Discrete and/or Longitudinal Data (small/big) analysis and The Behrens-Fisher problem

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Discrete data in the form of counts or proportions often arise in many fields of study, such as, epidemiology, biostatistics, medical and public health sciences, environmental studies and social sciences. These data often encounter over-dispersion (variance is larger than what can be predicted by a simple model, such as, the binomial or the Poisson model) and zero-inflation (more zero counts than what can be predicted by a simple model). Regression analysis of discrete data can be further complicated by the existence of missing values in the response variable and/or in the explanatory variables (covariates). If the missingness does not depend on observed data, then the missing data are called missing completely at random (MCAR). If the missing data mechanism depends only on observed data, then the data are missing at random (MAR). The MAR is also known as ignorable missing. That is, in this case, the missing data mechanism can be ignored. If the missing data mechanism depends on both observed and unobserved data, that is, failure to observe a value depends on the value that would have been observed, then the data are called missing not at random (MNAR) in which case the missingness is nonignorable. Longitudinal data (count/binary/continuous/survival) are frequently encountered in many subject-matter areas such as epidemiology, biostatistics, medical and public health sciences, environmental studies and social sciences. Longitudinal studies are characterized by observing the same variables repeatedly over a period of time. Usually the subjects are assumed to be independent, while the collected observations of the same subject are correlated.Further, model selection (selecting regression variables that contribution most) procedures in large (big) data sets with many explanatory variables is important, as in practice interpreting results from a simple model is much easier. In this research I we will develop estimation procedures in discrete data regression models (involving over-dispersion, zero-inflation, missing responses, measurement errors in covariates), model selection, and small sample bias correction of parameter estimates in longitudinal set up or otherwise.In many applied fields sometimes it is necessary to compare effectiveness of one procedure over another (two drugs, two teaching methods, two fertilizers etc.). For example, under two biologically different conditions we are often interested in identifying differentially expressed genes. It is often the case that the assumption of equal variances of the two groups is violated for many genes where a large number of them are required to be filtered or ranked. In these cases exact tests are unavailable. In this research I plan to develop approximate procedures and compare them with existing procedures under different assumptions regarding the data distribution (normal, negative binomial, beta-binomial, Weibull, Gamma etc.).