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 N be the number of observations in the (i, j) cell, π be the probability that a particular observation is in that cell and μ be the expected value of an observation in that cell. We assume that the {N } have a joint multinomial distribution with parameters n and {π }. Then μ̄ . = Σ /Σ is the expected value of a randomly chosen observation in the ith row. Hence, we consider testing that the μ̄ . are equal. With the {π } unknown, there is no obvious sum of squares and F-ratio computed by the widely available statistical packages for testing this hypothesis. Let Ȳ .. be the sample mean of the observations in the ith row. We show that Ȳ .. is an MLE of μ̄ ., is consistent and is conditionally unbiased. We then find the asymptotic joint distribution of the Ȳ .. and use it to construct a sensible asymptotic size α test of the equality of the μ̄ . and asymptotic simultaneous (1 - α) confidence intervals for contrasts in the μ̄ ..