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The MIXED Procedure

Example 41.3: Plotting the Likelihood

The data for this example are from Hemmerle and Hartley (1973) and are also used as an example for the VARCOMP procedure. The response variable consists of measurements from an oven experiment, and the model contains a fixed effect A and random effects B and A*B.

The SAS code is as follows: 

   data hh;
      input a b y @@;
      datalines;
   1 1 237   1 1 254    1 1 246
   1 2 178   1 2 179
   2 1 208   2 1 178    2 1 187
   2 2 146   2 2 145    2 2 141
   3 1 186   3 1 183
   3 2 142   3 2 125    3 2 136
   ;

   ods output ParmSearch=parms;
   proc mixed data=hh asycov mmeq mmeqsol covtest;
      class a b;
      model y = a / outp=predicted;
      random b a*b;
      lsmeans a;
      parms (17 to 20 by .1) (.3 to .4 by .005) (1.0);
   run;

   proc print data=predicted;
   run;

The ASYCOV option in the PROC statement requests the asymptotic variance matrix of the covariance parameter estimates. This matrix is the observed inverse Fisher information matrix, which equals 2H-1, where H is the Hessian matrix of the objective function evaluated at the final covariance parameter estimates. The MMEQ and MMEQSOL options in the PROC statement request that the mixed model equations and their solution be displayed.

The OUTP= option in the MODEL statement produces the data set predicted, containing the predicted values. Least-squares means (LSMEANS) are requested for A. The PARMS and ODS statements are used to construct a data set containing the likelihood surface.

The results from this analysis are shown in Output 41.3.1.

Output 41.3.1: Plotting the Likelihood

The Mixed Procedure

Model Information
Data Set WORK.HH
Dependent Variable y
Covariance Structure Variance Components
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Containment


The "Model Information" table lists details about this variance components model.

The Mixed Procedure

Class Level Information
Class Levels Values
a 3 1 2 3
b 2 1 2


The "Class Level Information" table lists the levels for A and B.

The Mixed Procedure

Dimensions
Covariance Parameters 3
Columns in X 4
Columns in Z 8
Subjects 1
Max Obs Per Subject 16
Observations Used 16
Observations Not Used 0
Total Observations 16


The "Dimensions" table reveals that X is 16 ×4 and Z is 16 ×8. Since there are no SUBJECT= effects, PROC MIXED considers the data effectively to be from one subject with 16 observations.

The Mixed Procedure

Parameter Search
CovP1 CovP2 CovP3 Variance Res Log Like -2 Res Log Like
17.0000 0.3000 1.0000 80.1400 -52.4699 104.9399
17.0000 0.3050 1.0000 80.0466 -52.4697 104.9393
17.0000 0.3100 1.0000 79.9545 -52.4694 104.9388
17.0000 0.3150 1.0000 79.8637 -52.4692 104.9384
17.0000 0.3200 1.0000 79.7742 -52.4691 104.9381
17.0000 0.3250 1.0000 79.6859 -52.4690 104.9379
17.0000 0.3300 1.0000 79.5988 -52.4689 104.9378
17.0000 0.3350 1.0000 79.5129 -52.4689 104.9377
17.0000 0.3400 1.0000 79.4282 -52.4689 104.9377
17.0000 0.3450 1.0000 79.3447 -52.4689 104.9378
... ... ... ... ... ...
20.0000 0.3550 1.0000 78.2003 -52.4683 104.9366
20.0000 0.3600 1.0000 78.1201 -52.4684 104.9368
20.0000 0.3650 1.0000 78.0409 -52.4685 104.9370
20.0000 0.3700 1.0000 77.9628 -52.4687 104.9373
20.0000 0.3750 1.0000 77.8857 -52.4689 104.9377
20.0000 0.3800 1.0000 77.8096 -52.4691 104.9382
20.0000 0.3850 1.0000 77.7345 -52.4693 104.9387
20.0000 0.3900 1.0000 77.6603 -52.4696 104.9392
20.0000 0.3950 1.0000 77.5871 -52.4699 104.9399
20.0000 0.4000 1.0000 77.5148 -52.4703 104.9406


Only a portion of the "Parameter Search" table is shown because the full listing has 651 rows.

