A wide variety of techniques exist for conditional inference on exponential families arising from discrete distributions. Normal theory methods, which rely on the approximate multivariate normality of the joint distribution of summary statistics from the data set, are often inaccurate for small data sets, and their quality can often be poor for summaries that indicate large parameter effects. They also ignore discreteness in the data. More sophisticated approximation techniques, known as saddlepoint techniques, are often used in cases when normal theory methods are inadequate. These techniques often do not account for discreteness in data, and hence are suboptimal in their unmodified forms. Exact inferential techniques are also available, but these techniques apply only to a limited number of models, require proprietary software, and fail when sample size reaches a moderate size. Extensions to this software that employ Monte Carlo techniques for larger sample sizes are not yet commercially available. These Monte Carlo techniques have the further disadvantage of delivering a variety of results for the same data set. The techniques proposed use saddlepoint approximations in a way that accounts for discreteness in the data while avoiding most of the computationally intractable aspects of exact calculations. Some of the projects proposed in this grant application involve new approximations, such as for approximating higher--dimensional distribution functions, and others involve modifications to existing approximations to avoid numerical instabilities. Other projects involve formulating confidence regions to make accurate calibration easy, and modifying the conditioning event to obtain a more powerful analysis, and performing diagnostics to ensure that the proper approximations are used. These methods will be general enough to apply to any canonical exponential family supported on a lattice, and hence to any generalized linear model with canonical link, observations supported on a lattice, and design matrix whose entries are confined to a lattice. Examples of models that will be accommodated are logistic regression, Poisson regression including log linear models for contingency tables, and multinomial models. Regression models with more exotic error structures, including positive Poisson and negative binomial distributions, will also be accommodated.
This proposed research is intended to aid in statistical inference on multiple parameters, in the presence of other nuisance parameters that are not of direct interest, when the distribution modeled is discrete. For example, the probability that a cancer patient will stay in remission can be modeled as a function of a variety of factors. Some of these effects, like which treatment a patient received or whether the patient had other cancer--related pathologies, may generalize to other populations, and others, like the effect of a particular center where the patient was treated, may not generalize. Thus one might be interested in describing the possible values that the parameters of interested take on, without being required to simultaneously estimate the remaining parameters. Typically one treats information associated with nuisance parameters as held fixed, and performs inference conditionally on this information. That is, one assesses the the evidence concerning the parameter of interest by comparing experimental results to the population of possible results such that the information about nuisance parameters is held fixed. The research agenda proposed here presents methods for doing these calculations, which balance high computational costs of exact methods against potential inaccuracies of approximations, and introduces and combines new methods for both exact and approximate calculations. These new methods will make the analysis of small discrete data sets, commonly occurring in applied sciences, quicker and more accurate.
