A measurement (distance in millimeters from the center of the pituitary to the pterygo-maxillary fissure) was made on each of 11 girls and 16 boys at ages 8, 10, 12 and 14 years.
Note that these data are perfectly balanced and complete, as can be seen in Figure A.1.
Figure A.1 also illustrates that the distance from the pituitary to the pterygomaxillary fissure increases with age and seems bigger for boys. In addition, the growth of this measurement seems heterogeneous between children. Zheng (2000) analyzed these data considering 5 different models containing a random intercept and 3 different models con-taining a random intercept and a random slope for age. We fit the same models as Zheng (2000) to these data and numbered them from 1 to 8, from the simplest to the most complex model. Our results for rc, R2V C, Drand, Prand, c, r2, R2 and ρ2 are the same as those published by Zheng (2000) and by Orelien and Edwards (2008).
Fori= 1, . . . ,27, j = 1, . . . ,4, our model 8 is yij ∼ N(αi+βiageij, σy2) αi
βi
!
∼ N
γ0+γ1sexi δ0+δ1sexi
!
, σα2 ρσασβ ρσασβ σ2β
!!
.
Its fixed part contains the predictors age, sex and the cross-level interaction between age and sex. And its random part contains a random intercept and a random slope for age.
The other models are obtained by restriction on model 8:
Model 1 contains only a fixed intercept and a random intercept (γ1 = 0; βi = 0 ∀i).
Model 2 additionally contains, compared with model 1, the predictor age (γ1 = 0;
βi =β ∀i).
Age (yr)
Distance from pituitary to pterygomaxillary fissure (mm)
16 18 20 22 24 26 28
8 910 12 14 F10
8 910 12 14 F09
8 910 12 14 F06
8 910 12 14 F01
8 910 12 14 F05
8 910 12 14 F07
8 910 12 14 F02
8 910 12 14 F08
8 910 12 14 F03
8 910 12 14 F04
8 910 12 14 F11
Age (yr)
Distance from pituitary to pterygomaxillary fissure (mm)
20 25 30
8 9 10 12 14
M16 M05
8 9 10 12 14
M02 M11
8 9 10 12 14
M07 M08
8 9 10 12 14
M03 M12
M13
8 9 10 12 14
M14 M09
8 9 10 12 14
M15 M06
8 9 10 12 14
M04 M01
8 9 10 12 14
20 25 30 M10
Figure A.1: Orthodontic data for the 11 girls (F for female) at the top and for the 16 boys (M for male) at the bottom.
Model 3 additionally contains, compared with model 1, the predictor sex (βi = 0 ∀i).
Model 4 additionally contains, compared with model 2, the predictor sex (βi =β ∀i).
Model 5 additionally contains, compared with model 4, the cross-level interaction (βi = δ0+δ1sexi ∀i).
Model 6 corresponds to model 8 without the predictor sex and the cross-level interaction (γ1 =δ1 = 0).
Model 7 corresponds to model 8 without the cross-level interaction (δ1 = 0).
The results are presented in Table A.1. For the measures requiring a null model, we additionally fit the null model 0 that contains only a fixed intercept.
The measures evaluating model adequacy (categories A and A&S), except the measures λy, R2α, λα, R2β and λβ, give similar results and bigger values are observed for models 6, 7 and 8. The random slope hence improves the adjustment of the model to the data.