The Mixed Procedure

Iteration History
Iteration Evaluations -2 Res Log Like Criterion
1 2 104.93416367 0.00000000

Convergence criteria met.


Convergence is quick because PROC MIXED starts from the best value from the grid search.

The Mixed Procedure

Covariance Parameter Estimates
Cov Parm Estimate Standard Error Z Value Pr Z
b 1464.36 2098.01 0.70 0.2426
a*b 26.9581 59.6570 0.45 0.3257
Residual 78.8426 35.3512 2.23 0.0129


The preceding table lists the variance components estimates. Note that B is much more variable than A*B.

The Mixed Procedure

Asymptotic Covariance Matrix of Estimates
Row Cov Parm CovP1 CovP2 CovP3
1 b 4401640 1.2831 -273.32
2 a*b 1.2831 3558.96 -502.84
3 Residual -273.32 -502.84 1249.71


The asymptotic covariance matrix also reflects the large variability of B relative to A*B.

The Mixed Procedure

Fit Statistics
Res Log Likelihood -52.5
Akaike's Information Criterion -55.5
Schwarz's Bayesian Criterion -53.5
-2 Res Log Likelihood 104.9

PARMS Model Likelihood Ratio
Test
DF Chi-Square Pr > ChiSq
2 0.00 1.0000


The PARMS likelihood ratio test (LRT) compares the best model from the grid search with the final fitted model. Since these models are nearly the same, the LRT is not significant.

The Mixed Procedure

Mixed Model Equations
Row Effect a b Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col10 Col11 Col12 Col13
1 Intercept     0.2029 0.06342 0.07610 0.06342 0.1015 0.1015 0.03805 0.02537 0.03805 0.03805 0.02537 0.03805 36.4143
2 a 1   0.06342 0.06342     0.03805 0.02537 0.03805 0.02537         13.8757
3 a 2   0.07610   0.07610   0.03805 0.03805     0.03805 0.03805     12.7469
4 a 3   0.06342     0.06342 0.02537 0.03805         0.02537 0.03805 9.7917
5 b   1 0.1015 0.03805 0.03805 0.02537 0.1022   0.03805   0.03805   0.02537   21.2956
6 b   2 0.1015 0.02537 0.03805 0.03805   0.1022   0.02537   0.03805   0.03805 15.1187
7 a*b 1 1 0.03805 0.03805     0.03805   0.07515           9.3477
8 a*b 1 2 0.02537 0.02537       0.02537   0.06246         4.5280
9 a*b 2 1 0.03805   0.03805   0.03805       0.07515       7.2676
10 a*b 2 2 0.03805   0.03805     0.03805       0.07515     5.4793
11 a*b 3 1 0.02537     0.02537 0.02537           0.06246   4.6802
12 a*b 3 2 0.03805     0.03805   0.03805           0.07515 5.1115


The mixed model equations are analogous to the normal equations in the standard linear model. For this example, rows 1 -4 correspond to the fixed effects, rows 5 -12 correspond to the random effects, and Col13 corresponds to the dependent variable.

