean_rxyi,
  mean_rtpa,
  mean_qxi,
  mean_qyi,
  mean_ut,
  var_res
)

estimate_var_rho_int_bvirr(mean_qxa, mean_qya, mean_ux, mean_uy, var_res)

estimate_var_rho_int_bvdrr(
  mean_rxyi,
  mean_rtpa,
  mean_qxa,
  mean_qya,
  mean_ux,
  mean_uy,
  var_res
)

estimate_var_rho_int_rb(
  mean_rxyi,
  mean_rtpa,
  mean_qx,
  mean_qy,
  mean_ux,
  var_res
)

Arguments

Value

A vector of non-linear estimates of the variance of rho.

Notes

estimate_var_rho_int_meas and estimate_var_rho_int_bvirr do not make use of numeric integration because they are linear functions.

References

Law, K. S., Schmidt, F. L., & Hunter, J. E. (1994). Nonlinearity of range corrections in meta-analysis: Test of an improved procedure. Journal of Applied Psychology, 79(3), 425–438. doi:10.1037/0021-9010.79.3.425

Taylor Series Approximation of variance of \rho corrected for psychometric artifacts

Description

Functions to estimate the variance of \rho corrected for psychometric artifacts. These functions use Taylor series approximations (i.e., the delta method) to estimate the variance in observed effect sizes predictable from the variance in artifact distributions based on the partial derivatives.

The available Taylor-series functions include:

• estimate_var_rho_tsa_meas: Variance of \rho corrected for measurement error only

• estimate_var_rho_tsa_uvdrr: Variance of \rho corrected for univariate direct range restriction (i.e., Case II) and measurement error

• estimate_var_rho_tsa_bvdrr: Variance of \rho corrected for bivariate direct range restriction and measurement error

• estimate_var_rho_tsa_uvirr: Variance of \rho corrected for univariate indirect range restriction (i.e., Case IV) and measurement error

• estimate_var_rho_tsa_bvirr: Variance of \rho corrected for bivariate indirect range restriction (i.e., Case V) and measurement error

• estimate_var_rho_tsa_rb1: Variance of \rho corrected using Raju and Burke's TSA1 correction for direct range restriction and measurement error

• estimate_var_rho_tsa_rb2: Variance of \rho corrected using Raju and Burke's TSA2 correction for direct range restriction and measurement error. Note that a typographical error in Raju and Burke's article has been corrected in this function so as to compute appropriate partial derivatives.

Usage

estimate_var_rho_tsa_meas(
  mean_rtp,
  var_rxy,
  var_e,
  mean_qx = 1,
  var_qx = 0,
  mean_qy = 1,
  var_qy = 0,
  ...
)

estimate_var_rho_tsa_uvdrr(
  mean_rtpa,
  var_rxyi,
  var_e,
  mean_ux = 1,
  var_ux = 0,
  mean_qxa = 1,
  var_qxa = 0,
  mean_qyi = 1,
  var_qyi = 0,
  ...
)

estimate_var_rho_tsa_bvdrr(
  mean_rtpa,
  var_rxyi,
  var_e = 0,
  mean_ux = 1,
  var_ux = 0,
  mean_uy = 1,
  var_uy = 0,
  mean_qxa = 1,
  var_qxa = 0,
  mean_qya = 1,
  var_qya = 0,
  ...
)

estimate_var_rho_tsa_uvirr(
  mean_rtpa,
  var_rxyi,
  var_e,
  mean_ut = 1,
  var_ut = 0,
  mean_qxa = 1,
  var_qxa = 0,
  mean_qyi = 1,
  var_qyi = 0,
  ...
)

estimate_var_rho_tsa_bvirr(
  mean_rtpa,
  var_rxyi,
  var_e = 0,
  mean_ux = 1,
  var_ux = 0,
  mean_uy = 1,
  var_uy = 0,
  mean_qxa = 1,
  var_qxa = 0,
  mean_qya = 1,
  var_qya = 0,
  sign_rxz = 1,
  sign_ryz = 1,
  ...
)

estimate_var_rho_tsa_rb1(
  mean_rtpa,
  var_rxyi,
  var_e,
  mean_ux = 1,
  var_ux = 0,
  mean_rxx = 1,
  var_rxx = 0,
  mean_ryy = 1,
  var_ryy = 0,
  ...
)

estimate_var_rho_tsa_rb2(
  mean_rtpa,
  var_rxyi,
  var_e,
  mean_ux = 1,
  var_ux = 0,
  mean_qx = 1,
  var_qx = 0,
  mean_qy = 1,
  var_qy = 0,
  ...
)

Arguments

Details

######## Measurement error only ########

The attenuation formula for measurement error is

\rho_{XY}=\rho_{TP}q_{X}q_{Y}

where \rho_{XY} is an observed correlation, \rho_{TP} is a true-score correlation, and q_{X} and q_{Y} are the square roots of reliability coefficients for X and Y, respectively.

The Taylor series approximation of the variance of \rho_{TP} can be computed using the following linear equation,

var_{\rho_{TP}} \approx \left[var_{r_{XY}}-var_{e}-\left(b_{1}^{2}var_{q_{X}}+b_{2}^{2}var_{q_{Y}}\right)\right]/b_{3}^{2}

where b_{1}, b_{2}, and b_{3} are first-order partial derivatives of the attenuation formula with respect to q_{X}, q_{Y}, and \rho_{TP}, respectively. The first-order partial derivatives of the attenuation formula are:

b_{1}=\frac{\partial\rho_{XY}}{\partial q_{X}}=\rho_{TP}q_{Y}

b_{2}=\frac{\partial\rho_{XY}}{\partial q_{Y}}=\rho_{TP}q_{X}

b_{3}=\frac{\partial\rho_{XY}}{\partial\rho_{TP}}=q_{X}q_{Y}

######## Univariate direct range restriction (UVDRR; i.e., Case II) ########

The UVDRR attenuation procedure may be represented as

\rho_{XY_{i}}=\frac{\rho_{TP_{a}}q_{Y_{i}}q_{X_{a}}u_{X}}{\sqrt{\rho_{TP_{a}}^{2}q_{X_{a}}^{2}\left(u_{X}^{2}-1\right)+1}}

The attenuation formula can also be represented as:

\rho_{XY_{i}}=\rho_{TP_{a}}q_{Y_{i}}q_{X_{a}}u_{X}A

where

A=\frac{1}{\sqrt{\rho_{TP_{a}}^{2}q_{X_{a}}^{2}\left(u_{X}^{2}-1\right)+1}}

The Taylor series approximation of the variance of \rho_{TP_{a}} can be computed using the following linear equation,

var_{\rho_{TP_{a}}} \approx \left[var_{r_{XY_{i}}}-var_{e}-\left(b_{1}^{2}var_{q_{X_{a}}}+b_{2}^{2}var_{q_{Y_{i}}}+b_{3}^{2}var_{u_{X}}\right)\right]/b_{4}^{2}

where b_{1}, b_{2}, b_{3}, and b_{4} are first-order partial derivatives of the attenuation formula with respect to q_{X_{a}}, q_{Y_{i}}, u_{X}, and \rho_{TP_{a}}, respectively. The first-order partial derivatives of the attenuation formula are:

b_{1}=\frac{\partial\rho_{XY_{i}}}{\partial q_{X_{a}}}=\rho_{TP_{a}}q_{Y_{i}}u_{X}A^{3}

b_{2}=\frac{\partial\rho_{XY_{i}}}{\partial q_{Y_{i}}}=\frac{\rho_{XY_{i}}}{q_{Y_{i}}}

b_{3}=\frac{\partial\rho_{XY_{i}}}{\partial u_{X}}=-\rho_{TP_{a}}q_{Y_{i}}q_{X_{a}}\left(\rho_{TP_{a}}^{2}q_{X_{a}}^{2}-1\right)A^{3}

b_{4}=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{TP_{a}}}=q_{Y_{i}}q_{X_{a}}u_{X}A^{3}

######## Univariate indirect range restriction (UVIRR; i.e., Case IV) ########

Under univariate indirect range restriction, the attenuation formula yielding \rho_{XY_{i}} is:

\rho_{XY_{i}}=\frac{u_{T}q_{X_{a}}}{\sqrt{u_{T}^{2}q_{X_{a}}^{2}+1-q_{X_{a}}^{2}}}\frac{u_{T}\rho_{TP_{a}}}{\sqrt{u_{T}^{2}\rho_{TP_{a}}^{2}+1-\rho_{TP_{a}}^{2}}}

The attenuation formula can also be represented as:

