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Two-way model with random cell sizes. (English) Zbl 1334.62134

Summary: 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 \(N_{ij}\) be the number of observations in the \((i,j)\) cell, \(\pi _{ij}\) be the probability that a particular observation is in that cell and \(\mu _{ij}\) 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 \(\{\pi _{ij}\}\). Then \({\overline{\mu}}_{i.}= \sum_j\pi_{ij}\mu_{ij}/\sum_j\pi_{ij}\) is the expected value of a randomly chosen observation in the ith row. Hence, we consider testing that the \(\overline\mu_{i.}\) are equal. With the \(\{\pi _{ij}\}\) unknown, there is no obvious sum of squares and F-ratio computed by the widely available statistical packages for testing this hypothesis. Let \(\overline Y_{i..}\) be the sample mean of the observations in the ith row. We show that \(\overline Y_{i..}\) is an MLE of \(\overline \mu_{i.}\) is consistent and is conditionally unbiased. We then find the asymptotic joint distribution of the \(\overline Y_{i..}\) and use it to construct a sensible asymptotic size \(\alpha \) test of the equality of the \(\overline \mu_{i.}\) and asymptotic simultaneous \((1 - \alpha )\) confidence intervals for contrasts in the \(\overline \mu_{i.}\).

MSC:

62J10 Analysis of variance and covariance (ANOVA)
62H15 Hypothesis testing in multivariate analysis
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References:

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