The POWER Procedure 
LOGISTIC Statement 
The LOGISTIC statement performs power and sample size analyses for the likelihood ratio chisquare test of a single predictor in binary logistic regression, possibly in the presence of one or more covariates. All predictor variables are assumed to be independent of each other. So, this analysis is not applicable to studies with correlated predictors—for example, most observational studies (as opposed to randomized studies).
Table 68.2 summarizes categories of options available in the LOGISTIC statement.
Task 
Options 

Define analysis 

Specify analysis information 

Specify effects 

Specify sample size 

Specify power 

Control sample size rounding 

Specify computational method 

Control ordering in output 
Table 68.3 summarizes the valid result parameters in the LOGISTIC statement.
Analyses 
Solve For 
Syntax 

TEST=LRCHI 
Power 

Sample size 
specifies the level of significance of the statistical test. The default is 0.05, corresponding to the usual 0.05 100% = 5% level of significance. See the section Specifying Value Lists in Analysis Statements for information about specifying the numberlist.
specifies the distributions of any predictor variables in the model but not being tested, using labels specified with the VARDIST= option. The distributions are assumed to be independent of each other and of the tested predictor. If this option is omitted, then the tested predictor specified by the TESTEDPREDICTOR= option is assumed to be the only predictor in the model. See the section Specifying Value Lists in Analysis Statements for information about specifying the groupednamelist.
specifies the odds ratios for the covariates in the full model (including variables in the TESTPREDICTOR= and COVARIATES= options). The ordering of the values corresponds to the ordering in the COVARIATES= option. If the response variable is coded as for success and for failure, then the odds ratio for each covariate is the odds of when divided by the odds of when , where and are determined from the DEFAULTUNIT= and UNITS= options. Values must be greater than zero. See the section Specifying Value Lists in Analysis Statements for information about specifying the groupednumberlist.
specifies the regression coefficients for the covariates in the full model including the test predictor (as specified by the TESTPREDICTOR= option). The ordering of the values corresponds to the ordering in the COVARIATES= option. See the section Specifying Value Lists in Analysis Statements for information about specifying the groupednumberlist.
specifies the default number of categories (or "bins") into which the distribution for each predictor variable is divided in internal calculations. Higher values increase computational time and memory requirements but generally lead to more accurate results. Each test predictor or covariate that is absent from the NBINS= option derives its bin number from the DEFAULTNBINS= option. The default value is DEFAULTNBINS=10.
There are two variable distributions for which the number of bins can be overridden internally:
For an ordinal distribution, the number of ordinal values is always used as the number of bins.
For a binomial distribution, if the requested number of bins is larger than , where n is the sample size parameter of the binomial distribution, then exactly bins are used.
specifies the default change in the predictor variables assumed for odds ratios specified with the COVODDSRATIOS= and TESTODDSRATIO= options. Each test predictor or covariate that is absent from the UNITS= option derives its change value from the DEFAULTUNIT= option. The value must be nonzero. The default value is DEFAULTUNIT=1. This option can be used only if at least one of the COVODDSRATIOS= and TESTODDSRATIO= options is used.
Valid specifications for changespec are as follows:
defines the odds ratio as the ratio of the response variable odds when to the odds when for any constant .
defines the odds ratio as the ratio of the odds when to the odds when (or , if SD is preceded by a minus sign (–)) for any constant , where is the standard deviation of (as determined from the VARDIST= option).
defines the odds ratio as the ratio of the odds when to the odds when for any constant , where is the standard deviation of (as determined from the VARDIST= option).
defines the odds ratio as the ratio of the odds when is equal to its percentile to the odds when is equal to its percentile (where the percentiles are determined from the distribution specified in the VARDIST= option). Values for p1 and p2 must be strictly between 0 and 1.
specifies the intercept in the full model (including variables in the TESTPREDICTOR= and COVARIATES= options). See the section Specifying Value Lists in Analysis Statements for information about specifying the numberlist.
