A common task in high-throughput biology is to test for differences in means between two samples across thousands of features (e.g., genes or proteins), often with only a handful of replicates per sample. Moderated t-tests handle this problem by assuming normality and equal variances, and by applying the empirical partially Bayes principle: a prior is posited and estimated for the nuisance parameters (variances) but not for the primary parameters (means). This approach has been highly successful in genomics, yet the equal variance assumption is often violated in practice. Meanwhile, Welch's unequal variance t-test with few replicates suffers from inflated type-I error and low power. Taking inspiration from moderated t-tests, we extend the empirical partially Bayes paradigm to two-sample testing with unequal variances. We develop two procedures: one that models the ratio of the two sample-specific variances and another that models the two variances jointly, with prior distributions estimated by nonparametric maximum likelihood. Our empirical partially Bayes methods yield p-values that are asymptotically uniform as the number of features grows while the number of replicates remains fixed, ensuring asymptotic type-I error control. Simulations and applications to genomic data demonstrate substantial gains in power.

翻译：在高通量生物学中，一项常见任务是对两个样本在数千个特征（如基因或蛋白质）上的均值差异进行检验，而每个样本通常仅有少量重复观测。调节t检验通过假设正态性和等方差性，并应用经验性部分贝叶斯原理来处理这一问题：对冗余参数（方差）设定并估计先验分布，但对主要参数（均值）则不设定先验。该方法在基因组学中取得了巨大成功，然而等方差假设在实践中常被违反。同时，针对少量重复观测的韦尔奇不等方差t检验则存在I类错误膨胀和检验功效低下的问题。受调节t检验的启发，我们将经验性部分贝叶斯范式推广至不等方差条件下的双样本检验。我们提出了两种方法：一种对两个样本特异性方差的比值进行建模，另一种则对方差进行联合建模，其先验分布通过非参数最大似然估计得到。我们的经验性部分贝叶斯方法所产生的p值，在特征数量增长而重复观测数保持固定时，具有渐近均匀性，从而保证了渐近的I类错误控制。模拟实验及在基因组数据上的应用均表明，该方法能显著提升检验功效。