Chapter 1 Introduction
4.7 Some developmental notes on GEE over timetime
4.7 Some developmental notes on GEE over
longitudinal data.
4. An alternative to GEE is the alternating logistic regressions (ALR) proposed by Carey, Zeger and Diggle (1993), but not of interest in the current work.
5. Le Cessie and Van Houwelingen (1994) suggested an approximation to the true likelihood by means of a pseudo-likelihood (PL) function that is easier to evaluate and to maximize. Both GEE and PL give con- sistent and asymptotically normal estimators provided an empirically corrected variance estimator which we have called the sandwich esti- mator is used. GEE is well suited only to marginal models while PL can be used for marginal models (Geys, Molenberghs and Lipsitz, 1998) and conditional models (Geys, Molenberghs and Ryan, 1997, 1999).
6. Wang and Lin (2005) investigate the impacts of misspecifing the vari- ance function which is known to be a function of the mean. They state that in the framework of GEE, the correct specification of the variance function can improve the estimation efficiency even if the correlation structure is misspecified. However misspecification of the variance func- tion impacts much more on the estimators for within cluster covariates than for cluster level covariates and also if the variance function is misspecified, the correct choice of the correlation structure may not necessarily improve estimation efficiency.
7. Mainstream statistical software packages such as SAS (PROC GEN- MOD), STATA(XTGEE command) and GENSTAT has the methodol- ogy of the GEE described above in-built .
4.7.1 Application of fitting GEE models to the RSV data set
A series of various models were fitted using the ‘Proc Genmod’ procedure in SAS by changing the correlation structure within individual responses and then assessing the main effects. The model that was first fitted included all the main effects terms. Only those terms that were found to be significant were retained with the suitable correlation structure. The main effects terms that we consider are: age, dt, prev, actipass and timemonth. These variables were described in detail in Chapter 1. All the interaction terms were assessed by sequentially adding them to the full model of main effects one at a time and then assessing the p-values of the Wald test of the model but none of the interaction terms were found to be significant. Hence they are not reported here. The results are summarized below.
Exchangeable Independent AR(1)
Parameter Est. Std. Error Pr>|Z| Est. Std. Error Pr>|Z| Est. Std. Error Pr>|Z|
Intercept -5.0329 1.4454 0.001 -5.0363 1.4479 0.001 -5.0337 1.458 0.001
age 0 -0.9253 1.2175 0.447 -0.9197 1.2194 0.451 -0.9261 1.2343 0.453
age 1 -0.6499 1.0714 0.544 -0.647 1.0736 0.547 -0.6011 1.0824 0.579
age 2 -0.2792 1.0337 0.787 -0.276 1.0356 0.790 -0.241 1.0437 0.817
age 3 -0.0714 0.9744 0.942 -0.0689 0.9759 0.944 -0.0305 0.9835 0.975
age 4 -0.6709 0.9491 0.480 -0.669 0.9502 0.481 -0.6499 0.9579 0.498
age 5 -2.6057 1.298 0.045 -2.6025 1.2979 0.045 -2.5411 1.2919 0.049
age 6 -1.5989 1.0086 0.113 -1.596 1.0087 0.114 -1.566 1.0105 0.121
age 7 -2.2518 1.1538 0.051 -2.25 1.1538 0.051 -2.2603 1.1692 0.053
age 8 -1 0.5944 0.093 -0.9989 0.5946 0.093 -0.96 0.5969 0.108
age 9 -0.7399 0.5124 0.149 -0.7389 0.5126 0.149 -0.7361 0.5189 0.156
age 10 -0.3234 0.4528 0.475 -0.3221 0.4528 0.477 -0.2992 0.4574 0.513
age 11 -0.5684 0.4612 0.218 -0.5685 0.4612 0.218 -0.5365 0.4637 0.247
age 12 0.000 0.000 . 0.000 0.000 . 0.000 0.000 .
dt 0.0008 0.0084 0.919 0.0009 0.0084 0.919 0.0014 0.0082 0.866
prev 44.6065 8.1063 < .0001 44.5942 8.1055 < .0001 43.8948 8.1214 < .0001
timemonth -0.0457 0.1044 0.662 -0.0454 0.1046 0.664 -0.0437 0.1053 0.678
actipass 0 2.2345 0.1768 < .0001 2.2341 0.1769 < .0001 2.2049 0.1759 < .0001
actipass 1 0.000 0.000 . 0.000 0.000 . 0.000 0.000 .
