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

For the analysis of caries experience in seven-year old children the association between the presence or absence of caries experience among deciduous molars within each child is explored. Some of the 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 to disappear once conditioned on the caries experience status of the other deciduous molars, covariates and random effects. However, using discrete models for multivariate binary data, conditioning does not remove the diagonal association. When the association is explored on a latent scale, e.g., by a multivariate probit model, then conditional independence can be concluded. This contrast is confirmed when using other models on the (observed) binary scale and on the latent scale. Depending on the point of view, the differences in conditional independence might be seen as a consequence of different types of measurements or as a consequence of different models. An example shows that the results and conclusions can be markedly different with important consequences on model building. The explanation for this result is exemplified mathematically and illustrated using dental data from the Signal-Tandmobiel®® study.