\rho_{XY_{i}}=q_{X_{a}}q_{Y_{i}}\rho_{TP_{a}}u_{T}^{2}AB

where

A=\frac{1}{\sqrt{u_{T}^{2}q_{X_{a}}^{2}+1-q_{X_{a}}^{2}}}

and

B=\frac{1}{\sqrt{u_{T}^{2}\rho_{TP_{a}}^{2}+1-\rho_{TP_{a}}^{2}}}

The Taylor series approximation of the variance of \rho_{TP_{a}} can be computed using the following linear equation,

var_{\rho_{TP_{a}}} \approx \left[var_{r_{XY_{i}}}-var_{e}-\left(b_{1}^{2}var_{q_{X_{a}}}+b_{2}^{2}var_{q_{Y_{i}}}+b_{3}^{2}var_{u_{T}}\right)\right]/b_{4}^{2}

where b_{1}, b_{2}, b_{3}, and b_{4} are first-order partial derivatives of the attenuation formula with respect to q_{X_{a}}, q_{Y_{i}}, u_{T}, and \rho_{TP_{a}}, respectively. The first-order partial derivatives of the attenuation formula are:

b_{1}=\frac{\partial\rho_{XY_{i}}}{\partial q_{X_{a}}}=\frac{\rho_{XY_{i}}}{q_{X_{a}}}-\rho_{XY_{i}}q_{X_{a}}B^{2}\left(u_{T}^{2}-1\right)

b_{2}=\frac{\partial\rho_{XY_{i}}}{\partial q_{Y_{i}}}=\frac{\rho_{XY_{i}}}{q_{Y_{i}}}

b_{3}=\frac{\partial\rho_{XY_{i}}}{\partial u_{T}}=\frac{2\rho_{XY_{i}}}{u_{T}}-\rho_{XY_{i}}u_{T}q_{X_{a}}^{2}B^{2}-\rho_{XY_{i}}u_{T}\rho_{TP_{a}}^{2}A^{2}

b_{4}=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{TP_{a}}}=\frac{\rho_{XY_{i}}}{\rho_{TP_{a}}}-\rho_{XY_{i}}\rho_{TP_{a}}A^{2}\left(u_{T}^{2}-1\right)

######## Bivariate direct range restriction (BVDRR) ########

Under bivariate direct range restriction, the attenuation formula yielding \rho_{XY_{i}} is:

\rho_{XY_{i}}=\frac{A+\rho_{TP_{a}}^{2}q_{X_{a}}q_{Y_{a}}-\frac{1}{q_{X_{a}}q_{Y_{a}}}}{2\rho_{TP_{a}}u_{X}u_{Y}}

where

A=\sqrt{\left(\frac{1}{q_{X_{a}}q_{Y_{a}}}-\rho_{TP_{a}}^{2}q_{X_{a}}q_{Y_{a}}\right)^{2}+4\rho_{TP_{a}}u_{X}^{2}u_{Y}^{2}}

The Taylor series approximation of the variance of \rho_{TP_{a}} can be computed using the following linear equation,

var_{\rho_{TP_{a}}} \approx \left[var_{r_{XY_{i}}}-var_{e}-\left(b_{1}^{2}var_{q_{X_{a}}}+b_{2}^{2}var_{q_{Y_{i}}}+b_{3}^{2}var_{u_{X}}+b_{4}^{2}var_{u_{Y}}\right)\right]/b_{5}^{2}

where b_{1}, b_{2}, b_{3}, b_{4}, and b_{5} are first-order partial derivatives of the attenuation formula with respect to q_{X_{a}}, q_{Y_{a}}, u_{X}, u_{Y}, and \rho_{TP_{a}}, respectively. First, we define terms to simplify the computation of partial derivatives:

B=\left(\rho_{TP_{a}}^{2}q_{X_{a}}^{2}q_{Y_{a}}^{2}+q_{X_{a}}q_{Y_{a}}A-1\right)

C=2\rho_{TP_{a}}q_{X_{a}}^{2}q_{Y_{a}}^{2}u_{X}u_{Y}A

The first-order partial derivatives of the attenuation formula are:

b_{1}=\frac{\partial\rho_{XY_{i}}}{\partial q_{X_{a}}}=\frac{\left(\rho_{TP_{a}}^{2}q_{X_{a}}^{2}q_{Y_{a}}^{2}+1\right)B}{q_{X_{a}}C}

b_{2}=\frac{\partial\rho_{XY_{i}}}{\partial q_{Y_{i}}}=\frac{\left(\rho_{TP_{a}}^{2}q_{X_{a}}^{2}q_{Y_{a}}^{2}+1\right)B}{q_{Y_{a}}C}

b_{3}=\frac{\partial\rho_{XY_{i}}}{\partial u_{X}}=-\frac{\left(\rho_{TP_{a}}q_{X_{a}}q_{Y_{a}}-1\right)\left(\rho_{TP_{a}}q_{X_{a}}q_{Y_{a}}+1\right)B}{u_{X}C}

b_{4}=\frac{\partial\rho_{XY_{i}}}{\partial u_{Y}}=-\frac{\left(\rho_{TP_{a}}q_{X_{a}}q_{Y_{a}}-1\right)\left(\rho_{TP_{a}}q_{X_{a}}q_{Y_{a}}+1\right)B}{u_{Y}C}

b_{5}=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{TP_{a}}}=\frac{\left(\rho_{TP_{a}}^{2}q_{X_{a}}^{2}q_{Y_{a}}^{2}+1\right)B}{\rho_{TP_{a}}C}

######## Bivariate indirect range restriction (BVIRR; i.e., Case V) ########

Under bivariate indirect range restriction, the attenuation formula yielding \rho_{XY_{i}} is:

\rho_{XY_{i}}=\frac{\rho_{TP_{a}}q_{X_{a}}q_{Y_{a}}-\lambda\sqrt{\left|1-u_{X}^{2}\right|\left|1-u_{Y}^{2}\right|}}{u_{X}u_{Y}}

The Taylor series approximation of the variance of \rho_{TP_{a}} can be computed using the following linear equation,

var_{\rho_{TP_{a}}} \approx \left[var_{r_{XY_{i}}}-var_{e}-\left(b_{1}^{2}var_{q_{X_{a}}}+b_{2}^{2}var_{q_{Y_{i}}}+b_{3}^{2}var_{u_{X}}+b_{4}^{2}var_{u_{Y}}\right)\right]/b_{5}^{2}

where b_{1}, b_{2}, b_{3}, b_{4}, and b_{5} are first-order partial derivatives of the attenuation formula with respect to q_{X_{a}}, q_{Y_{a}}, u_{X}, u_{Y}, and \rho_{TP_{a}}, respectively. First, we define terms to simplify the computation of partial derivatives:

b_{1}=\frac{\partial\rho_{XY_{i}}}{\partial q_{X_{a}}}=\frac{\rho_{TP_{a}}q_{Y_{a}}}{u_{X}u_{Y}}

b_{2}=\frac{\partial\rho_{XY_{i}}}{\partial q_{Y_{i}}}=\frac{\rho_{TP_{a}}q_{X_{a}}}{u_{X}u_{Y}}

b_{3}=\frac{\partial\rho_{XY_{i}}}{\partial u_{X}}=\frac{\lambda\left(1-u_{X}^{2}\right)\sqrt{\left|1-u_{Y}^{2}\right|}}{u_{Y}\left|1-u_{X}^{2}\right|^{1.5}}-\frac{\rho_{XY_{i}}}{u_{X}}

b_{4}=\frac{\partial\rho_{XY_{i}}}{\partial u_{Y}}=\frac{\lambda\left(1-u_{Y}^{2}\right)\sqrt{\left|1-u_{X}^{2}\right|}}{u_{X}\left|1-u_{Y}^{2}\right|^{1.5}}-\frac{\rho_{XY_{i}}}{u_{Y}}

b_{5}=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{TP_{a}}}=\frac{q_{X_{a}}q_{Y_{a}}}{u_{X}u_{Y}}

######## Raju and Burke's TSA1 procedure ########

Raju and Burke's attenuation formula may be represented as

\rho_{XY_{i}}=\frac{\rho_{TP_{a}}u_{X}\sqrt{\rho_{XX_{a}}\rho_{YY_{a}}}}{\sqrt{\rho_{TP_{a}}^{2}\rho_{XX_{a}}\rho_{YY_{a}}u_{X}^{2}-\rho_{TP_{a}}^{2}\rho_{XX_{a}}\rho_{YY_{a}}+1}}

The Taylor series approximation of the variance of \rho_{TP_{a}} can be computed using the following linear equation,

var_{\rho_{TP_{a}}} \approx \left[var_{r_{XY_{i}}}-var_{e}-\left(B^{2}var_{\rho_{YY_{a}}}+C^{2}var_{\rho_{XX_{a}}}+D^{2}var_{u_{X}}\right)\right]/A^{2}

where A, B, C, and D are first-order partial derivatives of the attenuation formula with respect to \rho_{TP_{a}}, \rho_{XX_{a}}, \rho_{YY_{a}}, and u_{X}, respectively. The first-order partial derivatives of the attenuation formula are:

A=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{TP_{a}}}=\frac{\rho_{XY_{i}}}{\rho_{TP_{a}}}+\frac{\rho_{XY_{i}\left(1-u_{X}^{2}\right)}^{3}}{\rho_{TP_{a}}u_{X}^{2}}

B=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{YY_{a}}}=\frac{1}{2}\left(\frac{\rho_{XY_{i}}}{\rho_{YY_{a}}}+\frac{\rho_{XY_{i}\left(1-u_{X}^{2}\right)}^{3}}{\rho_{YY_{a}}u_{X}^{2}}\right)

C=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{XX_{a}}}=\frac{1}{2}\left(\frac{\rho_{XY_{i}}}{\rho_{XX_{a}}}+\frac{\rho_{XY_{i}\left(1-u_{X}^{2}\right)}^{3}}{\rho_{XX_{a}}u_{X}^{2}}\right)

D=\frac{\partial\rho_{XY_{i}}}{\partial u_{X}}=\frac{\rho_{XY_{i}}-\rho_{XY_{i}}^{3}}{u_{X}}

######## Raju and Burke's TSA2 procedure ########

Raju and Burke's attenuation formula may be represented as

\rho_{XY_{i}}=\frac{\rho_{TP_{a}}q_{X_{a}}q_{Y_{a}}u_{X}}{\sqrt{\rho_{TP_{a}}^{2}q_{X_{a}}^{2}q_{Y_{a}}^{2}u_{X}^{2}-\rho_{TP_{a}}^{2}q_{X_{a}}^{2}q_{Y_{a}}^{2}+1}}