Furthermore, the proportion of variation in the data explained by the model is higher for models including the predictor age (models 2 and 4 to 8). Additionally, the measures of Gelman and Pardoe (2006) summarize the explained variation and the pooling at each
“level” of the considered models. For the first “level,” the pooling factor λy is close to zero when within-group sample sizes are reasonably large. In this example, there are only four repeated measures, which seem too few, as λy is not close to zero. For the second
“level,”R2α is equal to zero for models 1, 2 and 6, as the predictor sex is not included. For models 3, 4, 5, 7 and 8,R2αmeasures the proportion of the variation among children that is explained by the predictor sex, and this proportion is highest for model 3, which does not contain the predictor age. Thus, when allowing for the inclusion of the predictor age, R2α is smaller, indicating that there is less variation among children to explain. The pooling factor λα is less than 0.5 when no random slope is included in the model, meaning that there is more within-child information than population-level information. In other words, the estimates of γ0 and γ1 are closer to the 27 predictions ˆαi obtained by least squares for the no-pooling modelαi =γ0+γ1sexi+ei than to the common estimates obtained for the complete-pooling model. In contrast, λα is bigger than 0.5 when a random slope for age is included in the model, indicating that the population-level information is higher than the within-child information. In other words, the children’s mean estimates are pooled by about 60% towards the regression line predicting the children’s sample means from sex.
For the third “level,” Rβ2 is equal to zero for models 6 and 7, as sex is not included at that “level.” For model 8, R2β indicates that 66% of the variation in the age effects across children is explained by sex. The pooling factor λβ is bigger than 0.5 for models 6, 7 and 8, indicating that the children’s slope estimates are pooled around 60%, 70% and 80%, respectively, towards the regression line predicting the children’s sample slopes from sex.
The measures in category F&S underscore the relevance of age, sex and their inter-action. However, the difference between the models containing age and sex and those containing additionally the interaction is quite small, and, on grounds of parsimony, we should not include the interaction, as advised by Orelien and Edwards (2008). The neg-ative value of R22 for model 6 indicates a misspecification of the fixed part of the model, which suggests the inclusion of the predictor sex.
For the comparison of the fixed effects structure, we perform a LRT built on -2LL ML.
Given that age and sex appear to be necessary, we compare models 4 and 5, and also models 7 and 8 to test the need of the inclusion of the interaction between age and
Table A.1: Measures for models of children’s orthodontic data. For the measures com-puted with several null models, (0) indicates that the considered null model is model 0 and (1) indicates that the considered null model is model 1. The dashed line separates null models. m.=marginal; c.=conditional.
Model
Measure Category 0 1 2 3 4 5 6 7 8
Ry2 A 0.416 0.759 0.416 0.759 0.771 0.781 0.775 0.780
Rα2 A 0 0 0.337 0.270 0.152 0 0.088 0.213
Rβ2 A 0 0 0.657
λy A 0.192 0.233 0.175 0.228 0.246 0.288 0.290 0.268
λα A 0.264 0.106 0.374 0.180 0.173 0.692 0.675 0.553
λβ A 0.588 0.737 0.811
Drand A 0.530 0.815 0.521 0.814 0.828 0.861 0.863 0.859
c A 0.784 0.900 0.777 0.896 0.899 0.916 0.917 0.912
R2(0) A 0.530 0.815 0.521 0.814 0.828 0.861 0.863 0.859
ρ2(0) A 0.520 0.809 0.513 0.808 0.822 0.852 0.853 0.851
R2(1) A 0.607 -0.020 0.603 0.633 0.704 0.708 0.701
ρ2(1) A 0.602 -0.016 0.599 0.629 0.692 0.694 0.689
RT2 A 0.530 0.815 0.521 0.814 0.828 0.861 0.863 0.859
c. rc,a A&S 0.641 0.890 0.636 0.888 0.897 0.919 0.919 0.917
c. R2V C,a (0) A&S 0.526 0.812 0.512 0.808 0.821 0.858 0.859 0.854
c. R2V C,a (1) A&S 0.600 -0.039 0.592 0.619 0.699 0.700 0.689