specifies the number of categories (or "bins") into which the distribution for each predictor variable (identified by its name from the VARDIST= option) is divided in internal calculations. Higher values increase computational time and memory requirements but generally lead to more accurate results. Each predictor variable that is absent from the NBINS= option derives its bin number from the DEFAULTNBINS= option.
There are two variable distributions for which the NBINS= value can be overridden internally:
For an ordinal distribution, the number of ordinal values is always used as the number of bins.
For a binomial distribution, if the requested number of bins is larger than , where n is the sample size parameter of the binomial distribution, then exactly bins are used.
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.
specifies the sample size or requests a solution for the sample size with a missing value (NTOTAL=.). Values must be at least one. See the section Specifying Value Lists in Analysis Statements for information about specifying the numberlist.
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 LOGISTIC 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 LOGISTIC statement.
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 numberlist.
specifies the response probability in the full model when all predictor variables (including variables in the TESTPREDICTOR= and COVARIATES= options) are equal to their means. The log odds of this probability are equal to the intercept in the full model where all predictor are centered at their means. If the response variable is coded as for success and for failure, then this probability is equal to the mean of in the full model when all Xs are equal to their means. Values must be strictly between zero and one. See the section Specifying Value Lists in Analysis Statements for information about specifying the numberlist.
specifies the likelihood ratio chisquare test of a single model parameter in binary logistic regression. This is the default test option.
specifies the odds ratio for the predictor variable being tested in the full model (including variables in the TESTPREDICTOR= and COVARIATES= options). If the response variable is coded as for success and for failure, then the odds ratio for the being tested is the odds of when divided by the odds of when , where and are determined from the DEFAULTUNIT= and UNITS= options. Values must be greater than zero. See the section Specifying Value Lists in Analysis Statements for information about specifying the numberlist.
specifies the distribution of the predictor variable being tested, using labels specified with the VARDIST= option. This distribution is assumed to be independent of the distributions of the covariates as defined in the COVARIATES= option. See the section Specifying Value Lists in Analysis Statements for information about specifying the namelist.
specifies the regression coefficient for the predictor variable being tested in the full model including the covariates specified by the COVARIATES= option. See the section Specifying Value Lists in Analysis Statements for information about specifying the numberlist.
specifies the changes in the predictor variables assumed for odds ratios specified with the COVODDSRATIOS= and TESTODDSRATIO= options. Each predictor variable whose name (from the VARDIST= option) is absent from the UNITS option derives its change value from the DEFAULTUNIT= option. This option can be used only if at least one of the COVODDSRATIOS= and TESTODDSRATIO= options is used.
Valid specifications for changespec are as follows:
defines the odds ratio as the ratio of the response variable odds when to the odds when for any constant .
defines the odds ratio as the ratio of the odds when to the odds when (or , if SD is preceded by a minus sign (–)) for any constant , where is the standard deviation of (as determined from the VARDIST= option).
defines the odds ratio as the ratio of the odds when to the odds when for any constant , where is the standard deviation of (as determined from the VARDIST= option).
defines the odds ratio as the ratio of the odds when is equal to its percentile to the odds when is equal to its percentile (where the percentiles are determined from the distribution specified in the VARDIST= option). Values for p1 and p2 must be strictly between 0 and 1.
Each unit value must be nonzero.
defines a distribution for a predictor variable.
For the VARDIST= option,
identifies the variable distribution in the output and with the COVARIATES= and TESTPREDICTOR= options.
specifies the distributional form of the variable.
specifies one or more parameters associated with the distribution.
