A full Bayesian calibration model for assessing age in adults by means of pulp/tooth area ratio in periapical radiography

Abstract

The Bayesian approach is being a fundamental tool in forensic and legal field where inferences and decisions are made. In this study, a full Bayesian calibration model was developed to make probabilistic inferences about age estimation in a reference sample of 891 periapical X-rays of upper and lower canines. These teeth belonged to both deceased and living adult subjects, aged between 20 and 86 years, coming from five different countries (Turkey, Italy, Portugal, Japan and Mexico). For this purpose, the narrowing of pulp chamber due to the apposition of secondary dentine was analysed by means of the pulp/tooth area ratio. To determine the agreement of the method, intra- and inter-observer differences for measuring process were calculated by means of the intraclass correlation coefficient (ICC) analysis. Observer error tests showed excellent agreement between observers and between repeated assessments. According to the results of the ANCOVA, neither nationality nor sex was associated to the secondary dentine apposition while it is associated with individual’s age. The results of the present study indicated that the concept of probability is intrinsically linked to the assessment of age in a forensic context, and the Bayesian approach could be considered a robust tool to overtake the bias generated by traditional regression models, thus helping the decision-making process in a legal framework.

Appendix

The first step in Bayesian analysis is to choose a probability model for the observed data. Suppose that, for each individual, the observations can be summarised by his age, t, and by the value, x, of RA_u or RA_l. In our Bayesian calibration approach, the probability model for a typical observation, (x | t, θ), is assumed to be normal with mean μ = μ(t, β) and variance σ²:

$$ p\left(x\ |\ t,\boldsymbol{\theta} \right)=\frac{1}{\sqrt{2\pi {\sigma}^2}}\exp \left\{-\frac{{\left(x-\mu \left(t,\boldsymbol{\beta} \right)\right)}^2}{2{\sigma}^2}\right\} $$

(A1)

A linear function was used to model the expected value in (A1):

$$ \mu =\mu \left(t,\boldsymbol{\beta} \right)={\beta}_0+{\beta}_1\cdotp t $$

(A2)

The vector of the model parameters, θ, consists of the coefficients, β, of the linear function and the model variance σ². Vector θ = (β, σ²) is supported by parameter space Θ ⊆ ℝ² × (0, +∞) and considers a vector of three random variables, the joint prior distribution of which is h(θ). We assume that:

• (A3) observations are independent and identically distributed with the probability model for observed data of the form p(x_i| t_i, θ), i = 1, …n, with unknown vector of parameters θ;

• (A4) given age u and θ, the new observation y is independent of the observed data and follows the same probability model.

With these assumptions, given observations t and x, the posterior distribution for θ may be written as:

$$ h\left(\boldsymbol{\theta}\ |\ \boldsymbol{t},\boldsymbol{x}\right)=\frac{h\left(\boldsymbol{\theta} \right)p\left(\boldsymbol{x}\ |\ \boldsymbol{t},\boldsymbol{\theta} \right)}{\underset{\Theta}{\int }h\left(\boldsymbol{\theta} \right)p\left(\boldsymbol{x}\ |\ \boldsymbol{t},\boldsymbol{\theta} \right)d\boldsymbol{\theta}}=\kern0.5em \frac{h\left(\boldsymbol{\theta} \right)\prod \limits_{i=1}^np\left({\boldsymbol{x}}_{\boldsymbol{i}}\ |\ {t}_i,\boldsymbol{\theta} \right)}{\underset{\Theta}{\int }h\left(\boldsymbol{\theta} \right)\prod \limits_{i=1}^np\left({\boldsymbol{x}}_{\boldsymbol{i}}\ |\ {t}_i,\boldsymbol{\theta} \right)d\boldsymbol{\theta}}. $$

Lastly, the calibrating distribution may be written as:

$$ f\left(u\ |\ y,\boldsymbol{t},\boldsymbol{x}\right)=\frac{p(u)\phi \left(y\ |\ u,\boldsymbol{t},\boldsymbol{x}\right)}{\underset{0}{\overset{+\infty }{\int }}p(u)\phi \left(y\ |\ u,\boldsymbol{t},\boldsymbol{x}\right) du} $$

(A5)

where ϕ(y | u, t, x) is the predictive distribution:

$$ \phi \left(y\ |\ u,\boldsymbol{t},\boldsymbol{x}\right)=\underset{\Theta}{\int }p\left(y\ |\ u,\boldsymbol{\theta} \right)\ h\left(\boldsymbol{\theta}\ |\ \boldsymbol{t},\boldsymbol{x}\right)d\boldsymbol{\theta} $$

(A6)

and p(u) is the prior distribution of age.

Taking into account that no prior information is available about the model parameters, we chose an uninformative prior distribution for them. In addition, in view of the frequent lack of prior age information in forensic age estimation and paleodemography, an improper uniform prior for age distribution, p(u) was selected for the model.

Under the assumptions A1–A4 on our Bayesian model, Bucci et al. [26] proved that the posterior predictive distribution results in noncentral Student’s t distribution and, considering that we assume p(u) improper uniform prior, p(u), for age distribution, for a given new value of the dental maturity, y, the calibrating distribution, f(u |y, t, x), results a truncated noncentral Student’s t distribution with n-2 degree of freedom.