The POWER Procedure

Analyses in the ONEWAYANOVA Statement

One-Degree-of-Freedom Contrast (TEST=CONTRAST)

The hypotheses are

StartLayout 1st Row 1st Column upper H 0 colon 2nd Column c 1 mu 1 plus midline-horizontal-ellipsis plus c Subscript upper G Baseline mu Subscript upper G Baseline equals c 0 2nd Row 1st Column upper H 1 colon 2nd Column StartLayout Enlarged left-brace 1st Row 1st Column c 1 mu 1 plus midline-horizontal-ellipsis plus c Subscript upper G Baseline mu Subscript upper G Baseline not-equals c 0 comma 2nd Column two hyphen sided 2nd Row 1st Column c 1 mu 1 plus midline-horizontal-ellipsis plus c Subscript upper G Baseline mu Subscript upper G Baseline greater-than c 0 comma 2nd Column upper one hyphen sided 3rd Row 1st Column c 1 mu 1 plus midline-horizontal-ellipsis plus c Subscript upper G Baseline mu Subscript upper G Baseline less-than c 0 comma 2nd Column lower one hyphen sided EndLayout EndLayout

where G is the number of groups, StartSet c 1 comma ellipsis comma c Subscript upper G Baseline EndSet are the contrast coefficients, and c 0 is the null contrast value.

The test is the usual F test for a contrast in one-way ANOVA. It assumes normal data with common group variances and requires upper N greater-than-or-equal-to upper G plus 1 and n Subscript i Baseline greater-than-or-equal-to 1.

O’Brien and Muller (1993, Section 8.2.3.2) give the exact power 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 upper G comma delta squared right-parenthesis greater-than-or-equal-to upper F Subscript 1 minus alpha Baseline left-parenthesis 1 comma upper N minus upper G right-parenthesis right-parenthesis comma 2nd Column two hyphen sided 2nd Row 1st Column upper P left-parenthesis t left-parenthesis upper N minus upper G comma delta right-parenthesis greater-than-or-equal-to t Subscript 1 minus alpha Baseline left-parenthesis upper N minus upper G right-parenthesis right-parenthesis comma 2nd Column upper one hyphen sided 3rd Row 1st Column upper P left-parenthesis t left-parenthesis upper N minus upper G comma delta right-parenthesis less-than-or-equal-to t Subscript alpha Baseline left-parenthesis upper N minus upper G right-parenthesis right-parenthesis comma 2nd Column lower one hyphen sided EndLayout

where

delta equals upper N Superscript one-half Baseline left-parenthesis StartFraction sigma-summation Underscript i equals 1 Overscript upper G Endscripts c Subscript i Baseline mu Subscript i Baseline minus c 0 Over sigma left-parenthesis sigma-summation Underscript i equals 1 Overscript upper G Endscripts StartFraction c Subscript i Superscript 2 Baseline Over w Subscript i Baseline EndFraction right-parenthesis Superscript one-half Baseline EndFraction right-parenthesis
Overall F Test (TEST=OVERALL)

The hypotheses are

StartLayout 1st Row 1st Column upper H 0 colon 2nd Column mu 1 equals mu 2 equals midline-horizontal-ellipsis equals mu Subscript upper G Baseline 2nd Row 1st Column upper H 1 colon 2nd Column mu Subscript i Baseline not-equals mu Subscript j Baseline for some i comma j EndLayout

where G is the number of groups.

The test is the usual overall F test for equality of means in one-way ANOVA. It assumes normal data with common group variances and requires upper N greater-than-or-equal-to upper G plus 1 and n Subscript i Baseline greater-than-or-equal-to 1.

O’Brien and Muller (1993, Section 8.2.3.1) give the exact power as

normal p normal o normal w normal e normal r equals upper P left-parenthesis upper F left-parenthesis upper G minus 1 comma upper N minus upper G comma lamda right-parenthesis greater-than-or-equal-to upper F Subscript 1 minus alpha Baseline left-parenthesis upper G minus 1 comma upper N minus upper G right-parenthesis right-parenthesis

where the noncentrality is

lamda equals upper N left-parenthesis StartFraction sigma-summation Underscript i equals 1 Overscript upper G Endscripts w Subscript i Baseline left-parenthesis mu Subscript i Baseline minus mu overbar right-parenthesis squared Over sigma squared EndFraction right-parenthesis

and

mu overbar equals sigma-summation Underscript i equals 1 Overscript upper G Endscripts w Subscript i Baseline mu Subscript i
Last updated: December 09, 2022