The POWER Procedure

PAIREDMEANS Statement

Subsections:

Summary of Options
Dictionary of Options
Restrictions on Option Combinations
Option Groups for Common Analyses

PAIREDMEANS <options>;

The PAIREDMEANS statement performs power and sample size analyses for t tests, equivalence tests, and confidence interval precision involving paired samples.

Summary of Options

Table 77.14 summarizes the options available in the PAIREDMEANS statement.

Table 77.16: PAIREDMEANS Statement Options

Option	Description
Define analysis
CI=	Specifies an analysis of precision of the confidence interval for the mean difference
DIST=	Specifies the underlying distribution assumed for the test statistic
TEST=	Specifies the statistical analysis
Specify analysis information
ALPHA=	Specifies the significance level
LOWER=	Specifies the lower equivalence bound
NULLDIFF=	Specifies the null mean difference
NULLRATIO=	Specifies the null mean ratio
SIDES=	Specifies the number of sides and the direction of the statistical test or confidence interval
UPPER=	Specifies the upper equivalence bound
Specify effects
HALFWIDTH=	Specifies the desired confidence interval half-width
MEANDIFF=	Specifies the mean difference
MEANRATIO=	Specifies the geometric mean ratio, $\gamma _2 / \gamma _1$
PAIREDMEANS=	Specifies the two paired means
Specify variability
CORR=	Specifies the correlation between members of a pair
CV=	Specifies the common coefficient of variation
PAIREDCVS=	Specifies the coefficient of variation for each member of a pair
PAIREDSTDDEVS=	Specifies the standard deviation of each member of a pair
STDDEV=	Specifies the common standard deviation
Specify sample size
NFRACTIONAL	Enables fractional input and output for sample sizes
NPAIRS=	Specifies the number of pairs
Specify power and related probabilities
POWER=	Specifies the desired power of the test
PROBTYPE=	Specifies the type of probability for the PROBWIDTH= option
PROBWIDTH=	Specifies the probability of obtaining a confidence interval half-width less than or equal to the value specified by the HALFWIDTH=
Control ordering in output
OUTPUTORDER=	Controls the output order of parameters

Table 77.17 summarizes the valid result parameters for different analyses in the PAIREDMEANS statement.

Table 77.17: Summary of Result Parameters in the PAIREDMEANS Statement

Analyses	Solve For	Syntax
TEST= DIFF	Power	POWER= .
	Sample size	NPAIRS= .
TEST= RATIO	Power	POWER= .
	Sample size	NPAIRS= .
TEST= EQUIV_DIFF	Power	POWER= .
	Sample size	NPAIRS= .
TEST= EQUIV_RATIO	Power	POWER= .
	Sample size	NPAIRS= .
CI= DIFF	Prob(width)	PROBWIDTH= .
	Sample size	NPAIRS= .

Dictionary of Options

ALPHA=number-list

specifies the level of significance of the statistical test. The default is 0.05, corresponding to the usual 0.05 $\times$ 100% = 5% level of significance. If the CI= and SIDES= 1 options are used, then the value must be less than 0.5. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

CI CI=DIFF

specifies an analysis of precision of the confidence interval for the mean difference. Instead of power, the relevant probability for this analysis is the probability of achieving a desired precision. Specifically, it is the probability that the half-width of the observed confidence interval will be at most the value specified by the HALFWIDTH= option. If neither the CI= option nor the TEST= option is used, the default is TEST= DIFF.

