Ordinal Data Analysis via Graphical Models

Background. Undirected graphical models or Markov random fields (MRFs) are very popular for modeling multivariate probability distributions. A considerable amount of work on MRFs has focused on modeling continuous variables and unordered categorical variables also called as nominal variables. However, data from many real world applications involve ordered categorical variables also called as ordinal variables (e.g., movie ratings on Netflix which can be ordered from 1 to 5 stars). While one can model ordinal variables using models designed for continuous or nominal variables, this can result in incorrect inferences about the variables. While, recent work has designed graphical models for modeling ordinal data, the proposed estimator for learning this model involves optimization of a difficult non-convex problem which is both computationally expensive and doesn’t come with statistical guarantees. Aim. Given multivariate ordinal data, we aim to estimate the joint probability distribution and the conditional dependency structure in the data. To this end, we provide a new estimator for ordinal probit model (a graphical model for ordinal data), that is computationally efficient and which comes with statistical guarantees. Data. We analyze HINTS-FDA dataset, which is a survey on how people access and use smoking and cancer related information and how they perceive risks of smoking. Results. We apply our estimator on HINTS-FDA data to understand the smoking behavior of people and their perceptions of smoking risks. Our analysis suggests that people who smoke, perceive smoking as less harmful than people who don’t smoke and the lack of awareness of smoking risks could be a reason why many people smoke. Conclusion. We have proposed a new estimator for learning ordinal probit model that is computationally tractable and can be easily scaled to large datasets. We empirically corroborated the superior performance of our estimator for the probit model.