The Mixed Procedure

Mixed Model Equations Solution
Row Effect a b Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col10 Col11 Col12 Col13
1 Intercept     761.84 -29.7718 -29.6578   -731.14 -733.22 -0.4680 0.4680 -0.5257 0.5257 -12.4663 -14.4918 159.61
2 a 1   -29.7718 59.5436 29.7718   -2.0764 2.0764 -14.0239 -12.9342 1.0514 -1.0514 12.9342 14.0239 53.2049
3 a 2   -29.6578 29.7718 56.2773   -1.0382 1.0382 0.4680 -0.4680 -12.9534 -14.0048 12.4663 14.4918 7.8856
4 a 3                            
5 b   1 -731.14 -2.0764 -1.0382   741.63 722.73 -4.2598 4.2598 -4.7855 4.7855 -4.2598 4.2598 26.8837
6 b   2 -733.22 2.0764 1.0382   722.73 741.63 4.2598 -4.2598 4.7855 -4.7855 4.2598 -4.2598 -26.8837
7 a*b 1 1 -0.4680 -14.0239 0.4680   -4.2598 4.2598 22.8027 4.1555 2.1570 -2.1570 1.9200 -1.9200 3.0198
8 a*b 1 2 0.4680 -12.9342 -0.4680   4.2598 -4.2598 4.1555 22.8027 -2.1570 2.1570 -1.9200 1.9200 -3.0198
9 a*b 2 1 -0.5257 1.0514 -12.9534   -4.7855 4.7855 2.1570 -2.1570 22.5560 4.4021 2.1570 -2.1570 -1.7134
10 a*b 2 2 0.5257 -1.0514 -14.0048   4.7855 -4.7855 -2.1570 2.1570 4.4021 22.5560 -2.1570 2.1570 1.7134
11 a*b 3 1 -12.4663 12.9342 12.4663   -4.2598 4.2598 1.9200 -1.9200 2.1570 -2.1570 22.8027 4.1555 -0.8115
12 a*b 3 2 -14.4918 14.0239 14.4918   4.2598 -4.2598 -1.9200 1.9200 -2.1570 2.1570 4.1555 22.8027 0.8115


This solution matrix results from sweeping all but the last row of the mixed model equations matrix. The final column contains a solution vector for the fixed and random effects. The first four rows correspond to fixed effects and the last eight to random effects.

The Mixed Procedure

Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
a 2 2 28.00 0.0345


The A factor is significant at the 5% level.

The Mixed Procedure

Least Squares Means
Effect a Estimate Standard Error DF t Value Pr > |t|
a 1 212.82 27.6014 2 7.71 0.0164
a 2 167.50 27.5463 2 6.08 0.0260
a 3 159.61 27.6014 2 5.78 0.0286


The significance of A appears to be from the difference between its first level and its other two levels.

Obs a b y Pred StdErrPred DF Alpha Lower Upper Resid
1 1 1 237 242.723 4.72563 10 0.05 232.193 253.252 -5.7228
2 1 1 254 242.723 4.72563 10 0.05 232.193 253.252 11.2772
3 1 1 246 242.723 4.72563 10 0.05 232.193 253.252 3.2772
4 1 2 178 182.916 5.52589 10 0.05 170.603 195.228 -4.9159
5 1 2 179 182.916 5.52589 10 0.05 170.603 195.228 -3.9159
6 2 1 208 192.670 4.70076 10 0.05 182.196 203.144 15.3297
7 2 1 178 192.670 4.70076 10 0.05 182.196 203.144 -14.6703
8 2 1 187 192.670 4.70076 10 0.05 182.196 203.144 -5.6703
9 2 2 146 142.330 4.70076 10 0.05 131.856 152.804 3.6703
10 2 2 145 142.330 4.70076 10 0.05 131.856 152.804 2.6703
11 2 2 141 142.330 4.70076 10 0.05 131.856 152.804 -1.3297
12 3 1 186 185.687 5.52589 10 0.05 173.374 197.999 0.3134
13 3 1 183 185.687 5.52589 10 0.05 173.374 197.999 -2.6866
14 3 2 142 133.542 4.72563 10 0.05 123.013 144.072 8.4578
15 3 2 125 133.542 4.72563 10 0.05 123.013 144.072 -8.5422
16 3 2 136 133.542 4.72563 10 0.05 123.013 144.072 2.4578


The preceding output lists the predicted values from the model. These values are the sum of the fixed effects estimates and the empirical best linear unbiased predictors (EBLUPs) of the random effects. It is often useful to plot predicted values and residuals to assess the adequacy of the model, using another SAS procedure to generate plots and diagnostic measures.

To plot the likelihood surface using the G3D procedure from SAS/GRAPH software, use the following source: 

   proc g3d data=parms;
      plot CovP1*CovP2 = ResLogLike 
           / ctop=red cbottom=blue caxis=black;
   run;
The results from this plot are shown in Output 41.3.2. The peak of the surface is the REML estimates for the B and A*B variance components.

Output 41.3.2: Plot of Likelihood Surface
mixx3k.gif (6069 bytes)

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