Abstract
We have created the tool BioAlder as an age prediction model based on the systems Greulich and Pyle (hand) and the Demirjian’s grading of the third molar tooth. The model compiles information from studies representing a total of 17,151 individuals from several parts of the world. The model offers a solution where issues as groupwise data format and age mimicry bias are bypassed. The model also provides a solution for combining the two grading systems, hand and tooth, to one combined age prediction result assuming independency. We have tested our model of age prediction and the independency assumption on a separate data set from Lebanon with 254 young individuals. The prediction intervals of BioAlder covered most of the data points; however, we observed some outliers. Our analyses indicate at least a weak dependency between the two methods.
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Acknowledgments
We thank Jayakumar Jayaraman, Simon Camilleri, Rick R. van Rijn, Eugénia Cunha, Abdul Mueed Zafar, Bernhard Knell, and Ivan Galić for providing data, and Thore Egeland and Torbjørn Wisløff for useful discussions. We also want to thank the reviewers for their useful comments.
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Appendices
Appendix 1. Description of the type 4 model
The type 4 format presents tables (see Table 2 in the paper for an example) with rows giving the mean and the standard deviation (SD) of the n individuals’ skeletal ages (SA_{mean} and SA_{sd}) and the individuals’ chronological ages (CA_{mean} and CA_{sd}), and the correlation coefficient (Corr) between the pairwise skeletal ages (SA) and chronological ages (CAs). For each row in the type 4 tables, we create a type 4 model. This consists of two steps. In the first step, we fit a discrete distribution for SA, having expectation SA_{mean} and variance SA_{sd} (where the outcome of the discrete stages is given in the GP atlas). In the second step, we fit a normal distribution of chronological for a given observed SA. Last, the SA and CA are generated for each of the n individuals based on simulations from the model in step 1 used to generate the SAs, and then from the model in step 2 used to generate CAs (given the generated SAs from step 1).
Step 1: Modeling the discrete distribution for skeletal age
To construct the discrete distribution of SA, we assume that it is a categorized version of an underlying (latent) normal distribution of skeletal age: The probability for a particular discrete skeletal age is the probabilities of being within “the midpoint between the current and previous SA” and “the midpoint between current and the next SA” (based on the normal distribution). For SA “17” for instance, the probability is equal to the area under the normal distribution from 16.5 to 17.5 years. The normal distribution was fitted such that the expectation and the SD of the discrete distribution are equal to SA_{mean} and SA_{sd}, respectively. The discrete distribution of SA is then defined and is further used to generate SA for the individuals (step 1). Mathematical description of this model is found below.
Mathematical details
The model we described here is called a “continuous latent response variable”: We first assume that variable X is normally distributed with the unknown parameters expectation μ and SD σ, X~N(μ, σ^{2}). The probability of observing SA s is then defined by using the cumulative distribution of X, F_{X}, where s^{+} is the defined SA after s and s^{−} is the defined SA before s (defined in the GP atlas). The interval we consider for a given SA s is \( \left[\frac{s+{s}^{}}{2},\frac{s+{s}^{}}{2}\right] \), which gives the probability: \( P\left(S=s\mu, \sigma \right)={F}_X\left(\frac{s+{s}^{+}}{2}\right){F}_X\left(\frac{s+{s}^{+}}{2}\right) \). Here, \( {F}_X\left(\frac{s+{s}^{+}}{2}\right)=1 \) for the last defined SA and \( {F}_X\left(\frac{s+{s}^{}}{2}\right)=0 \) for the first defined SA. We fit the probability model for the discrete SAs by choosing the parameters μ and σ such that E[S = s μ, σ] = ∑_{s}s × P(S = s μ, σ) = SA_{mean} and \( \mathrm{Var}\left[S=s\mu, \sigma \right]={\sum}_s{\left(sS{A}_{\mathrm{mean}}\right)}^2\times P\left(S=s\mu, \sigma \right)=S{A}_{\mathrm{sd}}^2 \).
Step 2: Modeling the conditional distribution of chronological age given skeletal age
In the second step, we assume that chronological age (CA) conditioned on a specific skeletal age (SA) s is normally distributed with mean and variance given as E[CA │ SA = s] = CA _ mean + (CA _ sd)/(SA _ sd ) × Corr × (s − SA _ mean) and \( \mathrm{Var}\left[ CAS=s\right]=\left(1\mathrm{Cor}{\mathrm{r}}^2\right)\times C{A}_{\mathrm{sd}}^2 \). Here, the correlation coefficient (Corr) between SA and CA is utilized to potentially reduce the simulated variability by conditioning on SA (if Corr is different from zero).
