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Mixed Model (LME)

Mixed Model (LME). pH vs. CO 2 in Eyes. Structure of data. 18 subjects 2 visits per subject Two measurements per visit (one per eye). The levels of CO 2 were randomly assigned to the 4 measurements. Data. id co2 ph prph visit eye 1 1 0 7.432 76.4 1 1

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Mixed Model (LME)

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  1. Mixed Model (LME) pH vs. CO2 in Eyes

  2. Structure of data • 18 subjects • 2 visits per subject • Two measurements per visit (one per eye). • The levels of CO2 were randomly assigned to the 4 measurements.

  3. Data id co2 ph prph visit eye 1 1 0 7.432 76.4 1 1 2 1 5 6.956 59.4 1 2 3 1 3 7.232 80.3 2 1 4 1 7 6.982 74.2 2 2 5 2 0 7.594 67.8 1 1 6 2 7 6.929 61.1 1 2 7 2 3 7.230 73.4 2 1 8 2 5 7.073 65.9 2 2 9 3 3 7.329 87.6 1 1 10 3 7 6.963 55.8 1 2

  4. Look at Data

  5. Model 1 - OLS • First model is as follows: for the ith individual, the jth visit, kth eye. • Xijk =CO2 • How does one interpret the parameters, 0, 1, and 2?

  6. Model 1, cont. • E(eijk)=0, cov(eijk, eij’k’)=0, jj’ or kk. • Var(eijk)= 2e . • What does V0 look like for this model?

  7. Model 1 using SAS ********************************************************; * PROC MIXED, Model 1; ********************************************************; procmixeddata=rep2001.phlong; class id; model ph = co2 co22 / s; estimate'3' co2 3 co22 9 / cl; estimate'5' co2 5 co22 25 / cl; estimate'7' co2 7 co22 49 / cl; run;

  8. Model 1 Results using SAS Covariance Parameter Estimates Cov Parm Estimate Residual 0.007792 Fit Statistics -2 Res Log Likelihood -117.1 AIC (smaller is better) -115.1 AICC (smaller is better) -115.0 BIC (smaller is better) -112.9 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 7.6462 0.02119 67 360.81 <.0001 CO2 -0.1373 0.01379 67 -9.96 <.0001 CO22 0.007339 0.001910 67 3.84 0.0003

  9. Model 1- Estimate Command Results Estimates Standard Label Estimate Error DF t Value Pr > |t| Alpha Lower Upper 3 -0.3458 0.02549 67 -13.57 <.0001 0.05 -0.3967 -0.2949 5 -0.5028 0.02733 67 -18.40 <.0001 0.05 -0.5574 -0.4483 7 -0.6011 0.02865 67 -20.98 <.0001 0.05 -0.6583 -0.5439

  10. Model 2 • How does one interpret the parameters, 0, 0i, and 1, 2?

  11. Model 2, cont. • E(eijk)=0, cov(eijk, eij’k’)=0, jj’ or kk’ • Var(eijk)= 2e • Var(0i)= 2id. • cov(eijk, 0i)= 0. • What does V0 look like for this model?

  12. Model 2 using SAS ********************************************************; * PROC MIXED, Model 2; ********************************************************; procmixeddata=rep2001.phlong; class id; model ph = co2 co22 / s; random int / sub=id; estimate'3' co2 3 co22 9 / cl; estimate'5' co2 5 co22 25 / cl; estimate'7' co2 7 co22 49 / cl; run;

  13. Model 2 Results using SAS Covariance Parameter Estimates Cov Parm Subject Estimate Intercept ID 0.005052 Residual 0.002696 Fit Statistics -2 Res Log Likelihood -152.4 AIC (smaller is better) -148.4 AICC (smaller is better) -148.2 BIC (smaller is better) -146.6 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 7.6458 0.02098 17 364.50 <.0001 CO2 -0.1373 0.008115 50 -16.92 <.0001 CO22 0.007344 0.001123 50 6.54 <.0001

