CI for the IQR

Description

This function calculates bootstrap CIs (by default "bca") for the population interquartile range (IQR), i.e., the difference between first and third quartile.

Usage

ci_IQR(
  x,
  probs = c(0.025, 0.975),
  type = "bootstrap",
  boot_type = c("bca", "perc", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Value

An object of class "cint", see ci_mean() for details.

Examples

x <- rnorm(100)
ci_IQR(x, R = 999)  # Use larger R

CI for the NCP of the Chi-Squared Distribution

Description

This function calculates CIs for the non-centrality parameter (NCP) of the \chi^2-distribution. A positive lower (1 - \alpha) \cdot 100\%-confidence limit for the NCP goes hand-in-hand with a significant association test at level \alpha.

Usage

ci_chisq_ncp(
  x,
  probs = c(0.025, 0.975),
  correct = TRUE,
  type = c("chi-squared", "bootstrap"),
  boot_type = c("bca", "perc", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Details

By default, CIs are computed by Chi-squared test inversion. This can be unreliable for very large test statistics. The default bootstrap type is "bca".

Value

An object of class "cint", see ci_mean() for details.

References

Smithson, M. (2003). Confidence intervals. Series: Quantitative Applications in the Social Sciences. New York, NY: Sage Publications.

See Also

ci_cramersv()

Examples

ci_chisq_ncp(mtcars[c("am", "vs")])
ci_chisq_ncp(mtcars[c("am", "vs")], type = "bootstrap", R = 999)  # Use larger R

CI for Correlation Coefficients

Description

This function calculates CIs for a population correlation coefficient. For Pearson correlation, "normal" CIs are available (by stats::cor.test()). Also bootstrap CIs are supported (by default "bca", and the only option for rank correlations).

Usage

ci_cor(
  x,
  y = NULL,
  probs = c(0.025, 0.975),
  method = c("pearson", "kendall", "spearman"),
  type = c("normal", "bootstrap"),
  boot_type = c("bca", "perc", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Value

An object of class "cint", see ci_mean() for details.

Examples

ci_cor(iris[1:2])
ci_cor(iris[1:2], type = "bootstrap", R = 999)  # Use larger R
ci_cor(iris[1:2], method = "spearman", type = "bootstrap", R = 999)  # Use larger R

CI for the Population Cramer's V

Description

This function calculates CIs for the population Cramer's V. By default, a parametric approach based on the non-centrality parameter (NCP) of the chi-squared distribution is utilized. Alternatively, bootstrap CIs are available (default "bca"), also by boostrapping CIs for the NCP and then mapping the result back to Cramer's V.

Usage

ci_cramersv(
  x,
  probs = c(0.025, 0.975),
  type = c("chi-squared", "bootstrap"),
  boot_type = c("bca", "perc", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  test_adjustment = TRUE,
  ...
)

Arguments

Details

A positive lower (1 - \alpha) \cdot 100\%-confidence limit for the NCP goes hand-in-hand with a significant association test at level \alpha. In order to allow such test approach also with Cramer's V, if the lower bound for the NCP is 0, we round down to 0 the lower bound for Cramer's V as well. Without this slightly conservative adjustment, the lower limit for V would always be positive since the CI for V is found by \sqrt{(\textrm{CI for NCP} + \textrm{df})/(n \cdot (k - 1))}, where k is the smaller number of levels in the two variables (see Smithson, p.40). Use test_adjustment = FALSE to switch off this behaviour. Note that this is also a reason to bootstrap V via NCP instead of directly bootstrapping V.

Further note that no continuity correction is applied for 2x2 tables, and that large chi-squared test statistics might provide unreliable results with method "chi-squared", see stats::pchisq().

Value

An object of class "cint", see ci_mean() for details.

References

Smithson, M. (2003). Confidence intervals. Series: Quantitative Applications in the Social Sciences. New York, NY: Sage Publications.

