# The MIXED Procedure

### Example 65.5 Random Coefficients

This example comes from a pharmaceutical stability data simulation performed by Obenchain (1990). The observed responses are replicate assay results, expressed in percent of label claim, at various shelf ages, expressed in months. The desired mixed model involves three batches of product that differ randomly in intercept (initial potency) and slope (degradation rate). This type of model is also known as a hierarchical or multilevel model (Singer, 1998; Sullivan, Dukes, and Losina, 1999).

The SAS statements are as follows:

data rc;
input Batch Month @@;
Monthc = Month;
do i = 1 to 6;
input Y @@;
output;
end;
datalines;
1   0  101.2 103.3 103.3 102.1 104.4 102.4
1   1   98.8  99.4  99.7  99.5    .     .
1   3   98.4  99.0  97.3  99.8    .     .
1   6  101.5 100.2 101.7 102.7    .     .
1   9   96.3  97.2  97.2  96.3    .     .
1  12   97.3  97.9  96.8  97.7  97.7  96.7
2   0  102.6 102.7 102.4 102.1 102.9 102.6
2   1   99.1  99.0  99.9 100.6    .     .
2   3  105.7 103.3 103.4 104.0    .     .
2   6  101.3 101.5 100.9 101.4    .     .
2   9   94.1  96.5  97.2 95.6     .     .
2  12   93.1  92.8  95.4 92.2   92.2  93.0
3   0  105.1 103.9 106.1 104.1 103.7 104.6
3   1  102.2 102.0 100.8  99.8    .     .
3   3  101.2 101.8 100.8 102.6    .     .
3   6  101.1 102.0 100.1 100.2    .     .
3   9  100.9  99.5 102.2 100.8    .     .
3  12   97.8  98.3  96.9  98.4  96.9  96.5
;


proc mixed data=rc;
class Batch;
model Y = Month / s;
random Int Month / type=un sub=Batch s;
run;


In the DATA step, Monthc is created as a duplicate of Month in order to enable both a continuous and a classification version of the same variable. The variable Monthc is used in a subsequent analysis

In the PROC MIXED statements, Batch is listed as the only classification variable. The fixed effect Month in the MODEL statement is not declared as a classification variable; thus it models a linear trend in time. An intercept is included as a fixed effect by default, and the S option requests that the fixed-effects parameter estimates be produced.

The two random effects are Int and Month, modeling random intercepts and slopes, respectively. Note that Intercept and Month are used as both fixed and random effects. The TYPE=UN option in the RANDOM statement specifies an unstructured covariance matrix for the random intercept and slope effects. In mixed model notation, is block diagonal with unstructured 22 blocks. Each block corresponds to a different level of Batch, which is the SUBJECT= effect. The unstructured type provides a mechanism for estimating the correlation between the random coefficients. The S option requests the production of the random-effects parameter estimates.

The results from this analysis are shown in Output 65.5.1Output 65.5.9. The "Unstructured" covariance structure in Output 65.5.1 applies to here.

Output 65.5.1: Model Information in Random Coefficients Analysis

The Mixed Procedure

Model Information
Data Set WORK.RC
Dependent Variable Y
Covariance Structure Unstructured
Subject Effect Batch
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Containment

Batch is the only classification variable in this analysis, and it has three levels (Output 65.5.2).

Output 65.5.2: Random Coefficients Analysis (continued)

Class Level Information
Class Levels Values
Batch 3 1 2 3

The "Dimensions" table in Output 65.5.3 indicates that there are three subjects (corresponding to batches). The 24 observations not used correspond to the missing values of Y in the input data set.

Output 65.5.3: Random Coefficients Analysis (continued)

Dimensions
Covariance Parameters 4
Columns in X 2
Columns in Z per Subject 2
Subjects 3
Max Obs per Subject 28

Number of Observations
Number of Observations Read 108
Number of Observations Used 84
Number of Observations Not Used 24

As Output 65.5.4 shows, only one iteration is required for convergence.

Output 65.5.4: Random Coefficients Analysis (continued)

Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 367.02768461
1 1 350.32813577 0.00000000

 Convergence criteria met.

