Espinosa Brito, Javier Alejandro; (2020) Using monotonicity constraints for the treatment of ordinal data in regression analysis. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

A regression model is proposed for the analysis of an ordinal response variable depending on a set of multiple covariates containing ordinal and potentially other types of variables. The ordinal predictors are not treated as nominal-scaled variables, and neither transformed into interval-scaled variables. Therefore, the information provided by the order of their categories is neither ignored nor overstated. The proportional odds cumulative logit model (POCLM, see McCullagh (1980)) is used for the ordinal response, and constrained maximum likelihood estimation is used to account for the ordinality of covariates. Ordinal predictors are coded by dummy variables. The parameters associated with the categories of the ordinal predictor(s) are constrained, enforcing them to be monotonic (isotonic or antitonic). A monotonicity direction classification procedure (MDCP) is proposed for classifying the monotonicity direction of the coefficients of the ordinal predictors, also providing information whether observations are compatible with both or no monotonicity direction. The MDCP consists of three steps, which offers two instances of decisions to be made by the researcher. Asymptotic theory of the constrained MLE (CMLE) for the POCLM is discussed. Some results of the asymptotic theory of the unconstrained MLE developed by Fahrmeir and Kaufmann (1985) are made explicit for the POCLM. These results are further adapted to extend the analysis of asymptotic theory to the constrained case. Asymptotic existence and strong consistency of the CMLE for the POCLM are proved. Asymptotic normality is also discussed. Different scenarios are identified in the analysis of confidence regions of the CMLE for the POCLM, which leads to the definition of three alternative confidence regions. Their results are compared through simulations in terms of their coverage probability. Similarly, different scenarios are identified in the analysis of confidence intervals of the CMLE and alternative definitions are provided. However, the fact that monotonicity is a feature of a parameter vector rather than of a singular parameter value becomes a problem for their computation, which is also discussed. Two monotonicity tests for the set of parameters of an ordinal predictor are proposed. One of them is based on a Bonferroni correction of the confidence intervals associated with the parameters of an ordinal predictor, and the other uses the analysis of confidence regions. Six constrained estimation methods are proposed depending on different approaches for making the decision of imposing the monotonicity constraints to the parameters of an ordinal predictor or not. Each one of them uses the steps of the MDCP or one of the two monotonicity tests. The constrained estimation methods are compared to the unconstrained proportional odds cumulative logit model through simulations under several settings. The results of using different scoring systems that transform ordinal variables into interval-scaled variables in regression analysis are compared to the ones obtained when using the proposed constrained regression methods based on simulations. The constrained model is applied to real data explaining a 10-Points Likert scale quality of life self-assessment variable by ordinal and other predictors.

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