title: A new framework of statistical inferences based on the valid joint sampling distribution of the observed counts in an incomplete contingency table.
abstract: Some existing confidence interval methods and hypothesis testing methods in the analysis of a contingency table with incomplete observations in both margins entirely depend on an underlying assumption that the sampling distribution of the observed counts is a product of independent multinomial/binomial distributions for complete and incomplete counts. However, it can be shown that this independency assumption is incorrect and can result in unreliable conclusions because of the under-estimation of the uncertainty. Therefore, the first objective of this paper is to derive the valid joint sampling distribution of the observed counts in a contingency table with incomplete observations in both margins. The second objective is to provide a new framework for analyzing incomplete contingency tables based on the derived joint sampling distribution of the observed counts by developing a Fisher scoring algorithm to calculate maximum likelihood estimates of parameters of interest, the bootstrap confidence interval methods, and the bootstrap testing hypothesis methods. We compare the differences between the valid sampling distribution and the sampling distribution under the independency assumption. Simulation studies showed that average/expected confidence-interval widths of parameters based on the sampling distribution under the independency assumption are shorter than those based on the new sampling distribution, yielding unrealistic results. A real data set is analyzed to illustrate the application of the new sampling distribution for incomplete contingency tables and the analysis results again confirm the conclusions obtained from the simulation studies.
authors.affiliation: 1 Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong, P. R. China.
authors.forename: Guo-Liang
authors.initials: GL
authors.lastname: Tian
authors.affiliation: 2 Department of Statistics, Yunnan University, Kunming, P. R. China.
authors.forename: Hui-Qiong
authors.initials: HQ
authors.lastname: Li
mesh_terms: D000465:Algorithms; D016010:Binomial Distribution; D002648:Child; D016001:Confidence Intervals; D003627:Data Interpretation, Statistical; D005260:Female; D006801:Humans; D016013:Likelihood Functions; D009035:Mothers / Q000706:statistics & numerical data; D009422:Nervous System Diseases / Q000150:complications; D012107:Research Design; D012135:Respiratory Sounds / Q000209:etiology; D012907:Smoking / Q000009:adverse effects; D014481:United States / Q000453:epidemiology
keywords: Bootstrap; confidence interval; incomplete contingency tables; sampling distribution; test statistics