For gene expression data analysis, an important task is to identify genes that are differentially expressed between two or more groups. Nevertheless, as biological experiments are often measured with a relatively small number of samples, how to accurately estimate the variances of gene expression becomes a challenging issue. To tackle this problem, we introduce a regularized $t$ distribution and derive its statistical properties including the probability density function and the moment generating function. The noncentral regularized $t$ distribution is also introduced for computing the statistical power of hypothesis testing. For practical applications, we apply the regularized $t$ distribution to establish the null distribution of the regularized $t$ statistic, and then formulate it as a regularized $t$-test for detecting the differentially expressed genes. Simulation studies and real data analysis show that our regularized $t$-test performs much better than the Bayesian $t$-test in the “limma” package, in particular when the sample sizes are small.

中文翻译：

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正则化 t$$ t $$ 分布：定义、属性和应用

对于基因表达数据分析，一项重要任务是识别两个或多个组之间差异表达的基因。然而，由于生物实验通常使用相对较少的样本进行测量，如何准确估计基因表达的方差成为一个具有挑战性的问题。为了解决这个问题，我们引入了正则化$t$分布并导出其统计特性，包括概率密度函数和矩生成函数。非中心正则化$t$还引入分布来计算假设检验的统计功效。对于实际应用，我们应用正则化$t$分布以建立正则化的零分布$t$统计量，然后将其表示为正则化$t$-检测差异表达基因的测试。模拟研究和真实数据分析表明我们的正则化$t$-测试比贝叶斯测试表现好得多$t$- 在“ limma ”包中进行测试，特别是当样本量较小时。