Type: Package
Title: Confidence Intervals for Coefficient Alpha and Standardized Alpha
Version: 1.0.1
Description: Calculate confidence intervals for alpha and standardized alpha using asymptotic theory or the studentized bootstrap, with or without transformations. Supports the asymptotic distribution-free method of Maydeu-Olivares, et al. (2007) <doi:10.1037/1082-989X.12.2.157>, the pseudo-elliptical method of Yuan & Bentler (2002) <doi:10.1007/BF02294845>, and the normal method of van Zyl et al. (1999) <doi:10.1007/BF02296146>, for both coefficient alpha and standardized alpha.
License: MIT + file LICENSE
URL: https://jonasmoss.github.io/alphaci/
Depends: R (≥ 3.5.0)
Imports: future.apply, matrixcalc
Suggests: covr, extraDistr, knitr, lavaan, psychTools, rmarkdown, testthat (≥ 3.0.0)
VignetteBuilder: knitr
Config/testthat/edition: 3
Encoding: UTF-8
RoxygenNote: 7.3.1
NeedsCompilation: no
Packaged: 2024-02-05 09:18:02 UTC; A2010578
Author: Jonas Moss ORCID iD [aut, cre]
Maintainer: Jonas Moss <jonas.moss.statistics@gmail.com>
Repository: CRAN
Date/Publication: 2024-02-05 12:20:09 UTC

Confidence intervals for alpha and standardized alpha

Description

Calculate confidence intervals for coefficient alpha (Cronbach, 1951) and standardized alpha (Falk & Savalei, 2011) using asymptotic methods or the studentized bootstrap. alphaci constructs confidence intervals for coefficient alpha and alphaci_std for standardized alpha.

Usage

alphaci(
  x,
  type = c("adf", "elliptical", "normal"),
  transform = "none",
  parallel = FALSE,
  conf_level = 0.95,
  alternative = c("two.sided", "greater", "less"),
  bootstrap = FALSE,
  n_reps = 1000
)

alphaci_std(
  x,
  type = c("adf", "elliptical", "normal"),
  transform = "none",
  parallel = FALSE,
  conf_level = 0.95,
  alternative = c("two.sided", "greater", "less"),
  bootstrap = FALSE,
  n_reps = 1000
)

Arguments

x

Input data data can be converted to a matrix using as.matrix. Rows containing missing values are ignored.

type

Type of confidence interval. Either adf, elliptical, or normal.

transform

One of "none", "log", "fisher", and ⁠"arcsin⁠. Defaults to "none".

parallel

If TRUE, makes calculations under the assumption of a parallel model. Defaults to FALSE.

conf_level

Confidence level. Defaults to 0.95.

alternative

A character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less".

bootstrap

If TRUE, performs a studentized bootstrap with n_reps repetitions. Defaults to FALSE.

n_reps

Number of bootstrap samples if bootstrap = TRUE. Ignored if bootstrap = FALSE. Defaults to 1000.

Details

The methods accept handle missing data using stats::na.omit, i.e., rows containing missing data are removed. The bootstrap option uses the studentized bootstrap (Efron, B. 1987), which is second order correct. Both functions makes use of future.apply when bootstrapping.

The type variables defaults to adf, asymptotically distribution-free, which is consistent when the fourth moment is finite (Maydeu-Olivares et al. 2007). The normal option assumes normality. (van Zyl et al. 1999), and is not concistent for models with excess kurtosis unequal to 0. The elliptical option assumes an elliptical or pseudo-elliptical distribution of the data. The resulting confidence intervals are corrected variants of the normal theory intervals with a kurtosis correction (Yuan & Bentler 2002). The common kurtosis parameter is calculated using the unbiased sample kurtosis (Joanes, 1998). All these methods have analogues for standardized alpha, which can be derived using the methods of Hayashi & Kamata (2005) and Neudecker (2006).

Value

A vector of class alphaci containing the confidence end points. The arguments of the function call are included as attributes.

References

Falk, C. F., & Savalei, V. (2011). The relationship between unstandardized and standardized alpha, true reliability, and the underlying measurement model. Journal of Personality Assessment, 93(5), 445-453. https://doi.org/10.1080/00223891.2011.594129

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297-334. https://doi.org/10.1007/BF02310555#'

Efron, B. (1987). Better Bootstrap Confidence Intervals. Journal of the American Statistical Association, 82(397), 171-185. https://doi.org/10.2307/2289144

Maydeu-Olivares, A., Coffman, D. L., & Hartmann, W. M. (2007). Asymptotically distribution-free (ADF) interval estimation of coefficient alpha. Psychological Methods, 12(2), 157-176. https://doi.org/10.1037/1082-989X.12.2.157

van Zyl, J. M., Neudecker, H., & Nel, D. G. (2000). On the distribution of the maximum likelihood estimator of Cronbach's alpha. Psychometrika, 65(3), 271-280. https://doi.org/10.1007/BF02296146

Yuan, K.-H., & Bentler, P. M. (2002). On robustness of the normal-theory based asymptotic distributions of three reliability coefficient estimates. Psychometrika, 67(2), 251-259. https://doi.org/10.1007/BF02294845

Joanes, D. N., & Gill, C. A. (1998). Comparing measures of sample skewness and kurtosis. Journal of the Royal Statistical Society: Series D (The Statistician), 47(1), 183-189. https://doi.org/10.1111/1467-9884.00122

Hayashi, K., & Kamata, A. (2005). A note on the estimator of the alpha coefficient for standardized variables under normality. Psychometrika, 70(3), 579-586. https://doi.org/10.1007/s11336-001-0888-1

