Analysis of Repeated Measures Data Ramon C. Littell

Effect on growth regulators on chrysanthemum plantsData courtesy James Barrett and Terril NellMixed Model Repeated Measures Analysesproc print data=mumsuni;Obs blk trt plt chem time ht1 1 3 1bonzi 1 3.02 1 3 1bonzi 2 4.03 1 3 1bonzi 3 5.54 1 3 1bonzi 4 19.55 1 3 1bonzi 5 33.06 1 3 1bonzi 6 44.57 1 3 2bonzi 1 2.58 1 3 2bonzi 2 3.09 1 3 2bonzi 3 5.510 1 3 2bonzi 4 18.011 1 3 2bonzi 5 33.512 1 3 2bonzi 6 46.513 1 9 1sumag 1 1.014 1 9 1sumag 2 2.015 1 9 1sumag 3 2.016 1 9 1sumag 4 6.017 1 9 1sumag 5 14.018 1 9 1sumag 6 26.519 1 9 2sumag 1 4.020 1 9 2sumag 2 4.521 1 9 2sumag 3 7.022 1 9 2sumag 4 17.023 1 9 2sumag 5 31.524 1 9 2sumag 6 44.025 2 3 1bonzi 1 1.026 2 3 1bonzi 2 2.027 2 3 1bonzi 3 4.028 2 3 1bonzi 4 17.529 2 3 1bonzi 5 35.530 2 3 1bonzi 6 47.031 2 3 2bonzi 1 4.032 2 3 2bonzi 2 4.533 2 3 2bonzi 3 11.534 2 3 2bonzi 4 29.035 2 3 2bonzi 5 41.536 2 3 2bonzi 6 54.037 2 9 1sumag 1 3.038 2 9 1sumag 2 4.039 2 9 1sumag 3 5.540 2 9 1sumag 4 17.5

Effect on growth regulators on chrysanthemum plantsData courtesy James Barrett and Terril NellMixed Model Repeated Measures AnalysesMixed Model analysis of repeated measures data incorporates the covariance into theanalysis, resulting in efficient and valid analyses. The first step is to model the covariancestructure. It is usually to begin with unstructured covariance to examine the covariancematrix for patterns.The MIXED procedure uses syntax similar to the GLM procedure. A major distinction isthat only fixed effects appear in the model statement.The repeated statement is used to define the covariance structure. The MIXED procedureemploys likelihood methods to fit the model and compute inferential statistics.proc mixed data=mumsuni; class blk chem plt time;model ht = chem time chem*time / ddfm=kr;repeated time / sub=plt(chem time) type=un r rcorr;The Mixed ProcedureModel InformationData SetWORK.MUMSUNIDependent Variable htCovariance Structure UnstructuredSubject Effectplt(blk*chem)Estimation MethodREMLResidual Variance Method NoneFixed Effects SE Method Prasad-Rao-Jeske-Kackar-HarvilleDegrees of Freedom Method Kenward-RogerClass Level InformationClass LevelsValuesblk 51 2 3 4 5chem 2bonzi sumagplt 21 2time 61 2 3 4 5 6Iteration HistoryIteration Evaluations -2 Res Log Like Criterion0 1 669.213999901 1 356.89670434 0.00000000This is good news!Convergence criteria met.

Effect on growth regulators on chrysanthemum plantsData courtesy James Barrett and Terril NellMixed Model Repeated Measures AnalysesCovariance Matrix from r option in repeated statement:Estimated R Matrix for plt(blk*chem) 1 1 bonziRow Col1 Col2 Col3 Col4 Col5 Col61 0.8681 0.7806 1.6236 3.4014 4.3750 4.17362 0.7806 0.8111 1.5667 3.4639 4.7583 4.64443 1.6236 1.5667 4.5361 10.2403 12.5236 11.87224 3.4014 3.4639 10.2403 27.1181 34.4472 32.99035 4.3750 4.7583 12.5236 34.4472 50.6736 49.31256 4.1736 4.6444 11.8722 32.9903 49.3125 49.5417Interpretation:• The variance of height is .8681 at time 1, .8111 at time 2, 4.536 at time 3, etc.• The covariance is .7806 between heights at times 1 and 2, 1.623 between times 1 and 3,1.566 between times 2 and 3, etc.• General pattern:- variances increase with time- covariances increase with timeCorrelation Matrix from rcorr option in repeated statement:Interpretation:Estimated R Correlation Matrix forplt(blk*chem) 1 1 bonziRow Col1 Col2 Col3 Col4 Col5 Col61 1.0000 0.9302 0.8182 0.7011 0.6596 0.63642 0.9302 1.0000 0.8168 0.7386 0.7422 0.73273 0.8182 0.8168 1.0000 0.9233 0.8260 0.79204 0.7011 0.7386 0.9233 1.0000 0.9293 0.90015 0.6596 0.7422 0.8260 0.9293 1.0000 0.98426 0.6364 0.7327 0.7920 0.9001 0.9842 1.0000• The correlation is .9302 between heights at times 1 and 2, .8182 between times 1 and 3,.8162 between times 2 and 3, etc.• General pattern:- correlations decrease with time interval- correlations of equal time lag are similar

