The classic growth models fit so far place a common, but sometimes unre- alistic, assumption on the behavior of the r i j , the within-person residuals over time. Were we to fit a model in which only the intercepts varied across persons (which we will do in a minute), we would be assuming a compound symmetric error covariance matrix for each person. When we fit a model in which the slopes vary as well, we introduce heteroscedasticity into this error covariance matrix (which can be seen through the inclusion of the effect of TIME in the random portion of the model in Equation 6 . 7 ~ ) .
How realistic are such assumptions? One of the strengths of PROC MIXED is that it allows you to compare different structures for the er- ror covariance matrix. Instead of the intercepts and slopes as outcomes model in Equations 6.7a to 6.7c, consider the following simpler model for observations over time:
where
SAS PROC M I X E D 159 In this model, the intercepts and growth rates are assumed to be constant across people. But, the model introduces a different type of complexity: the residual observations within persons (after controlling for the linear effect of TIME) are correlated through the within-person error variance-covariance matrix C. By considering alternative structures for C (that ideally derive from theory), and by comparing the goodness of fit of resulting models, you can determine what type of structure is most appropriate for the data at hand.
Many different types of error-covariance structures are possible. If there are only three waves of data, it is worth exploring only a few of these pos- sibilities because there are so few data for each person. With additional observations per person (in this example we have four), additional struc- tures for the C matrix (called the R matrix in the language of PROC MIXED) are possible. The interested reader is referred to pages 92-102 in the SASSAS System for Mixed Models, the PROC MIXED documentation, and the helpful paper by Wolfinger (1996) devoted to this topic.
The structure of the within-person error covariance matrix is specified using a REPEATED statement. To fit the model in Equation 6.15 under the assumption that C is compound symmetric we write:
proc mixed noclprint covtest noitprint;
class id wave;
model y = time/s notest;
repeated wave/subject=id type=cs r;
run ;
Notice that I have added a second CLASS variable (WAVE) to indicate the time-structured nature of the data within person and I have used WAVE on the REPEATED statement. By including WAVE on the CLASSification statement, SAS treats this variable as a series of dummies. By not includ- ing TIME on the CLASSification statement, SAS treats this predictor as continuous. In this representation, then, we have two ways of including the temporal information.
The REPEATED statement uses syntax similar to the RANDOM state- ment to specify the structure of the error variance-covariance matrix. How- ever, the matrix being described exists at level-l, not level-2. The variable specified on the REPEATED statement must be categorical (although it need not be equal interval). The SUBJECT=ID option continues t o tell SAS that the matrix should have separate blocks, one for each subject.
Once again, the TYPE = option is crucial, for it specifies the form of each of the within-person error variance-covariance submatrices. However, when using a REPEATED statement, we do not necessarily routinely invoke the unstructured specification usually used on the RANDOM statement. Al- though the UNstructured option is still viable, other possibilities include the compound symmetry specification (CS) shown here and AR(1) for au- toregressive with a lag of 1. Indeed, the possibility of alternative structures
Assumption Compound symmetry
A R P ) Unstructured
N parameters AIC SBC -2RLL 2 -652.17 -655.10 1300.34 2 -636.73 -641.66 1273.47 10 -641.71 -656.35 1263.42 How do we decide which model to adopt? Although a smaller -2RLL statistic indicates better fit, improvement often requires the use of addi- tional parameters. This is where the AIC and SBC statistics come in. Both take the log-likelihood (half of the negated -2RLL) and penalize it for the number of parameters estimated, with the SBC exacting a higher penalty for increased complexity. The larger the AIC and SBC statistics, the bet- ter the fit. (Note that, when their values are negative, as they are here, lower numbers in absolute values are preferred.) Although there are no for- mal tests for comparing these statistics across models, Raftery (1995) offers some rough rules of thumb for differences in SBC: 0 t o 2 suggests ‘Lweak”
evidence of a better model; 2 to 6 suggests “positive” evidence; 6 to 10 sug- gests “strong” evidence; and any difference in excess of 10 suggests “very strong” evidence. Readers interested in learning more comparing models using these statistics should consult Raftery (1995).
What do we find when we compare error structures for the opposites naming data? From the perspective of the -2RLL statistic, the totally unstructured C yields the smallest value. The estimated variance covariance matrix from this model is:
8f 8 1 2 8 1 3 8 1 4 1,308 977 921
977
921 1,120 1,018 1,018 1,289 1,081 ”) (6.16)
But, this superior value of -2RLL requires 10 parameters, many more than we might need given the structure of the estimated variance covariance ma- trix. In examining this matrix, notice that it has similar variances along the diagonal and the off-diagonal elements decrease for covariances between errors that are further spaced in time. This type of structure is exactly that
6 4 1 6 4 2 8 4 3 564 856 1,081 1,415
SAS PROC MIXED 161
Table 6.6
Results of Assuming Compound Symmetry for the Within-Person Covariance Structure
Estimated R Matrix for ID 1
Row COll c012 Co13 ‘2014
1 1280.71 904.81 904.81 904.81 2 904.81 1280.71 904.81 904.81 3 904.81 904.81 1280.71 904.81 4 904.81 904.81 904.81 1280.71 Covariance Parameter Estimates
Cov Parm Subject Estimate
cs ID 904.81
Residual 375.90
Fitting Information Res Log Likelihood
Akaike’s Information Criterion Schwarz’s Bayesian Criterion
- 2 Res Log Likelihood
Solution for Fixed Effects
Standard Effect Estimate Error Intercept 164.37 5.7766
Time 26.9600 1.4656
Standard Z
Error Value Pr Z 0.0002 52.1281 7.21 < .OOOl
242.59 3.73
-650.2 -652.2 -653.7 1300.3
DF t Value Pr > It1 34 28.45 <. 0001 104 18.40 <. 0001
specified by the lagged autoregressive structure. As a result, it is not sur- prising that the AR(1) model, which uses only two parameters, yields AIC and SBC statistics that are considerably superior than the totally unstruc- tured model even though the -2RLL statistic is actually larger (worse). The AR(1) model estimates C to be:
1,324 1,092 901
1,092 1,324 1,092 901 1,092 1,324 1,092 ”)
743 901 1,092 1,324
which is quite similar t o the unstructured estimate, but with only two parameters, c2 and p. These analyses suggest that the AR(1) structure provides a better fit t o the data. Were this an actual analysis, however, we would also consider alternative structures before stopping a t this conclu- sion.
