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SBNTEST procedure

Calculates the sample size for binomial tests (R.W. Payne & D.A. Murray).


PRINT = string token What to print (replication, power); default repl, powe
PRMETHOD = string token Method to be used to calculate the probabilities for the binomial test (angular, normalapproximation, exact); default norm
PROBABILITY = scalar Significance level for the test; default 0.05
POWER = scalar The required power (i.e. probability of detection) of the test; default 0.9
TMETHOD = string token Type of test to be done (onesided, twosided); default ones
NULL = scalar Probability under the null hypothesis for the one-sample test; default 0.5
RATIOREPLICATION = scalar Ratio of replication sample2:sample1 (i.e. the size of sample 2 should be RATIOREPLICATION times the size of sample 1); default 1
REPLICATION = variate Replication values for which to calculate and print or save the power; default * takes 11 replication values centred around the required number of replicates


P1 = scalars Probability to detect in sample 1
P2 = scalars Probability to detect in sample 2
NREPLICATES = scalars Saves the required number of replicates
VREPLICATION = variates Numbers of replicates for which powers have been calculated
VPOWER = variates Power (i.e. probability of detection) for the various numbers of replicates


SBNTEST calculates the number of replicates (or sample size) required for a binomial test. A one-sample binomial test assesses the evidence that the probability of success within a sample differs from some specific value. The probability that needs to be detected is specified by the P1 parameter, and the value from which it needs to be distinguished (i.e. the value under the null hypothesis) is specified by the NULL option. If NULL is not set, the default is 0.5. Alternatively, a two-sample test assess the evidence that probabilities within two samples are different. The anticipated probability within the first sample is then specified by the P1 parameter, and the probability within the second sample (from which it must be distinguished) is specified by the P2 parameter.

The PRMETHOD option defines the type of binomial test that is to be done. The normalapproximation setting relates to a test based on the Normal approximation to the binomial distribution (see the BNTEST procedure), while the angular setting is for a test using an angular transformation of the probabilities. The final setting, exact, is available only for the one-sample test and assumes an exact test using the binomial distribution.

The significance level for the test is specified by the PROBABILITY option (default 0.05 i.e. 5%). The required probability for detection of the difference between the probabilities (that is, the power of the test) is specified by the POWER option (default 0.9). It is generally assumed that the sizes of the samples in the two-sample test should be equal. However, you can set the RATIOREPLICATION option to a scalar, R say, to indicate that the size of the second sample should be R times the size of the first sample. By default, SBNTEST assumes a one-sided test is to be used, but you can set option TMETHOD=twosided to take a two-sided test instead. The NREPLICATES parameter allows you to save the required size of the first sample.

The PRINT option controls printed output, with settings:

    replication to print the required number of replicates in each sample (i.e. the size of each sample);
    power to print a table giving the power (i.e. probability of detection) provided by a range of numbers of replicates.

By default both are printed.

The replications and corresponding powers can also be saved, in variates, using the VREPLICATION and VPOWER parameters. The REPLICATION option can specify the replication values for which to calculate and print or save the power; if this is not set, the default is to take 11 replication values centred around the required number of replicates.




When PRMETHOD=normalapproximation, the distribution of the probability in sample i is approximated by a Normal distribution with mean pi and variance pi(1-pi)/ni, where pi is the binomial probability and ni is the sample size. With PRMETHOD=angular, the probability is transformed to radians by an angular distribution, and the variance is then √(0.25/ni). For PRMETHOD=exact, the calculations are done using the CUBINOMIAL and EDBINOMIAL functions (one-sample test only).

See also

Procedure: BNTEST.

Commands for: Design of experiments.


        '1) one-sample test, probability to detect 0.7.'; STYLE=meta,plain
SBNTEST [PRINT=replication,power; TMETHOD=twosided] 0.7
CAPTION '2) two-sample test, probabilities 0.4 and 0.6.'
SBNTEST [PRINT=replication,power; TMETHOD=twosided] 0.4; P2=0.6
Updated on June 18, 2019

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