CORR=number-list

specifies the correlation between members of a pair. For tests that assume lognormal data (DIST= LOGNORMAL, or TEST= RATIO or TEST= EQUIV_RATIO), values of the CORR= option are restricted to the range $(\rho _ L, \rho _ U)$ , where

$\begin{align*} \rho _ L & = \frac{\exp \left(-\left[ \log (\mr{CV}_1^2+1) \log (\mr{CV}_2^2+1) \right]^\frac {1}{2} \right) - 1}{\mr{CV}_1 \mr{CV}_2} \\ \rho _ U & = \frac{\exp \left(\left[ \log (\mr{CV}_1^2+1) \log (\mr{CV}_2^2+1) \right]^\frac {1}{2}\right) - 1}{\mr{CV}_1 \mr{CV}_2} \end{align*}$

and $\mr{CV}_1$ are the $\mr{CV}_2$ coefficient of variation values specified by the CV= or PAIREDCVS= option. See Paired t Test for Mean Ratio with Lognormal Data (TEST=RATIO) for more information about this restriction on correlation values. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

CV=number-list

specifies the coefficient of variation that is assumed to be common to both members of a pair. The coefficient of variation is defined as the ratio of the standard deviation to the mean on the original data scale. You can use this option only with DIST= LOGNORMAL. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

DIST=LOGNORMAL | NORMAL

specifies the underlying distribution assumed for the test statistic. NORMAL corresponds the normal distribution, and LOGNORMAL corresponds to the lognormal distribution. The default value (also the only acceptable value in each case) is NORMAL for TEST= DIFF, TEST= EQUIV_DIFF, and CI= DIFF; and LOGNORMAL for TEST= RATIO and TEST= EQUIV_RATIO.

HALFWIDTH=number-list

specifies the desired confidence interval half-width. The half-width is defined as the distance between the point estimate and a finite endpoint. This option can be used only with the CI= DIFF analysis. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

LOWER=number-list

specifies the lower equivalence bound for the mean difference or mean ratio, in the original scale (whether DIST= NORMAL or DIST= LOGNORMAL). This option can be used only with the TEST= EQUIV_DIFF and TEST= EQUIV_RATIO analyses. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

MEANDIFF=number-list

specifies the mean difference, defined as the mean of the difference between the second and first members of a pair, $\mu _2 - \mu _1$ . This option can be used only with the TEST= DIFF and TEST= EQUIV_DIFF analyses. When TEST= EQUIV_DIFF, the mean difference is interpreted as the treatment mean minus the reference mean. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

MEANRATIO=number-list

specifies the geometric mean ratio, defined as $\gamma _2 / \gamma _1$ . This option can be used only with the TEST= RATIO and TEST= EQUIV_RATIO analyses. When TEST= EQUIV_RATIO, the mean ratio is interpreted as the treatment mean divided by the reference mean. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

NFRACTIONAL NFRAC

enables fractional input and output for sample sizes. See the section Sample Size Adjustment Options for information about the ramifications of the presence (and absence) of the NFRACTIONAL option.

NPAIRS=number-list

specifies the number of pairs or requests a solution for the number of pairs with a missing value (NPAIRS= .). For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

NULLDIFF=number-list NULLD=number-list

specifies the null mean difference. The default value is 0. This option can be used only with the TEST= DIFF analysis. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

NULLRATIO=number-list NULLR=number-list

specifies the null mean ratio. The default value is 1. This option can be used only with the TEST= RATIO analysis. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

OUTPUTORDER=INTERNAL | REVERSE | SYNTAX

controls how the input and default analysis parameters are ordered in the output. OUTPUTORDER= INTERNAL (the default) arranges the parameters in the output according to the following order of their corresponding options:

SIDES=
NULLDIFF=
NULLRATIO=
LOWER=
UPPER=
ALPHA=
PAIREDMEANS=
MEANDIFF=
MEANRATIO=
HALFWIDTH=
STDDEV=
PAIREDSTDDEVS=
CV=
PAIREDCVS=
CORR=
NPAIRS=
POWER=
PROBTYPE=
PROBWIDTH=

The OUTPUTORDER= SYNTAX option arranges the parameters in the output in the same order in which their corresponding options are specified in the PAIREDMEANS statement. The OUTPUTORDER= REVERSE option arranges the parameters in the output in the reverse of the order in which their corresponding options are specified in the PAIREDMEANS statement.

