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 fixedeffects 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 randomeffects parameter estimates.
The results from this analysis are shown in Output 41.5.1.
Output 41.5.1: Random Coefficients Analysis
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 
ModelBased 
Degrees of Freedom Method 
Containment 

The "Unstructured" covariance structure applies to G
here.
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.
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.
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.
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.
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 Ratio Test 
DF 
ChiSquare 
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.
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.
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.
Type 3 Tests of Fixed Effects 
Effect 
Num DF 
Den DF 
F Value 
Pr > F 
Month 
1 
2 
19.41 
0.0478 

The Fstatistic in the "Type 3 Tests of Fixed Effects"
table is the square of the tstatistic 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
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 
ModelBased 
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 NewtonRaphson 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.
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.
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 betterfitting covariance model impacts the standard
errors of the fixed effects parameter estimates more than
the estimates themselves.
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.
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.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.