**5. Application: periodontal probing depth**

In this section, an example is given in which a multilevel generalized linear model is used for data from a cross-sectional study conducted by Romero-Castro et al. [10]. This study was carried out among adults who reside in the state of Guerrero, Mexico, and who went to the external dental clinical service of the Dental School of the Autonomous University of Guerrero (UAGro) in search of treatment, during the period from August 2015 to February 2016. The protocol was approved (registration no. CB005/2015) by the ethic committee at UAGro.

The goal of this multilevel analysis was to determine the clinical factors associated with the depth of periodontal probing.

Thirty-two teeth were examined in each of the 116 patients. Probing pocket depth was recorded at six sites in each tooth, that is, *mesiobuccal, mid buccal, distobuccal, mesiolingual, mid lingual,* and *distolingual* locations of each tooth. Pocket depth was recorded by use of Florida probe in the six sites. The response variable was *probing*

## *Bayesian Multilevel Modeling in Dental Research DOI: http://dx.doi.org/10.5772/intechopen.108442*

*depth* measured in millimeters; that is, probing depth is a continuous variable and greater than zero ( ≥0). The data set consisted of 18,358 observations.

The independent variables, except the age, were all dichotomous: *bleeding, mobility, plaque, calculus, insulin resistance (fasting plasma glucose* > *100 mg=dL), smoking, root remnants, and mismatched restorations*, where 0 indicated absence and 1 presence.

**Figure 1** and **Table 1** show the three levels of the data and the variables at each level. The first level corresponded to the probing sites where the independent variables *bleeding* and *furcation* and the response variable *probing depth* were measured. Level two corresponded to the dental piece, that is, teeth that only had the independent variable *mobility*, and level three corresponded to the patients, measuring the independent variables *age, plaque, calculus, insulin resistance, smoking, root remnants*, and *mismatched restorations*.

A first data analysis was done using a three-level multilevel model assuming a normal distribution for the probing depth. The frequentist fit had two problems: the residuals did not have a normal distribution and the numerical method to obtain the estimates did not converge.

The minimum of probing depth was 0.2 mm, Q1 was 0.8 mm, Q2 was 1.2 mm, Q3 was 1.8 and the maximum was 9 mm. In addition, its distribution was asymmetric to the right (skewness = 1.6 and kurtosis = 8.0). Therefore, it was assumed that probing depth had gamma distribution with mean *μ* and variance *μ*<sup>2</sup>*=α*:

$$f(\boldsymbol{y}) = \frac{(a/\mu)^a}{\Gamma(a)} \boldsymbol{y}^{a-1} \exp\left(-\frac{a\boldsymbol{y}}{\mu}\right) \tag{44}$$

It is well known that gamma regression belongs to the generalized linear model family. But as the data studied is of hierarchical nature, the appropriate model is the

#### **Figure 1.**

*Multilevel structure of the probing depth of 1 tooth out of 32 teeth for each patient.*


**Table 1.** *Independent variables on levels.* multilevel generalized linear model. Given that the response variable is non-negative, the link function used was the natural logarithm to get expected probing depth greater than zero.
