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, π ij be the probability that a particular observation is in that cell and μ ij be the expected value of an observation in that cell. We assume that the { N ij } have a joint multinomial distribution with parameters n and { π ij } . Then μ ¯ i . = ∑ j π ij μ ij / ∑ j π ij 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 { π ij } 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 . .