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 μ̄i . = Σ jπijμij /Σ jπij is the expected value of a randomly chosen observation in the ith row. Hence, we consider testing that the μ̄i . 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 Ȳi .. be the sample mean of the observations in the ith row. We show that Ȳi .. is an MLE of μ̄i ., is consistent and is conditionally unbiased. We then find the asymptotic joint distribution of the Ȳi .. and use it to construct a sensible asymptotic size α test of the equality of the μ̄i . and asymptotic simultaneous (1 - α) confidence intervals for contrasts in the μ̄i ..