Appendix 2. Description of modeling stage probabilities
We did model selection based on four different variants of ordinal regression models:
 1)
Proportionalodds cumulative model with probit link

2)
Proportionalodds cumulative model with logit link

3)
Continuationratio model with probit link

4)
Continuationratio model with logit link
For j = J (last stage), we have P(Y ≤ J θ, a) = 1 and P(Y = J Y ≥ J, θ, a) = 1. The link function \( \mathrm{logit}(x)=\log \left(\frac{x}{1x}\right) \), while the link function probit is cumulative standard normal distribution.
Here, Y is a discrete stochastic variable with the stage outcomes 1, …, J which an individual with chronological age x may have. θ = (α_{1}, …, α_{J − 1}, β, λ) represents the model parameters we want to estimate (optimize) based on the observations y (vector of stages) and x (vector of ages). Here, we assume that λ takes the values 0.1, 0.2, …, 1.0., whereas the other variables are continuous. We used the VGAM R package with the vglm function to fit the models [1,2,3,4] based on maximum likelihood estimates and the predict function to calculate the stage probabilities for a given age:
 1)
vglm(y~I(x^λ), cumulative(link = “probit”,parallel = T,reverse = F)
 2)
vglm(y~I(x^λ), cumulative(link = “logit”,parallel = T,reverse = F)
 3)
vglm(y~I(x^λ), cratio(link = “probit”,parallel = T,reverse = F)
 4)
vglm(y~I(x^λ), cratio(link = “logit”,parallel = T,reverse = F)
Since all variants of the transition models [1,2,3,4] have the same number of parameters, the final model was chosen as the one which gave the best fit with the observed data. The value of the maximized likelihood function was used to measure this. Importantly, the model with best fit may not describe the observations adequately. This could further be investigated by a goodnessoffit test as demonstrate in [14].
Since some of the observed data were generated, the candidate model that gave the best fit over the 100 generated complete datasets was chosen. The following models were selected for each of the methods and sexes:
Maleshand: proportionalodds cumulative model with logit link with λ = 1.0
Malestooth: continuationratio model with probit link with λ = 0.7
Femaleshand: proportionalodds cumulative model with logit link with λ = 1.0
Femalestooth: continuationratio model with probit link with λ = 0.7
Based on these model types, we fitted the stage probabilities (for ages 7.00, 7.01, …, 27.00) for each of the 100 generated complete datasets that the tool results are based on (i.e., the parameters (α_{1}, …, α_{J − 1}, β) are estimated for each of the generated datasets).
Appendix 3. Bivariate distribution of proportionalodds cumulative model with probit link function
In this section, we derive the bivariate distribution Pr(S^{H} = i, S^{T} = j x), the probability that an individual with age x have stage S^{H} = i for hand and stage S^{T} = j for tooth, for the proportionalodds cumulative model with a probit link function. For such a model, the underlying latent continuous variable for the skeletal ages and tooth stages (Y^{H}, Y^{T}) are normally distributed with means μ_{H} = − β_{H} ∗ x, μ_{T} = − β_{T} ∗ x and a given covariance structure with correlation coefficient ρ:
We let the parameters \( {\alpha}_1^H,{\alpha}_2^H,..,{\alpha}_{I1}^H \) be rules which form a stage discretization for hand (I stages), whereas the parameters \( {\alpha}_1^T,{\alpha}_2^T,..,{\alpha}_{J1}^T \) form a stage discretization for tooth (J stages):
 a.
Assign as hand stage i if \( {Y}^H\in \left({\alpha}_{i1}^H,{\alpha}_i^H\right] \) for i = 1, …, I (here \( {\alpha}_0^H=\infty \) and \( {\alpha}_I^H=\infty \Big) \)
 b.
Assign as tooth stage j if \( {Y}^T\in \left({\alpha}_{j1}^T,{\alpha}_j^T\right] \) for j = 1, …, J (here \( {\alpha}_0^T=\infty \) and \( {\alpha}_J^T=\infty \Big) \)
We have that Pr(S = j x) = Pr(Y ∈ (α_{j − 1}, α_{j}]) = P( Y ≤ α_{j}) − P( Y ≤ α_{j − 1}) = P( Y − μ ≤ α_{j} − μ) − P( Y − μ ≤ α_{j − 1} − μ). Since Y – μ~N(0, 1), the probability that an individual with age x has stage S = s is given as
Pr(S = j x) = Φ(α_{j} + β × x) − Φ(α_{j − 1} + β × x) for j = 2, …, J − 1.
