compute_dmod_npar: Function for computing non-parametric d_Mod effect sizes for...

In psychmeta: Psychometric Meta-Analysis Toolkit

Function for computing non-parametric d_Mod effect sizes for a single focal group

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

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.

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 = sum(n_i * [X_i * (b_1_1 - b_1_2) + b_0_1 - b_0_2]) / (SD_Y_1 * sum(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 ith 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 = sum(n_i * |X_i * (b_1_1 - b_1_2) + b_0_1 - b_0_2]| / (SD_Y_1 * sum(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 = 1/SD_Y_1 * Min[X * (b_1_1 - b_1_2) + b_0_1 - b_0_2].

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 = 1/SD_Y_1 * Max[X * (b_1_1 - b_1_2) + b_0_1 - b_0_2].

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)

psychmeta documentation built on Aug. 26, 2022, 5:14 p.m.