OPTION 8

Standard Statistical Problems Using Normal Approximations

Ref: Lachin JM. Introduction to sample size determination and power analysis for clinical trials. Controlled Clinical Trials 1981; 2:93-113.

This program performs power calculations for one or two sample experiments having outcome responses of means, proportions or correlation coefficients. Power calculations are based on the difference between the alternative and null hypothesis parameter values with the alternative value of the parameter considered larger than the null value. Computations for the detectable alternative hypothesis difference may "breakdown" in those cases (proportions and correlations) when the upper limit of 1 is exceeded; the user may switch the hypothesized values for complete calculations. The program can compute any one of the following:

1. One-sided significance level.
2. Power of the experiment.
3. Detectable difference.
4. Required sample size,

when the user supplies values for the other three.

Means:

Calculations are performed assuming that Student's t test will be used to test that a mean is equal to some a priori specified value against an alternative value. The variances are assumed equal under both alternatives, and the variance estimate must be specified by the user. Computations are done using IMSL subroutines for calculations associated with both the central and non-central t-distributions. Solutions are computed using iterative search techniques.

Proportions:

Calculations are performed assuming that the proportion of events in a sample size of N is normally distributed with mean, p, and variance p(1-p)/N. For the one-sample test, po vs p1, the variance under the null is po(1-po)/N and the variance under the alternative is p1(1-p1)/N. For the two sample problem, po vs p1, the variance under the null is computed using the weighted average of po and p1 to compute the single variance; while the variance under the alternative is computed using the two variances computed from po and p1 separately. The end result is that sample size calculations do not depend on which p is specified as the "null" and which is the "alternative".

Correlations:

The calculations for correlations employ Fisher's arctanh transformation:

C(r)=.5[ln(1+r)-ln(1-r)]

The assumption is that if a sample correlation, r, based on N observations is distributed about an actual correlation value (parameter) p, then C(r) is normally distributed with mean, C(p), and variance, 1/(N-3).

Calculations are then made using these normal theory approximations.