# Imprecise Probability Models for Logistic Regression

## Date

2012-10-30

## Authors

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## Type

## Degree Level

Doctoral

## Abstract

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.

## Description

## Keywords

Imprecise probability model, imprecise Dirichlet model, imprecise
logit-normal model, aggregation property, ImpLogit model, interval estimate

## Citation

## Degree

Doctor of Philosophy (Ph.D.)

## Department

Mathematics and Statistics

## Program

Mathematics