The Taylor series approximation of the variance of \rho_{TP_{a}} can be computed using the following linear equation,

var_{\rho_{TP_{a}}} \approx \left[var_{r_{XY_{i}}}-var_{e}-\left(F^{2}var_{q_{Y_{a}}}+G^{2}var_{q_{X_{a}}}+H^{2}var_{u_{X}}\right)\right]/E^{2}

where E, F, G, and H are first-order partial derivatives of the attenuation formula with respect to \rho_{TP_{a}}, q_{X_{a}}, q_{Y_{a}}, and u_{X}, respectively. The first-order partial derivatives of the attenuation formula (with typographic errors in the original article corrected) are:

E=\frac{\partial\rho_{XY_{i}}}{\partial\rho_{TP_{a}}}=\frac{\rho_{XY_{i}}}{\rho_{TP_{a}}}+\frac{\rho_{XY_{i}\left(1-u_{X}^{2}\right)}^{3}}{\rho_{TP_{a}}u_{X}^{2}}

F=\frac{\partial\rho_{XY_{i}}}{\partial q_{Y_{a}}}=\frac{\rho_{XY_{i}}}{q_{Y_{a}}}+\frac{\rho_{XY_{i}\left(1-u_{X}^{2}\right)}^{3}}{q_{Y_{a}}u_{X}^{2}}

G=\frac{\partial\rho_{XY_{i}}}{\partial q_{X_{a}}}=\frac{\rho_{XY_{i}}}{q_{X_{a}}}+\frac{\rho_{XY_{i}\left(1-u_{X}^{2}\right)}^{3}}{q_{X_{a}}u_{X}^{2}}

H=\frac{\partial\rho_{XY_{i}}}{\partial u_{X}}=\frac{\rho_{XY_{i}}-\rho_{XY_{i}}^{3}}{u_{X}}

Value

Vector of meta-analytic variances estimated via Taylor series approximation.

Notes

A typographical error in Raju and Burke's article has been corrected in estimate_var_rho_tsa_rb2 so as to compute appropriate partial derivatives.

References

Dahlke, J. A., & Wiernik, B. M. (2020). Not restricted to selection research: Accounting for indirect range restriction in organizational research. Organizational Research Methods, 23(4), 717–749. doi:10.1177/1094428119859398

Hunter, J. E., Schmidt, F. L., & Le, H. (2006). Implications of direct and indirect range restriction for meta-analysis methods and findings. Journal of Applied Psychology, 91(3), 594–612. doi:10.1037/0021-9010.91.3.594

Raju, N. S., & Burke, M. J. (1983). Two new procedures for studying validity generalization. Journal of Applied Psychology, 68(3), 382–395. doi:10.1037/0021-9010.68.3.382

Examples

estimate_var_rho_tsa_meas(mean_rtp = .5, var_rxy = .02, var_e = .01,
                 mean_qx = .8, var_qx = .005,
                 mean_qy = .8, var_qy = .005)
estimate_var_rho_tsa_uvdrr(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
                  mean_ux = .8, var_ux = .005,
                  mean_qxa = .8, var_qxa = .005,
                  mean_qyi = .8, var_qyi = .005)
estimate_var_rho_tsa_bvdrr(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
                  mean_ux = .8, var_ux = .005,
                  mean_uy = .8, var_uy = .005,
                  mean_qxa = .8, var_qxa = .005,
                  mean_qya = .8, var_qya = .005)
estimate_var_rho_tsa_uvirr(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
                  mean_ut = .8, var_ut = .005,
                  mean_qxa = .8, var_qxa = .005,
                  mean_qyi = .8, var_qyi = .005)
estimate_var_rho_tsa_bvirr(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
                  mean_ux = .8, var_ux = .005,
                  mean_uy = .8, var_uy = .005,
                  mean_qxa = .8, var_qxa = .005,
                  mean_qya = .8, var_qya = .005,
                  sign_rxz = 1, sign_ryz = 1)
estimate_var_rho_tsa_rb1(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
                mean_ux = .8, var_ux = .005,
                mean_rxx = .8, var_rxx = .005,
                mean_ryy = .8, var_ryy = .005)
estimate_var_rho_tsa_rb2(mean_rtpa = .5, var_rxyi = .02, var_e = .01,
                mean_ux = .8, var_ux = .005,
                mean_qx = .8, var_qx = .005,
                mean_qy = .8, var_qy = .005)

Taylor Series Approximation of effect-size variances corrected for psychometric artifacts

Description

Functions to estimate the variances corrected for psychometric artifacts. These functions use Taylor series approximations (i.e., the delta method) to estimate the corrected variance of an effect-size distribution.

The available Taylor-series functions include:

• estimate_var_tsa_meas: Variance of \rho corrected for measurement error only

• estimate_var_tsa_uvdrr: Variance of \rho corrected for univariate direct range restriction (i.e., Case II) and measurement error

• estimate_var_tsa_bvdrr: Variance of \rho corrected for bivariate direct range restriction and measurement error

• estimate_var_tsa_uvirr: Variance of \rho corrected for univariate indirect range restriction (i.e., Case IV) and measurement error

• estimate_var_tsa_bvirr: Variance of \rho corrected for bivariate indirect range restriction (i.e., Case V) and measurement error

• estimate_var_tsa_rb1: Variance of \rho corrected using Raju and Burke's TSA1 correction for direct range restriction and measurement error

• estimate_var_tsa_rb2: Variance of \rho corrected using Raju and Burke's TSA2 correction for direct range restriction and measurement error. Note that a typographical error in Raju and Burke's article has been corrected in this function so as to compute appropriate partial derivatives.

Usage

estimate_var_tsa_meas(mean_rtp, var = 0, mean_qx = 1, mean_qy = 1, ...)

estimate_var_tsa_uvdrr(
  mean_rtpa,
  var = 0,
  mean_ux = 1,
  mean_qxa = 1,
  mean_qyi = 1,
  ...
)

estimate_var_tsa_bvdrr(
  mean_rtpa,
  var = 0,
  mean_ux = 1,
  mean_uy = 1,
  mean_qxa = 1,
  mean_qya = 1,
  ...
)

estimate_var_tsa_uvirr(
  mean_rtpa,
  var = 0,
  mean_ut = 1,
  mean_qxa = 1,
  mean_qyi = 1,
  ...
)

estimate_var_tsa_bvirr(
  mean_rtpa,
  var = 0,
  mean_ux = 1,
  mean_uy = 1,
  mean_qxa = 1,
  mean_qya = 1,
  sign_rxz = 1,
  sign_ryz = 1,
  ...
)

estimate_var_tsa_rb1(
  mean_rtpa,
  var = 0,
  mean_ux = 1,
  mean_rxx = 1,
  mean_ryy = 1,
  ...
)

estimate_var_tsa_rb2(
  mean_rtpa,
  var = 0,
  mean_ux = 1,
  mean_qx = 1,
  mean_qy = 1,
  ...
)

Arguments

Value

Vector of variances corrected for mean artifacts via Taylor series approximation.

Notes

A typographical error in Raju and Burke's article has been corrected in estimate_var_tsa_rb2() so as to compute appropriate partial derivatives.

References

Dahlke, J. A., & Wiernik, B. M. (2020). Not restricted to selection research: Accounting for indirect range restriction in organizational research. Organizational Research Methods, 23(4), 717–749. doi:10.1177/1094428119859398

Hunter, J. E., Schmidt, F. L., & Le, H. (2006). Implications of direct and indirect range restriction for meta-analysis methods and findings. Journal of Applied Psychology, 91(3), 594–612. doi:10.1037/0021-9010.91.3.594

Raju, N. S., & Burke, M. J. (1983). Two new procedures for studying validity generalization. Journal of Applied Psychology, 68(3), 382–395. doi:10.1037/0021-9010.68.3.382

Examples

estimate_var_tsa_meas(mean_rtp = .5, var = .02,
                 mean_qx = .8,
                 mean_qy = .8)
estimate_var_tsa_uvdrr(mean_rtpa = .5, var = .02,
                  mean_ux = .8,
                  mean_qxa = .8,
                  mean_qyi = .8)
estimate_var_tsa_bvdrr(mean_rtpa = .5, var = .02,
                  mean_ux = .8,
                  mean_uy = .8,
                  mean_qxa = .8,
                  mean_qya = .8)
estimate_var_tsa_uvirr(mean_rtpa = .5, var = .02,
                  mean_ut = .8,
                  mean_qxa = .8,
                  mean_qyi = .8)
estimate_var_tsa_bvirr(mean_rtpa = .5, var = .02,
                  mean_ux = .8,
                  mean_uy = .8,
                  mean_qxa = .8,
                  mean_qya = .8,
                  sign_rxz = 1, sign_ryz = 1)
estimate_var_tsa_rb1(mean_rtpa = .5, var = .02,
                mean_ux = .8,
                mean_rxx = .8,
                mean_ryy = .8)
estimate_var_tsa_rb2(mean_rtpa = .5, var = .02,
                mean_ux = .8,
                mean_qx = .8,
                mean_qy = .8)

Filter a list to remove NULL entries

Description

Filter a list to remove NULL entries

Usage

filter_listnonnull(x)

Arguments

Value

A list with NULL entries removed.

Filter meta-analyses

Description

Filter psychmeta meta-analysis objects based on specified criteria.

