Behavior of convergence in logistic regression models - Assessing the drop of the Kolmogorov distance between the sampling distribution and the asymptotic distribution of estimators and test statistics in logistic regression analysis

Professorship/Faculty: Fakultät Sozial- und Wirtschaftswissenschaften: Abschlussarbeiten 
Author(s): Nold, Mariana Saskia
Publisher Information: Bamberg : opus
Year of publication: 2014
Pages / Size: 117 S. : graph. Darst.
Supervisor(s): Rässler, Susanne; Heinze, Georg
Language(s): English
Bamberg, Univ., Diss.
Licence: German Act on Copyright 
URN: urn:nbn:de:bvb:473-opus4-100306
Document Type: Doctoralthesis
Using classical inference, hypothesis tests and confidence intervals are often based on large-sample assumptions, which are said to hold if the sample size is large enough.
The weakness of this approach is, that the researcher does not know what sample size is required for this purpose in a concrete situation.
A common problem that encounters in statistics is the procedure of modeling the relationship between explanatory variables and a binary response. Here logistic regression analysis often represents the appropriate method. This method is used to estimate the probability or odds of occurrence of the binary response in dependence of explanatory variables. But, what is the sample size to be large enough to base statistical conclusions on asymptotic properties?
The type of convergence, with which we are dealing here, is convergence in law, in the following denoted as L-convergence. If the limiting distribution of a statistic is continuous, then L-convergence is equivalent to convergence with respect to the Kolmogorov distance.
Therefore, the Kolmogorov distance is an effective tool for discussing the behavior of L-convergence.
The present work uses an autogenerated process that involves the classical theory of logistic regression analysis to explore the behavior of L-convergence by means of the Kolmogorov distance. Based on the Kolmogorov distance two methods are developed in order to investigate the behavior of L-convergence and its impacts on statistical conclusions. The first serves to extend the spectrum of methods to discuss the impacts of the Firth-penalization, the second to use the classical inference
as a more deliberate method with respect to asymptotic properties.
The first method consists of the distance-sample-size-diagram and the accuracy-diagram. The distance-sample-size-diagram represents the mean approximate Kolmogorov distance as a function of the predefined sample size. The predefined sample size is displayed on the horizontal axis and the mean approximate Kolmogorov distance between the statistic of interest and its limiting distribution on the vertical axis. This is a fruitful graphical representation of the behavior of L-convergence in dependence of the rate at which empirical information accrues. Finally the accuracy-diagram presents the actual accuracy function of a confidence interval and its reference derived from asymptotics. This diagram complements the distance-sample-size-diagram as a tool to study the impact of penalizations.
The second method, the p-value-uniform-diagram, shows the actual empirical cumulative distribution function of the p-values of a statical test and the cumulative distribution function of the uniform distribution as the reference of the former. A deviation from this reference indicates that L-convergence is not reached.
SWD Keywords: Regressionsanalyse ; Konvergenz ; Kolmogorov-System ; Statistik ; Online-Publikation
Keywords: Kolmogorov distance, logistic regression analysis, asymptotic properties, Firth-penalization
DDC Classification: 310 Statistics 
RVK Classification: QH 234   
Release Date: 5. May 2014

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