Multiple Linear Regression. Concept and uses. Model and assumptions. Intrinsically linear models. Model development and validation. Problem areas. Non-normality. Heterogeneous variance. Correlated errors. Influential points and outliers. Model inadequacies. Collinearity. Errors in X variables. September 19, 2018 AGR206

Concept & Uses. Did you know? REGRESSION ANOVA Did you know? Description restricted to data set. Did biomass increase with pH in the sample? Prediction of Y. How much biomass we expect to find in certain soil conditions? Extrapolation for new conditions: can we predict biomass in other estuaries? Estimation and understanding. How much does biomass change per unit change in pH and controlling for other factors? Control of process: requires causality. Can we create sites with certain biomass by changing the pH? September 19, 2018 AGR206

Body fat example in JMP. Three variables (X1, X3, X3) were measured to predict body fat % (Y) in people. Random sample of people. Y was measured by an expensive and very accurate method (assume it reveals true %fat). X1: thickness of triceps skinfold X2: thigh circumference X3: midarm circumference. Bodyfat.jmp September 19, 2018 AGR206

Ho’s or values “of interest” Does thickness of triceps skinfold contribute significantly to predict fat content? What is the CI for fat content for a person whose X’s have been measured? Do I have more or less fat than last summer? Do I have more fat than recommended? September 19, 2018 AGR206

Model and Assumptions. Linear, additive model to relate Y to p independent variables. Note: here, p is number of variables, but some authors use p for number of parameters, which is one more than variables due to the intercept. Yi=b0+ b1 Xi1+…+ bp Xip+ei where ei are normal and independent random variables with common variance s2. In matrix notation the model and solution are exactly the same as for SLR: Y= Xb + e b=(X’X)-1(X’Y) All equations from SLR apply without change. September 19, 2018 AGR206

Linear models Linear, and intrinsically linear models. Linearity refers to the parameters. The model can involve any function of X’s for as long as they do not have parameters that have to be adjusted. A linear model does not always produce a hyperplane. Yi=b0+ b1 f1(Xi1)+…+ bp fp(Xi1)+ei Polynomial regression. Is a special case where the functions are powers of X. September 19, 2018 AGR206

Matrix Equations September 19, 2018 AGR206

Extra Sum of Squares Effects of order of entry on SS. The 4 types of SS. Partial correlation. September 19, 2018 AGR206

Extra Sum of Squares:body fat September 19, 2018 AGR206

Response plane and error The response surface in more than 3D is a hyperplane. X1 X2 Yi E{Yi} Y September 19, 2018 AGR206

Model development What variables to include. Depends on objective: descriptive -> no need to reduce number of variables. Prediction and estimation of Yhat: OK to reduce for economical use. Estimation of b and understanding: sensitive to deletions; may bias MSE and b. No real solution other than getting more data from better experiment. (Sorry!) September 19, 2018 AGR206

Variable Selection Effects of elimination of variables: MSE is positively biased unless true b for variables eliminated is 0. bhat and Yhat are biased unless previous condition or variables eliminated are orthogonal to those retained. Variance of estimated parameters and predictions is usually lower. There are conditions for which MSE for reduced model (including variance and bias2) is smaller. September 19, 2018 AGR206

Criteria for variable selection R2 - Coefficient of determination. R2 = SSReg/SSTotal MSE or MSRes - Mean squared residuals. if all X’s in it estimates s2. R2adj - Adjusted R2. R2adj = 1-MSE/MSTo = =1-[(n-1)/(n-p)] (SSE/SSTo) Mallow’s Cp Cp=[SSRes/MSEFull] + 2 p- n (p=number of parameters) September 19, 2018 AGR206

Example September 19, 2018 AGR206

Checking assumptions. Note that although we have many X’s, errors are still in a single dimension. Residual analysis is performed as for SLR, sometimes repeated over different X’s. Normality. Use proc univ normal option. Transform. Homogeneity of variance. Plot error vs. each X. Transform. Weighted least squares. Independence of errors. Adequacy of model. Plots errors. LOF. Influence and outliers. Use influence option in proc reg. Collinearity. Use collinoint option of proc reg. September 19, 2018 AGR206

code for PROC REG data s00.spart2; set s00.spartina; colin=2*ph+0.5*acid+sal+rannor(23); run; proc reg data=s00.spart2; model bmss= colin h2s sal eh7 ph acid p k ca mg na mn zn cu nh4 / r influence vif collinoint stb partial; run; model colin=ph sal acid; September 19, 2018 AGR206

Spartina ANOVA output Model: MODEL1 Dependent Variable: BMSS Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 15 16369583.2 1091305.552 11.297 0.0001 Error 29 2801379.9 96599.307 C Total 44 19170963.2 Root MSE 310.80429 R-square 0.8539 Dep Mean 1000.80000 Adj R-sq 0.7783 C.V. 31.05558 September 19, 2018 AGR206

Parameters and VIF September 19, 2018 AGR206 Parameter Estimates Parameter Standard T for H0: Standardized Variable DF Estimate Error Parameter=0 Prob > |T| Estimate INTERCEP 1 3809.233562 3038.081 1.254 0.2199 0.00000000 COLIN 1 -178.317065 58.718 -3.037 0.0050 -1.06227792 H2S 1 0.336242 2.656 0.127 0.9001 0.01563626 SAL 1 150.513276 61.960 2.429 0.0216 0.84818417 EH7 1 2.288694 1.785 1.282 0.2099 0.12813770 PH 1 486.417077 306.756 1.586 0.1237 0.91891994 ACID 1 -24.816449 109.856 -0.226 0.8229 -0.09422943 P 1 0.153015 2.417 0.063 0.9500 0.00639498 K 1 -0.733250 0.439 -1.668 0.1061 -0.33059243 CA 1 -0.137163 0.111 -1.230 0.2286 -0.35706572 MG 1 -0.318586 0.243 -1.308 0.2010 -0.45340287 NA 1 -0.005294 0.022 -0.239 0.8127 -0.05520175 MN 1 -4.279887 4.836 -0.885 0.3835 -0.15872971 ZN 1 -26.270852 19.452 -1.351 0.1873 -0.32953283 CU 1 346.606818 99.295 3.491 0.0016 0.54452366 NH4 1 0.539373 3.061 0.176 0.8614 0.03862822 Variance Inflation 0.00000000 24.28364757 3.02785626 24.19556405 1.98216733 66.64921013 34.53131689 2.02507775 7.79660017 16.72702792 23.82835726 10.57323219 6.38589662 11.81574077 4.82931410 9.53842459 September 19, 2018 AGR206