Usage

filter_ma(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  ...
)

filter_meta(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  ...
)

Arguments

Value

A psychmeta meta-analysis object with analyses matching the specified criteria.

Examples

ma_obj <- ma_r(ma_method = "ic", rxyi = rxyi, n = n, rxx = rxxi, ryy = ryyi,
               construct_x = x_name, construct_y = y_name, sample_id = sample_id, citekey = NULL,
               moderators = moderator, data = data_r_meas_multi,
               impute_artifacts = FALSE, clean_artifacts = FALSE)
ma_obj <- ma_r_ad(ma_obj, correct_rr_x = FALSE, correct_rr_y = FALSE)

filter_ma(ma_obj, analyses="all")
filter_ma(ma_obj, analyses=list(construct="X"), match="all")
filter_ma(ma_obj, analyses=list(construct="X", k_min=21), match="any")
filter_ma(ma_obj, analyses=list(construct="X", k_min=21), match="all")

Create long-format datasets in simulate_r_database

Description

Create long-format datasets in simulate_r_database

Usage

format_long(
  x,
  param,
  sample_id,
  var_names,
  show_applicant,
  decimals = 2,
  noalpha = FALSE
)

Arguments

Value

A dataframe of results

Format numbers for presentation

Description

A function to format numbers and logical values as characters for display purposes. Includes control over formatting of decimal digits, leading zeros, sign characters, and characters to replace logical, NA, NaN, and Inf values. Factors are converted to strings. Strings are returned verbatim.

Usage

format_num(x, digits = 2L, decimal.mark = getOption("OutDec"),
              leading0 = "conditional", drop0integer = FALSE,
              neg.sign = "\u2212", pos.sign = "figure",
              big.mark = "\u202F", big.interval = 3L,
              small.mark = "\u202F", small.interval = 3L,
              na.mark = "\u2014", lgl.mark = c("+", "\u2212"),
              inf.mark = c("+\u221E", "\u2212\u221E") )

Arguments

Examples

# format_num() converts numeric values to characters with the specified formatting options.
# By default, thousands digit groups are separated by thin spaces, negative signs are replaced
# with minus signs, and positive signs and leading zeros are replaced with figure spaces
# (which have the same width as numbers and minus signs). These options ensure that all
# results will align neatly in columns when tabled.
format_num(x = c(10000, 1000, 2.41, -1.20, 0.41, -0.20))

# By default, format_num() uses your computer locale's default decimal mark as
# the decimal point. To force the usage of "." instead (e.g., for submission to
# a U.S. journal), set decimal.mark = ".":
format_num(x = .41, decimal.mark = ".")

# By default, format_num() separates groups of large digits using thin spaces.
# This is following the international standard for scientific communication (SI/ISO 31-0),
# which advises against using "." or "," to separate digits because doing so can lead
# to confusion for human and computer readers because "." and "," are also used
# as decimal marks in various countries. If you prefer to use commas to separate
# large digit groups, set big.mark = ",":
format_num(x = 10000, big.mark = ",")

Create wide-format datasets in simulate_r_database

Description

Create wide-format datasets in simulate_r_database

Usage

format_wide(
  x,
  param,
  sample_id,
  var_names,
  show_applicant,
  decimals = 2,
  noalpha = FALSE
)

Arguments

Value

A dataframe of results

Generate a list of references included in meta-analyses

Description

This function generates a list of studies contributing to a meta-analysis

Usage

generate_bib(
  ma_obj = NULL,
  bib = NULL,
  title.bib = NULL,
  style = "apa",
  additional_citekeys = NULL,
  file = NULL,
  output_dir = getwd(),
  output_format = c("word", "html", "pdf", "text", "odt", "rmd", "biblatex", "citekeys"),
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  save_build_files = FALSE,
  header = list(),
  ...
)

Arguments

Value

A list containing a tibble of bibtex reference data. Additionally, a reference list formatted in the requested style and output_format is exported (or printed if file is "console").

See Also

Other output functions: metabulate_rmd_helper(), metabulate()

Examples

## Not run: 
## Run a meta-analysis using ma_r() and include a citekey argument to provide
## citation information for each source contributing to the meta-analyses.
ma_obj <- ma_r(ma_method = "ic", rxyi = rxyi, n = n, rxx = rxxi, ryy = ryyi,
               construct_x = x_name, construct_y = y_name, sample_id = sample_id,
               moderators = moderator, citekey = citekey, data = data_r_meas_multi)

## Next, use generate_bib() to generate the bibliography for the retained studies.
## The bib argument is the BibTeX or BibLaTeX .bib file containing the full
## reference information for each of the citekeys included in the meta-analysis database.
generate_bib(ma_obj, bib = system.file("templates/sample_bibliography.bib", package="psychmeta"),
             file = "sample bibliography", output_dir = tempdir(), output_format = "word")

## End(Not run)

Generate a system of folders from a file path to a new directory

Description

This function is intended to be helpful in simulations when directories need to be created and named according to values that are used or created within the simulation.

Usage

generate_directory(path)

Arguments

Value

Creates a system of folders to a new directory.

Extract results from a psychmeta meta-analysis object

Description

Functions to extract specific results from a meta-analysis tibble. This family of functions harvests information from meta-analysis objects and returns it as lists or tibbles that are easily navigable.

Available functions include:

• get_stuff: Wrapper function for all other "get_" functions.

• get_metatab: Retrieve list of meta-analytic tables.

• get_ad: Retrieve list of artifact-distribution objects or a summary table of artifact descriptive statistics.

• get_plots: Retrieve list of meta-analytic plots.

• get_escalc: Retrieve list of escalc objects (i.e., effect-size data) for use with metafor.

• get_metafor: Alias for get_escalc.

• get_followup: Retrieve list of follow-up analyses.

• get_leave1out: Retrieve list of leave-one-out meta-analyses (special case of get_followup).

• get_cumulative: Retrieve list of cumulative meta-analyses (special case of get_followup).

• get_bootstrap: Retrieve list of bootstrap meta-analyses (special case of get_followup).

• get_metareg: Retrieve list of meta-regression analyses (special case of get_followup).

• get_heterogeneity: Retrieve list of heterogeneity analyses (special case of get_followup).

• get_matrix: Retrieve a tibble of matrices summarizing the relationships among constructs (only applicable to meta-analyses with multiple constructs).

Usage

get_stuff(
  ma_obj,
  what = c("metatab", "escalc", "metafor", "ad", "followup", "heterogeneity",
    "leave1out", "cumulative", "bootstrap", "metareg", "matrix", "plots"),
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  ma_method = c("bb", "ic", "ad"),
  correction_type = c("ts", "vgx", "vgy"),
  moderators = FALSE,
  as_ad_obj = TRUE,
  inputs_only = FALSE,
  ad_type = c("tsa", "int"),
  follow_up = c("heterogeneity", "leave1out", "cumulative", "bootstrap", "metareg"),
  plot_types = c("funnel", "forest", "leave1out", "cumulative"),
  ...
)

get_escalc(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  moderators = TRUE,
  ...
)

get_metafor(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  moderators = TRUE,
  ...
)

get_metatab(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  ma_method = c("bb", "ic", "ad"),
  correction_type = c("ts", "vgx", "vgy"),
  ...
)

get_ad(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  ma_method = c("ad", "ic"),
  ad_type = c("tsa", "int"),
  as_ad_obj = FALSE,
  inputs_only = FALSE,
  ...
)

get_followup(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  follow_up = c("heterogeneity", "leave1out", "cumulative", "bootstrap", "metareg"),
  ...
)

get_heterogeneity(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  ...
)

get_leave1out(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  ...
)

get_cumulative(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  ...
)

get_bootstrap(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  ...
)

get_metareg(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  ...
)

get_matrix(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  ...
)

get_plots(
  ma_obj,
  analyses = "all",
  match = c("all", "any"),
  case_sensitive = TRUE,
  plot_types = c("funnel", "forest", "leave1out", "cumulative"),
  ...
)

Arguments

Value

Selected set of results.

Examples

## Not run: 
## Run meta-analysis:
ma_obj <- ma_r(ma_method = "ic", rxyi = rxyi, n = n, rxx = rxxi, ryy = ryyi,
               construct_x = x_name, construct_y = y_name,
               sample_id = sample_id, citekey = NULL,
               moderators = moderator, data = data_r_meas_multi,
               impute_artifacts = FALSE, clean_artifacts = FALSE)
ma_obj <- ma_r_ad(ma_obj, correct_rr_x = FALSE, correct_rr_y = FALSE)

## Run additional analyses:
ma_obj <- heterogeneity(ma_obj)
ma_obj <- sensitivity(ma_obj, boot_iter = 10, boot_ci_type = "norm")
ma_obj <- metareg(ma_obj)
ma_obj <- plot_funnel(ma_obj)
ma_obj <- plot_forest(ma_obj)

## View summary:
summary(ma_obj)

## Extract selected analyses:
get_metatab(ma_obj)
get_matrix(ma_obj)
get_escalc(ma_obj)
get_bootstrap(ma_obj)
get_cumulative(ma_obj)
get_leave1out(ma_obj)
get_heterogeneity(ma_obj)
get_metareg(ma_obj)
get_plots(ma_obj)
get_ad(ma_obj, ma_method = "ic", as_ad_obj = TRUE)
get_ad(ma_obj, ma_method = "ic", as_ad_obj = FALSE)

## Same extractions as above, but using get_stuff() and the "what" argument:
get_stuff(ma_obj, what = "metatab")
get_stuff(ma_obj, what = "matrix")
get_stuff(ma_obj, what = "escalc")
get_stuff(ma_obj, what = "bootstrap")
get_stuff(ma_obj, what = "cumulative")
get_stuff(ma_obj, what = "leave1out")
get_stuff(ma_obj, what = "heterogeneity")
get_stuff(ma_obj, what = "metareg")
get_stuff(ma_obj, what = "plots")
get_stuff(ma_obj, what = "ad", ma_method = "ic", as_ad_obj = TRUE)
get_stuff(ma_obj, what = "ad", ma_method = "ic", as_ad_obj = FALSE)

## End(Not run)

Supplemental heterogeneity statistics for meta-analyses

Description

This function computes a variety of supplemental statistics for meta-analyses. The statistics here are included for interested users. It is strongly recommended that heterogeneity in meta-analysis be interpreted using the SD_{res}, SD_{\rho}, and SD_{\delta} statistics, along with corresponding credibility intervals, which are reported in the default ma_obj output (Wiernik et al., 2017).