The Estimate column in Output 65.5.5 lists the estimated elements of the unstructured 22 matrix comprising the blocks of . Note that the random coefficients are negatively correlated.

Output 65.5.5: Random Coefficients Analysis (continued)

Covariance Parameter Estimates
Cov Parm Subject Estimate
UN(1,1) Batch 0.9768
UN(2,1) Batch -0.1045
UN(2,2) Batch 0.03717
Residual   3.2932

The null model likelihood ratio test indicates a significant improvement over the null model consisting of no random effects and a homogeneous residual error (Output 65.5.6).

Output 65.5.6: Random Coefficients Analysis (continued)

Fit Statistics
-2 Res Log Likelihood 350.3
AIC (Smaller is Better) 358.3
AICC (Smaller is Better) 358.8
BIC (Smaller is Better) 354.7

Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
3 16.70 0.0008

The fixed-effects estimates represent the estimated means for the random intercept and slope, respectively (Output 65.5.7).

Output 65.5.7: Random Coefficients Analysis (continued)

Solution for Fixed Effects
Effect Estimate Standard
Error
DF t Value Pr > |t|
Intercept 102.70 0.6456 2 159.08 <.0001
Month -0.5259 0.1194 2 -4.41 0.0478

The random-effects estimates represent the estimated deviation from the mean intercept and slope for each batch (Output 65.5.8). Therefore, the intercept for the first batch is close to , while the intercepts for the other two batches are greater than 102.7. The second batch has a slope less than the mean slope of –0.526, while the other two batches have slopes greater than –0.526.

Output 65.5.8: Random Coefficients Analysis (continued)

Solution for Random Effects
Effect Batch Estimate Std Err Pred DF t Value Pr > |t|
Intercept 1 -1.0010 0.6842 78 -1.46 0.1474
Month 1 0.1287 0.1245 78 1.03 0.3047
Intercept 2 0.3934 0.6842 78 0.58 0.5669
Month 2 -0.2060 0.1245 78 -1.65 0.1021
Intercept 3 0.6076 0.6842 78 0.89 0.3772
Month 3 0.07731 0.1245 78 0.62 0.5365

The F statistic in the "Type 3 Tests of Fixed Effects" table in Output 65.5.9 is the square of the t statistic used in the test of Month in the preceding "Solution for Fixed Effects" table (compare Output 65.5.7 and Output 65.5.9). Both statistics test the null hypothesis that the slope assigned to Month equals 0, and this hypothesis can barely be rejected at the 5% level.

Output 65.5.9: Random Coefficients Analysis (continued)

Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
Month 1 2 19.41 0.0478

It is also possible to fit a random coefficients model with error terms that follow a nested structure (Fuller and Battese, 1973). The following SAS statements represent one way of doing this:

proc mixed data=rc;
class Batch Monthc;
model Y = Month / s;
random Int Month Monthc / sub=Batch s;
run;


The variable Monthc is added to the CLASS and RANDOM statements, and it models the nested errors. Note that Month and Monthc are continuous and classification versions of the same variable. Also, the TYPE=UN option is dropped from the RANDOM statement, resulting in the default variance components model instead of correlated random coefficients. The results from this analysis are shown in Output 65.5.10.

Output 65.5.10: Random Coefficients with Nested Errors Analysis

The Mixed Procedure

Model Information
Data Set WORK.RC
Dependent Variable Y
Covariance Structure Variance Components
Subject Effect Batch
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Containment

Class Level Information
Class Levels Values
Batch 3 1 2 3
Monthc 6 0 1 3 6 9 12

Dimensions
Covariance Parameters 4
Columns in X 2
Columns in Z per Subject 8
Subjects 3
Max Obs per Subject 28

Number of Observations
Number of Observations Read 108
Number of Observations Used 84
Number of Observations Not Used 24

Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 367.02768461
1 4 277.51945360 .
2 1 276.97551718 0.00104208
3 1 276.90304909 0.00003174
4 1 276.90100316 0.00000004
5 1 276.90100092 0.00000000

 Convergence criteria met.

