Motivated by questions arising from the field of statistical genetics, we consider the problem of testing main, nested and interaction effects in unbalanced factorial designs. Based on the concept of composite linear rank statistics, a new notion of weighted rank is proposed. Asymptotic normality of weighted linear rank statistics is established under mild conditions and consistent estimators are developed for the corresponding limiting covariance structure. A unified framework to employ weighted rank to construct test statistics for main, nested and interaction effects in unbalanced factorial designs is established. The proposed tests statistics are applicable to unbalanced designs with arbitrary cell replicates that are greater than one per cell. The limiting distributions under both the null hypotheses and Pitman alternatives are derived. Monte Carlo simulations are conducted to confirm the validity and power of the proposed tests. Genetic data sets from a simulated backcross study are analyzed to demonstrate the application of the proposed tests in quantitative trait loci mapping.
Group for Research in Decision Analysis