compute_dmod_npar: Function for computing non-parametric $d_{M o d}$ effect sizes for a single focal group

Description

This function computes non-parametric $d_{M o d}$ 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

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_x

Focal group's vector of predictor scores.

referent_sd_y

Referent group's criterion standard deviation.

Value

A vector of effect sizes ( $d_{M o d_{S i g n e d}}$ , $d_{M o d_{U n s i g n e d}}$ , $d_{M o d_{U n d e r}}$ , $d_{M o d_{O v e r}}$ ), proportions of under- and over-predicted criterion scores, minimum and maximum differences (i.e., $d_{M o d_{U n d e r}}$ and $d_{M o d_{O v e r}}$ ), and the scores associated with minimum and maximum differences.

Details

The $d_{M o d_{S i g n e d}}$ effect size (i.e., the average of differences in prediction over the range of predictor scores) is computed as $d_{M o d_{S i g n e d}} = \frac{\sum_{i = 1}^{m} n_{i} [X_{i} (b_{1_{1}} - b_{1_{2}}) + b_{0_{1}} - b_{0_{2}}]}{S D_{Y_{1}} \sum_{i = 1}^{m} n_{i}},$ where

$S D_{Y_{1}}$ 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 $i^{t h}$ 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 $d_{M o d_{U n d e r}}$ and $d_{M o d_{O v e r}}$ effect sizes are computed using the same equation as $d_{M o d_{S i g n e d}}$ , but $d_{M o d_{U n d e r}}$ is the weighted average of all scores in the area of underprediction (i.e., the differences in prediction with negative signs) and $d_{M o d_{O v e r}}$ is the weighted average of all scores in the area of overprediction (i.e., the differences in prediction with negative signs).

The $d_{M o d_{U n s i g n e d}}$ effect size (i.e., the average of absolute differences in prediction over the range of predictor scores) is computed as $d_{M o d_{U n s i g n e d}} = \frac{\sum_{i = 1}^{m} n_{i} | X_{i} (b_{1_{1}} - b_{1_{2}}) + b_{0_{1}} - b_{0_{2}} |}{S D_{Y_{1}} \sum_{i = 1}^{m} n_{i}} .$

The $d_{M i n}$ effect size (i.e., the smallest absolute difference in prediction observed over the range of predictor scores) is computed as $d_{M i n} = \frac{1}{S D_{Y_{1}}} M i n [| X (b_{1_{1}} - b_{1_{2}}) + b_{0_{1}} - b_{0_{2}} |] .$

The $d_{M a x}$ effect size (i.e., the largest absolute difference in prediction observed over the range of predictor scores)is computed as $d_{M a x} = \frac{1}{S D_{Y_{1}}} M a x [| X (b_{1_{1}} - b_{1_{2}}) + b_{0_{1}} - b_{0_{2}} |] .$ Note: When $d_{M i n}$ and $d_{M a x}$ 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.

Examples

Run this code

# NOT RUN {
# 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)
# }