Choices for distributional forms and their parameters are as follows:
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.
is a beta distribution with shape parameters and and optional location parameters and . The values of and must be greater than 0, and must be less than . The default values for and are 0 and 1, respectively.
is a binomial distribution with probability of success and number of independent Bernoulli trials . The value of must be greater than 0 and less than 1, and must be an integer greater than 0.
is an exponential distribution with scale , which must be greater than 0.
is a gamma distribution with shape and scale . The values of and must be greater than 0.
is a Laplace distribution with location and scale . The value of must be greater than 0.
is a logistic distribution with location and scale . The value of must be greater than 0.
is a lognormal distribution with location and scale . The value of must be greater than 0.
is a normal distribution with mean and standard deviation . The value of must be greater than 0.
is a Poisson distribution with mean . The value of must be greater than 0.
is a uniform distribution on the interval , where .
To specify the intercept in the full model, choose one of the following two parameterizations:
intercept (using the INTERCEPT= options)
Prob() when all predictors are equal to their means (using the RESPONSEPROB= option)
To specify the effect associated with the predictor variable being tested, choose one of the following two parameterizations:
odds ratio (using the TESTODDSRATIO= options)
regression coefficient (using the TESTREGCOEFFS= option)
To describe the effects of the covariates in the full model, choose one of the following two parameterizations:
odds ratios (using the COVODDSRATIOS= options)
regression coefficients (using the COVREGCOEFFS= options)
This section summarizes the syntax for the common analyses supported in the LOGISTIC statement.
You can express effects in terms of response probability and odds ratios, as in the following statements:
proc power; logistic vardist("x1a") = normal(0, 2) vardist("x1b") = normal(0, 3) vardist("x2") = poisson(7) vardist("x3a") = ordinal((5 0 5) : (.3 .4 .3)) vardist("x3b") = ordinal((5 0 5) : (.4 .3 .3)) testpredictor = "x1a" "x1b" covariates = "x2"  "x3a" "x3b" responseprob = 0.15 testoddsratio = 1.75 covoddsratios = (2.1 1.4) ntotal = 100 power = .; run;
The VARDIST= options define the distributions of the predictor variables. The TESTPREDICTOR= option specifies two scenarios for the test predictor distribution, Normal(10,2) and Normal(10,3). The COVARIATES= option specifies two covariates, the first with a Poisson(7) distribution. The second covariate has an ordinal distribution on the values –5, 0, and 5 with two scenarios for the associated probabilities: (.3, .4, .3) and (.4, .3, .3). The response probability in the full model with all variables equal to zero is specified by the RESPONSEPROB= option as 0.15. The odds ratio for a unit decrease in the tested predictor is specified by the TESTODDSRATIO= option to be 1.75. Corresponding odds ratios for the two covariates in the full model are specified by the COVODDSRATIOS= option to be 2.1 and 1.4. The POWER=. option requests a solution for the power at a sample size of 100 as specified by the NTOTAL= option.
Default values of the TEST= and ALPHA= options specify a likelihood ratio test of the first predictor with a significance level of 0.05. The default of DEFAULTUNIT=1 specifies that all odds ratios are defined in terms of unit changes in predictors. The default of DEFAULTNBINS=10 specifies that each of the three predictor variables is discretized into a distribution with 10 categories in internal calculations.
You can also express effects in terms of regression coefficients, as in the following statements:
proc power; logistic vardist("x1a") = normal(0, 2) vardist("x1b") = normal(0, 3) vardist("x2") = poisson(7) vardist("x3a") = ordinal((5 0 5) : (.3 .4 .3)) vardist("x3b") = ordinal((5 0 5) : (.4 .3 .3)) testpredictor = "x1a" "x1b" covariates = "x2"  "x3a" "x3b" intercept = 6.928162 testregcoeff = 0.5596158 covregcoeffs = (0.7419373 0.3364722) ntotal = 100 power = .; run;
The regression coefficients for the tested predictor (TESTREGCOEFF=0.5596158) and covariates (COVREGCOEFFS=(0.7419373 0.3364722)) are determined by taking the logarithm of the corresponding odds ratios. The intercept in the full model is specified as –6.928162 by the INTERCEPT= option. This number is calculated according to the forumula at the end of Analyses in the LOGISTIC Statement, which expresses the intercept in terms of the response probability, regression coefficients, and predictor means:
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