Usage

heterogeneity(
  ma_obj,
  es_failsafe = NULL,
  conf_level = attributes(ma_obj)$inputs$conf_level,
  var_res_ci_method = c("profile_var_es", "profile_Q", "normal_logQ"),
  ...
)

Arguments

Value

ma_obj with heterogeneity statistics added. Included statistics include:

Results are reported using computation methods described by Schmidt and Hunter. For barebones and individual-correction meta-analyses, results are also reported using computation methods described by DerSimonian and Laird, outlier-robust computation methods, and, if weights from metafor are used, heterogeneity results from metafor.

References

Becker, B. J. (2005). Failsafe N or file-drawer number. In H. R. Rothstein, A. J. Sutton, & M. Borenstein (Eds.), Publication bias in meta-analysis: Prevention, assessment and adjustments (pp. 111–125). Wiley. doi:10.1002/0470870168.ch7

Higgins, J. P. T., & Thompson, S. G. (2002). Quantifying heterogeneity in a meta-analysis. Statistics in Medicine, 21(11), 1539–1558. doi:10.1002/sim.1186

Lin, L., Chu, H., & Hodges, J. S. (2017). Alternative measures of between-study heterogeneity in meta-analysis: Reducing the impact of outlying studies. Biometrics, 73(1), 156–166. doi:10.1111/biom.12543

Schmidt, F. (2010). Detecting and correcting the lies that data tell. Perspectives on Psychological Science, 5(3), 233–242. doi:10.1177/1745691610369339

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Sage. doi:10.4135/9781483398105. pp. 15, 414, 426, 533–534.

Wiernik, B. M., Kostal, J. W., Wilmot, M. P., Dilchert, S., & Ones, D. S. (2017). Empirical benchmarks for interpreting effect size variability in meta-analysis. Industrial and Organizational Psychology, 10(3). doi:10.1017/iop.2017.44

Examples

## Correlations
ma_obj <- ma_r_ic(rxyi = rxyi, n = n, rxx = rxxi, ryy = ryyi, ux = ux,
                  correct_rr_y = FALSE, data = data_r_uvirr)
ma_obj <- ma_r_ad(ma_obj, correct_rr_y = FALSE)
ma_obj <- heterogeneity(ma_obj = ma_obj)
ma_obj$heterogeneity[[1]]$barebones
ma_obj$heterogeneity[[1]]$individual_correction$true_score
ma_obj$heterogeneity[[1]]$artifact_distribution$true_score

## d values
ma_obj <- ma_d_ic(d = d, n1 = n1, n2 = n2, ryy = ryyi,
                  data = data_d_meas_multi)
ma_obj <- ma_d_ad(ma_obj)
ma_obj <- heterogeneity(ma_obj = ma_obj)
ma_obj$heterogeneity[[1]]$barebones
ma_obj$heterogeneity[[1]]$individual_correction$latentGroup_latentY
ma_obj$heterogeneity[[1]]$artifact_distribution$latentGroup_latentY

Impute missing and impossible artifact values

Description

Impute missing and impossible artifact values

Usage

impute_artifacts(
  sample_id = NULL,
  construct_id = NULL,
  measure_id = NULL,
  art_vec,
  cat_moderator_matrix,
  impute_method = "bootstrap_mod",
  art_type = "rel",
  n_vec = rep(1, length(art_vec))
)

Arguments

Value

A vector of artifacts that includes imputed values.

Examples

# art_vec <- c(.6, .7, NA, .8, .9, NA)
# cat_moderator_matrix <- matrix(c(rep(1, 3), rep(2, 3)))
# art_type <- "rel"
# n_vec <- c(50, 200, 100, 50, 200, 100)
#
# ## Compute unweighted means
# impute_artifacts(art_vec = art_vec, cat_moderator_matrix = cat_moderator_matrix,
#                 impute_method = "unwt_mean_full", art_type = art_type, n_vec = n_vec)
# impute_artifacts(art_vec = art_vec, cat_moderator_matrix = cat_moderator_matrix,
#                 impute_method = "unwt_mean_mod", art_type = art_type, n_vec = n_vec)
#
# ## Compute weighted means
# impute_artifacts(art_vec = art_vec, cat_moderator_matrix = cat_moderator_matrix,
#                 impute_method = "wt_mean_full", art_type = art_type, n_vec = n_vec)
# impute_artifacts(art_vec = art_vec, cat_moderator_matrix = cat_moderator_matrix,
#                 impute_method = "wt_mean_mod", art_type = art_type, n_vec = n_vec)
#
# ## Simulate from distribution with the mean and variance of the observed artifacts
# impute_artifacts(art_vec = art_vec, cat_moderator_matrix = cat_moderator_matrix,
#                 impute_method = "simulate_full", art_type = art_type, n_vec = n_vec)
# impute_artifacts(art_vec = art_vec, cat_moderator_matrix = cat_moderator_matrix,
#                 impute_method = "simulate_mod", art_type = art_type, n_vec = n_vec)
#
# ## Sample random values from the observed distribution of artifacts
# impute_artifacts(art_vec = art_vec, cat_moderator_matrix = cat_moderator_matrix,
#                 impute_method = "bootstrap_mod", art_type = art_type, n_vec = n_vec)
# impute_artifacts(art_vec = art_vec, cat_moderator_matrix = cat_moderator_matrix,
#                 impute_method = "bootstrap_full", art_type = art_type, n_vec = n_vec)
#
# ## If all values are missing from a moderator category, the program will run
# ## full-data imputation on the remaining missing values:
# impute_artifacts(art_vec = c(NA, NA, NA, .7, .8, .9), cat_moderator_matrix = cat_moderator_matrix,
#                 impute_method = "bootstrap_mod", art_type = art_type, n_vec = n_vec)

Confidence limits of tau

Description

Note that this interval does not incorporate uncertainty in artifact estimates, so the interval will be somewhat conservative when applied to individual-correction or artifact-distribution meta-analyses.

Usage

limits_tau(
  var_es,
  var_pre,
  k,
  method = c("profile_var_es", "profile_Q", "normal_logQ"),
  conf_level = 0.95,
  var_unbiased = TRUE
)

Arguments

Value

The confidence limits of tau

Examples

limits_tau(var_es = 0.008372902, var_pre = 0.004778935, k = 20)

Confidence limits of tau-squared

Description

Note that this interval does not incorporate uncertainty in artifact estimates, so the interval will be somewhat conservative when applied to individual-correction or artifact-distribution meta-analyses.

Usage

limits_tau2(
  var_es,
  var_pre,
  k,
  method = c("profile_var_es", "profile_Q", "normal_logQ"),
  conf_level = 0.95,
  var_unbiased = TRUE
)

Arguments

Value

The confidence limits of tau-squared

Examples

limits_tau2(var_es = 0.008372902, var_pre = 0.004778935, k = 20)

Compute linear regression models and generate "lm" objects from covariance matrices.

Description

Compute linear regression models and generate "lm" objects from covariance matrices.

Usage

lm_mat(
  formula,
  cov_mat,
  mean_vec = rep(0, ncol(cov_mat)),
  n = Inf,
  se_beta_method = c("lm", "normal"),
  ...
)

matrixreg(
  formula,
  cov_mat,
  mean_vec = rep(0, ncol(cov_mat)),
  n = Inf,
  se_beta_method = c("lm", "normal"),
  ...
)

matreg(
  formula,
  cov_mat,
  mean_vec = rep(0, ncol(cov_mat)),
  n = Inf,
  se_beta_method = c("lm", "normal"),
  ...
)

lm_matrix(
  formula,
  cov_mat,
  mean_vec = rep(0, ncol(cov_mat)),
  n = Inf,
  se_beta_method = c("lm", "normal"),
  ...
)

Arguments

Value

An object with the class "lm_mat" that can be used with summary, print, predict, and anova methods.