PAIREDCVS=grouped-number-list

specifies the coefficient of variation for each member of a pair. Unlike the CV= option, the PAIREDCVS= option supports different values for each member of a pair. The coefficient of variation is defined as the ratio of the standard deviation to the mean on the original data scale. Values must be nonnegative (unless both are equal to zero, which is permitted). This option can be used only with DIST= LOGNORMAL. For information about specifying the grouped-number-list, see the section Specifying Value Lists in Analysis Statements.

PAIREDMEANS=grouped-number-list PMEANS=grouped-number-list

specifies the two paired means, in the original scale. The means are arithmetic if DIST= NORMAL and geometric if DIST= LOGNORMAL. This option cannot be used with the CI= DIFF analysis. When TEST= EQUIV_DIFF, the means are interpreted as the reference mean (first) and the treatment mean (second). For information about specifying the grouped-number-list, see the section Specifying Value Lists in Analysis Statements.

PAIREDSTDDEVS=grouped-number-list PAIREDSTDS=grouped-number-list PSTDDEVS=grouped-number-list PSTDS=grouped-number-list

specifies the standard deviation of each member of a pair. Unlike the STDDEV= option, the PAIREDSTDDEVS= option supports different values for each member of a pair. This option can be used only with DIST= NORMAL. For information about specifying the grouped-number-list, see the section Specifying Value Lists in Analysis Statements.

POWER=number-list

specifies the desired power of the test or requests a solution for the power with a missing value (POWER= .). The power is expressed as a probability, a number between 0 and 1, rather than as a percentage. This option cannot be used with the CI= DIFF analysis. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

PROBTYPE=keyword-list

specifies the type of probability for the PROBWIDTH= option. A value of CONDITIONAL (the default) indicates the conditional probability that the confidence interval half-width is at most the value specified by the HALFWIDTH= option, given that the true mean difference is captured by the confidence interval. A value of UNCONDITIONAL indicates the unconditional probability that the confidence interval half-width is at most the value specified by the HALFWIDTH= option. you can use the alias GIVENVALIDITY for CONDITIONAL. The PROBTYPE= option can be used only with the CI= DIFF analysis. For information about specifying the keyword-list, see the section Specifying Value Lists in Analysis Statements.

CONDITIONAL: width probability conditional on interval containing the mean
UNCONDITIONAL: unconditional width probability

PROBWIDTH=number-list

specifies the desired probability of obtaining a confidence interval half-width less than or equal to the value specified by the HALFWIDTH= option. A missing value (PROBWIDTH= .) requests a solution for this probability. The type of probability is controlled with the PROBTYPE= option. Values are expressed as probabilities (for example, 0.9) rather than percentages. This option can be used only with the CI= DIFF analysis. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

SIDES=keyword-list

specifies the number of sides (or tails) and the direction of the statistical test or confidence interval. For information about specifying the keyword-list, see the section Specifying Value Lists in Analysis Statements. Valid keywords and their interpretation for the TEST= analyses are as follows:

1: one-sided with alternative hypothesis in same direction as effect
2: two-sided
U: upper one-sided with alternative greater than null value
L: lower one-sided with alternative less than null value

For confidence intervals, SIDES= U refers to an interval between the lower confidence limit and infinity, and SIDES= L refers to an interval between minus infinity and the upper confidence limit. For both of these cases and SIDES= 1, the confidence interval computations are equivalent. The SIDES= option cannot be used with the TEST= EQUIV_DIFF and TEST= EQUIV_RATIO analyses. The default value is 2.

STDDEV=number-list STD=number-list

specifies the standard deviation assumed to be common to both members of a pair. This option can be used only with DIST= NORMAL. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

TEST=DIFF | EQUIV_DIFF | EQUIV_RATIO | RATIO TEST

specifies the statistical analysis. TEST or TEST= DIFF (the default) specifies a paired t test on the mean difference. TEST= EQUIV_DIFF specifies an additive equivalence test of the mean difference by using a two one-sided tests (TOST) analysis (Schuirmann, 1987). TEST= EQUIV_RATIO specifies a multiplicative equivalence test of the mean ratio by using a TOST analysis. TEST= RATIO specifies a paired t test on the geometric mean ratio. If neither the TEST= option nor the CI= option is used, the default is TEST= DIFF.

