power2(noncent2,numDF,denomDF,alpha), noncent2 >= 0, 0 < alpha < 1, numDF >0, denomDF > 0; some or all arguments may be vectors |

power2(noncen2,numDF,denomDF,alpha) computes the power for an F test with numDF numerator degrees of freedom, denomDF denominator degrees of freedom, a significance level of alpha, and noncentrality parameter noncen2. The noncentrality parameter noncen2 = sum(n_i*(effect_i)^2)/sigma^2, where n_i and effect_i = mu_i - mu_all are the sample size and treatment effect for group i, with mu_i = treatment i mean and mu_all = sum(n_i*mu_i)/sum(n_i). An more mathematical definition is noncen2 = numDF*(E[Numerator MS]/E[denominator MS] - 1); Note that this differs from the n=1 non-centrality parameter expected by power() which does not include a sample size. For example, you will get the same answers from Cmd> power2(nrep*noncen,ngrp-1,(nrep-1)*ngrp,alpha) and Cmd> power(noncen,ngrp,nrep,alpha) power2(n*mu_a^2/sigma^2,1,n-1,alpha) computes the power against the alternative hypothesis H_a: mu = mu_a of a single-sample two-tail t-test of H_0: mu = 0 based on a sample of size n. To compute the power of a one-tail t-test, see cumstu:"non_central_t". Some or all of the arguments of power2() may be vectors, in which case all non-scalars must be the same length, which will also be the length of the result. For example, you can compute a power as a function of noncen2 by, say Cmd> power2(run(0,100)*1.2,10,20,.05) If nrep was computed as samplesize(noncen,ngrp,alpha,pwr), the value of power2(nrep*noncen,ngrp-1,(nrep-1)*ngrp,alpha) should be approximately equal to pwr, but no smaller. power2() is useful for computing the power of a test for a contrast, or for interaction and related effects where the error degrees of freedom are complicated functions of n. See also power() and samplesize().

Gary Oehlert 2003-01-15