Chapter 1 Introduction
5.4 Analysis and Application to the RSV data
5.4.2 Adaptive and Non-adaptive Gaussian Quadra- tureture
variables are significant at the 5% significance level as well as ‘age’ tending towards significance and the difference between the age 5 and age 12 groups are significant here as well with respect to the disease process.
5.4.2 Adaptive and Non-adaptive Gaussian Quadra-
Gaussian Quadrature
Effect Q= 3 Q= 5 Q= 20
Intercept -5.04(1.488) -5.72(1.584) -5.466(1.384) beta0 -0.92(1.244) -0.94(1.862) -0.91(1.35) beta1 -0.65(1.082) -0.68(1.985) -0.69(1.857) beta2 -0.28(1.081) -0.27(1.056) -0.28(1.045) beta3 -0.07(1.001) -0.08(0.991) -0.07(0.984) beta4 -0.67(0.981) -0.69(0.967) -0.67(0.991) beta5 -2.71(1.381) -2.12(1.354) -2.74(1.311) beta6 -1.61(1.021) -1.59(1.044) -1.60(1.058) beta7 -2.23(1.184) -2.13(1.192) 2.20(1.188) beta8 0.99(0.624) -1.05(0.652) -1.01(0.652) beta9 -0.76(0.521) -0.74(0.601) -0.75(0.504) beta10 -0.33(0.456) -0.42(0.498) -0.32(0.416) beta11 -.57(0.481) -0.54(0.453) -0.57(0.485) beta13 -0.0008(0.009) 0.0009(0.007) -0.0008(0.006) beta14 47.2(7.995) 49.3(8.994) 46.1(8.774) beta15 2.31(0.168) 2.33(0.183) 2.38(0.199) beta16 -0.05(0.109) -0.04(0.137) -0.05(0.108)
τ 1.03(0.0114) 1.01(0.018) 1.03(0.013)
−2` 2243.7 2242.2 2243.9
Table 5.15: Solution for the fixed effects-Gaussian quadrature
Adaptive Gaussian Quadrature
Effect Q= 3 Q= 5 Q= 20
Intercept -5.02(1.433) -5.71(1.498) -5.786(1.354) beta0 -0.92(1.244) -0.94(1.862) -0.91(1.145) beta1 -0.65(1.012) -0.68(1.385) -0.68(1.017) beta2 -0.28(1.051) -0.23(1.034) -0.26(1.026) beta3 -0.07(0.991) -0.08(0.988) -0.07(0.999) beta4 -0.66(0.989) -0.69(0.997) -0.67(0.982) beta5 -2.61(1.341) -2.12(1.369) -2.72(1.311) beta6 -1.61(1.071) -1.59(1.022) -1.63(1.758) beta7 -2.24(1.191) -2.13(1.189) 2.20(1.198) beta8 0.99(0.674) -1.02(0.652) -1.01(0.699) beta9 -0.74(0.501) -0.74(0.561) -0.75(0.540) beta10 -0.33(0.459) -0.42(0.498) -0.32(0.446) beta11 -0.57(0.498) -0.54(0.422) -0.57(0.494) beta13 -0.0008(0.007) 0.0009(0.009) -0.0008(0.006) beta14 43.8(8.395) 49.3(8.994) 46.1(8.214) beta15 2.22(0.178) 2.23(0.183) 2.41(0.189) beta16 -0.05(0.109) -0.03(0.158) -0.05(0.104)
τ 1.03(0.0115) 1.01(0.018) 1.03(0.013)
−2` 2243.8 2242.8 2243.8
Table 5.16: Solution for the fixed effects-adaptive Gaussian quadrature In this particular case there seems not to be much differences between the adaptive and non-adaptive Gaussian quadrature estimates. The standard errors are also consistent with those from the GLIMMIX procedure. It must also be stated that there is a correct maximum to the likelihood for these models. If different quadrature methods lead to different answers, then at most one can be lading to the correct MLE. Different choices of quadrature starting values, and convergence criteria should be used until one is able to consistently obtain the same correct MLE’s. Differences among the estimates merely indicate lack of, or inappropriate, convergence. Next a generalized linear mixed model with only the two variables, ‘prev’ and ’actipass’ was
fitted since they were the only significant variables. The results are given below for different covariance structure models as shown in Tables 5.17, 5.18 and 5.19. The solution for the fixed effects for all the different covariance
Covariance Structure Estimate Standard Error
Unstructured UN(1,1) No convergence No convergence Residual(VC) No convergence No convergence Compound symmetry Var(child) 7.187E-6 2.885E-6
CS(child) -6.71E-6 .
