The POWER Procedure

Analyses in the CUSTOM Statement

The first of the following subsections specifies notation for the analyses in the CUSTOM statement, and the remaining subsections show power formulas and special cases equivalent to analyses elsewhere in PROC POWER or PROC GLMPOWER for each type of test statistic distribution.

Notation

Table 35 displays notation for the analyses in the CUSTOM statement.

Table 35: Common Notation

Symbol Description Associated Option
N Total sample size NTOTAL=
p 1 Test degrees of freedom TESTDF=
p Model degrees of freedom MODELDF=
lamda Superscript star Primary noncentrality for chi squared or F distribution PRIMNC=
delta Superscript star Primary noncentrality for t or normal distribution PRIMNC=
rho Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Partial correlation between p 1 greater-than-or-equal-to 1 variables in a set upper X 1 and a variable Y, adjusting for p minus p 1 variables in a (possibly empty) set upper X Subscript negative 1 and assuming multivariate normality CORR=
m 1 Scalar multiplier of primary noncentrality PNCMULT=
m 2 Scalar multiplier of critical value CRITMULT=
alpha Significance level ALPHA=


Noncentral Chi-Square Distribution (DIST=CHISQUARE)

The power is computed as

normal p normal o normal w normal e normal r equals upper P left-parenthesis chi squared left-parenthesis p 1 comma upper N m 1 lamda Superscript star Baseline right-parenthesis greater-than-or-equal-to m 2 chi Subscript 1 minus alpha Superscript 2 Baseline left-parenthesis p 1 right-parenthesis right-parenthesis

The sample size is computed by numerically inverting the power formula.

The logistic regression analysis in the LOGISTIC statement is a special case in which p 1 equals 1 and m 2 equals 1. The two-sided Wilcoxon-Mann-Whitney test (TWOSAMPLEWILCOXON) is a special case in which m 1 equals 1.

Correlation Coefficient Distribution Assuming Multivariate Normal Data (DIST=CORR)

The distribution of the correlation coefficient given the value of rho Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript is shown in (Johnson, Kotz, and Balakrishnan 1995, Chapter 32). The one-sided cases are restricted to p 1 equals 1.

The power is computed as

normal p normal o normal w normal e normal r equals StartLayout Enlarged left-brace 1st Row 1st Column upper P left-bracket upper R Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Superscript 2 Baseline greater-than-or-equal-to StartStartFraction upper F Subscript 1 minus alpha Baseline left-parenthesis p 1 comma upper N minus p right-parenthesis OverOver upper F Subscript 1 minus alpha Baseline left-parenthesis p 1 comma upper N minus p right-parenthesis plus StartFraction upper N minus p Over p 1 EndFraction EndEndFraction vertical-bar rho Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Baseline right-bracket 2nd Column two hyphen sided 2nd Row 1st Column upper P left-bracket upper R Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Baseline greater-than-or-equal-to StartFraction t Subscript 1 minus alpha Baseline left-parenthesis upper N minus p minus 1 right-parenthesis Over left-parenthesis t Subscript 1 minus alpha Superscript 2 Baseline left-parenthesis upper N minus p minus 1 right-parenthesis plus upper N minus p minus 1 right-parenthesis Superscript one-half Baseline EndFraction vertical-bar rho Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Baseline right-bracket 2nd Column upper one hyphen sided 3rd Row 1st Column upper P left-bracket upper R Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Baseline less-than-or-equal-to StartFraction t Subscript alpha Baseline left-parenthesis upper N minus p minus 1 right-parenthesis Over left-parenthesis t Subscript alpha Superscript 2 Baseline left-parenthesis upper N minus p minus 1 right-parenthesis plus upper N minus p minus 1 right-parenthesis Superscript one-half Baseline EndFraction vertical-bar rho Subscript upper Y upper X 1 vertical-bar upper X Sub Subscript negative 1 Subscript Baseline right-bracket 2nd Column lower one hyphen sided EndLayout

The sample size is computed by numerically inverting the power formula.

The Pearson correlation analysis that assumes joint multivariate normality (ONECORR DIST=T MODEL=RANDOM) and the multiple regression analysis that assumes joint multivariate normal predictors (MULTREG MODEL=RANDOM) are special cases.