Table 4.1: Model based standard errors and estimates GEE
Exchangeable Independent AR(1)
Parameter Est. Std. Error Pr>|Z| Est. Std. Error Pr>|Z| Est. Std. Error Pr>|Z|
Intercept -5.033 1.165 < .0001 -5.036 1.165 < .0001 -5.034 1.161 < .0001
age 0 -0.925 1.230 0.452 -0.920 1.229 0.454 -0.926 1.226 0.450
age 1 -0.650 0.906 0.473 -0.647 0.906 0.475 -0.601 0.902 0.505
age 2 -0.279 0.858 0.745 -0.276 0.858 0.748 -0.241 0.857 0.779
age 3 -0.071 0.801 0.929 -0.069 0.801 0.932 -0.031 0.800 0.970
age 4 -0.671 0.746 0.368 -0.669 0.746 0.370 -0.650 0.749 0.385
age 5 -2.606 1.194 0.029 -2.603 1.194 0.029 -2.541 1.172 0.030
age 6 -1.599 0.869 0.066 -1.596 0.869 0.066 -1.566 0.860 0.069
age 7 -2.252 1.040 0.030 -2.250 1.039 0.030 -2.260 1.033 0.029
age 8 -1.000 0.606 0.099 -0.999 0.607 0.100 -0.960 0.606 0.113
age 9 -0.740 0.561 0.187 -0.739 0.561 0.188 -0.736 0.554 0.184
age 10 -0.323 0.473 0.494 -0.322 0.473 0.496 -0.299 0.473 0.527
age 11 -0.568 0.443 0.199 -0.569 0.443 0.199 -0.537 0.447 0.231
age 12 0.000 0.000 . 0.000 0.000 . 0.000 0.000 .
dt 0.001 0.011 0.937 0.001 0.011 0.937 0.001 0.010 0.893
prev 44.607 6.554 < .0001 44.594 6.552 < .0001 43.895 6.527 < .0001
timemonth -0.046 0.085 0.589 -0.045 0.085 0.592 -0.044 0.084 0.603
actipass 0 2.235 0.181 < .0001 2.234 0.181 < .0001 2.205 0.178 < .0001
actipass 1 0.000 0.000 . 0.000 0.000 . 0.000 0.000 .
Table 4.2: Empirical based standard errors and estimates GEE
The algorithm for the unstructured correlation matrix option did not con- verge and the results are omitted. The results of the model based estimates and standard errors are not very different between the three correlation struc- tures. The magnitude of the estimates are somewhat similar. Moreover, we see that the model based and the empirical parameter estimates are not very
different in magnitude. This is a feature of GEE because the choice between naive and empirical only affects the estimation of the covariance matrix of the regression parameter β. The output for the correlation between two re- peated measurement for the exchangeable correlation matrix was found to be −0.00035. A possible reason why the unstructured correlation matrix did not produce convergence is because the observations can not be aligned that is the observations were not equally spaced. Table 4.1 and 4.2 shows that for the model and empirical based estimates that at the 5% significance level there were significant differences between age group 5 relative to age group 12 and mildly between age group 7 relative to age group 12 in deter- mining whether a child is infected or not. The variables prevalence (prev) and type of sampling (actipass), whether a child was actively or passively sampled (actipass 0 versus actipass 1) were both significant at the 5% level in influencing whether a child is infected or not. The full results are tab- ulated in Tables 4.1 and 4.2 for the types of standard errors and the three correlation structures. It is also worthwhile noting that the exchangeable and independent correlation structures have their empirical standard errors slightly closer to the model based standard errors than the AR(1) correlation structure. The estimated GEE correlation matrices are all essentially inde- pendent, so we expect to see no appreciable differences among the columns of Table 4.1 and 4.2. It is however interesting that the sandwich estimator appears to be picking up dependence not captured by the working correla- tion matrices given the estimated correlation parameters. It is necessary to reiterate that the unstructured correlation matrix is found to be unsuitable in this scientific setting and is dropped.
Correlation Type Source DF Chi-Square Pr> Chi-Sq
Exchangeable age 12 30.39 0.0024
dt 1 0.01 0.9379
prev 1 23.32 < .0001
timemonth 1 0.3 0.5860
actipass 1 61.86 <.0001
Independent age 12 30.39 0.0024
dt 1 0.01 0.9378
prev 1 23.32 < .0001
timemonth 1 0.29 0.5882
actipass 1 61.81 <.0001
AR(1) age 12 30.54 0.0023
dt 1 0.02 0.8974
prev 1 22.94 < .0001
timemonth 1 0.27 0.6008
actipass 1 62.00 <.0001 Table 4.3: Score statistics for Type III GEE
The type III score statistics show that the age, prev and actipass variables to be significant at the 5% level in all three correlation structures. The magnitude of the estimates do not differ by vast amounts from each other in the three correlation structures.