Generalized Additive Models And Its Application In Medical Fields

The Regression analysis is employed extensively in clinical and medicalresearches. However, due to the complexity of life phenomena especially inexploring researches, we often fail to make sure some terms in the regressionequation. Thus, those classical regression methods can not work well whileGeneralized Additive Models (GAM) is a good alternative to solve this problem.The author introduced GAM, meanwhile he discussed the influence of outliersin Y space and concurvity to the result. To overcome the influence, heintroduced several methods and gave some medical examples. To attack on the dimensionality, Friedman and Stuetzle came up withProjection-Pursuit Regression that is thought of as a rudiment of GAM in 1981.To reduce dimensionality, this method projects predicted variable onto the samedirection. Afterwards, Hastie and Tibshirani applied Additive Models intoGeneralized Linear Models in 1984 .Thus, Generalized Additive Models cameinto being. In many environments when the parametric regression equation failsto reflect the relationship between outcome variables and predict variables well,we can change the expected forms of outcome variables into g(μ) whereμ=E(Y/X₁,.... ,X_P).Now this model can be written: g(μ)=α+β₁X₁+ ... +β_PX_Pcalled Generalized Linear Models GLM. Meanwhile, we can also usenonparametric forms f (x) to reflect their relationship. This model can bewritten: E(Y / X ) = f (x) , it may be extended to some predict variables:E(Y/X₁,.... ,X_P) =α + f(X₁)+ .... + f (X_P) called Additive Models. If thesetwo forms are incorporated, this model can be g(μ) = α + f(X₁)+ ..... + f (X_P)called Generalize Additive Models GAM. Therefore Additive Models and GLMare combined to GAM. GAM need few known conditions about data so theymay be used widely, especially in Poisson and Logistic regression. However,there are some drawbacks in them: â‘  This kind of model can not involve theinteraction in predict variables; â‘¡ They are only an approach to the real curvefaces without high accuracy when many predict variables. Nonetheless, as thissort of model is superior they are used in medical fields more widely and morewidely. There are several problems influencing their accuracy, for exampleoutliers in Y axis may lead to a far value from the real value. So it is necessaryto introduce the classic robust method into GAM. But because of the differentforms of models, the classic robust method fails to apply to GAM ,it has to bemodified. We should adjust the method of weighting according to the process ofestimating GAM and change the square sum of residuals into deviances of GAM,etc. In data, concurvity is a usual and inevitable problem and is similar tocollinearity in linear regression models. It is a nonparametric form ofcollinearity in GAM. Standard error of parametric terms will be underestimatedand the outcome of the model is not unique because of its existence. Then I typeerror will rise and the backfitting process to fit Additive Models is influenced byinitial function. To overcome the effect of concurvity, we have several ways tochoose. In this papers, the author introduced nonparametric conditionalbootstrap and gam.exact function and carried out researches by simulations andsome examples. Meanwhile, this papers compared the outcomes of GAM withthat of GLM. All the proceedings of estimation and outcomes in this papers are achievedby S-plus.