Effect on growth regulators on chrysanthemum plantsData courtesy James Barrett and Terril NellMixed Model Repeated Measures AnalysesSelecting a Covariance StructureThe next step is to select a covariance structure from those with the characteristics identified inthe “unstructured” covariance and correlation matrices. One such candidate is heterogeneousautoregressive.proc mixed data=mumsuni; class blk chem plt time;model ht = chem time chem*time / ddfm=kr;repeated time / sub=plt(chem time) type=arh(1) r rcorr;The Mixed ProcedureModel InformationData SetWORK.MUMSUNIDependent Variable htCovariance Structure Heterogeneous AutoregressiveSubject Effectplt(blk*chem)Estimation MethodREMLResidual Variance Method NoneFixed Effects SE Method Prasad-Rao-Jeske-Kackar-HarvilleDegrees of Freedom Method Kenward-RogerIteration HistoryIteration Evaluations -2 Res Log Like Criterion0 1 669.213999901 2 390.36247704 0.041374902 1 389.73086224 0.021287893 1 387.28045811 0.004561454 1 386.77198610 0.000575825 1 386.71272264 0.000014986 1 386.71128525 0.000000017 1 386.71128394 0.00000000Interpretation:Convergence criteria met.The estimation algorithm converges in 7 steps; more good news.

Effect on growth regulators on chrysanthemum plantsData courtesy James Barrett and Terril NellMixed Model Repeated Measures AnalysesSelecting a Covariance StructureEstimated R Matrix for plt(blk*chem) 1 1 bonziRow Col1 Col2 Col3 Col4 Col5 Col61 1.0498 0.9780 2.0913 4.3011 5.0151 4.44982 0.9780 1.0623 2.2716 4.6720 5.4475 4.83353 2.0913 2.2716 5.6636 11.6484 13.5819 12.05114 4.3011 4.6720 11.6484 27.9341 32.5708 28.89995 5.0151 5.4475 13.5819 32.5708 44.2811 39.29046 4.4498 4.8335 12.0511 28.8999 39.2904 40.6491Estimated R Correlation Matrix forplt(blk*chem) 1 1 bonziRow Col1 Col2 Col3 Col4 Col5 Col61 1.0000 0.9261 0.8576 0.7942 0.7355 0.68122 0.9261 1.0000 0.9261 0.8576 0.7942 0.73553 0.8576 0.9261 1.0000 0.9261 0.8576 0.79424 0.7942 0.8576 0.9261 1.0000 0.9261 0.85765 0.7355 0.7942 0.8576 0.9261 1.0000 0.92616 0.6812 0.7355 0.7942 0.8576 0.9261 1.0000Interpretation:Fit Statistics-2 Res Log Likelihood 386.7AIC (smaller is better) 400.7AICC (smaller is better) 401.8BIC (smaller is better) 407.7The arh(1) covariance and correlation matrices are similar to the “unstructured” matrices. TheAICC fit index is 401.8 and the BIC fit index is 407.7.Type 3 Tests of Fixed EffectsEffect Num Den F Value Pr > FDF DFchem 1 18.6 2.39 0.1393time 5 35.9 274.95

Effect on growth regulators on chrysanthemum plantsData courtesy James Barrett and Terril NellMixed Model Repeated Measures AnalysesSelecting a Covariance StructureAnother covariance structure that is often useful is the Toeplitz structure.proc mixed data=mumsuni; class blk chem plt time;model ht = chem time chem*time / ddfm=kr;repeated time / sub=plt(chem time) type=toep r rcorr;The Mixed ProcedureModel InformationData SetWORK.MUMSUNIDependent Variable htCovariance Structure ToeplitzSubject Effectplt(blk*chem)Estimation MethodREMLResidual Variance Method ProfileFixed Effects SE Method Prasad-Rao-Jeske-Kackar-HarvilleDegrees of Freedom Method Kenward-RogerIteration HistoryIteration Evaluations -2 Res Log Like Criterion0 1 669.213999901 2 521.60761870 17800.2291992 1 514.30158009 3121.58896403 1 508.26590893 999.488427064 1 505.47449670 0.023182285 3 503.09397952 0.002476596 1 502.66411897 0.000168727 1 502.63720490 0.000001068 1 502.63704334 0.00000000Interpretation:Convergence criteria met.The estimation algorithm converges in 8 steps; even more good news.