Residual(VC) 0.8814 0.01306
Power Var(child) 4.745E-7 2.885E-6
SP(POW)(child) 0.000 0.000
Residual(VC) 0.8814 0.01306
Spherical Var(child) 4.745E-7 2.885E-6
SP(SPH)(child) 0.000 0.000
Residual(VC) 0.8814 0.01306
Gaussian Var(child) 4.745E-7 2.885E-6
SP(GAU)(child) 0.000 0.000
Residual(VC) 0.8814 0.01306
Table 5.17: Covariance parameter estimates in a random effects model-prev and actipass
structure models are:
Effect Estimate Standard Error Pr>|t|
Intercept -5.9583 0.1796 <0.0001 Prev 50.7801 4.8589 <0.0001 Actipass 0 2.0576 0.1608 <0.0001 Actipass 1 0.000 0.000 0.000
Table 5.18: Parameter estimates and standard errors of the fixed effects-using prev and actipass in a random effects model
Effect F-Value P-value Prev 109.22 <0.0001 Actipass 163.74 <0.0001 Timemonth 0.18 0.6701
Table 5.19: Type III Effects for random effects model-prev and actipass
Fitting the above model as a random intercept model yielded the following results summarized in Tables 5.20, 5.21 and 5.22:: The solution for the fixed effects for all the different covariance structure models are:
Once again the random intercept model estimates are very similar to those of the random effects estimates. The above model was also fitted as a GLMM but using PROC NLMIXED. The fitted model was:
rsvpos=β00+β0prev+β1actipass+childef f ect(τ) The sample program is:
proc nlmixed data =lisa qpoints=20 tech=nmsimp; parms beta00=-5.9 beta0=50.7 beta1=2.03 tau2=0.85;
teta=beta00+b+beta0*prev+beta1*actipass; expteta=exp(teta);
p=expteta/(1+expteta); model rsvpos~binary(p); random b~normal(0,tau2) subject=rsv; run;
The results for adaptive Gaussian quadrature and non-adaptive Gaussian quadrature are given below in Tables 5.23 and 5.24 :
The results here, not surprisingly are similar to those achieved by using Proc GLIMMIX. It is important to stress that each log-likelihood equals the
Covariance Structure Estimate Standard Error
Unstructured UN(1,1) 0.2086 0.1638
Residual(VC) 0.8782 0.01308 Compound symmetry Var(child) 0.1053 0.1638
CS(child) 0.1034 .
Residual(VC) 0.8782 0.01308
Power Var(child) 0.2086 0.1638
SP(POW)(child) 0.000 0.000 Residual(VC) 0.8782 0.01308
Spherical Var(child) 0.2086 0.1638
SP(SPH)(child) 1.000 0.000 Residual(VC) 0.8782 0.01308
Gaussian Var(child) 0.2086 0.1638
SP(GAU)(child) 1.000 0.000 Residual(VC) 0.8782 0.01308
Table 5.20: Covariance parameter estimates in a random intercept model- prev and actipass
Effect Estimate Standard Error Pr>|t|
Intercept -5.9564 0.1802 <0.0001 Prev 50.6917 4.8703 <0.0001 Actipass 0 2.0601 0.1615 <0.0001 Actipass 1 0.000 0.000 0.000
Table 5.21: Parameter estimates and standard errors of the fixed effects-using prev and actipass in a random intercept model
Effect F-Value P-value Prev 108.33 <0.0001 Actipass 163.77 <0.0001 Timemonth 0.18 0.6701
Table 5.22: Type III Effects for random intercept model-prev and actipass maximum of the approximation to the model likelihood implying that log- likelihoods corresponding to different estimation procedures and/or different
Gaussian Quadrature
Effect Q= 3 Q= 5 Q= 20
Intercept -3.9555(0.1805) -3.6407(0.1764) -3.9257(0.1790) beta0 50.2189(5.7557) 52.7442(6.1598) 50.1865(5.703) beta1 -1.9171(0.1686) -2.1389(0.1853) -1.9715(0.1685)
τ 0.8700(0.0114) 0.7300(0.018) 0.7800(0.0190)
−2` 1494.7 1458.2 1494.6
Table 5.23: Solution for the fixed effects-gaussian quadrature using prev and actipass
Adaptive Gaussian Quadrature
Effect Q= 3 Q= 5 Q= 20
Intercept -3.9515(0.1807) -2.8854(0.1482) -3.9243(0.1789) beta0 50.5363(5.7469) 49.8533(6.1178) 50.1887(5.700) beta1 -1.9558(0.1696) -1.9917(0.1673) -1.9717(0.1684)
τ 0.8700(0.012) 0.8590(0.0114) 0.8700(0.0119)
−2` 1494.7 1420 1494.6
Table 5.24: Solution for the fixed effects-adaptive gaussian quadrature using prev and actipass
number of quadrature points are not necessarily comparable. This means that difference in log-likelihood values reflect the differences in the quality of the numerical approximations and thus higher log-likelihood values do not necessarily correspond to better approximations.
The random intercept and random effects models differed slightly in their covariance structure estimated but not in their parameter estimates, this can be expected. The adaptive and non-adaptive Gaussian results were similar to each other and in the current scientific setting the PROC GLIMMIX and PROC NLMIXED results are similar to each other. In the context of the RSV data, the ‘prev’ and ‘actipass’ variables are significant at the 5% level and are influential in contributing to a child’s RSV status. The ‘age’ variable
was tending to significance at the 5% level. There are significant differences in the ‘age 5 versus age 12’and ‘age 7 versus age 12’ groups at the 5% level.