Estimating sample sizes for binary, ordered categorical, and continuous outcomes in two group comparisons

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Abstract
Parameter definition Of all the parameters that have to be specified before the sample size can be determined the most critical is the effect size. Reducing the effect size by half will quadruple the required sample size. The effect size can be interpreted as a “clinically important difference,” but this is often difficult to quantify. A valuable attempt at classification was made by Burnand et al, who reviewed three major medical journals and looked for words such as “impressive difference,” “important difference,” “dramatic increase” and then calculated a standardised effect size.8 This provided a guide to the size of effect regarded as important by other authors. There are several ways of eliciting useful sample sizes: a Bayesian perspective has been given recently,9 along with an economic approach,10 and one based on patients' rather than clinicians' perceptions of benefit.11 In statistical significance tests one sets up a null hypothesis and, given the observed difference of interest, calculates the probability of observing the difference (or a more extreme one) under the null hypothesis. This yields the P value. If the P value is less than some prespecified level then we reject the null hypothesis. This level is known as the significance level (alpha). If we reject the null hypothesis when it is true we make a type I error, and we set (alpha), the significance level, to control the probability of doing this. If the null hypothesis is in fact false but we fail to reject it, we make a type II error, and the probability of a type II error is denoted as ß. The probability of rejecting the null hypothesis when it is false is termed the power and is defined as 1-ß. UNEQUAL ALLOCATION Given m, calculated assuming equal sized groups, let m' be the sample size in the first group and rm' the sample size in the second group. Then m' is given by m1=r+1/2rxm, (1) where r is the allocation ratio.