The POWER Procedure

TWOSAMPLEWILCOXON Statement

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

TWOSAMPLEWILCOXON <options> ;

The TWOSAMPLEWILCOXON statement performs power and sample size analyses for the Wilcoxon-Mann-Whitney test (also called the Wilcoxon rank-sum test, Mann-Whitney-Wilcoxon test, or Mann-Whitney U test) for two independent groups.

Note that the O’Brien-Castelloe approach to computing power for the Wilcoxon test is approximate, based on asymptotic behavior as the total sample size gets large. The quality of the power approximation degrades for small sample sizes; conversely, the quality of the sample size approximation degrades if the two distributions are far apart, so that only a small sample is needed to detect a significant difference. But this degradation is rarely a problem in practical situations, in which experiments are usually performed for relatively close distributions.

Summary of Options

Table 71.25 summarizes the options available in the TWOSAMPLEWILCOXON statement.

Table 71.25: TWOSAMPLEWILCOXON Statement Options

Option	Description
Define analysis
TEST=	Specifies the statistical analysis
Specify analysis information
ALPHA=	Specifies the significance level
SIDES=	Specifies the number of sides and the direction of the statistical test
Specify distributions
VARDIST=	Defines a distribution for a variable
VARIABLES=	Specifies the distributions of two or more variables
Specify sample size and allocation
GROUPNS=	Specifies the two group sample sizes
GROUPWEIGHTS=	Specifies the sample size allocation weights for the two groups
NFRACTIONAL	Enables fractional input and output for sample sizes
NPERGROUP=	Specifies the common sample size per group
NTOTAL=	Specifies the sample size
Specify power
POWER=	Specifies the desired power of the test
Specify computational options
NBINS=	Specifies the number of categories for each variable
Control ordering in output
OUTPUTORDER=	Controls the output order of parameters

Table 71.26 summarizes the valid result parameters in the TWOSAMPLEWILCOXON statement.

Table 71.26: Summary of Result Parameters in the TWOSAMPLEWILCOXON Statement

Analyses	Solve For	Syntax
TEST=WMW	Power	POWER=.
	Sample size	NTOTAL=.
		NPERGROUP=.

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. See the section Specifying Value Lists in Analysis Statements for information about specifying the number-list.

GROUPNS=grouped-number-list GNS=grouped-number-list

specifies the two group sample sizes. See the section Specifying Value Lists in Analysis Statements for information about specifying the grouped-number-list.

GROUPWEIGHTS=grouped-number-list GWEIGHTS=grouped-number-list

specifies the sample size allocation weights for the two groups. This option controls how the total sample size is divided between the two groups. Each pair of values for the two groups represents relative allocation weights. Additionally, if the NFRACTIONAL option is not used, the total sample size is restricted to be equal to a multiple of the sum of the two group weights (so that the resulting design has an integer sample size for each group while adhering exactly to the group allocation weights). Values must be integers unless the NFRACTIONAL option is used. The default value is (1 1), a balanced design with a weight of 1 for each group. See the section Specifying Value Lists in Analysis Statements for information about specifying the grouped-number-list.

NBINS=number-list

specifies the number of categories (or “bins”) each variable’s distribution is divided into (unless it is ordinal, in which case the categories remain intact) in internal calculations. Higher values increase computational time and memory requirements but generally lead to more accurate results. However, if the value is too high, then numerical instability can occur. The default value is 1000. See the section Specifying Value Lists in Analysis Statements for information about specifying the number-list.

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.

NPERGROUP=number-list NPERG=number-list

specifies the common sample size per group or requests a solution for the common sample size per group with a missing value (NPERGROUP=.). Use of this option implicitly specifies a balanced design. See the section Specifying Value Lists in Analysis Statements for information about specifying the number-list.

NTOTAL=number-list

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

OUTPUTORDER=INTERNAL OUTPUTORDER=REVERSE OUTPUTORDER=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 TWOSAMPLEWILCOXON 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 TWOSAMPLEWILCOXON statement.

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. See the section Specifying Value Lists in Analysis Statements for information about specifying the number-list.

SIDES=keyword-list

specifies the number of sides (or tails) and the direction of the statistical test. Valid keywords 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

The default value is 2.