Examples

## Generate data
S <- reshape_vec2mat(cov = c(.3 * 2 * 3,
                             .4 * 2 * 4,
                             .5 * 3 * 4),
                     var = c(2, 3, 4)^2,
                     var_names = c("X", "Y", "Z"))
mean_vec <- setNames(c(1, 2, 3), colnames(S))
dat <- data.frame(MASS::mvrnorm(n = 100, mu = mean_vec,
                                Sigma = S, empirical = TRUE))

## Compute regression models with lm
lm_out1 <- lm(Y ~ X, data = dat)
lm_out2 <- lm(Y ~ X + Z, data = dat)

## Compute regression models with lm_mat
matreg_out1 <- lm_mat(formula = Y ~ X, cov_mat = S, mean_vec = mean_vec, n = nrow(dat))
matreg_out2 <- lm_mat(formula = Y ~ X + Z, cov_mat = S, mean_vec = mean_vec, n = nrow(dat))

## Compare results of lm and lm_mat with one predictor
lm_out1
matreg_out1

## Compare summaries of lm and lm_mat with one predictor
summary(lm_out1)
summary(matreg_out1)

## Compare results of lm and lm_mat with two predictors
lm_out2
matreg_out2

## Compare summaries of lm and lm_mat with two predictors
summary(lm_out2)
summary(matreg_out2)

## Compare predictions made with lm and lm_mat
predict(object = matreg_out1, newdata = data.frame(X = 1:5))
predict(object = summary(matreg_out1), newdata = data.frame(X = 1:5))
predict(lm_out1, newdata = data.frame(X = 1:5))

## Compare predictions made with lm and lm_mat (with confidence intervals)
predict(object = matreg_out1, newdata = data.frame(X = 1:5),
        se.fit = TRUE, interval = "confidence")
predict(lm_out1, newdata = data.frame(X = 1:5),
        se.fit = TRUE, interval = "confidence")

## Compare predictions made with lm and lm_mat (with prediction intervals)
predict(object = matreg_out1, newdata = data.frame(X = 1:5),
        se.fit = TRUE, interval = "prediction")
predict(lm_out1, newdata = data.frame(X = 1:5),
        se.fit = TRUE, interval = "prediction")

## Compare model comparisons computed using lm and lm_mat objects
anova(lm_out1, lm_out2)
anova(matreg_out1, matreg_out2)

## Model comparisons can be run on lm_mat summaries, too:
anova(summary(matreg_out1), summary(matreg_out2))
## Or summaries and raw models can be mixed:
anova(matreg_out1, summary(matreg_out2))
anova(summary(matreg_out1), matreg_out2)


## Compare confidence intervals computed using lm and lm_mat objects
confint(object = lm_out1)
confint(object = matreg_out1)
confint(object = summary(matreg_out1))

confint(object = lm_out2)
confint(object = matreg_out2)
confint(object = summary(matreg_out2))

Meta-analysis of d values

Description

The ma_r_bb, ma_r_ic, and ma_r_ad functions implement bare-bones, individual-correction, and artifact-distribution correction methods for d values, respectively. The ma_d function is the master function for meta-analyses of d values - it facilitates the computation of bare-bones, artifact-distribution, and individual-correction meta-analyses of correlations for any number of group-wise contrasts and any number of dependent variables. When artifact-distribution meta-analyses are performed, ma_d will automatically extract the artifact information from a database and organize it into the requested type of artifact distribution object (i.e., either Taylor series or interactive artifact distributions). ma_d is also equipped with the capability to clean databases containing inconsistently recorded artifact data, impute missing artifacts (when individual-correction meta-analyses are requested), and remove dependency among samples by forming composites or averaging effect sizes and artifacts. The automatic compositing features in ma_d are employed when sample_ids and/or construct names are provided.

Usage

ma_d(
  d,
  n1,
  n2 = NULL,
  n_adj = NULL,
  sample_id = NULL,
  citekey = NULL,
  treat_as_r = FALSE,
  ma_method = c("bb", "ic", "ad"),
  ad_type = c("tsa", "int"),
  correction_method = "auto",
  group_id = NULL,
  group1 = NULL,
  group2 = NULL,
  group_order = NULL,
  construct_y = NULL,
  facet_y = NULL,
  measure_y = NULL,
  construct_order = NULL,
  wt_type = c("n_effective", "sample_size", "inv_var_mean", "inv_var_sample", "DL", "HE",
    "HS", "SJ", "ML", "REML", "EB", "PM"),
  correct_bias = TRUE,
  correct_rel = NULL,
  correct_rGg = FALSE,
  correct_ryy = TRUE,
  correct_rr = NULL,
  correct_rr_g = TRUE,
  correct_rr_y = TRUE,
  indirect_rr = NULL,
  indirect_rr_g = TRUE,
  indirect_rr_y = TRUE,
  rGg = NULL,
  pi = NULL,
  pa = NULL,
  ryy = NULL,
  ryy_restricted = TRUE,
  ryy_type = "alpha",
  k_items_y = NULL,
  uy = NULL,
  uy_observed = TRUE,
  sign_rz = NULL,
  sign_rgz = 1,
  sign_ryz = 1,
  moderators = NULL,
  cat_moderators = TRUE,
  moderator_type = c("simple", "hierarchical", "none"),
  supplemental_ads = NULL,
  data = NULL,
  control = control_psychmeta(),
  ...
)

ma_d_ad(
  ma_obj,
  ad_obj_g = NULL,
  ad_obj_y = NULL,
  correction_method = "auto",
  use_ic_ads = c("tsa", "int"),
  correct_rGg = FALSE,
  correct_ryy = TRUE,
  correct_rr_g = TRUE,
  correct_rr_y = TRUE,
  indirect_rr_g = TRUE,
  indirect_rr_y = TRUE,
  sign_rgz = 1,
  sign_ryz = 1,
  control = control_psychmeta(),
  ...
)

ma_d_bb(
  d,
  n1,
  n2 = rep(NA, length(d)),
  n_adj = NULL,
  sample_id = NULL,
  citekey = NULL,
  wt_type = c("n_effective", "sample_size", "inv_var_mean", "inv_var_sample", "DL", "HE",
    "HS", "SJ", "ML", "REML", "EB", "PM"),
  correct_bias = TRUE,
  moderators = NULL,
  cat_moderators = TRUE,
  moderator_type = c("simple", "hierarchical", "none"),
  data = NULL,
  control = control_psychmeta(),
  ...
)

ma_d_ic(
  d,
  n1,
  n2 = NULL,
  n_adj = NULL,
  sample_id = NULL,
  citekey = NULL,
  treat_as_r = FALSE,
  wt_type = c("n_effective", "sample_size", "inv_var_mean", "inv_var_sample", "DL", "HE",
    "HS", "SJ", "ML", "REML", "EB", "PM"),
  correct_bias = TRUE,
  correct_rGg = FALSE,
  correct_ryy = TRUE,
  correct_rr_g = FALSE,
  correct_rr_y = TRUE,
  indirect_rr_g = TRUE,
  indirect_rr_y = TRUE,
  rGg = NULL,
  pi = NULL,
  pa = NULL,
  ryy = NULL,
  ryy_restricted = TRUE,
  ryy_type = "alpha",
  k_items_y = NULL,
  uy = NULL,
  uy_observed = TRUE,
  sign_rgz = 1,
  sign_ryz = 1,
  moderators = NULL,
  cat_moderators = TRUE,
  moderator_type = c("simple", "hierarchical", "none"),
  supplemental_ads_y = NULL,
  data = NULL,
  control = control_psychmeta(),
  ...
)

Arguments

Details

The options for correction_method are:

• "auto": Automatic selection of the most appropriate correction procedure, based on the available artifacts and the logical arguments provided to the function. (default)

• "meas": Correction for measurement error only.

• "uvdrr": Correction for univariate direct range restriction (i.e., Case II). The choice of which variable to correct for range restriction is made using the correct_rr_x and correct_rr_y arguments.

• "uvirr": Correction for univariate indirect range restriction (i.e., Case IV). The choice of which variable to correct for range restriction is made using the correct_rr_x and correct_rr_y arguments.

• "bvdrr": Correction for bivariate direct range restriction. Use with caution: This correction is an approximation only and is known to have a positive bias.

• "bvirr": Correction for bivariate indirect range restriction (i.e., Case V).

• "rbOrig": Not recommended: Raju and Burke's version of the correction for direct range restriction, applied interactively. We recommend using "uvdrr" instead.

• "rbAdj": Not recommended: Raju and Burke's version of the correction for direct range restriction, applied interactively. Adjusted to account for range restriction in the reliability of the Y variable. We recommend using "uvdrr" instead.

• "rb1Orig": Not recommended: Raju and Burke's version of the correction for direct range restriction, applied using their TSA1 method. We recommend using "uvdrr" instead.

• "rb1Adj": Not recommended: Raju and Burke's version of the correction for direct range restriction, applied using their TSA1 method. Adjusted to account for range restriction in the reliability of the Y variable. We recommend using "uvdrr" instead.

• "rb2Orig": Not recommended: Raju and Burke's version of the correction for direct range restriction, applied using their TSA2 method. We recommend using "uvdrr" instead.

• "rb2Adj": Not recommended: Raju and Burke's version of the correction for direct range restriction, applied using their TSA2 method. Adjusted to account for range restriction in the reliability of the Y variable. We recommend using "uvdrr" instead.

Value

A nested tabular object of the class "ma_psychmeta". Components of output tables for bare-bones meta-analyses:

• Pair_ID: Unique identification number for each construct-contrast pairing.

• group_contrast: Name of the variable analyzed as the group-contrast variable.

• construct_y: Name of the variable analyzed as construct Y.

• analysis_id: Unique identification number for each analysis.

• analysis_type: Type of moderator analyses: Overall, Simple Moderator, or Hierarchical Moderator.

• k: Number of effect sizes meta-analyzed.

• N: Total sample size of all effect sizes in the meta-analysis.

• mean_d: Mean observed d value.

• var_d: Weighted variance of observed d values.

• var_e: Predicted sampling-error variance of observed d values.

• var_res: Variance of observed d values after removing predicted sampling-error variance.