For j = 1 and j = J, we have
Pr(S = 1 x) = Φ(α_{1} + β × x) and Pr(S = J x) = 1 − Φ(α_{j − 1} + β × x)
From the conditional formula
Pr(S^{H} = i, S^{T} = j x) = Pr(S^{H} = i x) × Pr (S^{T} = j S^{H} = i, x)
Hence, we need to derive Pr(S^{T} = j S^{H} = i, x):
The conditional distribution Y^{T} ∣ Y^{H} = y is a normal distribution:
So, given the latent continuous variable y for the hand method (which is unknown), we have \( \mathit{\Pr}\left({S}^T=j{Y}^H=y,x\right)={F}_{Y^T\mid {Y}^H}\left({\alpha}_j\right){F}_{Y^T\mid {Y}^H}\left({\alpha}_{j1}\right) \)
Here, \( F={F}_{Y^T\mid {Y}^H} \) is the cumulative density function of the conditional distribution Y^{T} ∣ Y^{H} = y, x. For our model (normal distribution), F(α_{j}) = \( \Phi \left(\frac{\alpha_j+{\beta}^T\ast x\rho \ast \left(y+{\beta}^H\ast x\right)}{\sqrt{1{\rho}^2}}\right) \) for j = 1, …, J − 1 where F(α_{j}) = 1 for j = J (last stage) and F(α_{j − 1}) = 0 for j = 1 (first stage). The formula was derived using conditional distribution rule of normal distributions. We can now derive Pr(S^{T} = j S^{H} = i, x):
Pr(S^{T} = j S^{H} = i, x) = \( {\int}_{\alpha_{i1}}^{\alpha_i}\Pr \left({S}^T=j,{Y}^H=y{S}^H=i,x\right)\mathrm{d}y\kern0.5em \)where Pr(S^{T} = j, Y^{H} = y S^{H} = i, x) = Pr (Y^{H} = y S^{H} = i, x) ∗ Pr (S^{T} = j Y^{H} = y, x)
Pr(S^{T} = j Y^{H} = y, x) is already given in equation above and \( \mathit{\Pr}\left({Y}^H=y{S}^H=i,x\right)=\mathrm{Norma}{\mathrm{l}}_{\left[{\alpha}_{i1},{\alpha}_i\right]}\ \left({\beta}^H\ast x,1\right) \)
The likelihood function
Based on the n observations, for i = 1, …, n, we have observed hand stage \( {s}_i^H \), the tooth stage \( {s}_i^T \), and the age x_{i}. By letting \( \theta =\left({\alpha}_1^H,\dots, {\alpha}_{I1}^H,{\beta}^H,{\alpha}_1^T,\dots, {\alpha}_{J1}^T,{\beta}^T,\rho \right) \) be the model parameters for the bivariate model, the likelihood function is given as
Appendix 4. Description of estimating confidence interval using bootstrap simulation
In this section, we describe how we estimate a 95% confidence interval of the correlation coefficient parameter ρ which is one of the model parameters θ for the bivariate distribution as defined in Appendix 3. Here, M is the number of bootstrap samples. For each sex, we did the following steps:
 1)
Optimize the likelihood function for the bivariate model based on the Lebanese observations to get the maximum likelihood estimates of the model parameters, \( \widehat{\theta} \).
 2)
Generate M independent datasets from the fitted model \( \mathit{\Pr}\left({S}^H,{S}^Tx,\widehat{\theta}\right) \).
 a.
Each dataset includes n individuals with the observed age x as in the Lebanese dataset but with generated hand and tooth stages.
 b.
Notice that some simulated datasets do not have all the hand and tooth stages found in the Lebanese dataset. For these scenarios, the number of parameters in the model is less.
 a.
 3)
For each of the M generated datasets, we estimated the model parameters using maximum likelihood estimates.
 4)
Calculate the 2.5% and 97.5% percentiles of the M estimated ρ correlation parameter values.
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Bleka, Ø., Rolseth, V., Dahlberg, P.S. et al. BioAlder: a tool for assessing chronological age based on two radiological methods. Int J Legal Med 133, 1177–1189 (2019). https://doi.org/10.1007/s0041401819595
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Keywords
 Age estimation
 Demirjian’s
 Greulich and Pyle
 Conditional dependency