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Change Point Problems in Generalized Linear Models

 Hongling Zhou, PhD Candidate, Johns Hopkins Department of Biostatistics

Statistical models incorporating change points are common in practice, especially in the area of biomedicine. This approach is appealing in that a specific parameter is introduced to count for the abrupt change in the response variable relating to a particular independent variable of interest. While the introduction of the change point parameter into statistical model gives more precise estimate for the threshold value rather than a simple cutoff point via conventional wisdom or experience, it also brings intriguing theoretical difficulties in both detection and estimation of such phenomenon. It has long been recognized that the standard asymptotic methods are not directly applicable regarding the detection and estimation of the unknown change point. The statistical challenge one encounters is that the likelihood function is not differentiable with respect to this change point parameter. In hypothesis testing aspect, the change point as a parameter is meaningless under the null hypothesis of no change. To this end, we propose a class of test statistic along with its asymptotic properties to alleviate the concerns stated above. We also present an asymptotic approximation to statistical power of the proposed test. In estimation, we propose a procedure for estimating the change point along with other regression coefficients under the generalized linear model framework. We show that the proposed estimators enjoy the conventional asymptotic properties including consistency and normality. Simulation work we conducted suggests that it performs well for the situations considered. The estimation of change point for correlated data is discussed as well using the generalized estimating equation approach. The proposed test statistic and the estimation procedure are applied to a case-control study aimed to examine the relationship between the risk of myocardial infarction and alcohol intake.

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