TEST=WMW

specifies the Wilcoxon-Mann-Whitney test for two independent groups This is the default test option.

VARDIST("label")=distribution (parameters)

defines a distribution for a variable.

For the VARDIST= option,

label: identifies the variable distribution in the output and with the VARIABLES= option.
distribution: specifies the distributional form of the variable.
parameters: specifies one or more parameters associated with the distribution.

Choices for distributional forms and their parameters are as follows:

ORDINAL ((values) : (probabilities)): is an ordered categorical distribution. The values are any numbers separated by spaces. The probabilities are numbers between 0 and 1 (inclusive) separated by spaces. Their sum must be exactly 1. The number of probabilities must match the number of values.
BETA (a, b <, l, r >): is a beta distribution with shape parameters a and b and optional location parameters l and r. The values of a and b must be greater than 0, and l must be less than r. The default values for l and r are 0 and 1, respectively.
BINOMIAL (p, n): is a binomial distribution with probability of success p and number of independent Bernoulli trials n. The value of p must be greater than 0 and less than 1, and n must be an integer greater than 0. If n = 1, then the distribution is binary.
EXPONENTIAL ( $\lambda$ ): is an exponential distribution with scale $\lambda$ , which must be greater than 0.
GAMMA (a, $\lambda$ ): is a gamma distribution with shape a and scale $\lambda$ . The values of a and $\lambda$ must be greater than 0.
LAPLACE ( $\theta$ , $\lambda$ ): is a Laplace distribution with location $\theta$ and scale $\lambda$ . The value of $\lambda$ must be greater than 0.
LOGISTIC ( $\theta$ , $\lambda$ ): is a logistic distribution with location $\theta$ and scale $\lambda$ . The value of $\lambda$ must be greater than 0.
LOGNORMAL ( $\theta$ , $\lambda$ ): is a lognormal distribution with location $\theta$ and scale $\lambda$ . The value of $\lambda$ must be greater than 0.
NORMAL ( $\theta$ , $\lambda$ ): is a normal distribution with mean $\theta$ and standard deviation $\lambda$ . The value of $\lambda$ must be greater than 0.
POISSON (m): is a Poisson distribution with mean m. The value of m must be greater than 0.
UNIFORM (l, r): is a uniform distribution on the interval l, r , where l r.

VARIABLES=grouped-name-list VARS=grouped-name-list

specifies the distributions of two or more variables, using labels specified with the VARDIST= option. See the section Specifying Value Lists in Analysis Statements for information about specifying the grouped-name-list.

Restrictions on Option Combinations

To specify the sample size and allocation, choose one of the following parameterizations:

sample size per group in a balanced design (using the NPERGROUP= option)
total sample size and allocation weights (using the NTOTAL= and GROUPWEIGHTS= options)
individual group sample sizes (using the GROUPNS= option)

Option Groups for Common Analyses

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

Wilcoxon-Mann-Whitney Test for Comparing Two Distributions

The following statements performs a power analysis for Wilcoxon-Mann-Whitney tests comparing an ordinal variable with each other type of distribution. Default values for the ALPHA=, NBINS=, SIDES=, and TEST= options specify a two-sided test with a significance level of 0.05 and the use of 1000 categories per distribution when discretization is needed.

proc power;
   twosamplewilcoxon
      vardist("myordinal") = ordinal ((0 1 2) : (.2 .3 .5))
      vardist("mybeta1") = beta (1, 2)
      vardist("mybeta2") = beta (1, 2, 0, 2)
      vardist("mybinomial") = binomial (.3, 3)
      vardist("myexponential") = exponential (2)
      vardist("mygamma") = gamma (1.5, 2)
      vardist("mylaplace") = laplace (1, 2)
      vardist("mylogistic") = logistic (1, 2)
      vardist("mylognormal") = lognormal (1, 2)
      vardist("mynormal") = normal (3, 2)
      vardist("mypoisson") = poisson (2)
      vardist("myuniform") = uniform (0, 2)
      variables = "myordinal" | "mybeta1" "mybeta2" "mybinomial"
                                "myexponential" "mygamma" "mylaplace"
                                "mylogistic" "mylognormal" "mynormal"
                                "mypoisson" "myuniform"
      ntotal = 40
      power = .;
run;