  14. Model 2- Estimate Command Results Estimates Standard Label Estimate Error DF t Value Pr > |t| Alpha Lower Upper 3 -0.3456 0.01484 50 -23.28 <.0001 0.05 -0.3755 -0.3158 5 -0.5025 0.01596 50 -31.50 <.0001 0.05 -0.5346 -0.4705 7 -0.6006 0.01681 50 -35.72 <.0001 0.05 -0.6343 -0.5668

  15. Model 3 • How does one interpret the parameters, 0, 0i, 0i jand 1, 2?

  16. Model 3, cont. • E(eijk)=0, cov(eijk, eij’k’)=0, jj’ or kk’ • Var(eijk)= 2e • Var(0i)= 2id • Var(0ij)= 2visit • cov(eijk, 0i)= 0, cov(eijk, 0ij)= 0, cov(0i, 0ij)= 0 • What does V0 look like for this model?

  17. Model 3 using SAS ********************************************************; * PROC MIXED, Model 3; ********************************************************; procmixeddata=rep2001.phlong; class id visit; model ph = co2 co22 / s; random id visit(id); estimate'3' co2 3 co22 9 / cl; estimate'5' co2 5 co22 25 / cl; estimate'7' co2 7 co22 49 / cl; run;

  18. Model 3 Results using SAS Covariance Parameter Estimates Cov Parm Estimate ID 0.004928 VISIT(ID) 0.000395 Residual 0.002433 Fit Statistics -2 Res Log Likelihood -152.8 AIC (smaller is better) -146.8 AICC (smaller is better) -146.5 BIC (smaller is better) -144.2 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 7.6446 0.02092 17 365.50 <.0001 CO2 -0.1365 0.008061 33 -16.94 <.0001 CO22 0.007262 0.001115 33 6.51 <.0001

  19. Model 3- Estimate Command Results Estimates Standard Label Estimate Error DF t Value Pr > |t| Alpha Lower Upper 3 -0.3440 0.01471 33 -23.39 <.0001 0.05 -0.3739 -0.3141 5 -0.5007 0.01575 33 -31.79 <.0001 0.05 -0.5327 -0.4686 7 -0.5992 0.01642 33 -36.49 <.0001 0.05 -0.6326 -0.5658

  20. Model 4 • How does one interpret the parameters, 0, 0i, 1i,2i and 1, 2?

  21. Model 4, cont. • E(eijk)=0, cov(eijk, eij’k’)=0, jj’ or kk’ • Var(eijk)= 2e • Var(0i)= 20, Var(1i)= 21, Var(2i)= 22 • cov(eijk, 0i)= 0, cov(eijk, 1i)= 0, cov(eijk, 2i)= 0. • cov(0i, 1i)= 01, cov(0i, 2i)= 02, cov(1i, 2i)= 12. • What does variance-covariance of random effects look like?

  22. Model 4 using SAS ********************************************************; * PROC MIXED, Model 4; ********************************************************; procmixeddata=rep2001.phlong; class id visit; model ph = co2 co22 / s; random int co2 co22 /sub=id; estimate'3' co2 3 co22 9 / cl; estimate'5' co2 5 co22 25 / cl; estimate'7' co2 7 co22 49 / cl; run;

  23. Model 4 Results using SAS Covariance Parameter Estimates Cov Parm Subject Estimate Intercept ID 0.005054 CO2 ID 0 CO22 ID 0 Residual 0.002696 Fit Statistics -2 Res Log Likelihood -152.4 AIC (smaller is better) -148.4 AICC (smaller is better) -148.2 BIC (smaller is better) -146.6 Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > |t| Intercept 7.6458 0.02098 17 364.47 <.0001 CO2 -0.1373 0.008114 17 -16.92 <.0001 CO22 0.007344 0.001123 16 6.54 <.0001

  24. Model 4 - Estimate Command Results Estimates Standard Label Estimate Error DF t Value Pr > |t| Alpha Lower Upper 3 -0.3456 0.01484 16 -23.29 <.0001 0.05 -0.3771 -0.3142 5 -0.5025 0.01595 16 -31.50 <.0001 0.05 -0.5363 -0.4687 7 -0.6006 0.01681 16 -35.72 <.0001 0.05 -0.6362 -0.5649

