Human dental age estimation using third molar developmental stages: does a Bayesian approach outperform regression models to discriminate between juveniles and adults?

Abstract

Dental age estimation methods based on the radiologically detected third molar developmental stages are implemented in forensic age assessments to discriminate between juveniles and adults considering the judgment of young unaccompanied asylum seekers. Accurate and unbiased age estimates combined with appropriate quantified uncertainties are the required properties for accurate forensic reporting. In this study, a subset of 910 individuals uniformly distributed in age between 16 and 22 years was selected from an existing dataset collected by Gunst et al. containing 2,513 panoramic radiographs with known third molar developmental stages of Belgian Caucasian men and women. This subset was randomly split in a training set to develop a classical regression analysis and a Bayesian model for the multivariate distribution of the third molar developmental stages conditional on age and in a test set to assess the performance of both models. The aim of this study was to verify if the Bayesian approach differentiates the age of maturity more precisely and removes the bias, which disadvantages the systematically overestimated young individuals. The Bayesian model offers the discrimination of subjects being older than 18 years more appropriate and produces more meaningful prediction intervals but does not strongly outperform the classical approaches.

Appendix: Bayesian approach

A multivariate ordinal regression model to obtain the likelihoods $f\left( {x_{i1}, ...,x_{i4} {\text{|age}}_i } \right)$:Formally, let x _ij denote the j-th third molar stage, j = 1,…,4, for subject i, with K possible values, then

$$ \log \left\{ {\frac{{P\left( {x_{ij} \le k} \right)}}{{1 - P\left( {x_{ij} \le k} \right)}}} \right\} = \alpha_{0k} + \alpha_{1k} U_{ij} + h\left( {{\text{age}}_i } \right) + b_i, $$

(3)

where α _0k α _0k are the K − 1 intercept terms to model the marginal frequencies in the K ordered categories of the stage. The left-hand side of the equation represents various logits, i.e., natural logarithms of a specific odds (the odds of observing a stage lower than a specific value k). Observe that if a developmental stage would only have two different values (say 1 and 2), the left-hand side would pertain to a single logit, yielding a binary regression model. A binary indicator U is valued 1 if the third molar is located in the upper jaw and 0 elsewhere. The α _1k quantify the difference in stage between upper and lower jaw. The subscript k indicates in the latter that the effect of jaw is allowed to be non-constant over the intercepts implicating that a proportional odds assumption is not made for this effect. A flexible function h(.) is used to relate age to the logit scale, more specifically, restricted cubic splines have been used [36]. The key idea is to allow non-linearity (on the logit scale) in a flexible way without over fitting the data. Finally, the b _i denotes the random subject effect, assumed to be normal distributed. By including this term in Eq. 3, each subject i is allowed to have its own stage level (on logit scale), hereby accounting for the correlation, which exists between the four repeated stage measures. The resulting model is a generalized linear mixed model, where the term mixed refers to the simultaneous presence of fixed effects (i.e., age and jaw) and a random effect (the b _i ). See for example Molenberghs and Verbeke [37]. Due to the low incidence of stages lower than or equal to 5, those stages are combined into one category. Moreover, no distinction is made between the location (left/right) of a stage. As such, a stage pattern “8 8 6 7” pertains to two stages equal to 8 in the upper jaw and one stage 6 (left or right) and one stage 7 (left or right) in the lower jaw. The generalized linear mixed model is fitted with the procedure PROC NLMIXED in the SAS 9.1 statistical package (SAS Institute, Cary, NC, USA), using adaptive Gaussian quadrature.

Once model Eq. 3 is fitted on the data, the likelihood $f\left( {x_{i1}, ...,x_{i4} {\text{|age}}_i } \right)$ $f\left( {x_{i1}, ...,x_{i4} {\text{|age}}_i } \right)$ can be calculated for all possible patterns (x _i1,…,x _i4) given a specific age. This has been done in steps of 0.1 years, hence the integral in the denominator of Eq. 2 is replaced by a sum over age intervals of 0.1 years and the posterior distribution in Eq. 2 will also have steps of 0.1 years as support points. For the prior distribution, a uniform distribution has been used, implying that each age-category within the considered range (16–22 years for the comparison of the approaches) is given the same (prior) probability.