Robust Estimate For Generalized Linear Models And Its Application In Medical Feilds

Generalized linear models are powerful and popular techniques for modeling a large variety of data. They allow to model the relationship between the predictors and a function of the mean of continuous and discrete response variables, especially for discrete ones. These models are widely used for analyzing the data in the field of medicine, economics and sociology. However, the classic maximum likelihood estimation is vulnerable to the effects of outliers, and even gets the erroneous conclusions. So it is very important to study robust estimations which are resistant to outliers.In this paper, the author reviewed and compared four robust methods for GLM: Mallows qusi-likelihood estimation, conditionally unbiased bound influence estimation, Mallows downweight leverage estimation and consistent misclass model estimation. Firstly we reviewed the basic theory of robust estimation, and then studied the robust properties of these four methods in details.In the simulation studies, the weight function in the direction to x's of Mqle was chosen in three ways: hat matrix, MVE, MCD. Mallows downweight estimation considered in two ways for weight function: Carroll and Huber function. The simulation studies are based on two popular models of GLM: logistic model and Poisson model, then we contaminate the samples in two ways with different proportions: in y's only and in x's and y's simultaneously. Through the simulation studies the author is to explore the ability against different types and proportions of outliers for different estimations of two models.From the simulation studies, we got the following conclusions:1. Compared to the classical MLE, these estimations are more resistant to the effects of outliers and fit the optimal model for bulk of data. In addition, they can be used to diagnose the outliers better. When there is no outliers in data, the robust estimators are as good as MLE; when MLE conditions are not satisfied, the robust estimators are much better than MLE.2. Among all the methods metioned in the thesis, Mqle based on MVE and MCD can resist outliers best either in logistic or in Poisson model. However Mqle based on hat matrix downweight function has lower breakdown point because of hat matrix's non-robustness.3. In Mallows downweight leverage estimation the definition of weight function is only based on x's, so it is invalid with 1% outliers in y's. But when there are outliers simultaneous in x's and y's, it have good behaviors. When the proportion of outliers increases, it can cause the perfect separation. This is because its weight function is assigned as 0 for the observation which is a outlier in x's, so this method will lost much information of sample in its downweight procedure.4. The performance of consistent misclass model estimation is worse than the former two methods. However when compared to MLE, it has better behaviors against outliers. Its shortcoming is downweight the normal observations.5. CUBIF can resist the outliers in both x's and y's simultaneously, but its behavior is not always good unless it is in Poisson model with outliers in y's.With the increase of the value of X for outliers ,the standard error of estimates will get larger.Finally, the author explored the practical application of these methods through two examples.