Package 'psychmeta'

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 [email protected]

Authors:

• Brenton M. Wiernik [email protected]

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

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)

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)

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)

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)

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)

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)

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)

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))

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)

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))

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)

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))

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)

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)

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)

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; [email protected]), Keke Lai (University of California–Merced)

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")

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)

Description

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

Arguments

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)

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()

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)

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).

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)

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