EXST 7015 - Statistical Inference II, Fall 2011 Lab 6: Multiple Linear Regression_Variable diagnostics OBJECTIVES In multiple regression however, a number of variables can be involved and regressed on one another (model: Y = β0 + β1X1+β2X2+ · · · + βpXP + ε) .The overall test of hypothesis of multiple linear regression is H0: β1=β2 = · · ·= βp =0 v.s. H1: at least one β ≠ 0. Rejection of H0 implies that at least one of the regressors, X1, X2, . . . , Xp, contributes significantly to the model. In the lab5, the problem of multicollinearity caused by highly correlated variables was introduced by indentify the diagnostic statistics of sequential parameter estimates, simple correlation, variance inflation factor (VIF) and condition index. In this lab, an extreme case of muticolinearlity will be presented to help you fully understand those statistics. Since there are more than one independent variables in the model of multiple linear regression, many of you have raised the question that which variables are more important than the others. By using partial SS F-test (Type II, III, IV) and t-test of regression coefficients, the larger the F-value or t-value (the smaller the P-value), the more significant of the variable to the model as you might be aware in lab5. In addition, standardized regression coefficients and partial R^2 will be discussed to help you evaluate the relative importance of individual variables in the model in this lab. Some of you might realize that the absolute value of regression coefficient is not a good predictor of relative importance of the variables. Why it happens is that, most often, the variables are not on the same scale or are of arbitrary scale, which leads to un-meaningful slope (Y units per X units). In such cases, the variables could be standardized with a mean=0 and variance=1. Then the standardized regression coefficients are obtained, which is the relative measurements of the importance of the variable. In multiple linear regression, R^2 for overall model is the proportion of variation in dependent variable explained by all independent variables included in the model (SSModel/SSTotal). Likewise, a partial R^2 could be calculated for each individual variable, which measures the marginal contribution of one independent variable when all the other variables are already included in model. However, such interpretation is not valid unless there is no problem of multicollinearity. LABORATORY INSTRUCTIONS Housekeeping Statements dm 'log; clear; output; clear'; options nodate nocenter pageno = 1 ls=78 ps=53; title1 'EXST7015 lab 2, Name, Section#'; ods rtf file = ‘c:/temp/lab2.rtf’; ods html file = ‘c:/temp/lab2.html’;Data set The data set is from Chapter 6, Problem 18 in “Introduction to Regession Analysis” by Abraham and Ledolter @ 2006 Thomson Brook. This data set came from an experiment to investigate the amount of drug retained in the liver of a rat. 19 rats were weighted and dosed. The dose was approximately 40mg/kg of body weight. It can be expected that the liver is strongly correlated with body weight. After a fixed length of time the rat was sacrificed, the liver weighted and the percentage of dose in the liver was determined. The variables are: bodyWT (body weight), liverWT (liver weight), DOSE and Y (Dose remained in liver).We will perform a multiple regression using Y as dependent variable and bodyWT, liverWT and DOSE as independent variables. data Liver; title2 'Multilinear regression_Variable Diagnostics'; input bodyWT liverWT dose Y; datalines; 176 6.5 0.88 0.42 176 9.5 0.88 0.25 190 9 1 0.56 176 8.9 0.88 0.23 200 7.2 1 0.23 167 8.9 0.83 0.32 188 8 0.94 0.37 195 10 0.98 0.41 176 8 0.88 0.33 165 7.9 0.84 0.38 158 6.9 0.8 0.27 148 7.3 0.74 0.36 149 5.2 0.75 0.21 163 8.4 0.81 0.28 170 7.2 0.85 0.34 186 6.8 0.94 0.28 146 7.3 0.73 0.3 181 9 0.9 0.37 149 6.4 0.75 0.46 ; Proc print data=liver; run; ; Multiple Linear Regression by using PROC REG Proc reg data=liver; title2 'Multiple Linear Regression_Variable diagonostics'; model Y=bodyWT liverWT dose/all influence collin; OUTPUT out=outdata1 p=Predicted r=resid lclm=lclm uclm=uclm lcl=ccl ucl=ucl; run; proc plot data=outdata1; title2 'Residual plot'; plot resid*predicted;run; proc univariate data=outdata1 normal plot; title2 'Normality test'; var resid; run; All: Specify this option in your model is equivalent to requesting all the following options: ACOV, CLB, CLI, CLM, CORRB, COVB, I, P, PCORR1, PCORR2, R, SCORR1, SCORR2, SEQB, SPEC, SS1, SS2, STB, TOL, VIF, and XPX. In this lab, we are particularly interested in the analysis performance by those in bold letters. Note that, while it is nice not having to memorize and type a lot of options, pages of possible irrelevant information are generated and you need to be able to navigate through to find what you need. STB: prints standardized regression coefficients. CORRB: prints the correlation matrix of estimates. PCORR1: requests partial R^2 type I {SeqSSX/(SeqSSXi+SSError)} PCORR2: requests partial R^2 type II{PartialSSXi/(PartialSSXi+SSError)} SCORR1: requests semi-partial R^2 type I (SeqSSX/SSTotal) SCORR2: requests semi-partial R^2 type II (PartialSSX/SSTotal) Carefully exam the output, you will find that Type I semi-partial R^2 sums to overall R^2 since the type I SS sums to the SSReg. In contrast, partial R^2 type II is not predictable since the type II SS may sum to more or less than SSReg. The values of partial R^2 type II follows similar trends of the t-value in t-test of regression coefficients. CLM: prints the 95% upper and lower confidence limits for the expected value of the dependent variable (mean) for each observation. CLI: requests the 95% upper and lower confidence limits for an individual predicted value. Collin: generates a number of collinearity diagnostics include condition indicies. If condition index exceeds 30, multicollinearity might be a problem. VIF: the value of VIF is expected to be 1 if the regressors are not correlated. If the value is much greater than 2, serious problems are suggested. SEQB: generate the sequential parameter estimates to exam whether there are large fluctuations as variable enters.LAB ASSIGNMENT Use PROC REG with appropriate options to fit the multiple linear model Y = β0 + β1bodyWT + β2liverWT+ β3DOSE + ε, and answer the following questions.