**1. Introduction**

The most widely used statistical methods in dental research are t-test, ANOVA (one, two and three factors), non-parametric tests, and regression models [1]. These methods assume that the observations of the studied variables are independent. Nested data structures are frequently found in dental research. An example is an experimental design in which multiple measurements are performed on the same individual. If, in addition to performing multiple measurements in an individual, we perform multiple measurements in each tooth, we will obtain a nested data structure. This nesting of the data results in grouped data. Typically, for clinical and dental data, contextual variables are measured in each individual (i.e., socioeconomic level,

educational level, etc.), and these characteristics can form another group of data. Considering the detection of bacterial plaque in each tooth of individuals who have a home with a high marginality index, two nested groups are distinguished, namely, teeth nested in individuals and individuals nested at group. The word "nested" can be understood as "within" or "contained in." It is to be expected that items from the same group may be more similar to each other than items from a different group; that is, measurements from one individual are expected to be more similar to each other in comparison with measurements from other individuals. This fact indicates that the assumption of independence does not apply to nested data. Multilevel models take into account the non-independence of the observations. One consequence of ignoring the dependence of observations is that the results of some tests may be statistically significant when, actually, they are not. Under the classical approach, the estimation of the parameters of a multilevel model is performed using maximum likelihood, which has optimal properties in many scenarios; however, problems such as noncompliance with model assumptions or lack of convergence of iterative methods can occur. The Bayesian approach has some advantages over the classical approach.

The purpose of this chapter is to show the minimum statistical theory for the use of multilevel models in dental research when the response variable is numerical. For this, we will remember the definitions of multilevel models and multilevel generalized linear models (MGLM), in addition to the main Bayesian concepts and their application to MGLM. We will use an adaptation of the bottom-up strategy to specify a multilevel model. Our adaptation proposal tries to use the Bayesian leave-one-out cross-validation (LOO-CV) between the different steps for the comparison of models. We will check the adequacy of the fit of the postulated model using posterior predictive density. Finally, we will provide an example of this model applied to a numerical response variable, such as periodontal probing.
