**Table 3.**

*Forward variable selection for the level-three variables\* .*

curves agree very well, the posterior predictive density fits very well with the distribution of the probing depth. Both distributions are clearly not symmetric, and they seem to follow a gamma distribution. Definitely, normal distribution was not an appropriate assumption for probing depth. In conclusion, the random slope model (49) had a good fit.

**Figure 2.** *Posterior predictive density of Model 49 with a random slope for bleeding between teeth and patient.*

**Figure 3** shows the histograms of the empirical posterior distributions of the parameters. Finally, the MCMC of Model 49 converged since all the Rhats were at most 1.01, and the trace plots of **Figure 3** show that the chains mix well.

## **5.2 Discussion**

In this example of probing depth, the variance at the tooth level (1.59) and the variance at the patient level (1.49) were significant (**Table 2**); that is, the mean of the dependent variable varied between teeth nested in patients, and the ICC at the tooth level (0.45) was higher than that at the patient level (0.42); that is, there was greater dependence between the measurements of the probing sites of different teeth than between measurements of the probing sites of different patients. This finding probes that using a multilevel model for these probing depth data was better than using a single-level model, and the former produced more accurate estimates and credible intervals. In addition, the random slope of bleeding was significant between teeth; that is, there was a positive relationship between probing depth and bleeding that varied between teeth in the patients (probing depth between teeth increased by an average of 1.28 mm if the site was bleeding). On the other hand,

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

calculus is a form of hardened dental plaque. In the random slope model (Eq. (49)), bleeding and calculus were significant parameters that estimated that, on average, the depth of bleeding probing sites was 1.14 mm greater than the sites that did not exhibit bleeding. On average, the probing depth of patients who had calculus on any of the teeth was 1.11 mm greater than of patients who did not have calculus. The plausible intervals for bleeding and calculus were 1ð Þ *:*07,1*:*22 and 1ð Þ *:*03,1*:*20 , respectively.

In the random slope model, mobility and smoking were not significantly associated with probing depth, but if we decide to give them an interpretation, we can say that, on average, the probing depth of patients who presented dental mobility was 1.03 mm greater than the probing depth of patients who did not have dental mobility. Similarly, smoking patients had, on average, a probing depth 0.98 mm greater than that in non-smoking patients. Different results and interpretations could be obtained from measuring the independent variables at levels other than those given in this example. Specifically, the variable calculus could have been measured at the tooth level. Before fitting the Bayesian multilevel model, we tried to estimate the multilevel model using restricted maximum likelihood; however, the numerical method did not converge. More practical examples using the *R Software* can be found at [13].
