Semiparametric Models for Correlated Data: The Quadratic Inference Function Approach

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This research focuses on a new statistical method for the analysis of correlated data, the quadratic inference function (QIF) approach (Qu, Lindsay & Li 2000). The QIF is built on a semiparametric framework defined by a set of mean zero estimating functions, but differs from the standard estimating function approach in that there are more equations than the number of unknown parameters. The QIF has advantages compared to the estimating function approach, such as not requiring the specification of the likelihood function. It also overcomes limitations of the estimating function approach such as a lack of objective functions and likelihood functions for testing. One of the main goals of the proposed project is to explore the QIF for robustness with respect to the consistency of estimators when mean zero assumptions are not satisfied. A second goal focuses on the missing data problem, which occurs often in longitudinal data. Testing whether missing data are ignorable is still a challenging problem in general. The goodness-of-fit test for the QIF appears to be a valid test for nonignorable missing data. The third goal is to test order restricted alternative hypotheses for correlated data using the QIF. Current existing testing tools are not satisfactory and are mainly based on the likelihood function for parametric models, and therefore are not applicable for correlated data where the likelihood function is difficult to formulate. The QIF is related to the empirical likelihood (Owen, 1988) which is popular for nonparametric models. The proposed project also illustrates the Edgeworth expansion of QIF and explores how to apply the bootstrap strategy to improve testing accuracy for small samples of correlated data.This research will have a significant impact and many applications in biostatistics, econometrics, and the environmental and social sciences where correlated data arise often. In particular, the QIF method substantially improves the estimation of regression parameters in generalized estimating equation settings (Liang & Zeger, 1986). Considering a real world example of air pollution for health impact assessment, even a slight difference in the regression parameter estimates can have a major impact on our health and environmental policies. Further, it is also the first effort to connect the generalized method of moments (Hansen, 1982) in econometrics to estimating functions in the statistics field. It attempts to answer a question frequently asked by econometricians: how to choose the most informative moment conditions with the lowest dimension possible. The research will also serve an educational purpose through developing a new course on longitudinal data and training of graduate students.