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(pm1) on
depression(dm1) controlling for emotion-focused
coping(em1) and avoidance coping(am1) using
the first-wave data subset in internal data depress.
Four ways will be present in the following:
Firstly, examine the unique effect of pm1 using
t-test. Model C (Compact model) regresses dm1 on
em1 and am1. Model A(Augmented model)
regresses dm1 on pm1, em1, and
am1.
# multiple regression
fitC <- lm(dm1 ~ em1 + am1, depress)
fitA <- lm(dm1 ~ pm1 + em1 + am1, depress)
summary(fitA)
#>
#> Call:
#> lm(formula = dm1 ~ pm1 + em1 + am1, data = depress)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.63018 -0.24748 -0.00681 0.21045 1.01320
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 2.10497 0.25517 8.249 1.25e-12 ***
#> pm1 -0.16705 0.04988 -3.349 0.00119 **
#> em1 0.19504 0.05712 3.415 0.00096 ***
#> am1 -0.06675 0.05992 -1.114 0.26822
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.337 on 90 degrees of freedom
#> Multiple R-squared: 0.2491, Adjusted R-squared: 0.224
#> F-statistic: 9.949 on 3 and 90 DF, p-value: 9.928e-06As shown, the partial regression coefficient of pm1 is
-0.16705, t(90) = -3.349, p = 0.00119.
Secondly, examine the unique effect of pm1 using hierarchical regression and its F-test. In SPSS, this F-test is presented as the F-test for R2 change.
anova(fitC, fitA)
#> Analysis of Variance Table
#>
#> Model 1: dm1 ~ em1 + am1
#> Model 2: dm1 ~ pm1 + em1 + am1
#> Res.Df RSS Df Sum of Sq F Pr(>F)
#> 1 91 11.498
#> 2 90 10.224 1 1.2743 11.217 0.001185 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1As 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 pm1. In the case that the df of F’s numerator is 1, F = t2, and t’s df equals to the df of F’s denominator.
Thirdly, examine the unique effect of pm1 using PRE.
print(compare_lm(fitC, fitA), digits = 3)
#> Baseline C A A vs. C
#> SSE 13.6 1.15e+01 1.02e+01 1.27427
#> df 93.0 9.10e+01 9.00e+01 1.00000
#> Number of parameters 1.0 3.00e+00 4.00e+00 1.00000
#> R_squared NA 1.55e-01 2.49e-01 0.09359
#> f_squared NA 1.84e-01 3.32e-01 0.12464
#> R_squared_adj NA 1.37e-01 2.24e-01 NA
#> PRE NA 1.55e-01 2.49e-01 0.11082
#> F(PA-PC,n-PA) NA 8.38e+00 9.95e+00 11.21719
#> p NA 4.58e-04 9.93e-06 0.00119
#> PRE_adj NA 1.37e-01 2.24e-01 0.10094
#> power_post NA 9.59e-01 9.97e-01 0.91202As shown, F (1, 90) = 11.217, p = 0.00119. The F-test of PRE is equivalent to the F-test of anova above.
Fourthly, examine the unique effect of pm1 using
residuals. Regress dm1 on em1 and
am1, and attain the residuals of dm1,
dm_res, which partials out the effect of em1
and am1 on dm1.
Regress pm1 on em1 and am1,
and attain the residuals of pm1, pm_res, which
partials out the effect of em1 and am1 on
pm1.
Correlate dm_res with pm_res, we attain the
partial correlation of dm1 and pm1.
dm_res <- lm(dm1 ~ em1 + am1, depress)$residuals
pm_res <- lm(pm1 ~ em1 + am1, depress)$residuals
resDat <- data.frame(dm_res, pm_res)
cor(dm_res, pm_res)
#> [1] -0.3329009As shown, the partial correlation of dm1 and
pm1 is -0.3329009.
Regress dm_res on pm_res, and we attain the
unique effect of pm1 on dm1.
summary(lm(dm_res ~ pm_res, data.frame(dm_res, pm_res)))
#>
#> Call:
#> lm(formula = dm_res ~ pm_res, data = data.frame(dm_res, pm_res))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.63018 -0.24748 -0.00681 0.21045 1.01320
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 5.153e-17 3.438e-02 0.000 1.00000
#> pm_res -1.670e-01 4.933e-02 -3.386 0.00104 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.3334 on 92 degrees of freedom
#> Multiple R-squared: 0.1108, Adjusted R-squared: 0.1012
#> F-statistic: 11.47 on 1 and 92 DF, p-value: 0.001044As shown, the regression coefficient of pm_res equals
the partial regression coefficients of pm1 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)
#> Baseline C A A vs. C
#> SSE 11.5 11.5 10.22400 1.27427
#> df 93.0 93.0 92.00000 1.00000
#> Number of parameters 1.0 1.0 2.00000 1.00000
#> R_squared NA 0.0 0.11082 0.11082
#> f_squared NA 0.0 0.12464 0.12464
#> R_squared_adj NA 0.0 0.10116 NA
#> PRE NA 0.0 0.11082 0.11082
#> F(PA-PC,n-PA) NA NA 11.46646 11.46646
#> p NA NA 0.00104 0.00104
#> PRE_adj NA 0.0 0.10116 0.10116
#> power_post NA NA 0.91784 0.91784Compare the PRE of pm_res with the PRE
of pm1. 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
pm1 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.