Covariance Parameter Estimates
Cov Parm Subject Estimate
Intercept Batch 0
Month Batch 0.01243
Monthc Batch 3.7411
Residual   0.7969

For this analysis, the Newton-Raphson algorithm requires five iterations and nine likelihood evaluations to achieve convergence. The missing value in the Criterion column in iteration 1 indicates that a boundary constraint has been dropped.

The estimate for the Intercept variance component equals 0. This occurs frequently in practice and indicates that the restricted likelihood is maximized by setting this variance component equal to 0. Whenever a zero variance component estimate occurs, the following note appears in the SAS log:

NOTE: Estimated G matrix is not positive definite.


The remaining variance component estimates are positive, and the estimate corresponding to the nested errors (MONTHC) is much larger than the other two.

A comparison of AIC and BIC for this model with those of the previous model favors the nested error model (compare Output 65.5.11 and Output 65.5.6). Strictly speaking, a likelihood ratio test cannot be carried out between the two models because one is not contained in the other; however, a cautious comparison of likelihoods can be informative.

Output 65.5.11: Random Coefficients with Nested Errors Analysis (continued)

Fit Statistics
-2 Res Log Likelihood 276.9
AIC (Smaller is Better) 282.9
AICC (Smaller is Better) 283.2
BIC (Smaller is Better) 280.2

The better-fitting covariance model affects the standard errors of the fixed-effects parameter estimates more than the estimates themselves (Output 65.5.12).

Output 65.5.12: Random Coefficients with Nested Errors Analysis (continued)

Solution for Fixed Effects
Effect Estimate Standard
Error
DF t Value Pr > |t|
Intercept 102.56 0.7287 2 140.74 <.0001
Month -0.5003 0.1259 2 -3.97 0.0579

The random-effects solution provides the empirical best linear unbiased predictions (EBLUPs) for the realizations of the random intercept, slope, and nested errors (Output 65.5.13). You can use these values to compare batches and months.

Output 65.5.13: Random Coefficients with Nested Errors Analysis (continued)

Solution for Random Effects
Effect Batch Monthc Estimate Std Err Pred DF t Value Pr > |t|
Intercept 1   0 . . . .
Month 1   -0.00028 0.09268 66 -0.00 0.9976
Monthc 1 0 0.2191 0.7896 66 0.28 0.7823
Monthc 1 1 -2.5690 0.7571 66 -3.39 0.0012
Monthc 1 3 -2.3067 0.6865 66 -3.36 0.0013
Monthc 1 6 1.8726 0.7328 66 2.56 0.0129
Monthc 1 9 -1.2350 0.9300 66 -1.33 0.1888
Monthc 1 12 0.7736 1.1992 66 0.65 0.5211
Intercept 2   0 . . . .
Month 2   -0.07571 0.09268 66 -0.82 0.4169
Monthc 2 0 -0.00621 0.7896 66 -0.01 0.9938
Monthc 2 1 -2.2126 0.7571 66 -2.92 0.0048
Monthc 2 3 3.1063 0.6865 66 4.53 <.0001
Monthc 2 6 2.0649 0.7328 66 2.82 0.0064
Monthc 2 9 -1.4450 0.9300 66 -1.55 0.1250
Monthc 2 12 -2.4405 1.1992 66 -2.04 0.0459
Intercept 3   0 . . . .
Month 3   0.07600 0.09268 66 0.82 0.4152
Monthc 3 0 1.9574 0.7896 66 2.48 0.0157
Monthc 3 1 -0.8850 0.7571 66 -1.17 0.2466
Monthc 3 3 0.3006 0.6865 66 0.44 0.6629
Monthc 3 6 0.7972 0.7328 66 1.09 0.2806
Monthc 3 9 2.0059 0.9300 66 2.16 0.0347
Monthc 3 12 0.002293 1.1992 66 0.00 0.9985

Output 65.5.14: Random Coefficients with Nested Errors Analysis (continued)

Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
Month 1 2 15.78 0.0579

The test of Month is similar to that from the previous model, although it is no longer significant at the 5% level (Output 65.5.14).