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Categorical data analysis becomes challenging when high-dimensional sparse covariates are involved, which is often the case for omics data. We introduce a statistical procedure based on multinomial logistic regression analysis for such scenarios, including variable screening, model selection, order selection for response categories, and variable selection. We perform our procedure on high-dimensional gene expression data with 801 patients, 2426 genes, and five types of cancerous tumors. As a result, we recommend three finalized models: one with 74 genes achieves extremely low cross-entropy loss and zero predictive error rate based on a five-fold cross-validation; and two other models with 31 and 4 genes, respectively, are recommended for prognostic multi-gene signatures. 

Sparse data with a high portion of zeros arise in various disciplines. Modeling sparse high-dimensional data is a challenging and growing research area. In this paper, we provide statistical methods and tools for analyzing sparse data in a fairly general and complex context. We utilize two real scientific applications as illustrations, including a longitudinal vaginal microbiome data and a high dimensional gene expression data. We recommend zero-inflated model selections and significance tests to identify the time intervals when the pregnant and non-pregnant groups of women are significantly different in terms of Lactobacillus species. We apply the same techniques to select the best 50 genes out of 2426 sparse gene expression data. The classification based on our selected genes achieves 100% prediction accuracy. Furthermore, the first four principal components based on the selected genes can explain as high as 83% of the model variability.