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 The MIXED Procedure

## Example 41.5: 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 code is 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 allow both a continuous and classification version of the same variable. The variable Monthc is used in a subsequent analysis.

In the PROC MIXED code, Batch is listed as the only classification variable. The fixed effect Month in the MODEL statement is not declared 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, G is block diagonal with unstructured 2×2 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 41.5.1.

Output 41.5.1: 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

The "Unstructured" covariance structure applies to G here.

 The Mixed Procedure

 Class Level Information Class Levels Values Batch 3 1 2 3

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

 The Mixed Procedure

 Dimensions Covariance Parameters 4 Columns in X 2 Columns in Z Per Subject 2 Subjects 3 Max Obs Per Subject 36 Observations Used 84 Observations Not Used 24 Total Observations 108

The "Dimensions" table 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.

 The Mixed Procedure

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

 Convergence criteria met.

Only one iteration is required for convergence.

 The Mixed Procedure

 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 estimated elements of the unstructured 2×2 matrix comprising the blocks of G are listed in the Estimate column. Note that the random coefficients are negatively correlated.

 The Mixed Procedure

 Fit Statistics Res Log Likelihood -175.2 Akaike's Information Criterion -179.2 Schwarz's Bayesian Criterion -177.4 -2 Res Log Likelihood 350.3

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

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

 The Mixed Procedure

 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 fixed effects estimates represent the estimated means for the random intercept and slope, respectively.

 The Mixed Procedure

 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 random effects estimates represent the estimated deviation from the mean intercept and slope for each batch. Therefore, the intercept for the first batch is close to 102.7 - 1 = 101.7, 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 larger than -0.526.

 The Mixed Procedure

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

The F-statistic in the "Type 3 Tests of Fixed Effects" table is the square of the t-statistic used in the test of Month in the preceding "Solution for Fixed Effects" table. 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.

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 code represents 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 41.5.2.

Output 41.5.2: 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 36 Observations Used 84 Observations Not Used 24 Total Observations 108

 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.

 The Mixed Procedure

 Fit Statistics Res Log Likelihood -138.5 Akaike's Information Criterion -141.5 Schwarz's Bayesian Criterion -140.1 -2 Res Log Likelihood 276.9

A comparison of AIC (-141.5) and SBC (-140.1) for this model with those of the previous model (-179.2 and -177.4, respectively) favors the nested error model. 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.

 The Mixed Procedure

 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 better-fitting covariance model impacts the standard errors of the fixed effects parameter estimates more than the estimates themselves.

 The Mixed Procedure

 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

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

 The Mixed Procedure

 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.

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