  25. Comparing Models

  26. Splus Appendix

  27. Model 1 using lm (OLS) > lm.ph.model1<-lm(ph~co2+co2^2,data=phc02long,na.action=na.omit) > summary(lm.ph.model1) Coefficients: Value Std. Error t value Pr(>|t|) (Intercept) 7.6462 0.0212 360.8133 0.0000 0 co2 -0.1373 0.0138 -9.9555 0.0000 1 I(co2^2) 0.0073 0.0019 3.8432 0.0003 2 Residual standard error: 0.08827 on 67 degrees of freedom e

  28. Model 2 using lme > lme.ph.model2<-lme(ph~co2+co2^2,random=~1 | id, data=phc02long, na.action=na.omit) > summary(lme.ph.model2) Linear mixed-effects model fit by REML Data: phc02long AIC BIC logLik -142.3814 -131.3579 76.19069 ------------------------------------------------------------ Random effects: Formula: ~ 1 | id (Intercept) Residual StdDev: 0.0710906 0.05191807 id e ------------------------------------------------------------ Fixed effects: ph ~ co2 + co2^2 Value Std.Error DF t-value p-value (Intercept) 7.645778 0.02097791 50 364.4681 <.0001 0 co2 -0.137281 0.00811416 50 -16.9187 <.0001 1 I(co2^2) 0.007344 0.00112340 50 6.5373 <.0001 2 Interpretation?

  29. Model 3 using lme > lme.ph.model3<-lme(ph~co2+co2^2,random=~1 | id/visit, data=phc02long, + na.action=na.omit) > summary(lme.ph.model3) Linear mixed-effects model fit by REML Data: phc02long AIC BIC logLik -140.8394 -127.6112 76.41968 Random effects: Formula: ~ 1 | id (Intercept) StdDev: 0.07019805 id Formula: ~ 1 | visit %in% id (Intercept) Residual StdDev: 0.01988396 0.04932218 visite (Intercept) 7.644635 0.02091566 33 365.4981 <.0001 0 co2 -0.136527 0.00806119 33 -16.9363 <.0001 1 I(co2^2) 0.007262 0.00111502 33 6.5131 <.0001 2 ------------------------------------------------------------

  30. Model 3 - alternative syntax > lme.ph.model3.alt<-lme(ph~co2+co2^2,random=list(~1 | id,~1 | visit), data=phc02long,na.action=na.omit) > summary(lme.ph.model3.alt) Linear mixed-effects model fit by REML Data: phc02long AIC BIC logLik -140.8394 -127.6112 76.41968 Random effects: Formula: ~ 1 | id (Intercept) StdDev: 0.07019805 id Formula: ~ 1 | visit %in% id (Intercept) Residual StdDev: 0.01988396 0.04932218 visite (Intercept) 7.644635 0.02091566 33 365.4981 <.0001 0 co2 -0.136527 0.00806119 33 -16.9363 <.0001 1 I(co2^2) 0.007262 0.00111502 33 6.5131 <.0001 2 ------------------------------------------------------------

  31. Model 4 using lme > lme.ph.model4<-lme(ph~co2+co2^2,random=~1+co2+co2^2 | id,data=phc02long,na.action=na.omit) > summary(lme.ph.model4) Linear mixed-effects model fit by REML Data: phc02long AIC BIC logLik -140.2228 -118.1759 80.11142 ------------------------------------------------------------ Random effects: Formula: ~ 1 + co2 + co2^2 | id Structure: General positive-definite StdDev Corr (Intercept) 0.100906871 (Intr) co2 0 co2 0.013648260 -0.885 1 01 I(co2^2) 0.001191092 0.656 -0.932 2 02 12 Residual 0.047338794 e ------------------------------------------------------------ Fixed effects: ph ~ co2 + co2^2 Value Std.Error DF t-value p-value (Intercept) 7.645563 0.02653446 50 288.1372 <.0001 0 co2 -0.137186 0.00808829 50 -16.9611 <.0001 1 I(co2^2) 0.007335 0.00106247 50 6.9033 <.0001 2

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