A conditional predictive p-value to compare a multinomial with an overdispersed multinomial in the analysis of T-cell populations

Abstract

Immunological experiments that record primary molecular sequences of T-cell receptors produce moderate to high-dimensional categorical data, some of which may be subject to extra-multinomial variation caused by technical constraints of cell-based assays. Motivated by such experiments in melanoma research, we develop a statistical procedure for testing the equality of two discrete populations, where one population delivers multinomial data and the other is subject to a specific form of overdispersion. The procedure computes a conditional-predictive p-value by splitting the data set into two, obtaining a predictive distribution for one piece given the other, and using the observed predictive ordinate to generate a p-value. The procedure has a simple interpretation, requires fewer modeling assumptions than would be required of a fully Bayesian analysis, and has reasonable operating characteristics as evidenced empirically and by asymptotic analysis.

Keywords: Bayesian p-value; Dirichlet multinomial; Double overdispersion; Fisher's exact test; HPRT assay; Mass culture experiments; Molecular sequence data; T-cell receptor.

Fig. 1.

Summary statistics for J region frequencies from Tables 1 and 2. Colored bars show empirical frequencies of each J region type, from various data sources. Boxplots summarize posterior analysis of the underlying proportions conditional on the MT (not WT) data. Types are arranged from top to bottom by increasing value of the posterior median proportion. Note that the boxplots track the green bars better than the red bars, since the SC data are less variable than the MC data. The hypothesis test asks if the common (vector over types) mean of the MT data differs significantly from the mean of the WT cells (blue).