See Also

cramersv(), ci_chisq_ncp()

Examples

# Example from Smithson, M., page 41
test_scores <- as.table(
  rbind(
    Private = c(6, 14, 17, 9),
    Public = c(30, 32, 17, 3)
  )
)
suppressWarnings(X2 <- stats::chisq.test(test_scores))
ci_cramersv(X2)

CI for the Non-Centrality Parameter of the F Distribution

Description

Based on the inversion principle, parametric CIs for the non-centrality parameter (NCP) Delta of the F distribution are calculated. To keep the input interface simple, we do not provide bootstrap CIs here.

Usage

ci_f_ncp(x, df1 = NULL, df2 = NULL, probs = c(0.025, 0.975))

Arguments

Details

A positive lower (1 - \alpha) \cdot 100\%-confidence limit for the NCP goes hand-in-hand with a significant F test at level \alpha. According to stats::pf(), the results might be unreliable for very large F values.

Value

An object of class "cint", see ci_mean() for details.

References

Smithson, M. (2003). Confidence intervals. Series: Quantitative Applications in the Social Sciences. New York, NY: Sage Publications.

See Also

ci_rsquared()

Examples

fit <- lm(Sepal.Length ~ ., data = iris)
ci_f_ncp(fit)
ci_f_ncp(fit, probs = c(0.05, 1))

CI for the Kurtosis

Description

This function calculates bootstrap CIs for the population kurtosis. Note that we use the version of the kurtosis that equals 3 under a normal distribution, i.e., we are not calculating the excess kurtosis. By default, bootstrap type "bca" is used.

Usage

ci_kurtosis(
  x,
  probs = c(0.025, 0.975),
  type = "bootstrap",
  boot_type = c("bca", "perc", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Value

An object of class "cint", see ci_mean() for details.

See Also

kurtosis(), ci_skewness()

Examples

x <- 1:20
ci_kurtosis(x, R = 999)  # Use larger R

CI for the MAD

Description

This function calculates bootstrap CIs (default: "bca") for the population median absolute deviation (MAD), see stats::mad() for more information.

Usage

ci_mad(
  x,
  probs = c(0.025, 0.975),
  constant = 1.4826,
  type = "bootstrap",
  boot_type = c("bca", "perc", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Value

An object of class "cint", see ci_mean() for details.

Examples

x <- rnorm(100)
ci_mad(x, R = 999)  # Use larger R

CI for the Population Mean

Description

This function calculates CIs for the population mean. By default, Student's t method is used. Alternatively, Wald and bootstrap CIs are available.

Usage

ci_mean(
  x,
  probs = c(0.025, 0.975),
  type = c("t", "Wald", "bootstrap"),
  boot_type = c("stud", "bca", "perc", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Details

The default bootstrap type for the mean is "stud" (bootstrap t) as it enjoys the property of being second order accurate and has a stable variance estimator (see Efron, p. 188).

Value

An object of class "cint" containing these components:

• parameter: Parameter specification.

• interval: CI for the parameter.

• estimate: Parameter estimate.

• probs: Lower and upper probabilities.

• type: Type of interval.

• info: Additional description.

References

1. Smithson, M. (2003). Confidence intervals. Series: Quantitative Applications in the Social Sciences. New York, NY: Sage Publications.

2. Efron, B. and Tibshirani R. J. (1994). An Introduction to the Bootstrap. Chapman & Hall/CRC.

Examples

x <- 1:100
ci_mean(x)
ci_mean(x, type = "bootstrap", R = 999, seed = 1)  # Use larger R

CI for the Population Mean Difference

Description

This function calculates CIs for the population value of mean(x) - mean(y). The default is Student's method with Welch's correction for unequal variances, but also bootstrap CIs are available.