UPPER=number-list

specifies the upper equivalence bound for the mean difference or mean ratio, in the original scale (whether DIST= NORMAL or DIST= LOGNORMAL). This option can be used only with the TEST= EQUIV_DIFF and TEST= EQUIV_RATIO analyses. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

Restrictions on Option Combinations

To define the analysis, choose one of the following parameterizations:

a statistical test (by using the TEST= option)
confidence interval precision (by using the CI= option)

To specify the means, choose one of the following parameterizations:

individual means (by using the PAIREDMEANS= option)
mean difference (by using the MEANDIFF= option)
mean ratio (by using the MEANRATIO= option)

To specify the coefficient of variation, choose one of the following parameterizations:

common coefficient of variation (by using the CV= option)
individual coefficients of variation (by using the PAIREDCVS= option)

To specify the standard deviation, choose one of the following parameterizations:

common standard deviation (by using the STDDEV= option)
individual standard deviations (by using the PAIREDSTDDEVS= option)

Option Groups for Common Analyses

This section summarizes the syntax for the common analyses supported in the PAIREDMEANS statement.

Paired t Test

You can express effects in terms of the mean difference and variability in terms of a correlation and common standard deviation, as in the following statements. Default values for the DIST= , SIDES= , NULLDIFF= , and ALPHA= options specify a two-sided test for no difference with a normal distribution and a significance level of 0.05.

proc power;
   pairedmeans test=diff
      meandiff = 7
      corr = 0.4
      stddev = 12
      npairs = 50
      power = .;
run;

You can also express effects in terms of individual means and variability in terms of correlation and individual standard deviations:

proc power;
   pairedmeans test=diff
      pairedmeans = 8 | 15
      corr = 0.4
      pairedstddevs = (7 12)
      npairs = .
      power = 0.9;
run;

Paired t Test of Mean Ratio with Lognormal Data

You can express variability in terms of correlation and a common coefficient of variation, as in the following statements. Defaults for the DIST= , SIDES= , NULLRATIO= and ALPHA= options specify a two-sided test of mean ratio = 1 assuming a lognormal distribution and a significance level of 0.05.

proc power;
   pairedmeans test=ratio
      meanratio = 7
      corr = 0.3
      cv = 1.2
      npairs = 30
      power = .;
run;

You can also express variability in terms of correlation and individual coefficients of variation:

proc power;
   pairedmeans test=ratio
      meanratio = 7
      corr = 0.3
      pairedcvs = 0.8 | 0.9
      npairs = 30
      power = .;
run;

Additive Equivalence Test for Mean Difference with Normal Data

The following statements demonstrate a sample size computation for a TOST equivalence test for a normal mean difference. Default values for the DIST= and ALPHA= options specify a normal distribution and a significance level of 0.05.

proc power;
   pairedmeans test=equiv_diff
      lower = 2
      upper = 5
      meandiff = 4
      corr = 0.2
      stddev = 8
      npairs = .
      power = 0.9;
run;

Multiplicative Equivalence Test for Mean Ratio with Lognormal Data

The following statements demonstrate a power computation for a TOST equivalence test for a lognormal mean ratio. Default values for the DIST= and ALPHA= options specify a lognormal distribution and a significance level of 0.05.

proc power;
   pairedmeans test=equiv_ratio
      lower = 3
      upper = 7
      meanratio = 5
      corr = 0.2
      cv = 1.1
      npairs = 50
      power = .;
run;

Confidence Interval for Mean Difference

By default CI= DIFF analyzes the conditional probability of obtaining the desired precision, given that the interval contains the true mean difference, as in the following statements. The defaults of SIDES= 2 and ALPHA= 0.05 specify a two-sided interval with a confidence level of 0.95.

proc power;
   pairedmeans ci = diff
      halfwidth = 4
      corr = 0.35
      stddev = 8
      npairs = 30
      probwidth = .;
run;