Theory And Applications Of Generalized Linear Model

Linear regression model is mainly applied to the case of that dependent variables(especially obey normal distribution) are continuous random variables. Nelder and Wedderburn(1972) extended linear regression model, proposed generalized linear model(GLM). This model connected the expectation of dependent variable to a linear function of independent variables by a known link function. and extended the distribution of the dependent variables to exponential family, which could solve these kinds of regression analysis problems that the dependent variables are common discrete or continuous random variables, especially for discrete random variables. So it is of great significance for the analysis of common attribute data or count data in biological, medical, economic and society.This paper is divided into theoretical part and applied part. The theory part of this thesis mainly introduces the basic concept of exponential family of distribution and the derivation of its mean and variance, the conception and properties of link function, the fundamental conception of generalized linear model. In aspect of parameter estimation, it gives the likelihood equation solution is ?1()T Tb-= X X X Y under special situation, and the iterative weighted least squares solution is ?1()T Tb-= X WX X WY in general situation. And in aspect of statistical inference, it mainly introduces the significance test made to the estimated parameters, include that the likelihood ratio test for a set of independent variables’ influence, and significance test for a single independent variable’s influence.The applied part select two practical applications. The first one is the application of generalized linear model in biological population of the relationship between female horseshoe crab habits and its body factors. Applying of Poisson model to analyze the relationship between the female horseshoe’s crab characters and average number of its followers. Given the estimated mathematical relationship between average number of followers and crustaceans width and color. It’s concluded that the carapace width for female horseshoe crab has a positive effect on the number of its followers, while the color depth of the horseshoe crab has negative impact on it, and if the carapace width and color are in the same situation, the thorns intact for female horseshoe crab tends to have more followers. Applying of Logistic model to analyze the relationship between female horseshoe crab’s body features and the probability of its followers. On the same level of color, carapace width increases each unit, the advantage for female horseshoe crab having follower increases by 58.1%.And on the same width, the advantage for female horseshoe crab having follower decreases by 39%.The second practical application is applying Logistic to analyze the relationship between the survival time of patients with acute lymphoid leukemia and some factors, such as patient’s condition and follow-up treatment, etc. The analysis shows that for the probability of survival time for one year or more than one year, the lymphatic invasion level has negative impact, and the follow-up treatment has positive impact.