Prand A&S 0.425 0.763 0.431 0.765 0.782 0.802 0.804 0.805
r2 (0) A&S 0.425 0.761 0.425 0.761 0.776 0.800 0.800 0.800
r2 (1) A&S 0.584 0.000 0.584 0.610 0.652 0.652 0.652
RTF,a2 A&S 0.420 0.759 0.420 0.759 0.774 0.798 0.798 0.798
R12 F&S 0.249 0.139 0.388 0.399 0.238 0.376 0.391
R22 F&S 0.000 0.242 0.242 0.242 -0.017 0.223 0.231
m. rc,a F&S -0.009 0.397 0.252 0.569 0.577 0.397 0.561 0.579
m. R2V C,a F&S -0.009 0.242 0.137 0.393 0.400 0.242 0.372 0.400
m. Drand F&S 0.000 0.256 0.153 0.409 0.423 0.256 0.409 0.423
m. Prand F&S 0.000 0.256 0.153 0.409 0.423 0.256 0.409 0.423
m. c F&S 0.5 0.682 0.620 0.738 0.734 0.682 0.738 0.734
m. R2 F&S 0.000 0.256 0.153 0.409 0.423 0.256 0.409 0.423
RF,a2 F&S -0.019 0.235 0.129 0.387 0.395 0.228 0.380 0.389
-2LL ML S 537.582 515.491 443.389 506.958 434.856 428.639 439.212 432.835 427.806
-2LL REML S 515.362 447.002 505.872 437.512 433.757 442.637 435.234 432.582
mAIC S 541.582 521.491 451.389 514.958 444.856 440.639 451.212 446.835 443.806
BIC S 546.946 529.537 462.118 525.687 458.267 456.732 467.304 465.610 465.263
cAIC ML S 502.562 411.503 501.325 410.996 404.902 406.255 407.274 406.179
cAIC REML S 502.502 411.734 501.135 411.226 405.468 405.986 406.893 405.506
DIC S 541.666 502.778 411.752 434.245 246.509 378.165 406.237 282.316 227.903
sex. With a significance level of 5%, we choose the models containing age, sex and their interaction (Compare models 4 and 5: LRT of 6.217 for 1 df,p= 0.013; Compare models 7 and 8: LRT of 5.029 for 1 df, p = 0.025). To test whether the model should include a random slope for age, we can consider either -2LL ML or -2LL REML to perform a LRT (cf. Section 1.3.2.1). Based on the results obtained with -2LL ML, we prefer the models without random slope (Compare models 2 and 6: LRT of 4.177 for 2 df,p= 0.062;
Compare models 4 and 7: LRT of 2.021 for 2 df,p= 0.182; Compare models 5 and 8: LRT of 0.833 for 2 df,p= 0.330). The results with -2LL REML give rise to the same conclusion.
For the other measures of model selection (category S), the information criteria, the most appropriate model is the one giving rise to the smallest value. According to the DIC, model 8 appears as the most appropriate. Similar values are obtained for models 4, 5, 7 and 8 with the mAIC, for models 4 and 5 with the BIC and for models 5 to 8 with the cAIC for the ML and REML estimations. Thus, selecting the most appropriate model depends on the measure considered, but predictors age and sex seem necessary predictors.
To summarize, the measures evaluating model adequacy show the improvement of the adjustment of the model to the data if a random slope for age is included; the indices comparing the fixed part of the model favor the inclusion of age and sex; the LRT points to the relevance of the inclusion of the interaction between age and sex and of the non-inclusion of a random slope; and with the information criteria, we observe that it is difficult to decide between models 4 to 8. Following the recommendations given in Section 1.6 (discussion of Chapter 1), we would consider the cAIC REML, the marginal Prand and Prandto respectively select the most appropriate model, select the best set of fixed effects, and evaluate the adjustment of the model at hand. We would thus conclude that models 5 to 8 are equivalently appropriate considering the cAIC REML, and that age and sex should be included in the model considering the marginalPrand. Combining these results, model 7 seems the most appropriate for these data and considering Prand, this model reduces by 80% the PQL compared to the null model containing a fixed intercept. In the end, according to model 7, the distance in millimeters from the center of the pituitary to the pterygomaxillary fissure increases with age and is bigger for boys. Moreover, the children are heterogeneous with respect to their average value for that distance at age 8 years and also with respect to the growth of that measurement between ages 8 and 14 years.