The POWER Procedure

Common Notation

Table 34 displays notation for some of the more common parameters across analyses. The Associated Syntax column shows examples of relevant analysis statement options, where applicable.

Table 34: Common Notation

Symbol Description Associated Syntax
alpha Significance level ALPHA=
N Total sample size NTOTAL=, NPAIRS=
n Subscript i Sample size in ith group NPERGROUP=, GROUPNS=
w Subscript i Allocation weight for ith group (standardized to sum to 1) GROUPWEIGHTS=
mu (Arithmetic) mean MEAN=
mu Subscript i (Arithmetic) mean in ith group GROUPMEANS=, PAIREDMEANS=
mu Subscript normal d normal i normal f normal f (Arithmetic) mean difference, mu 2 minus mu 1 or mu Subscript upper T Baseline minus mu Subscript upper R MEANDIFF=
mu 0 Null mean or mean difference (arithmetic) NULL=, NULLDIFF=
gamma Geometric mean MEAN=
gamma Subscript i Geometric mean in ith group GROUPMEANS=, PAIREDMEANS=
gamma 0 Null mean or mean ratio (geometric) NULL=, NULLRATIO=
sigma Standard deviation (or common standard deviation per group) STDDEV=
sigma Subscript i Standard deviation in ith group GROUPSTDDEVS=, PAIREDSTDDEVS=
sigma Subscript normal d normal i normal f normal f Standard deviation of differences
CV Coefficient of variation, defined as the ratio of the standard deviation to the (arithmetic) mean on the original data scale CV=, PAIREDCVS=
rho Correlation CORR=
mu Subscript upper T Baseline comma mu Subscript upper R Baseline Treatment and reference (arithmetic) means for equivalence test GROUPMEANS=, PAIREDMEANS=
gamma Subscript upper T Baseline comma gamma Subscript upper R Baseline Treatment and reference geometric means for equivalence test GROUPMEANS=, PAIREDMEANS=
theta Subscript upper L Lower equivalence bound LOWER=
theta Subscript upper U Upper equivalence bound UPPER=
t left-parenthesis nu comma delta right-parenthesis t distribution with df nu and noncentrality delta
upper F left-parenthesis nu 1 comma nu 2 comma lamda right-parenthesis F distribution with numerator df nu 1, denominator df nu 2, and noncentrality lamda
t Subscript p semicolon nu pth percentile of t distribution with df nu
upper F Subscript p semicolon nu 1 comma nu 2 pth percentile of F distribution with numerator df nu 1 and denominator df nu 2
normal upper B normal i normal n left-parenthesis upper N comma p right-parenthesis Binomial distribution with sample size N and proportion p


A "lower one-sided" test is associated with SIDES=L (or SIDES=1 with the effect smaller than the null value), and an "upper one-sided" test is associated with SIDES=U (or SIDES=1 with the effect larger than the null value).

Owen (1965) defines a function, known as Owen’s Q, that is convenient for representing terms in power formulas for confidence intervals and equivalence tests:

upper Q Subscript nu Baseline left-parenthesis t comma delta semicolon a comma b right-parenthesis equals StartStartFraction StartRoot 2 pi EndRoot OverOver normal upper Gamma left-parenthesis StartFraction nu Over 2 EndFraction right-parenthesis 2 Superscript StartFraction nu minus 2 Over 2 EndFraction Baseline EndEndFraction integral Subscript a Superscript b Baseline normal upper Phi left-parenthesis StartFraction t x Over StartRoot nu EndRoot EndFraction minus delta right-parenthesis x Superscript nu minus 1 Baseline phi left-parenthesis x right-parenthesis normal d x

where phi left-parenthesis dot right-parenthesis and normal upper Phi left-parenthesis dot right-parenthesis are the density and cumulative distribution function of the standard normal distribution, respectively.

Last updated: December 09, 2022