Conditional independence of multivariate binary data with an application in caries research

For the analysis of caries experience in seven year old children we explored the association between the presence or absence of caries experience among different deciduous molars within each child. Some of the observed high associations have an etiological basis (e.g., between symmetrically opponent molars), while others (diagonally opponent molars) are assumed to be the result of the transitivity of association and hence are believed to disappear once conditioned on the caries experience status of the other deciduous molars. However, using discrete models for multivariate binary data, conditioning on the caries experience of the other teeth present in the mouth and on the (un)known subject-specific characteristics did not remove the latter type of association. When the association was explored on a latent scale, say by a multivariate probit model, then the partial correlation matrix indicated conditional independence. This contrast was confirmed when using other models on the (observed) binary scale and on the latent scale. While it seems logical that conditional dependence partly depends on the chosen model, our example shows that the results and conclusions can be markedly different. The explanation for this surprising result is exemplified mathematically and illustrated using dental data from the Signal Tandmobiel study.