การเปรียบเทียบเกณฑ์การคัดเลือกตัวแบบความถดถอยพหุนามแบบติดกลุ่ม - พจนา แววสวัสดิ์ (จุฬาลงกรณ์มหาวิทยาลัย, 2000)
The objective of this study is to compare model selection criteria for nested polynomial regression model. The general model for polynomial regression is show as follows: Y = X β+£ where Y is an n x 1 random vector of dependent variables; X is an n x (p+1) fixed matrix of independent variables; β is a (p + 1)x1 unknown parametric vector of regression coefficients; £ is an n x 1 random vector of errors and £ is normally distributed with mean vector 0 and variance-covariance matrix 2I ; I is an n x n identity matrix ; 2 is the variance of random errors; n is the number of observed data and p is number of independent variables. The 3 stopping criteria of model selection are the partial F-test statistic using backward elimination (BW), Akaike’s Information Criterion (AIC), Bayesian Information Criterion (BIC) respectively. The BW method is started with putting all independent variables in the model and then eliminating each variable, one at a time, using the partial F-test statistic as the stopping criterion. The test statistic is as follows: [Equation] SSEj is the sum of square errors for the model j., SSEk is the sum of square errors for the model k, pk,pj is the number of parameters in model j and k. The AIC criterion uses the maximum likelihood method for each corresponding model as the stopping criterion. The test statistic is as follows: AIC = -2 log (MLk – MLj) +2(pk – pj); MLj,MLk is the maximum likelihood estimators form model j and k. pj, pk is the number of parameters in model j and k. The BIC criterion uses the posterior probability for each corresponding model as the stopping criterion. The test statistic is as follows: BIC = -2 log [Equation] β -j, βk is the maximum likelihood estimator for model j and k, Mj,Mk Is the model j and k. p (X β-k, Mk) is the posterior probability for the corresponding model. The compare is done by the model generation using a monte carto technique through S-plus 2000 code. The number of independent variables is 3 5 9 14 20 and 27 respectively. The random error are generated with normal distribution with me and 0 and standard deviation 5 10 20 and 25 respectively. The number of observed data varies form 35 50 75 and 100. The degree of collinearity among independent variables is changed from 0(orthogonality among independent variables), 0.5 to 1. Significant levels is 0.05 and 0.01. The averaqe of mean square error (AMSE) from each criterion, between the predicted value calculate from each selected model and the true values is use to compare these 3 model selection criteria. The result of the study show that when the independent variables are orthogonal, the predicted value from BIC criterion provide the minimum average of mean square error (AMSE), in all simulated cases. When independent variables are not orthogonal and they are interrelated, the nearly orthogonal situation can be done through data centering. This will result in making the AMSE for the predicted value from BIC minimum in all situations. However the nearly orthogonalized independent variables for such case will be appropriate only when the exponent of the between 1 to 3. When the model exponent of the model is from 4 to 6, the polynomial orthogonalization of the independent variables can be better achievable than the data centering and this will result in making the AMSE for the because the AMSE of centering data is highest orthogonal polynomial, the predicted value from BIC minimum in all cases as well.