Regularized t$$ t $$ distribution: definition, properties, and applications

Zongliang Hu, Yiping Yang, Gaorong Li and Tiejun Tong

Scandinavian Journal of Statistics, 2023, vol. 50, issue 4, 1884-1900

Abstract: 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$$ t $$ distribution and derive its statistical properties including the probability density function and the moment generating function. The noncentral regularized t$$ t $$ distribution is also introduced for computing the statistical power of hypothesis testing. For practical applications, we apply the regularized t$$ t $$ distribution to establish the null distribution of the regularized t$$ t $$ statistic, and then formulate it as a regularized t$$ t $$‐test for detecting the differentially expressed genes. Simulation studies and real data analysis show that our regularized t$$ t $$‐test performs much better than the Bayesian t$$ t $$‐test in the “limma” package, in particular when the sample sizes are small.

Date: 2023
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