An extension of the sign test for replicated measurements

An extension of the sign test is proposed for the case where there are paired groups or sets of replicated measurements; the individual measurements are unpaired between groups. A class of methods is investigated where the values of the groups of replicates are replaced by summary statistics (exceedances) whose distribution is independent, under the null hypothesis, of the underlying parametric structure. Possible adjustment for discontinuity of the underlying distribution (ties) is discussed and a normal approximation is presented. The proposed sign text extension, appropriate for small samples and large numbers of replicates, is used to compare the relative strengths of aqueous and lyophilized solutions of ragweed pollen at three dilutions for two types of reactions (erythematous and wheal area). Because correlations for individual reactions at two separate measurement times could not be estimated under the non-parametric procedures the analyses were performed for the cases where the responses at two measurement times were presumed to be (1) perfectly correlated, and (2) uncorrelated. Comparison of exact results with the normal approximation showed good correspondence for both cases, e.g., where n = 4 (n = number of subjects) and r = 8 (r = replicates) and n = 8 and r = 4. The analysis showed that the lyophilized product was superior to the aqueous at all solution strengths and for both measures of reactivity.