Imprecise Probability Models for Logistic Regression
Imprecise probability models are applied to logistic regression to produce interval estimates of regression parameters. The lengths of interval estimates are of main interest. Shorter interval estimates correspond to less imprecision in regression parameters estimates. This thesis applies imprecise probabilistic methods to the logit model. Imprecise logistic regression, briefly called ImpLogit model, is presented and established for the first time. ImpLogit model is applied based on an inferential paradigm that applies Bayes theorem to a family of prior distributions, yielding interval posterior probabilities. The so-called interval estimates of regression parameters are computed using Metropolis-Hastings algorithm. Two imprecise prior probability models are applied to 2-parameter ImpLogit model : the imprecise Dirichlet model (IDM) and the imprecise logit-normal model (ILnM). The 2-parameter ImpLogit model is fitted using real life dose-response data. This takes into account the cases of increasing, decreasing and mixed-belief ImpLogit models. The relation between the lengths of interval estimates of regression parameters and both of covariate values and imprecise prior hyperparameters, in 2-parameter ImpLogit model, is studied by simulation. Different designs are applied in order to investigate a way to shorten the lengths of interval estimates of regression parameters. Having covariate fixed values to surround the prior believed median value of the logistic distribution results in reducing the imprecision in interval estimates. Fixing covariate values around the prior believed median value in a short range increases the lengths of interval estimates. The number of fixed covariate values (say number of distinct dose levels in a dose-response experiment) affects the produced imprecision. A larger number of fixed covariate values increases the lengths of interval estimates. Therefore, a good design has a small number of fixed covariate values, located and spread out not in a short range. ImpLogit model designs that are recommended by the simulation study, are compared to optimal designs in the frequentist approach using Fisher information matrix (FIM). Designs in FIM agree with designs that reduce imprecision in 2-parameter ImpLogit model, in the necessity of having covariate values to be fixed around the prior believed median value of the logistic distribution, not in a short range.
Imprecise probability model, imprecise Dirichlet model, imprecise logit-normal model, aggregation property, ImpLogit model, interval estimate
Doctor of Philosophy (Ph.D.)
Mathematics and Statistics