psychmeta (version 2.3.4)

.integrate_dmod: Integration function for computing parametric signed or unsigned \(d_{Mod}\) effect sizes for a single focal group

Description

This internal function exists to support the compute_dmod_par function, but may also be useful as a bare-bones tool for computing signed and unsigned \(d_{Mod}\) effect sizes. Please note that this function does not include an option for re-scaling its result to compensate for cumulative densities smaller than 1.

Usage

.integrate_dmod(referent_int, referent_slope, focal_int, focal_slope,
  focal_mean_x, focal_sd_x, referent_sd_y, focal_min_x, focal_max_x,
  signed = TRUE)

Arguments

referent_int

Referent group's intercept.

referent_slope

Referent group's slope.

focal_int

Focal group's intercept.

focal_slope

Focal group's slope.

focal_mean_x

Focal group's predictor-score mean.

focal_sd_x

Focal group's predictor-score standard deviation.

referent_sd_y

Referent group's criterion standard deviation.

focal_min_x

Focal group's minimum predictor score.

focal_max_x

Focal group's maximum predictor score.

signed

Logical argument that indicates whether the function should compute \(d_{Mod_{Signed}}\) (TRUE; default) or \(d_{Mod_{Unsigned}}\) (FALSE).

Value

A \(d_{Mod_{Signed}}\) or \(d_{Mod_{Unsigned}}\) effect size, depending on the signed argument.

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

• \(b_{0_{1}}\) and \(b_{0_{2}}\) 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 \(d_{Mod_{Unsigned}}\) effect size (i.e., the average of absolute differences in prediction over the range of predictor scores) is computed as $$d_{Mod_{Unsigned}}=\frac{1}{SD_{Y_{1}}}\intop f_{2}(X)\left|X\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right|dX.$$

References

Nye, C. D., & Sackett, P. R. (2016). New effect sizes for tests of categorical moderation and differential prediction. Organizational Research Methods, https://doi.org/10.1177/1094428116644505.

Examples

# NOT RUN {
# Example for computing \eqn{d_{Mod_{Signed}}}{d_Mod_Signed}
.integrate_dmod(referent_int = -.05, referent_slope = .5,
              focal_int = -.05, focal_slope = .3,
              focal_mean_x = -.5, focal_sd_x = 1,
              referent_sd_y = 1, focal_min_x = -Inf, focal_max_x = Inf,
              signed = TRUE)

# Example for computing \eqn{d_{Mod_{Unsigned}}}{d_Mod_Unsigned}
.integrate_dmod(referent_int = -.05, referent_slope = .5,
              focal_int = -.05, focal_slope = .3,
              focal_mean_x = -.5, focal_sd_x = 1,
              referent_sd_y = 1, focal_min_x = -Inf, focal_max_x = Inf,
              signed = FALSE)
# }