The POWER Procedure

Common Notation

Table 71.29 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 71.29: Common Notation

Symbol	Description	Associated Syntax
$\alpha$	Significance level	ALPHA=
N	Total sample size	NTOTAL=, NPAIRS=
	Sample size in ith group	NPERGROUP=, GROUPNS=
	Allocation weight for ith group (standardized to sum to 1)	GROUPWEIGHTS=
$\mu$	(Arithmetic) mean	MEAN=
$\mu _ i$	(Arithmetic) mean in ith group	GROUPMEANS=, PAIREDMEANS=
$\mu _\mr {diff}$	(Arithmetic) mean difference, $\mu _2 - \mu _1$ or $\mu _ T - \mu _ R$	MEANDIFF=
$\mu _0$	Null mean or mean difference (arithmetic)	NULL=, NULLDIFF=
$\gamma$	Geometric mean	MEAN=
$\gamma _ i$	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=
$\sigma _ i$	Standard deviation in ith group	GROUPSTDDEVS=, PAIREDSTDDEVS=
$\sigma _\mr {diff}$	Standard deviation of differences
CV	Coefficient of variation, defined as the ratio of the standard deviation to the (arithmetic) mean	CV=, PAIREDCVS=
$\rho$	Correlation	CORR=
$\mu _ T, \mu _ R$	Treatment and reference (arithmetic) means for equivalence test	GROUPMEANS=, PAIREDMEANS=
$\gamma _ T, \gamma _ R$	Treatment and reference geometric means for equivalence test	GROUPMEANS=, PAIREDMEANS=
$\theta _ L$	Lower equivalence bound	LOWER=
$\theta _ U$	Upper equivalence bound	UPPER=
$t(\nu , \delta )$	t distribution with df $\nu$ and noncentrality $\delta$
$F(\nu _1, \nu _2, \lambda )$	F distribution with numerator df $\nu _1$ , denominator df $\nu _2$ , and noncentrality $\lambda$
$t_{p; \nu }$	pth percentile of t distribution with df $\nu$
$F_{p; \nu _1, \nu _2}$	pth percentile of F distribution with numerator df $\nu _1$ and denominator df $\nu _2$
$\mr {Bin}(N, p)$	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:

$Q_\nu (t, \delta ; a, b) = \frac{\sqrt {2 \pi }}{\Gamma (\frac{\nu }{2})2^{\frac{\nu -2}{2}}} \int _ a^ b \Phi \left(\frac{tx}{\sqrt {\nu }} - \delta \right) x^{\nu -1} \phi (x) \mr {d}x$

where $\phi (\cdot )$ and $\Phi (\cdot )$ are the density and cumulative distribution function of the standard normal distribution, respectively.