title: Two-way model with random cell sizes.
abstract: 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. LetNijbe the number of observations in the (i,j) cell,πijbe the probability that a particular observation is in that cell andμijbe the expected value of an observation in that cell. We assume that the {Nij} have a joint multinomial distribution with parameters n and {πij}. Thenμ̄i. = Σjπijμij/Σjπijis the expected value of a randomly chosen observation in theith row. Hence, we consider testing that theμ̄i. 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. LetȲi.. be the sample mean of the observations in theith row. We show thatȲi.. is an MLE ofμ̄i., is consistent and is conditionally unbiased. We then find the asymptotic joint distribution of theȲi.. and use it to construct a sensible asymptotic sizeαtest of the equality of theμ̄i. and asymptotic simultaneous (1 -α) confidence intervals for contrasts in theμ̄i..
authors.affiliation: Pennsylvania State University, State College, PA, United States.
authors.forename: Steven F
authors.initials: SF
authors.lastname: Arnold
authors.forename: Panagis G
authors.initials: PG
authors.lastname: Moschopoulos
keywords: Analysis of variance; Main effects; Multinomial cell sizes; Two-way model; Unbalanced data
references.citation: Am Sociol Rev. 1967 Dec;32(6):945-59
references.pmid: 6078852
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references.pmid: 7217475