The POWER Procedure

Analyses in the MULTREG Statement

Type III F Test in Multiple Regression (TEST=TYPE3)

Maxwell (2000) discusses a number of different ways to represent effect sizes (and to compute exact power based on them) in multiple regression. PROC POWER supports two of these, multiple partial correlation and in full and reduced models.

Let p denote the total number of predictors in the full model (excluding the intercept), and let Y denote the response variable. You are testing that the coefficients of predictors in a set are 0, controlling for all of the other predictors , which consists of variables.

The hypotheses can be expressed in two different ways. The first is in terms of , the multiple partial correlation between the predictors in and the response Y adjusting for the predictors in :

StartLayout 1st Row 1st Column upper H 0 colon 2nd Column rho Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Superscript 2 Baseline equals 0 2nd Row 1st Column upper H 1 colon 2nd Column rho Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Superscript 2 Baseline greater-than 0 EndLayout

The second is in terms of the multiple correlations in full () and reduced () nested models:

StartLayout 1st Row 1st Column upper H 0 colon 2nd Column rho Subscript upper Y vertical-bar left-parenthesis upper X 1 comma upper X Sub Subscript negative 1 Subscript right-parenthesis Superscript 2 Baseline minus rho Subscript upper Y vertical-bar upper X Sub Subscript negative 1 Subscript Superscript 2 Baseline equals 0 2nd Row 1st Column upper H 1 colon 2nd Column rho Subscript upper Y vertical-bar left-parenthesis upper X 1 comma upper X Sub Subscript negative 1 Subscript right-parenthesis Superscript 2 Baseline minus rho Subscript upper Y vertical-bar upper X Sub Subscript negative 1 Subscript Superscript 2 Baseline greater-than 0 EndLayout

Note that the squared values of and are the population values for full and reduced models.

The test statistic can be written in terms of the sample multiple partial correlation ,

upper F equals StartLayout Enlarged left-brace 1st Row 1st Column left-parenthesis upper N minus 1 minus p right-parenthesis StartFraction upper R Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Superscript 2 Baseline Over 1 minus upper R Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Superscript 2 Baseline EndFraction comma 2nd Column intercept 2nd Row 1st Column left-parenthesis upper N minus p right-parenthesis StartFraction upper R Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Superscript 2 Baseline Over 1 minus upper R Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Superscript 2 Baseline EndFraction comma 2nd Column no intercept EndLayout

or the sample multiple correlations in full () and reduced () models,

The test is the usual Type III F test in multiple regression:

Reject upper H 0 if StartLayout Enlarged left-brace 1st Row 1st Column upper F greater-than-or-equal-to upper F Subscript 1 minus alpha Baseline left-parenthesis p 1 comma upper N minus 1 minus p right-parenthesis comma 2nd Column intercept 2nd Row 1st Column upper F greater-than-or-equal-to upper F Subscript 1 minus alpha Baseline left-parenthesis p 1 comma upper N minus p right-parenthesis comma 2nd Column no intercept EndLayout

Although the test is invariant to whether the predictors are assumed to be random or fixed, the power is affected by this assumption. If the response and predictors are assumed to have a joint multivariate normal distribution, then the exact power is given by the following formula:

The distribution of (for any ) is given in Chapter 32 of Johnson, Kotz, and Balakrishnan (1995). Sample size tables are presented in Gatsonis and Sampson (1989).

If the predictors are assumed to have fixed values, then the exact power is given by the noncentral F distribution. The noncentrality parameter is

lamda equals upper N StartFraction rho Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Superscript 2 Baseline Over 1 minus rho Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Superscript 2 Baseline EndFraction

or equivalently,

lamda equals upper N StartFraction rho Subscript upper Y vertical-bar left-parenthesis upper X 1 comma upper X Sub Subscript negative 1 Subscript right-parenthesis Superscript 2 Baseline minus rho Subscript upper Y vertical-bar upper X Sub Subscript negative 1 Subscript Superscript 2 Baseline Over 1 minus rho Subscript upper Y vertical-bar left-parenthesis upper X 1 comma upper X Sub Subscript negative 1 Subscript right-parenthesis Superscript 2 Baseline EndFraction

The power is

StartLayout 1st Row 1st Column normal p normal o normal w normal e normal r 2nd Column equals StartLayout Enlarged left-brace 1st Row 1st Column upper P left-parenthesis upper F left-parenthesis p 1 comma upper N minus 1 minus p comma lamda right-parenthesis greater-than-or-equal-to upper F Subscript 1 minus alpha Baseline left-parenthesis p 1 comma upper N minus 1 minus p right-parenthesis right-parenthesis comma 2nd Column intercept 2nd Row 1st Column upper P left-parenthesis upper F left-parenthesis p 1 comma upper N minus p comma lamda right-parenthesis greater-than-or-equal-to upper F Subscript 1 minus alpha Baseline left-parenthesis p 1 comma upper N minus p right-parenthesis right-parenthesis comma 2nd Column no intercept EndLayout EndLayout

The minimum acceptable input value of N depends on several factors, as shown in Table 36.

Table 36: Minimum Acceptable Sample Size Values in the MULTREG Statement

Predictor Type	Intercept in Model?	?	Minimum N
Random	Yes	Yes	p + 3
Random	Yes	No	p + 2
Random	No	Yes	p + 2
Random	No	No	p + 1
Fixed	Yes	Yes or No	p + 2
Fixed	No	Yes or No	p + 1

Last updated: December 09, 2022