STAT 5200 Handout #25. R-Square & Design Matrix in Mixed Models

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1 STAT 5200 Handout #25 R-Square & Design Matrix in Mixed Models I. R-Square in Mixed Models (with Example from Handout #20): For mixed models, the concept of R 2 is a little complicated (and neither PROC MIXED nor PROC GLIMMIX report it). We can instead focus on the usual interpretation of R 2, the percent reduction in variability due to the model. Recall R 2 = 1 SSE / SSTOT, where SSE is the sum of squared residuals from the model, and SSTOT is the sum of squared deviations from the mean (which is the same as the sum of squared residuals from an intercept-only model). /* Define options */ ods html image_dpi=300 style=journal; /* Read in data (Handout 20 Example 1) */ data grass; input NSource $ Block Thatch cards; Urea Urea Urea Urea Urea Urea AmmSulph AmmSulph AmmSulph AmmSulph AmmSulph AmmSulph IBDU IBDU IBDU IBDU IBDU IBDU SCUrea SCUrea SCUrea SCUrea SCUrea SCUrea ; /* Fit this as a split-plot design */ proc mixed data=grass covtest method=type3; model Chlorophyll = NSource Thatch ; random Block Block*NSource; title1 'Fit Model with Factors'; /* Fit intercept-only design */ proc mixed data=grass covtest method=type3; model Chlorophyll = ; title1 'Fit Intercept-only Model'; 1

2 Fit Model with Factors Type 3 Analysis of Variance Source DF Sum of Squares NSource Thatch NSource*Thatch Block NSource*Block Covariance Parameter Estimates Cov Parm Estimate Standard Error Z Value Pr Z Block NSource*Block Residual Residual Fit Intercept-only Model Type 3 Analysis of Variance Source DF Sum of Squares Residual /* Calculate R-square */ data temp; SSE_Model = ; SSE_IntOnly = ; Rsquare = 1 - SSE_Model / SSE_IntOnly; proc print data=temp; var Rsquare; title1 'Percent reduction in variance due to model'; Percent reduction in variance due to model Obs Rsquare

3 II. Design Matrix (with Example from Handout #20): Most software (including SAS) use matrix forms to fit these models [ β = (X X) 1 X Y ] Vector of response variable Vector of fixed parameters Vector of random parameters Y = X β + Z γ + ε Design matrix for fixed Design matrix for random Vector of errors Cov(γ) = G Cov(ε) = R usually R = σ 2 I Design matrix usually refers to matrix X here: A row for each observation (experimental unit) A column for each specific [fixed ] parameter (including intercept µ) Usually first column is all 1 s ( intercept ) Other columns are 0/1 indicators of whether corresponding parameter (factor level) applies to that observation (row) Order of columns similar to order of parameter coefficients in CONTRAST statement (recall Handout #10), but sum-to-zero or last-one-zero [default] constraints apply and are built in here We can use PROC GLMMOD to look at these (pasted together, anyway). See SAS documentation The MIXED Procedure: Mixed Models Theory for details, including how to estimate β with matrix operations. (Note: SAS documentation and code sometimes refer to R-side and G-side or covariance structure. [See for example the RANDOM statement on p. 3 of Handout #23.] These refer to the R and G matrices above.) Much more on this in STAT 6000-level courses Theory of Linear Models and Generalized Linear Mixed Models. 3

4 /* Look at design matrix */ proc glmmod data=grass outparm=parm outdesign=design noprint; model Chlorophyll = NSource Thatch Block Block*NSource; proc print data=parm; title1 'Model Parameters'; Model Parameters Parameters µ N1 Obs _COLNUM_ EFFNAME NSource Block Thatch 1 1 Intercept 2 2 NSource AmmSulph Fixed Effects Parameters (β) N4 T1 T2 T3 NT NSource Urea 6 6 Thatch Thatch Thatch NSource*Thatch AmmSulph 2 NT NSource*Thatch Urea 8 Random Effects Parameters (γ) B1 B2 NB11 NB Block Block NSource*Block AmmSulph NSource*Block Urea 2 4

5 proc print data=design; title1 'Model Design Matrix'; Model Design Matrix Obs Chlorophyll Col1 Col2 Col3 Col4 Col5 Col29 Col NOTE: See _COLNUM_ on previous page to see definitions of column indicators: o Col2 = 1 if NSource=AmmSulph; 0 otherwise o Col5 = 1 if NSource=Urea; 0 otherwise Fixed parameters Col1-Col20 = X Random Effects parameters Col21-Col30 = Z 5