Multivariate Analysis of Data Sets with Missing Values: An Information Theory-Based Reliability Function

Missing values in complex biological data sets have significant impacts on our ability to correctly detect and quantify interactions in biological systems and to infer relationships accurately. In this article, we propose a useful metaphor to show that information theory measures, such as mutual information and interaction information, can be employed directly for evaluating multivariable dependencies even if data contain some missing values. The metaphor is that of thinking of variable dependencies as information channels between and among variables. In this view, missing data can be thought of as noise that reduces the channel capacity in predictable ways. We extract the available information in the data even if there are missing values and use the notion of channel capacity to assess the reliability of the result. This avoids the common practice—in the absence of prior knowledge of random imputation—of eliminating samples entirely, thus losing the information they can provide. We show how this reliability function can be implemented for pairs of variables, and generalize it for an arbitrary number of variables. Illustrations of the reliability functions for several cases are provided using simulated data.