The size of the Cochran–Armitage trend test in [formula omitted] contingency tables

Abstract

Although a number of studies showed that the Cochran–Armitage trend test does not preserve the nominal level, it is applied to only small sample cases, because the studies were conducted in small samples by simulation. Little is known about maintenance of the nominal level in infinite samples. Theoretical proof is needed to extend the results in small samples obtained by simulation into those in infinite samples. The purpose of this study is to investigate the sizes and the type I error rates of the Cochran–Armitage trend test, theoretically and numerically. Especially, we showed that the size (the supremum of the type I error rates over the nuisance parameter space) of the Cochran–Armitage trend test in large samples is always greater than or equal to the nominal level. That is, we proved lim n ⟶ ∞ sup 0 < p < 1 h ( p , n ) ⩾ α , where h ( p , n ) is the type I error rate of the Cochran–Armitage trend test depending on both the sample size n and the nuisance parameter p (common success rate). The essential idea of proof is to investigate the type I error rate when the Poisson approximation to the binomial distribution is more appropriate than the normal approximation. We further showed that the inequality holds strictly in many cases by computing the large-sample lower bounds of the sizes which are strictly greater than the nominal level. Then, the relationship between the large-sample lower bounds and the actual sizes in moderate sample sizes are investigated. Especially, the actual sizes in moderate sample sizes were computed by complete enumeration which does not involve any error. We also examine the properties of the sizes and the patterns of the type I error rates.