Vergelijking van een tweetal toetsen tegen verloop voor een aantal kansen

Two test statistics t and b, testing equality of probabilities pi of success in k different series against the hypothesis of trend with given numbers gi (weights) of the series, are compared. The first teststatistic, due to C. van Ee den en J, Hemelrijk [1] is with ni1 equal to the number of successes in ni trials and Defining trend by it appears that the teststatistic t gives rise to a test consistent for the complete set of alternatives τ≠ 0. The other teststatistic is b gives rise to a test which is not consistent for the general set of alternatives τ≠ 0 but for a rather important subset of these alternatives, i.e. those alternatives which show a lineair trend. Neither of the tests in necessarily unbiased. The asymptotic relative efficiency of test t with respect to b is equal to or lower than unity. (Equal in case n1= n2=…= nk with b = t; in this case b is also consistent for the set of alternatives τ≠ 0). The variances of the teststatistics can be estimated with Both tests are based on the approximately normal distribution of the teststatistics. To judge this approximation the 3rd and 4th cumulants of the distributions of the statistics are evaluated in terms of number of elements of a binomial distribution. It is concluded that in case of possible non‐lineair relationships the teststatistic t is preferable as it gives rise to a consistent, designfree test. In case a lineair relationship has to be tested against the hypothesis of no trend the teststatistic b has to be prefierred, especially if the number of trials in the series are very different. An example is discussed.