Two-way model with random cell sizes

We consider inference for row effects in the presence of possible interactions in a two-way fixed effects model when the numbers of observations are themselves random variables. Let Nij be the number of observations in the (i,j) cell, πij be the probability that a particular observation is in that cell and μij be the expected value of an observation in that cell. We assume that the {Nij} have a joint multinomial distribution with parameters n and {πij}. Then μ¯i.=∑jπijμij/∑jπij is the expected value of a randomly chosen observation in the ith row. Hence, we consider testing that the μ¯i. are equal. With the {πij} unknown, there is no obvious sum of squares and F-ratio computed by the widely available statistical packages for testing this hypothesis. Let Y¯i‥ be the sample mean of the observations in the ith row. We show that Y¯i‥ is an MLE of μ¯i., is consistent and is conditionally unbiased. We then find the asymptotic joint distribution of the Y¯i‥ and use it to construct a sensible asymptotic size α test of the equality of the μ¯i. and asymptotic simultaneous (1−α) confidence intervals for contrasts in the μ¯i..