Usage

ci_mean_diff(
  x,
  y,
  probs = c(0.025, 0.975),
  var.equal = FALSE,
  type = c("t", "bootstrap"),
  boot_type = c("stud", "bca", "perc", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Details

The default bootstrap type is "stud" (bootstrap t) as it has a stable variance estimator (see Efron, p. 188). Resampling is done within sample. When boot_type = "stud", the standard error is estimated by Welch's method if var.equal = FALSE (the default), and by pooling otherwise. Thus, var.equal not only has an effect for the classic Student approach (type = "t") but also for boot_type = "stud".

Value

An object of class "cint", see ci_mean() for details.

References

Efron, B. and Tibshirani R. J. (1994). An Introduction to the Bootstrap. Chapman & Hall/CRC.

Examples

x <- 10:30
y <- 1:30
ci_mean_diff(x, y)
t.test(x, y)$conf.int
ci_mean_diff(x, y, type = "bootstrap", R = 999)  # Use larger R

CI for the Population Median

Description

This function calculates CIs for the population median by calling ci_quantile().

Usage

ci_median(
  x,
  probs = c(0.025, 0.975),
  type = c("binomial", "bootstrap"),
  boot_type = c("bca", "perc", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Value

An object of class "cint", see ci_mean() for details.

See Also

ci_quantile()

Examples

ci_median(1:100)

CI for the Population Median Difference of two Samples

Description

This function calculates bootstrap CIs for the population value of median(x) - median(y) by calling ci_quantile_diff().

Usage

ci_median_diff(
  x,
  y,
  probs = c(0.025, 0.975),
  type = "bootstrap",
  boot_type = c("bca", "perc", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Value

An object of class "cint", see ci_mean() for details.

See Also

ci_quantile_diff()

Examples

x <- 10:30
y <- 1:30
ci_median_diff(x, y, R = 999)  # Use larger value for R

CI for the Odds Ratio

Description

This function calculates a CI for the odds ratio in a 2x2 table/matrix or a data frame with two columns. The CI is obtained through stats::fisher.test(). Bootstrap CIs are not available.

Usage

ci_oddsratio(x, probs = c(0.025, 0.975))

Arguments

Value

An object of class "cint", see ci_mean() for details.

See Also

oddsratio().

Examples

x <- cbind(c(10, 5), c(4, 4))
ci_oddsratio(x)

CI for a Population Proportion

Description

This function calculates CIs for a population proportion. By default, "Clopper-Pearson" CIs are calculated (via stats::binom.test()). Further possibilities are "Wilson" (without continuity correction), "Agresti-Coull" (using normal quantile instead of +2 correction), and "bootstrap" (by default "bca").

Usage

ci_proportion(
  x,
  n = NULL,
  probs = c(0.025, 0.975),
  type = c("Clopper-Pearson", "Agresti-Coull", "Wilson", "bootstrap"),
  boot_type = c("bca", "perc", "stud", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Details

Note that we use the formulas for the Wilson and Agresti-Coull intervals in https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval. They agree with binom::binom.confint(x, n, method = "ac"/"wilson").

Value

An object of class "cint", see ci_mean() for details.

References

1. Clopper, C. and Pearson, E. S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika. 26 (4).

2. Wilson, E. B. (1927). Probable inference, the law of succession, and statistical inference. Journal of the American Statistical Association, 22 (158).

3. Agresti, A. and Coull, B. A. (1998). Approximate is better than 'exact' for interval estimation of binomial proportions. The American Statistician, 52 (2).

Examples

x <- rep(0:1, times = c(50, 100))
ci_proportion(x)
ci_proportion(x, type = "Wilson")
ci_proportion(x, type = "Agresti-Coull")

CI for a Population Quantile

Description

This function calculates CIs for a population quantile. By default, distribution-free CIs based on the binomial distribution are calculated, see Hahn and Meeker. Alternatively, bootstrap CIs are available (default "bca").

Usage

ci_quantile(
  x,
  q = 0.5,
  probs = c(0.025, 0.975),
  type = c("binomial", "bootstrap"),
  boot_type = c("bca", "perc", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Value

An object of class "cint", see ci_mean() for details.

