By collecting multiple samples per subject, researchers can characterize intra-subject variation using physiologically relevant measurements such as gene expression profiling. This can yield important insights into fundamental biological questions ranging from cell type identity to tumour development. For each subject, the data measurements can be written as a matrix with the different subsamples (e.g. multiple tissues) indexing the columns and the genes indexing the rows. In this context, neither the genes nor the tissues are expected to be independent and straightforward application of traditional statistical methods that ignore this two-way dependence might lead to erroneous conclusions. Herein, we present a suite of tools embedded within the R/Bioconductor package HDTD for robustly estimating and performing hypothesis tests about the mean relationship and the covariance structure within the rows and columns. We illustrate the utility of HDTD by applying it to analyze data generated by the Genotype-Tissue Expression consortium.