Having established a method for specifying the structure of the within- person error covariance matrix, we may now consider what happens when we combine this specification with the intercepts and slopes as outcomes specification considered earlier. We allow the intercepts and slopes t o vary across people by writing:
p r o c mixed n o c l p r i n t c o v t e s t ; c l a s s i d wave;
model y = t i m e c c o v a r t i m e * c c o v a r / s o l u t i o n n o t e s t ddfm=bw;
random i n t e r c e p t t i m e /type=un s u b = i d g ; r e p e a t e d w a v e / t y p e = a r ( l ) s u b j e c t = i d r ; r u n ;
which yields the output shown in Table 6.7. Notice that we have added the option “g” t o the RANDOM statement, asking SAS to print the variance covariance matrix for the random effects in matrix form (in addition t o list form).
When interpreting this output, it is useful t o compare it with the sim- pler model in Table 6.5, which included the covariate and random effects for the intercepts and slopes, but which imposed no additional structure on the error covariance matrix (beyond the heteroscedastic structure of the intercepts and slopes as outcomes model). When we make these compar- isons, all signs point toward the conclusion that we do not need t o add the extra complexity of the autoregressive error structure, once the covariate has been taken i n t o account. I emphasize this last phrase because the error covariance structure within persons describes the behavior of the errors-in other words, what is “left over” after removing the other fixed and random effects in the model. In this instance, and in m a n y others, the autoregres- sive structure is no longer needed after other fixed and random effects are taken into account.
SAS’ PROC M I X E D 163 Table 6.7
Results of Assuming an Autoregressive Within-Person Covariance Matrix and Including a Person Level Covariate
Estimated R Matrix Row COll
1 141.37 2 -19.3631 3 2.6522 4 -0.3633 Estimated G Matrix
Row Effect 1 Intercept 2 time
for ID 1
c012 Co13 Co14
-19.3631 2.6522 -0.3633 141.37 -19.3631 2.6522 -19.3631 141.37 -19.3631 2.6522 -19.3631 141.37
ID COll c012
1 1258.10 -182.41 1 -182.41 110.94 Covariance Parameter Estimates
Standard Cov Parm Subject Estimate Error UN(1,l) ID 1258.10 333.25 UN(2.1) ID -182.41 84.5520 uN(2.2) ID 110.94 34.5299
AR(1) ID -0.1370 0.2589
Residual 141.37 36.3449
Z
Value Pr Z 3.78 < .OOOl -2.16 0.0310 3.21 0.0007 -0.53 0.5968
3.89 < .OOOl Fitting Information
Res Log Likelihood -630.0 Akaike’s Information Criterion -635.0 Schwarz’s Bayesian Criterion -638.9 -2 Res Log Likelihood 1260.0 Solution for Fixed Effects
Standard
Effect Est imate Error DF t Value Pr > It1 Intercept 164.42 6.1990 33 26.52 < . 0001 Time 26.9082 1.9775 103 13.61 < . 0001 Ccovar -0.1234 0.5034 33 -0.25 0.8079 Time*ccovar 0.4357 0.1606 103 2.71 0.0078
What evidence am I using t o reach this conclusion? First, consider the covariance estimate for the autoregressive parameter. We are unable t o reject the null hypothesis that this estimate, -0.14, could have been obtained from a population in which its true value was 0. Thus, there is little supporting evidence t o increase the complexity of C by adding off- diagonal elements. Second, when comparing the two models that include the covariate and its interaction with time, differing only in the inclusion of the autoregressive parameter, the -2RLL statistic improves only trivially, from -630.1 without this assumption t o -630.0 with this assumption. This improvement is so small that the AIC and SBC, which both penalize for the additional parameter, actually get worse. Therefore, despite the fact that there appears t o be an autoregressive error structure when the covariate is not included and the slopes are not treated as random, the need for this additional structure disappears when these features are added t o the model.
As this example shows, a range of models can be fit to the same data.
Experienced data analysts know that selecting among competing models can be tricky, especially when the number of observations per person is small. Were we conducting this analysis t o reach substantive conclusions about the relationship between the outcome and predictors, we would fit several additional models t o these data, including one with an AR(1) error covariance matrix and random intercepts. Readers interested in learning more about specifying the error covariance matrix and comparing results across models should consult VanLeeuwen (1997); Goldstein, Healy, and Rasbash (1994); and Wolfinger (1993, 1996).