Regression Analysis with Missing Data Using Interval Imputation

Regression analysis with missing data is a common problem in statistical modelling. Majority of the available methods use point imputation strategy to get rid of the missing entries. However, such methods rely on different observational assumptions. In this paper, we propose a novel approach based on interval imputation, that is, instead of a single value imputation, we replace the missing entries with the range of the variables obtained from the observational data. This way, we avoid any distributional assumption on the data and formulate our model based only on the information in hand. For estimation, we rely on the interval matrix algebra. We also introduce regularisation terms with Bayesian analysis which also allows us to incorporate our subjective belief and avoid singularity in the estimation process similar to ridge regression. We evaluate the maximum a posteriori estimates of these regression coefficients to obtain the approximate posterior bounds. Once we have these posterior bounds, we use cross validation to obtain mixing parameters between the lower and upper bounds of the posterior estimates for model fitting. Finally, we illustrate our method with real-life dataset and compare with other state of the art methods to showcase our methods applicability.