References

Hahn, G. and Meeker, W. (1991). Statistical Intervals. Wiley 1991.

See Also

ci_median()

Examples

x <- 1:100
ci_quantile(x, q = 0.25)

CI for the Population Quantile Difference of two Samples

Description

This function calculates bootstrap CIs for the population value of q-quantile(x) - q-quantile(y), by default using "bca" bootstrap. Resampling is done within sample.

Usage

ci_quantile_diff(
  x,
  y,
  q = 0.5,
  probs = c(0.025, 0.975),
  type = "bootstrap",
  boot_type = c("bca", "perc", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Value

An object of class "cint", see ci_mean() for details.

See Also

ci_median_diff()

Examples

x <- 10:30
y <- 1:30
ci_quantile_diff(x, y, R = 999)  # Use larger R

CI for the Population R-Squared

Description

This function calculates parametric CIs for the population R^2. It is based on CIs for the non-centrality parameter \Delta of the F distribution found by test inversion. Values of \Delta are mapped to R^2 by R^2 = \Delta / (\Delta + \textrm{df}_1 + \textrm{df}_2 + 1), where the \textrm{df}_j are the degrees of freedom of the F test statistic. A positive lower (1 - \alpha) \cdot 100\%-confidence limit for the R^2 goes hand-in-hand with a significant F test at level \alpha.

Usage

ci_rsquared(x, df1 = NULL, df2 = NULL, probs = c(0.025, 0.975))

Arguments

Details

According to stats::pf(), the results might be unreliable for very large F values. Note that we do not provide bootstrap CIs here to keep the input interface simple.

Value

An object of class "cint", see ci_mean() for details.

References

Smithson, M. (2003). Confidence intervals. Series: Quantitative Applications in the Social Sciences. New York, NY: Sage Publications.

See Also

ci_f_ncp()

Examples

fit <- lm(Sepal.Length ~ ., data = iris)
summary(fit)$r.squared
ci_rsquared(fit)
ci_rsquared(fit, probs = c(0.05, 1))

CI for the Population Std

Description

This function calculates CIs for the population standard deviation. They are derived from CIs for the variance by taking the square-root, see ci_var().

Usage

ci_sd(
  x,
  probs = c(0.025, 0.975),
  type = c("chi-squared", "bootstrap"),
  boot_type = c("bca", "perc", "stud", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Value

An object of class "cint", see ci_mean() for details.

See Also

ci_var()

Examples

x <- 1:100
ci_sd(x)
ci_sd(x, type = "bootstrap", R = 999)  # Use larger R

CI for the Skewness

Description

This function calculates bootstrap CIs for the population skewness. By default, bootstrap type "bca" is used.

Usage

ci_skewness(
  x,
  probs = c(0.025, 0.975),
  type = "bootstrap",
  boot_type = c("bca", "perc", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Value

An object of class "cint", see ci_mean() for details.

See Also

skewness(), ci_kurtosis()

Examples

x <- 1:20
ci_skewness(x, R = 999)  # Use larger R

CI for the Population Variance

Description

This function calculates CIs for the population variance.

Usage

ci_var(
  x,
  probs = c(0.025, 0.975),
  type = c("chi-squared", "bootstrap"),
  boot_type = c("bca", "perc", "stud", "norm", "basic"),
  R = 9999L,
  seed = NULL,
  ...
)

Arguments

Details

By default, classic CIs are calculated based on the chi-squared distribution, assuming normal distribution (see Smithson). Bootstrap CIs are also available (default: "bca"). We recommend them for the non-normal case.

The stud (bootstrap t) bootstrap uses the standard error of the sample variance given in Wilks.

Value

An object of class "cint", see ci_mean() for details.

References

1. Smithson, M. (2003). Confidence intervals. Series: Quantitative Applications in the Social Sciences. New York, NY: Sage Publications.

