For the balanced two-way layout of a count response variable Y classified by fixed or random factors A and B, we address the problems of (i) testing for individual and interactive effects on Y of two fixed factors, and (ii) testing for the effect of a fixed factor in the presence of a random factor and conversely. In case (i), we assume independent Poisson responses with µij= E(Y| A=i,B=j) = αiβjγij corresponding respectively to the multiplicative interactive and non-interactive cases. For case (ii) with factor A random, we derive a multivariate gamma-Poisson model by mixing on the random variable associated with each level of A. In each case Neyman C(α) score tests are derived. We present simulation results,and apply the interaction test to a data set, to evaluate and compare the size and power of the score test for interaction between two fixed factors, the competing Poisson-based likelihood ratio test, and the F-tests based on the assumptions that √Y+1 or log(Y+1) are approximately normal. Our results provide strong evidence that the normal-theory based F-tests typically are very far from nominal size, and that the likelihood ratio test is somewhat more liberal than the score test.
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