An important type of heterosis, known as hybrid vigor, refers to the enhancements in the phenotype of hybrid progeny relative to their inbred parents. Although hybrid vigor is extensively utilized in agriculture, its molecular basis is still largely unknown. In an effort to understand phenotypic heterosis at the molecular level, researchers are measuring transcript abundance levels of thousands of genes in parental inbred lines and their hybrid offspring using RNA sequencing (RNA-seq) technology. The resulting data allow researchers to search for evidence of gene expression heterosis as one potential molecular mechanism underlying heterosis of agriculturally important traits. The null hypotheses of greatest interest in testing for gene expression heterosis are composite null hypotheses that are difficult to test with standard statistical approaches for RNA-seq analysis. To address these shortcomings, we develop a hierarchical negative binomial model and draw inferences using a computationally tractable empirical Bayes approach to inference. We demonstrate improvements over alternative methods via a simulation study based on a maize experiment and then analyze that maize experiment with our newly proposed methodology. This article has supplementary material online.

When analyzing field data on consumer products, model-based approaches to inference require a model with sufficient flexibility to account for multiple kinds of failures. The causes of failure, while not interesting to the consumer per se, can lead to various observed lifetime distributions. Because of this, standard lifetime models, such as using a single Weibull or lognormal distribution, may be inadequate. Usually cause-of-failure information will not be available to the consumer and thus traditional competing risk analyses cannot be performed. Furthermore, when the information carried by lifetime data are limited by sample size, censoring, and truncation, estimates can be unstable and suffer from imprecision. These limitations are typical, for example, lifetime data for high-reliability products will naturally tend to be right-censored. In this article, we present a method for joint estimation of multiple lifetime distributions based on the generalized limited failure population (GLFP) model. This five-parameter model for lifetime data accommodates lifetime distributions with multiple failure modes: early failures (sometimes referred to in the literature as "infant mortality") and failures due to wearout. We fit the GLFP model to a heterogenous population of devices using a hierarchical modeling approach. Borrowing strength across subpopulations, our method enables estimation with uncertainty of lifetime distributions even in cases where the number of model parameters is larger than the number of observed failures. Moreover, using this Bayesian method, comparison of different product brands across the heterogenous population is straightforward because estimation of arbitrary functionals is easy using draws from the joint posterior distribution of the model parameters. Potential applications include assessment and comparison of reliability to inform purchasing decisions. Supplementary materials for this article are available online.

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