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Summary In this paper we consider a practical situation where information on two auxiliary variables related to the study variable is available at different levels. Following Kiregyera (1980, 1984) who has obtained a chain ratio‐to‐regression estimator and regression to regression estimator, we shall study several estimators that arise naturally in this context and compare them under the mean square error criterion. We extend these results to the case when multiple auxiliary information is available.

Adequate control of type I error rates will be necessary in the increasing genome‐wide search for interactive effects on complex traits. After observing unexpected variability in type I error rates from SNP‐by‐genome interaction scans, we sought to characterize this variability and test the ability of heteroskedasticity‐consistent standard errors to correct it. We performed 81 SNP‐by‐genome interaction scans using a product‐term model on quantitative traits in a sample of 1,053 unrelated European Americans from the NHLBI Family Heart Study, and additional scans on five simulated datasets. We found that the interaction‐term genomic inflation factor (lambda) showed inflation and deflation that varied with sample size and allele frequency; that similar lambda variation occurred in the absence of population substructure; and that lambda was strongly related to heteroskedasticity but not to minor non‐normality of phenotypes. Heteroskedasticity‐consistent standard errors narrowed the range of lambda, with HC3 outperforming HC0, but in individual scans tended to create new P‐value outliers related to sparse two‐locus genotype classes. We explain the lambda variation as a result of non‐independence of test statistics coupled with stochastic biases in test statistics due to a failure of the test to reach asymptotic properties. We propose that one way to interpret lambda is by comparison to an empirical distribution generated from data simulated under the null hypothesis and without population substructure. We further conclude that the interaction‐term lambda should not be used to adjust test statistics and that heteroskedasticity‐consistent standard errors come with limitations that may outweigh their benefits in this setting.

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