• sd_d: Square root of var_r.

• se_d: Standard error of mean_d.

• sd_e: Square root of var_e.

• sd_res: Square root of var_res.

• CI_LL_XX: Lower limit of the confidence interval around mean_d, where "XX" represents the confidence level as a percentage.

• CI_UL_XX: Upper limit of the confidence interval around mean_d, where "XX" represents the confidence level as a percentage.

• CR_LL_XX: Lower limit of the credibility interval around mean_d, where "XX" represents the credibility level as a percentage.

• CR_UL_XX: Upper limit of the credibility interval around mean_d, where "XX" represents the credibility level as a percentage.

Components of output tables for individual-correction meta-analyses:

• pair_id: Unique identification number for each construct-contrast pairing.

• group_contrast: Name of the variable analyzed as the group-contrast variable.

• construct_y: Name of the variable analyzed as construct Y.

• analysis_id: Unique identification number for each analysis.

• analysis_type: Type of moderator analyses: Overall, Simple Moderator, or Hierarchical Moderator.

• k: Number of effect sizes meta-analyzed.

• N: Total sample size of all effect sizes in the meta-analysis.

• mean_d: Mean observed d value.

• var_d: Weighted variance of observed d values.

• var_e: Predicted sampling-error variance of observed d values.

• var_res: Variance of observed d values after removing predicted sampling-error variance.

• sd_d: Square root of var_r.

• se_d: Standard error of mean_d.

• sd_e: Square root of var_e.

• sd_res: Square root of var_res.

• mean_delta: Mean artifact-corrected d value.

• var_d_c: Variance of artifact-corrected d values.

• var_e_c: Predicted sampling-error variance of artifact-corrected d values.

• var_delta: Variance of artifact-corrected d values after removing predicted sampling-error variance.

• sd_d_c: Square root of var_r_c.

• se_d_c: Standard error of mean_delta.

• sd_e_c: Square root of var_e_c.

• sd_delta: Square root of var_delta.

• CI_LL_XX: Lower limit of the confidence interval around mean_delta, where "XX" represents the confidence level as a percentage.

• CI_UL_XX: Upper limit of the confidence interval around mean_delta, where "XX" represents the confidence level as a percentage.

• CR_LL_XX: Lower limit of the credibility interval around mean_delta, where "XX" represents the credibility level as a percentage.

• CR_UL_XX: Upper limit of the credibility interval around mean_delta, where "XX" represents the credibility level as a percentage.

Components of output tables for artifact-distribution meta-analyses:

• pair_id: Unique identification number for each construct-contrast pairing.

• group_contrast: Name of the variable analyzed as the group-contrast variable.

• construct_y: Name of the variable analyzed as construct Y.

• analysis_id: Unique identification number for each analysis.

• analysis_type: Type of moderator analyses: Overall, Simple Moderator, or Hierarchical Moderator.

• k: Number of effect sizes meta-analyzed.

• N: Total sample size of all effect sizes in the meta-analysis.

• mean_d: Mean observed d value.

• var_d: Weighted variance of observed d values.

• var_e: Predicted sampling-error variance of observed d values.

• var_art: Amount of variance in observed d values that is attributable to measurement-error and range-restriction artifacts.

• var_pre: Total predicted artifactual variance (i.e., the sum of var_e and var_art).

• var_res: Variance of observed d values after removing predicted sampling-error variance and predicted artifact variance.

• sd_d: Square root of var_d.

• se_d: Standard error of mean_d.

• sd_e: Square root of var_e.

• sd_art: Square root of var_art.

• sd_pre: Square root of var_pre.

• sd_res: Square root of var_res.

• mean_delta: Mean artifact-corrected d value.

• var_d: Weighted variance of observed d values corrected to the metric of delta.

• var_e: Predicted sampling-error variance of observed d values corrected to the metric of delta.

• var_art: Amount of variance in observed d values that is attributable to measurement-error and range-restriction artifacts corrected to the metric of delta.

• var_pre: Total predicted artifactual variance (i.e., the sum of var_e and var_art) corrected to the metric of delta.

• var_delta: Variance of artifact-corrected d values after removing predicted sampling-error variance and predicted artifact variance.

• sd_d: Square root of var_d corrected to the metric of delta.

• se_d: Standard error of mean_d corrected to the metric of delta.

• sd_e: Square root of var_e corrected to the metric of delta.

• sd_art: Square root of var_art corrected to the metric of delta.

• sd_pre: Square root of var_pre corrected to the metric of delta.

• sd_delta: Square root of var_delta.

• CI_LL_XX: Lower limit of the confidence interval around mean_delta, where "XX" represents the confidence level as a percentage.

• CI_UL_XX: Upper limit of the confidence interval around mean_delta, where "XX" represents the confidence level as a percentage.

• CR_LL_XX: Lower limit of the credibility interval around mean_delta, where "XX" represents the credibility level as a percentage.

• CR_UL_XX: Upper limit of the credibility interval around mean_delta, where "XX" represents the credibility level as a percentage.

Note

The difference between "rb" methods with the "orig" and "adj" suffixes is that the original does not account for the impact of range restriction on criterion reliabilities, whereas the adjusted procedure attempts to estimate the applicant reliability information for the criterion. The "rb" procedures are included for posterity: We strongly recommend using the "uvdrr" procedure to appropriately correct for univariate range restriction.

References

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Sage. doi:10.4135/9781483398105. Chapter 4.

Law, K. S., Schmidt, F. L., & Hunter, J. E. (1994). Nonlinearity of range corrections in meta-analysis: Test of an improved procedure. Journal of Applied Psychology, 79(3), 425.

Dahlke, J. A., & Wiernik, B. M. (2020). Not restricted to selection research: Accounting for indirect range restriction in organizational research. Organizational Research Methods, 23(4), 717–749. doi:10.1177/1094428119859398

Raju, N. S., & Burke, M. J. (1983). Two new procedures for studying validity generalization. Journal of Applied Psychology, 68(3), 382. doi:10.1037/0021-9010.68.3.382

Examples

### Demonstration of ma_d ###
## The 'ma_d' function can compute multi-construct bare-bones meta-analyses:
ma_d(d = d, n1 = n1, n2 = n2, construct_y = construct, data = data_d_meas_multi)

## It can also perform multiple individual-correction meta-analyses:
ma_d(ma_method = "ic", d = d, n1 = n1, n2 = n2, ryy = ryyi,
     construct_y = construct, data = data_d_meas_multi)

## And 'ma_d' can also curate artifact distributions and compute multiple
## artifact-distribution meta-analyses:
ma_d(ma_method = "ad", d = d, n1 = n1, n2 = n2,
     ryy = ryyi, correct_rr_y = FALSE,
     construct_y = construct, data = data_d_meas_multi)


### Demonstration of ma_d_bb ###
## Example meta-analyses using simulated data:
ma_d_bb(d = d, n1 = n1, n2 = n2,
        data = data_d_meas_multi[data_d_meas_multi$construct == "Y",])
ma_d_bb(d = d, n1 = n1, n2 = n2,
        data = data_d_meas_multi[data_d_meas_multi$construct == "Z",])


### Demonstration of ma_d_ic ###
## Example meta-analyses using simulated data:
ma_d_ic(d = d, n1 = n1, n2 = n2, ryy = ryyi, correct_rr_y = FALSE,
        data = data_d_meas_multi[data_d_meas_multi$construct == "Y",])
ma_d_ic(d = d, n1 = n1, n2 = n2, ryy = ryyi, correct_rr_y = FALSE,
        data = data_d_meas_multi[data_d_meas_multi$construct == "Z",])

Second-order meta-analysis function for d values

Description

This function computes second-order meta-analysis function for d values. It supports second-order analyses of bare-bones, artifact-distribution, and individual-correction meta-analyses.

Usage

ma_d_order2(
  k,
  N = NULL,
  d = NULL,
  delta = NULL,
  var_d = NULL,
  var_d_c = NULL,
  ma_type = c("bb", "ic", "ad"),
  sample_id = NULL,
  citekey = NULL,
  moderators = NULL,
  moderator_type = "simple",
  construct_x = NULL,
  construct_y = NULL,
  construct_order = NULL,
  data = NULL,
  control = control_psychmeta(),
  ...
)

Arguments

Value

A nested tabular object of the class "ma_psychmeta".

Bare-bones meta-analysis of generic effect sizes

Description

This function computes bare-bones meta-analyses of any effect size using user-supplied effect error variances.

Usage

ma_generic(
  es,
  n,
  var_e,
  sample_id = NULL,
  citekey = NULL,
  construct_x = NULL,
  construct_y = NULL,
  group1 = NULL,
  group2 = NULL,
  wt_type = c("sample_size", "inv_var", "DL", "HE", "HS", "SJ", "ML", "REML", "EB", "PM"),
  moderators = NULL,
  cat_moderators = TRUE,
  moderator_type = c("simple", "hierarchical", "none"),
  data = NULL,
  control = control_psychmeta(),
  weights = NULL,
  ...
)

Arguments

Value

A nested tabular object of the class "ma_psychmeta".