Noncentral F Distribution (DIST=F)

The power is computed as

normal p normal o normal w normal e normal r equals upper P left-parenthesis upper F left-parenthesis p 1 comma upper N minus p comma upper N m 1 lamda Superscript star Baseline 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

The sample size is computed by numerically inverting the power formula.

The two-sided contrast in one-way analysis of variance (ONEWAYANOVA) is a special case in which p 1 equals 1 and m 1 equals 1. The following analyses are special cases in which m 1 equals 1:

  • multiple regression that assumes fixed predictors (MULTREG MODEL=FIXED)

  • overall F test in one-way analysis of variance (ONEWAYANOVA)

  • Type III F test in a fixed-effect univariate linear model (PROC GLMPOWER)

  • multivariate Type III F test in a fixed-effect multivariate linear model (PROC GLMPOWER)

Normal Distribution (DIST=NORMAL)

The power is computed as

normal p normal o normal w normal e normal r equals StartLayout Enlarged left-brace 1st Row 1st Column normal upper Phi left-parenthesis m 2 z Subscript alpha Baseline minus upper N Superscript one-half Baseline m 1 delta Superscript star Baseline right-parenthesis 2nd Column upper one hyphen sided 2nd Row 1st Column 1 minus normal upper Phi left-parenthesis m 2 z Subscript 1 minus alpha Baseline minus upper N Superscript one-half Baseline m 1 delta Superscript star Baseline right-parenthesis 2nd Column lower one hyphen sided 3rd Row 1st Column upper P left-parenthesis chi squared left-parenthesis 1 comma upper N left-parenthesis m 1 delta Superscript star Baseline right-parenthesis squared right-parenthesis greater-than-or-equal-to m 2 squared chi Subscript 1 minus alpha Superscript 2 Baseline left-parenthesis 1 right-parenthesis right-parenthesis 2nd Column two hyphen sided EndLayout

The sample size is computed by numerically inverting the power equation.

Cox proportional hazards regression (COXREG) and the log-rank test (TWOSAMPLESURVIVAL) are special cases in which m 1 equals 1 and m 2 equals 1. The Wilcoxon-Mann-Whitney test (TWOSAMPLEWILCOXON) is a special case in which m 1 equals 1. Other special cases occur among the tests for binomial proportions (ONESAMPLEFREQ, PAIREDFREQ, and TWOSAMPLEFREQ).

Noncentral T Distribution (DIST=T)

The power is computed as

normal p normal o normal w normal e normal r equals StartLayout Enlarged left-brace 1st Row 1st Column upper P left-parenthesis upper F left-parenthesis 1 comma upper N minus p comma upper N left-parenthesis m 1 delta Superscript star Baseline right-parenthesis squared right-parenthesis greater-than-or-equal-to upper F Subscript 1 minus alpha Baseline left-parenthesis 1 comma upper N minus p right-parenthesis right-parenthesis 2nd Column two hyphen sided 2nd Row 1st Column upper P left-parenthesis t left-parenthesis upper N minus p comma upper N Superscript one-half Baseline m 1 delta Superscript star Baseline right-parenthesis greater-than-or-equal-to t Subscript 1 minus alpha Baseline left-parenthesis upper N minus p right-parenthesis right-parenthesis 2nd Column upper one hyphen sided 3rd Row 1st Column upper P left-parenthesis t left-parenthesis upper N minus p comma upper N Superscript one-half Baseline m 1 delta Superscript star Baseline right-parenthesis less-than-or-equal-to t Subscript alpha Baseline left-parenthesis upper N minus p right-parenthesis right-parenthesis 2nd Column lower one hyphen sided EndLayout

The sample size is computed by numerically inverting the power formula.

The standard t tests in the ONESAMPLEMEANS and PAIREDMEANS statements are special cases in which p equals 1 and m 1 equals 1. The pooled t test (TWOSAMPLEMEANS) is a special case in which p equals 2 and m 1 equals 1. The Pearson correlation analysis that assumes fixed X values (ONECORR DIST=T MODEL=FIXED) is a special case in which m 1 equals 1 and p equals p Superscript star Baseline plus 2, where p Superscript star is the number of variables that are partialed out. The one-sided contrast in one-way analysis of variance (ONEWAYANOVA) is special case in which m 1 equals 1 and p 1 equals upper G, where G is the number of groups.

Last updated: December 09, 2022