Confidence intervals in regression that utilize uncertain prior information about a vector parameter

Submission note: A thesis submitted in total fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Engineering and Mathematical Sciences, Faculty of Science, Technology & Engineering, La Trobe University, Bundoora.

It is a common statistical practice to carry out preliminary statistical model selection and subsequent inference on the same data assuming that the selected model had been given a priori. This may lead to inaccurate or misleading results. Confidence intervals constructed with nominal coverage 1−α based on this false assumption can have poor coverage properties. In Chapter 2, we critically examine the coverage properties of resulting confidence intervals for stratum-specific odds ratios after a preliminary hypothesis test using a large sample approximation method and a Monte Carlo simulation method. This has been done in the context of a case-control study in which the aim is to assess the effect of a factor on disease occurrence. We carry out a preliminary test of homogeneity of the stratum-specific odds ratios and if the null hypothesis of homogeneity is rejected, we compute simultaneous confidence intervals for each of the stratum-specific odds ratios. We examine the statistical properties of this two-stage analysis and we show that the preliminary test of homogeneity of the stratum-specific odds ratios has a very harmful effect on the coverage probabilities of these confidence intervals. In Chapter 3 we describe a new confidence interval in regression that utilizes uncertain prior information. Suppose the parameter of interest is a specified linear combination of the components of the regression parameters. Preliminary statistical model selection may be motivated by a desire to utilize uncertain prior information about the regression parameters in the construction of the confidence interval for the parameter of interest. We use the form of the naive 1−α confidence interval to motivate a new 1−α confidence interval that utilizes uncertain prior information about a vector parameter. To an excellent approximation we show that this confidence interval has minimum coverage probability 1−α throughout the parameter space and it has expected length that is significantly less than the standard confidence interval when this prior information is correct and is also not too large when prior information is not correct. Also, this confidence interval reverts to the usual confidence interval when data happens to strongly contradict the prior information.