# Function for computing non-parametric $d_Mod$ effect sizes for a single focal group 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. ```r 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. ## Returns 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. ## Details The $d_Mod_Signed$ effect size (i.e., the average of differences in prediction over the range of predictor scores) is computed as $$ d_{Mod_{Signed}}=\frac{\sum_{i=1}^{m}n_{i}\left[X_{i}\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right]}{SD_{Y_{1}}\sum_{i=1}^{m}n_{i}},d_Mod_Signed = sum(n_i * [X_i * (b_1_1 - b_1_2) + b_0_1 - b_0_2]) / (SD_Y_1 * sum(n_i)), $$ 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 $$ d_{Mod_{Unsigned}}=\frac{\sum_{i=1}^{m}n_{i}\left|X_{i}\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right|}{SD_{Y_{1}}\sum_{i=1}^{m}n_{i}}.d_Mod_Unsigned = sum(n_i * |X_i * (b_1_1 - b_1_2) + b_0_1 - b_0_2]| / (SD_Y_1 * sum(n_i)). $$ The $d_Min$ effect size (i.e., the smallest absolute difference in prediction observed over the range of predictor scores) is computed as $$ d_{Min}=\frac{1}{SD_{Y_{1}}}Min\left[\left|X\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right|\right].d_Min = 1/SD_Y_1 * Min[X * (b_1_1 - b_1_2) + b_0_1 - b_0_2]. $$ The $d_Max$ effect size (i.e., the largest absolute difference in prediction observed over the range of predictor scores)is computed as $$ d_{Max}=\frac{1}{SD_{Y_{1}}}Max\left[\left|X\left(b_{1_{1}}-b_{1_{2}}\right)+b_{0_{1}}-b_{0_{2}}\right|\right].d_Max = 1/SD_Y_1 * Max[X * (b_1_1 - b_1_2) + b_0_1 - b_0_2]. $$ **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. ## Examples ```r # 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) ```

compute_dmod_npar function