Effect on growth regulators on chrysanthemum plantsData courtesy James Barrett and Terril NellMixed Model Repeated Measures AnalysesSelecting a Covariance StructureEstimated R Matrix for plt(blk*chem) 1 1 bonziRow Col1 Col2 Col3 Col4 Col5 Col61 22.0547 19.3033 14.4120 9.7663 7.1461 8.95522 19.3033 22.0547 19.3033 14.4120 9.7663 7.14613 14.4120 19.3033 22.0547 19.3033 14.4120 9.76634 9.7663 14.4120 19.3033 22.0547 19.3033 14.41205 7.1461 9.7663 14.4120 19.3033 22.0547 19.30336 8.9552 7.1461 9.7663 14.4120 19.3033 22.0547Estimated R Correlation Matrix forplt(blk*chem) 1 1 bonziRow Col1 Col2 Col3 Col4 Col5 Col61 1.0000 0.8752 0.6535 0.4428 0.3240 0.40602 0.8752 1.0000 0.8752 0.6535 0.4428 0.32403 0.6535 0.8752 1.0000 0.8752 0.6535 0.44284 0.4428 0.6535 0.8752 1.0000 0.8752 0.65355 0.3240 0.4428 0.6535 0.8752 1.0000 0.87526 0.4060 0.3240 0.4428 0.6535 0.8752 1.0000Fit Statistics-2 Res Log Likelihood 502.6AIC (smaller is better) 514.6AICC (smaller is better) 515.5BIC (smaller is better) 520.6Interpretation:The “toep” and “unstructured” correlation matrices are similar, but the covariance matricesare quite different due to heterogeneous variances, which are not accommodated by “toep.”The AICC fit index is 515.5 and the BIC fit index is 520.6 for toep, both larger than for arh(1).This indicates using the arh(1) covariance structure.

Effect on growth regulators on chrysanthemum plantsData courtesy James Barrett and Terril NellMixed Model Repeated Measures AnalysesThe next step is to use the selected covariance structure and compute inferential statistics. Forthis example, differences between the growth retardants at each time are of interest.proc mixed data=mumsuni; class blk chem plt time;model ht = chem time chem*time / ddfm=kr;repeated time / sub=plt(chem time) type=arh(1);estimate 'bonzi-sumag' chem 1 -1;estimate 'b-s time1' trt 1 -1 trt*time 1 0 0 0 0 0-1 0 0 0 0 0;estimate 'b-s time2' trt 1 -1 trt*time 0 1 0 0 0 00 -1 0 0 0 0;estimate 'b-s time3' trt 1 -1 trt*time 0 0 1 0 0 00 0 -1 0 0 0;estimate 'b-s time4' trt 1 -1 trt*time 0 0 0 1 0 00 0 0 -1 0 0;estimate 'b-s time5' trt 1 -1 trt*time 0 0 0 0 1 00 0 0 0 -1 0;estimate 'b-s time6' trt 1 -1 trt*time 0 0 0 0 0 10 0 0 0 0 -1;EstimatesLabel Estimate Standard DF t Value Pr > |t|Errorbonzi-sumag 2.4750 1.6025 18.6 1.54 0.1393b-s time1 -0.3500 0.4582 17.5 -0.76 0.4552b-s time2 -0.3000 0.4609 17.4 -0.65 0.5237b-s time3 -0.10000 1.0643 17.4 -0.09 0.9262b-s time4 2.8500 2.3636 18.3 1.21 0.2433b-s time5 5.9500 2.9759 19.6 2.00 0.0596b-s time6 6.8000 2.8513 20.5 2.38 0.0268Comparison of inferential results with ANOVA at each time:Time 1 2 3 4 5 6ANOVAChem p-value .48 .51 .93 .25 .10 .06Diff. (s.e.) -0.35(.48) -0.30(.44) -0.10(1.07) 2.85(2.36) 5.95(3.30) 6.80(3.24)MIXEDChem p-value .45 .52 .92 .24 .06 .03Diff. (s.e.) -0.35(.46) -0.30(.46) -0.10(1.06) 2.85(2.36) 5.95(2.97) 6.80(2.85)