The POWER Procedure

Common Notation

Table 77.29 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 77.29: Common Notation

Symbol

Description

Associated Syntax

$\alpha $

Significance level

ALPHA=

N

Total sample size

NTOTAL=, NPAIRS=

$n_ i$

Sample size in ith group

NPERGROUP=, GROUPNS=

$w_ i$

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 on the original data scale

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.