Partial Regression
In Keng: Knock Errors Off Nice Guesses

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(Keng)
data(depress)

Aiming to help researchers to understand the role of PRE in regression, this vignette will present several ways of examining the unique effect of problem-focused coping(cope_task1) on depression(depr1) controlling for emotion-focused coping(cope_emo1) and avoidance coping(cope_avo1) using the first-wave data subset in internal data depress.

Four ways will be present in the following:

Multiple regression with t-test.
Hierarchical regression with F-test.
The PRE of the single parameter, the partial regression coefficient of problem-focused coping.
One-predictor regression using the residuals.

Multiple regression with t-test

Firstly, examine the unique effect of cope_task1 using t-test. Model C (Compact model) regresses depr1 on cope_emo1 and cope_avo1. Model A(Augmented model) regresses depr1 on cope_task1, cope_emo1, and cope_avo1.

# multiple regression
fitC <- lm(depr1 ~ cope_emo1 + cope_avo1, depress)
fitA <- lm(depr1 ~ cope_task1 + cope_emo1 + cope_avo1, depress)
summary(fitA)

As shown, the partial regression coefficient of cope_task1 is -0.16705, t(90) = -3.349, p = 0.00119.

Hierarchical regression with F-test

Secondly, examine the unique effect of cope_task1 using hierarchical regression and its F-test. In SPSS, this F-test is presented as the F-test for R^2^ change.

anova(fitC, fitA)

As shown, F (1, 90) = 11.217, p = 0.001185. This F-test is equivalent to the t-test above, since they both examine the unique effect of cope_task1. In the case that the df of F's numerator is 1, F = t^2^, and t's df equals to the df of F's denominator.

The PRE of the single parameter

Thirdly, examine the unique effect of cope_task1 using PRE.

print(compare_lm(fitC, fitA), digits = 3)

As shown, F (1, 90) = 11.217, p = 0.00119. The F-test of PRE is equivalent to the F-test of anova above.

One-predictor regression using the residuals

Fourthly, examine the unique effect of cope_task1 using residuals. Regress depr1 on cope_emo1 and cope_avo1, and attain the residuals of depr1, dm_res, which partials out the effect of cope_emo1 and cope_avo1 on depr1.

Regress cope_task1 on cope_emo1 and cope_avo1, and attain the residuals of cope_task1, pm_res, which partials out the effect of cope_emo1 and cope_avo1 on cope_task1.

Correlate dm_res with pm_res, we attain the partial correlation of depr1 and cope_task1.

dm_res <- lm(depr1 ~ cope_emo1 + cope_avo1, depress)$residuals
pm_res <- lm(cope_task1 ~ cope_emo1 + cope_avo1, depress)$residuals
resDat <- data.frame(dm_res, pm_res)
cor(dm_res, pm_res)

As shown, the partial correlation of depr1 and cope_task1 is -0.3329009.

Regress dm_res on pm_res, and we attain the unique effect of cope_task1 on depr1.

summary(lm(dm_res ~ pm_res, data.frame(dm_res, pm_res)))

As shown, the regression coefficient of pm_res equals the partial regression coefficients of cope_task1 in fitA. However, their ts, as well as ps, are different. Why? Let's examine the unique effect of pm_res using PRE. Note that the F-test of one parameter's PRE is equivalent to the t-test of this parameter. In addition, Model A is relative to Model C. With your statistical purpose changing, the referents of Model C and Model A change.

fitC <- lm(dm_res ~ 1, resDat)
fitA <- lm(dm_res ~ pm_res, resDat)
print(compare_lm(fitC, fitA), digits = 3)

Compare the PRE of pm_res with the PRE of cope_task1. It's shown that two PREs are equivalent. However, df2s are different, which make Fs, as well as ps, different. In other words, though the unique effect of cope_task1 is constant, the compact models and augmented models used to evaluate its significance are different, which lead to different comparison conclusions (i.e., F-test and t-test results). Rethinking the F-test formula of PRE, we reach the following conclusion: With PRE being equal, the significance of PRE is determined by the df of Model C and the df-change of Model A against Model C.

Therefore, given the PRE of a specific set of predictor(s), the power of this specific set of predictor(s) are determined by the sample size n and the number of parameters [and hence the total number of predictor(s)] in the regression model. Similarly, given the PRE of a specific set of predictor(s), the required power for this specific set of predictor(s), and the number of parameters [and hence the total number of predictor(s)] in the regression model, we could compute the required sample size n.