Single nucleus ATAC-seq (snATAC-seq) experimental designs have become increasingly complex with multiple factors that might affect chromatin accessibility, including genotype, cell type, tissue of origin, sample location, batch, etc., whose compound effects are difficult to test by existing methods. In addition, current snATAC-seq data present statistical difficulties due to their sparsity and variations in individual sequence capture. To address these problems, we present a zero-adjusted statistical model, Probability model of Accessible Chromatin of Single cells (PACS), that can allow complex hypothesis testing of factors that affect accessibility while accounting for sparse and incomplete data. For differential accessibility analysis, PACS controls the false positive rate and achieves on average a 17% to 122% higher power than existing tools. We demonstrate the effectiveness of PACS through several analysis tasks including supervised cell type annotation, compound hypothesis testing, batch effect correction, and spatiotemporal modeling. We apply PACS to several datasets from a variety of tissues and show its ability to reveal previously undiscovered insights in snATAC-seq data.