# The POWER Procedure

### ONESAMPLEMEANS Statement

Subsections:

• ONESAMPLEMEANS <options>;

The ONESAMPLEMEANS statement performs power and sample size analyses for t tests, equivalence tests, and confidence interval precision involving one sample.

#### Summary of Options

Table 89.12 summarizes the options available in the ONESAMPLEMEANS statement.

Table 89.12: ONESAMPLEMEANS Statement Options

Option

Description

Define analysis

Specifies an analysis of precision of the confidence interval for the mean

Specifies the underlying distribution assumed for the test statistic

Specifies the statistical analysis

Specify analysis information

Specifies the significance level

Specifies the lower equivalence bound for the mean

Specifies the null mean

Specifies the number of sides and the direction of the statistical test

Specifies the upper equivalence bound for the mean

Specify effect

Specifies the desired confidence interval half-width

Specifies the mean

Specify variability

Specifies the coefficient of variation

Specifies the standard deviation

Specify sample size

Enables fractional input and output for sample sizes

Specifies the sample size

Specify power and related probabilities

Specifies the desired power of the test

Specifies the type of probability for the PROBWIDTH= option

Specifies the probability of obtaining a confidence interval half-width less than or equal to the value specified by HALFWIDTH=

Control ordering in output

Controls the output order of parameters

Table 89.13 summarizes the valid result parameters for different analyses in the ONESAMPLEMEANS statement.

Table 89.13: Summary of Result Parameters in the ONESAMPLEMEANS Statement

Analyses

Solve For

Syntax

TEST= T DIST= NORMAL

Power

Sample size

Alpha

Mean

Standard Deviation

TEST= T DIST= LOGNORMAL

Power

Sample size

TEST= EQUIV

Power

Sample size

CI= T

Prob(width)

Sample size

#### Dictionary of Options

ALPHA=number-list

specifies the level of significance of the statistical test or requests a solution for alpha by specifying a missing value (ALPHA= .). The default is 0.05, which corresponds to the usual 0.05 100% = 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=T

specifies an analysis of precision of the confidence interval for the mean. 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 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= T.

CV=number-list

specifies the coefficient of variation, 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 is NORMAL.

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= T 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. This option can be used only with the TEST= EQUIV analysis. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

MEAN=number-list

specifies the mean, in the original scale, or requests a solution for the mean by specifying a missing value (MEAN= .). The mean is arithmetic if DIST= NORMAL and geometric if DIST= LOGNORMAL. This option can be used only with the TEST= T and TEST= EQUIV analyses. 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.

NTOTAL=number-list

specifies the sample size or requests a solution for the sample size by specifying a missing value (NTOTAL= .). For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

NULLMEAN=number-list
NULLM=number-list

specifies the null mean, in the original scale (whether DIST= NORMAL or DIST= LOGNORMAL). The default value is 0 when DIST= NORMAL and 1 when DIST= LOGNORMAL. This option can be used only with the TEST= T 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:

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

POWER=number-list

specifies the desired power of the test or requests a solution for the power by specifying a missing value (POWER= .). The power is expressed as a probability, a number between 0 and 1, rather than as a percentage. This option can be used only with the TEST= T and TEST= EQUIV analyses. 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 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= T 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= T 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

specifies a one-sided test, with the alternative hypothesis in the same direction as the effect.

2

specifies a two-sided test.

U

specifies an upper one-sided test, with the alternative hypothesis indicating a mean greater than the null value.

L

specifies a lower one-sided test, with the alternative hypothesis indicating a mean less than the 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 can be used only with the TEST= T and CI= T analyses. By default, SIDES=2.

STDDEV=number-list
STD=number-list

specifies the standard deviation, or requests a solution for the standard deviation by specifying a missing value (STDDEV= .). You can use this option only with DIST= NORMAL. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.

TEST=EQUIV | T
TEST

specifies the statistical analysis. TEST= EQUIV specifies an equivalence test of the mean by using a two one-sided tests (TOST) analysis (Schuirmann 1987). TEST or TEST= T (the default) specifies a t test on the mean. If neither the TEST= option nor the CI= option is used, the default is TEST= T.

UPPER=number-list

specifies the upper equivalence bound for the mean, in the original scale (whether DIST= NORMAL or DIST= LOGNORMAL). This option can be used only with the TEST= EQUIV analysis. 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)

#### Option Groups for Common Analyses

This section summarizes the syntax for the common analyses that are supported in the ONESAMPLEMEANS statement.

##### One-Sample t Test

The following statements demonstrate a power computation for the one-sample t test. Default values for the DIST= , SIDES= , NULLMEAN= , and ALPHA= options specify a two-sided test for zero mean with a normal distribution and a significance level of 0.05.

proc power;
onesamplemeans test=t
mean = 7
stddev = 3
ntotal = 50
power = .;
run;

##### One-Sample t Test with Lognormal Data

The following statements demonstrate a sample size computation for the one-sample t test for lognormal data. Default values for the SIDES= , NULLMEAN= , and ALPHA= options specify a two-sided test for unit mean with a significance level of 0.05.

proc power;
onesamplemeans test=t dist=lognormal
mean = 7
cv = 0.8
ntotal = .
power = 0.9;
run;

##### Equivalence Test for Mean of Normal Data

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

proc power;
onesamplemeans test=equiv
lower = 2
upper = 7
mean = 4
stddev = 3
ntotal = 100
power = .;
run;

##### Equivalence Test for Mean of Lognormal Data

The following statements demonstrate a sample size computation for the TOST equivalence test for a lognormal mean. The default of ALPHA= 0.05 specifies a significance level of 0.05.

proc power;
onesamplemeans test=equiv dist=lognormal
lower = 1
upper = 5
mean = 3
cv = 0.6
ntotal = .
power = 0.85;
run;

##### Confidence Interval for Mean

By default CI= T analyzes the conditional probability of obtaining the desired precision, given that the interval contains the true mean, 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;
onesamplemeans ci = t
halfwidth = 14
stddev = 8
ntotal = 50
probwidth = .;
run;