Neudecker, H. (2006). On the Asymptotic Distribution of the Natural Estimator of Cronbach's Alpha with Standardised Variates under Nonnormality, Ellipticity and Normality. In P. Brown, S. Liu, & D. Sharma (Eds.), Contributions to Probability and Statistics: Applications and Challenges (pp. 167-171). World Scientific. https://doi.org/10.1142/9789812772466_0013

Examples

library("alphaci")
library("psychTools")
x <- bfi[, 1:5]
x[, 1] <- 7 - x[, 1] # Reverse-coded item.
alphaci(x)
alphaci_std(x)

# Calculate confidence intervals with other options.
library("lavaan")
x <- lavaan::HolzingerSwineford1939[1:20, 7:9]
results <- c(
  alphaci(x, type = "adf", parallel = FALSE),
  alphaci(x, type = "adf", parallel = TRUE),
  alphaci(x, type = "elliptical", parallel = FALSE),
  alphaci(x, type = "elliptical", parallel = TRUE),
  alphaci(x, type = "normal", parallel = FALSE),
  alphaci(x, type = "normal", parallel = TRUE)
)

Calculates asymptotic confidence intervals.

Description

Calculates asymptotic confidence intervals.

Usage

ci_asymptotic(est, sd, n, transformer, quants)

ci_boot(
  x,
  est,
  sd,
  type,
  transformer,
  parallel,
  quants,
  n_reps,
  standardized = FALSE
)

Arguments

est, sd

The estimate and estimated standard deviation.

n

Number of observations.

transformer

A transformer object.

quants

Quantiles for the confidence interval.

x

Data to estimate alpha on.

type

Type of confidence interval. Either adf, elliptical, or normal.

parallel

If TRUE, makes calculations under the assumption of a parallel model. Default to FALSE.

n_reps

Number of bootstrap samples if bootstrap = TRUE. Ignored if bootstrap = FALSE.

standardized

If TRUE, calculates the standardized alpha. Calculates coefficient alpha otherwise.

Transform lambda and sigma to a covariance matrix.

Description

Transform lambda and sigma to a covariance matrix.

Usage

covmat(lambda, sigma)

Arguments

lambda

Vector of loadings.

sigma

Vector of standard deviations.

Value

The covariance matrix implied by lambda and sigma.

Gamma matrix

Description

Calculate the gamma matrix from a matrix of observations.

Usage

gamma_mat(x, sigma, type = "adf")

Arguments

x

A numeric matrix of observations.

sigma

Covariance matrix of the data.

type

One of adf, normal and elliptical.

Value

The sample estimate of the gamma matrix.

The gs vector used in the asymptotic variance of standardized alpha.

Description

The gs vector used in the asymptotic variance of standardized alpha.

Usage

gs(phi)

Arguments

phi

Correlation matrix.

Value

The gs vector.

Calculate unbiased sample kurtosis.

Description

Calculate unbiased sample kurtosis.

Usage

kurtosis(x)

Arguments

x

Matrix of valus.

Value

Unbiased sample kurtosis.

Calculate kurtosis correction

Description

Calculate kurtosis correction

Usage

kurtosis_correction(x, type)

Arguments

x

Matrix of values

type

The type of correction, either "normal" or "elliptical".

Calculate limits of a confidence interval.

Description

Calculate limits of a confidence interval.

Usage

limits(alternative, conf_level)

Arguments

alternative

Alternative choosen.

conf_level

Confidence level.

Psi matrix

Description

Calculate the psi matrix from a matrix of observations.

Usage

psi_mat(x, sigma, type = "adf")

Arguments

x

A numeric matrix of observations.

sigma

Covariance matrix.

type

One of adf, normal and elliptical.

Value

The sample estimate of the psi matrix.

Reliability coefficients

Description

The congeneric reliability and standardized reliability; also the bias b so that omega - alpha = b.

Usage

alpha_bias(sigma, lambda, w = rep(1, length(lambda)))

omega(sigma, lambda)

omega_std(sigma, lambda)

alpha(sigma, lambda)

alpha_std(sigma, lambda)

Arguments

sigma

For alpha and alpha_std, either apositive definite covariance matrix or a vector of standard deviations. A vector of standard deviations for omega and omega_std.

lambda

Vector of loadings.

Value

The congeneric reliability standardized reliability, coefficient alpha, standardized, orsigma coefficient alpha.

Simulate observations from a congeneric one-factor model.

Description

Simulate observations from a one-factor model with specified latent variable and error variable distributions. The error terms are not correlated, hence the model is congeneric.

Usage

simulate_congeneric(
  n,
  k,
  sigma = 1,
  lambda = 1,
  latent = stats::rnorm,
  error = stats::rnorm
)

Arguments

n

Number of observations.

k

Size of matrix or number of testlets.

sigma

Vector of error standard deviations. Repeated to have k elements.

lambda

Vector of factor loadings. Repeated to have k elements.

latent

Distribution of the latent variable.

error

Distribution of the error variable.

Value

Covariance matrix.

Studentized bootstrap estimates using transformers.

Description

Studentized bootstrap estimates using transformers.

Usage

studentized_boots(n_reps, x, type, parallel, transformer, standardized = FALSE)

Arguments

n_reps

Number of bootstrap repetitions.

x

Data to estimate alpha on.

type

Type of confidence interval. Either adf, elliptical, or normal.

parallel

If TRUE, makes calculations under the assumption of a parallel model.

transformer

A transformer object.

standardized

If TRUE, calculates the standardized alpha. Calculates coefficient alpha otherwise.

Value

Studentized bootstrap estimates.

Thurstone weights

Description

Thurstone weights

Usage

thurstone(lambda, sigma)

Arguments

lambda

Vector of loadings.

sigma

Vector of standard deviations.

Value

The Thurstone weights.

Trace of matrix

Description

Trace of matrix

Usage

tr(mat)

Arguments

mat

A square matrix.

Value

Trace of the matrix.