Examples

es <- c(.3, .5, .8)
n <- c(100, 200, 150)
var_e <- 1 / n
ma_obj <- ma_generic(es = es, n = n, var_e = var_e)
ma_obj
summary(ma_obj)

Meta-analysis of correlations

Description

The ma_r_bb, ma_r_ic, and ma_r_ad functions implement bare-bones, individual-correction, and artifact-distribution correction methods for correlations, respectively. The ma_r function is the master function for meta-analyses of correlations - it facilitates the computation of bare-bones, artifact-distribution, and individual-correction meta-analyses of correlations for any number of construct pairs. When artifact-distribution meta-analyses are performed, ma_r will automatically extract the artifact information from a database and organize it into the requested type of artifact distribution object (i.e., either Taylor series or interactive artifact distributions). ma_r is also equipped with the capability to clean databases containing inconsistently recorded artifact data, impute missing artifacts (when individual-correction meta-analyses are requested), and remove dependency among samples by forming composites or averaging effect sizes and artifacts. The automatic compositing features in ma_r are employed when sample_ids and/or construct names are provided.

Usage

ma_r(
  rxyi,
  n,
  n_adj = NULL,
  sample_id = NULL,
  citekey = NULL,
  ma_method = c("bb", "ic", "ad"),
  ad_type = c("tsa", "int"),
  correction_method = "auto",
  construct_x = NULL,
  construct_y = NULL,
  facet_x = NULL,
  facet_y = NULL,
  measure_x = NULL,
  measure_y = NULL,
  construct_order = NULL,
  wt_type = c("sample_size", "inv_var_mean", "inv_var_sample", "DL", "HE", "HS", "SJ",
    "ML", "REML", "EB", "PM"),
  correct_bias = TRUE,
  correct_rel = NULL,
  correct_rxx = TRUE,
  correct_ryy = TRUE,
  correct_rr = NULL,
  correct_rr_x = TRUE,
  correct_rr_y = TRUE,
  indirect_rr = NULL,
  indirect_rr_x = TRUE,
  indirect_rr_y = TRUE,
  rxx = NULL,
  rxx_restricted = TRUE,
  rxx_type = "alpha",
  k_items_x = NULL,
  ryy = NULL,
  ryy_restricted = TRUE,
  ryy_type = "alpha",
  k_items_y = NULL,
  ux = NULL,
  ux_observed = TRUE,
  uy = NULL,
  uy_observed = TRUE,
  sign_rz = NULL,
  sign_rxz = 1,
  sign_ryz = 1,
  moderators = NULL,
  cat_moderators = TRUE,
  moderator_type = c("simple", "hierarchical", "none"),
  supplemental_ads = NULL,
  data = NULL,
  control = control_psychmeta(),
  ...
)

ma_r_ad(
  ma_obj,
  ad_obj_x = NULL,
  ad_obj_y = NULL,
  correction_method = "auto",
  use_ic_ads = c("tsa", "int"),
  correct_rxx = TRUE,
  correct_ryy = TRUE,
  correct_rr_x = TRUE,
  correct_rr_y = TRUE,
  indirect_rr_x = TRUE,
  indirect_rr_y = TRUE,
  sign_rxz = 1,
  sign_ryz = 1,
  control = control_psychmeta(),
  ...
)

ma_r_bb(
  r,
  n,
  n_adj = NULL,
  sample_id = NULL,
  citekey = NULL,
  wt_type = c("sample_size", "inv_var_mean", "inv_var_sample", "DL", "HE", "HS", "SJ",
    "ML", "REML", "EB", "PM"),
  correct_bias = TRUE,
  moderators = NULL,
  cat_moderators = TRUE,
  moderator_type = c("simple", "hierarchical", "none"),
  data = NULL,
  control = control_psychmeta(),
  ...
)

ma_r_ic(
  rxyi,
  n,
  n_adj = NULL,
  sample_id = NULL,
  citekey = NULL,
  wt_type = c("sample_size", "inv_var_mean", "inv_var_sample", "DL", "HE", "HS", "SJ",
    "ML", "REML", "EB", "PM"),
  correct_bias = TRUE,
  correct_rxx = TRUE,
  correct_ryy = TRUE,
  correct_rr_x = TRUE,
  correct_rr_y = TRUE,
  indirect_rr_x = TRUE,
  indirect_rr_y = TRUE,
  rxx = NULL,
  rxx_restricted = TRUE,
  rxx_type = "alpha",
  k_items_x = NULL,
  ryy = NULL,
  ryy_restricted = TRUE,
  ryy_type = "alpha",
  k_items_y = NULL,
  ux = NULL,
  ux_observed = TRUE,
  uy = NULL,
  uy_observed = TRUE,
  sign_rxz = 1,
  sign_ryz = 1,
  moderators = NULL,
  cat_moderators = TRUE,
  moderator_type = c("simple", "hierarchical", "none"),
  supplemental_ads_x = NULL,
  supplemental_ads_y = NULL,
  data = NULL,
  control = control_psychmeta(),
  ...
)

Arguments

Details

The options for correction_method are:

• "auto": Automatic selection of the most appropriate correction procedure, based on the available artifacts and the logical arguments provided to the function. (default)

• "meas": Correction for measurement error only.

• "uvdrr": Correction for univariate direct range restriction (i.e., Case II). The choice of which variable to correct for range restriction is made using the correct_rr_x and correct_rr_y arguments.

• "uvirr": Correction for univariate indirect range restriction (i.e., Case IV). The choice of which variable to correct for range restriction is made using the correct_rr_x and correct_rr_y arguments.

• "bvdrr": Correction for bivariate direct range restriction. Use with caution: This correction is an approximation only and is known to have a positive bias.

• "bvirr": Correction for bivariate indirect range restriction (i.e., Case V).

• "rbOrig": Not recommended: Raju and Burke's version of the correction for direct range restriction, applied interactively. We recommend using "uvdrr" instead.

• "rbAdj": Not recommended: Raju and Burke's version of the correction for direct range restriction, applied interactively. Adjusted to account for range restriction in the reliability of the Y variable. We recommend using "uvdrr" instead.

• "rb1Orig": Not recommended: Raju and Burke's version of the correction for direct range restriction, applied using their TSA1 method. We recommend using "uvdrr" instead.

• "rb1Adj": Not recommended: Raju and Burke's version of the correction for direct range restriction, applied using their TSA1 method. Adjusted to account for range restriction in the reliability of the Y variable. We recommend using "uvdrr" instead.

• "rb2Orig": Not recommended: Raju and Burke's version of the correction for direct range restriction, applied using their TSA2 method. We recommend using "uvdrr" instead.

• "rb2Adj": Not recommended: Raju and Burke's version of the correction for direct range restriction, applied using their TSA2 method. Adjusted to account for range restriction in the reliability of the Y variable. We recommend using "uvdrr" instead.

Value

A nested tabular object of the class "ma_psychmeta". Components of output tables for bare-bones meta-analyses:

• pair_id: Unique identification number for each construct pairing.

• construct_x: Name of the variable analyzed as construct X.

• construct_y: Name of the variable analyzed as construct Y.

• analysis_id: Unique identification number for each analysis.

• analysis_type: Type of moderator analyses: Overall, Simple Moderator, or Hierarchical Moderator.

• k: Number of effect sizes meta-analyzed.

• N: Total sample size of all effect sizes in the meta-analysis.

• mean_r: Mean observed correlation.

• var_r: Weighted variance of observed correlations.

• var_e: Predicted sampling-error variance of observed correlations.

• var_res: Variance of observed correlations after removing predicted sampling-error variance.

• sd_r: Square root of var_r.

• se_r: Standard error of mean_r.

• sd_e: Square root of var_e.

• sd_res: Square root of var_res.

• CI_LL_XX: Lower limit of the confidence interval around mean_r, where "XX" represents the confidence level as a percentage.

• CI_UL_XX: Upper limit of the confidence interval around mean_r, where "XX" represents the confidence level as a percentage.

• CR_LL_XX: Lower limit of the credibility interval around mean_r, where "XX" represents the credibility level as a percentage.

• CR_UL_XX: Upper limit of the credibility interval around mean_r, where "XX" represents the credibility level as a percentage.

Components of output tables for individual-correction meta-analyses:

• pair_id: Unique identification number for each construct pairing.

• construct_x: Name of the variable analyzed as construct X.

• construct_y: Name of the variable analyzed as construct Y.

• analysis_id: Unique identification number for each analysis.

• analysis_type: Type of moderator analyses: Overall, Simple Moderator, or Hierarchical Moderator.

• k: Number of effect sizes meta-analyzed.

• N: Total sample size of all effect sizes in the meta-analysis.

• mean_r: Mean observed correlation.

• var_r: Weighted variance of observed correlations.

• var_e: Predicted sampling-error variance of observed correlations.

• var_res: Variance of observed correlations after removing predicted sampling-error variance.

• sd_r: Square root of var_r.

• se_r: Standard error of mean_r.

• sd_e: Square root of var_e.

• sd_res: Square root of var_res.

• mean_rho: Mean artifact-corrected correlation.

• var_r_c: Variance of artifact-corrected correlations.

• var_e_c: Predicted sampling-error variance of artifact-corrected correlations.

• var_rho: Variance of artifact-corrected correlations after removing predicted sampling-error variance.

• sd_r_c: Square root of var_r_c.

• se_r_c: Standard error of mean_rho.

• sd_e_c: Square root of var_e_c.

• sd_rho: Square root of var_rho.

• CI_LL_XX: Lower limit of the confidence interval around mean_rho, where "XX" represents the confidence level as a percentage.

• CI_UL_XX: Upper limit of the confidence interval around mean_rho, where "XX" represents the confidence level as a percentage.

• CR_LL_XX: Lower limit of the credibility interval around mean_rho, where "XX" represents the credibility level as a percentage.

• CR_UL_XX: Upper limit of the credibility interval around mean_rho, where "XX" represents the credibility level as a percentage.