2. S.S. Wilks (1962), Mathematical Statistics, Wiley & Sons.

See Also

ci_sd()

Examples

x <- 1:100
ci_var(x)
ci_var(x, type = "bootstrap", R = 999)  # Use larger R

Cramer's V

Description

This function calculates Cramer's V, a measure of association between two categorical variables.

Usage

cramersv(x)

Arguments

Details

Cramer's V is a scaled version of the chi-squared test statistic \chi^2 and takes values in [0, 1]. It is calculated as \sqrt{\chi^2 / (n \cdot (k - 1))}, where n is the number of observations, and k is the smaller of the number of levels of the two variables.

Yates continuity correction is never applied. So in the 2x2 case, if x is the result of stats::chisq.test(), make sure no continuity correction was applied. Otherwise, results can be inconsistent.

Value

A numeric vector of length one.

References

Cramer, Harald. 1946. Mathematical Methods of Statistics. Princeton: Princeton University Press, page 282 (Chapter 21. The two-dimensional case).

See Also

ci_cramersv()

Examples

cramersv(mtcars[c("am", "vs")])

Type Check

Description

Checks if an object inherits class "cint".

Usage

is.cint(x)

Arguments

Value

A logical vector of length one.

Examples

is.cint(ci_proportion(5, 20))
is.cint(c(1, 2))

Pearson's Measure of Kurtosis

Description

Defined as the ratio of the 4th central moment and the squared second central moment. Under perfect normality, the kurtosis equals 3. Put differently, we do not show "excess kurtosis" but rather kurtosis.

Usage

kurtosis(z, na.rm = TRUE)

Arguments

Value

Numeric vector of length 1.

See Also

moment(), skewness()

Examples

kurtosis(1:10)
kurtosis(rnorm(1000))

Sample Moments

Description

Calculates central or non-central sample moments.

Usage

moment(z, p = 1, central = TRUE, na.rm = TRUE)

Arguments

Value

Numeric vector of length 1.

See Also

skewness(), kurtosis()

Examples

moment(1:10, p = 1)
moment(1:10, p = 1, central = FALSE)
moment(1:10, p = 2) / stats::var(1:10)

Odds Ratio

Description

This function calculates the odds ratio of a 2x2 table/matrix, or a data frame with two columns.

Usage

oddsratio(x)

Arguments

Details

The numerator equals the ratio of the top left entry and the bottom left entry of the 2x2 table, while the denominator equals the ratio of the top right entry and the bottom right entry. The result is usually slightly different from the one of stats::fisher.test(), which is based on the ML estimate of the odds ratio.

Value

A numeric vector of length one.

See Also

ci_oddsratio()

Examples

tab <- cbind(c(10, 5), c(4, 4))
oddsratio(tab)

Print "cint" Object

Description

Print method for an object of class "cint".

Usage

## S3 method for class 'cint'
print(x, digits = getOption("digits"), ...)

Arguments

Value

Invisibly, the input is returned.

Examples

ci_mean(1:100)

Standard errors

Description

Functions to calculate standard errors of different statistics. The availability of a standard error (or statistic proportional to it) allows to apply "stud" (bootstrap t) bootstrap.

Usage

se_mean(z, na.rm = TRUE, ...)

se_mean_diff(z, y, na.rm = TRUE, var.equal = FALSE, ...)

se_var(z, na.rm = TRUE, ...)

se_proportion(z, na.rm = TRUE, ...)

Arguments

Value

A numeric vector of length one.

Examples

se_mean(1:100)

Sample Skewness

Description

Calculates sample skewness. A value of 0 refers to a perfectly symmetric distribution.

Usage

skewness(z, na.rm = TRUE)

Arguments

Value

Numeric vector of length 1.

See Also

moment(), kurtosis()

Examples

skewness(1:10)
skewness(rexp(100))