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Open Access 03.01.2024 | Original reports

Combination of several predictors for forensic age estimation

verfasst von: Michael Hubig, Daniel Wittschieber, Thomas Hunold, Holger Muggenthaler, Sebastian Schenkl, Gita Mall

Erschienen in: Rechtsmedizin | Ausgabe 2/2024

Abstract

Background

Nearly all practical forensic age diagnostics studies estimate the probability distribution of the age conditional on the developmental status of a certain anatomical feature. Given such a probability distribution, the probability of a person exceeding a certain legal age threshold is computed. In court, forensic experts are often asked to summarize the probabilities obtained by evaluating different age indicators of the same person.

Objective

The present study demonstrates computation of the age probability distribution conditional on the conjunction of several different age indicators given the age probability distributions conditional on the development status of the respective single anatomical features.

Material and methods

Data from two distinctively different studies on age estimation were used to join their probability information via Bayes’ theorem. Each of the cited studies is based on the development status of only one of two different anatomical structures: third molar and clavicular epiphysis.

Results

We derive general formulae for Bayesian information joining in forensic age estimation. Posterior distributions of age class, given the simultaneous statuses of the two anatomical features are generated. Finally, the study presents the technique on an artificial case example.

Conclusion

Bayes’ theorem can be used in forensic age estimations to combine information from several different anatomical features to yield more precise probability values of age given development status data of several distinctly different anatomical features. Conditional stochastic independence of the single age indicators as used in our article has to be scrutinized and is not generally recommendable.

Code availability: The program code is provided in the electronic supplementary as a Maple-sheet written in Maple 2022.2.

Supplementary Information

The online version of this article (https://doi.org/10.1007/s00194-023-00672-7) contains supplementary material, which is available to authorized users.

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Introduction

Most forensic studies estimate age distributions conditional on a certain anatomical structures’ development stage as hand and wrist bones (see e.g., [1]), medial clavicular epiphysis (see e.g., [2, 3]) and third molars (see e.g., [4], except e.g., [5, 6]). Leading experts of the DGRM (German Forensic Medicine Society) Study Group on Forensic Age Diagnostics (AGFAD) recommend using the maximum diversity of age indicators for age estimations [7‐10], proposing [11] usage of the maximum age over all indicator minimum ages. Up to now there is no European consensus on how to compute the final results of an age estimation. Our study merges age-diagnostic probabilities using Bayes’ theorem, which is a well-known way to join age information from different anatomical markers.

Table 1 presents a concise sample of approaches to the problem of age estimation using several different markers and/or using Bayesian estimation techniques. The parameters displayed are first author, year of publication, anatomical markers used, methods and sample size.

Table 1

State of research: studies dealing with Bayesian methods and/or multiple anatomical markers

No., year, first author	Anatomical markers, method	Sample size
[12] 1985 Lovejoy, Meindl et al.	Marker M_k: pubic symphyseal face, auricular surface, proximal femur, dental wear, suture closure	261
[12] 1985 Lovejoy, Meindl et al.	Summary age by multifactorial aging method: single factor M_k analysis	261
[13] 1993 Bedford, Russel et al.	Marker M_k: auricular surface, proximal femur, clavicula	55
[13] 1993 Bedford, Russel et al.	Summary age by multifactorial aging method vs. single indicator, blinded test	55
[14] 1996 Lucy, Aykroyd et al.	Marker M_k: periodontal recession, translucency, root roughness multiple linear legression vs. Bayesian prediction using stochastic independence conditional on age (Eq. 8) assumption, prior: age proportion in sample	71
[15] 2002 Boldsen, Milner et al.,	Marker M_k: pubic symphysis, iliac portion of the sacroiliac joint, suture segments	92f, 94m
[15] 2002 Boldsen, Milner et al.,	Bayesian approach: transition analysis, prior estimation: continuation ratio, Eq. 8 with dependency corrected prediction intervals	92f, 94m
[16] 2005 Braga, Heuze et al.	Marker M_k: left permanent mandibular teeth (except 3rd molar)	668f, 456m
[16] 2005 Braga, Heuze et al.	Correspondence analysis and linear regression (CAR) vs. Bayesian prediction with uniform priors, with joint probability age-conditional distribution via dental mineralization sequence	668f, 456m
[17] 2007 Bhat, Kamath	Marker M_k: 3rd molar mandibular root development, wrist: fusion ossification centers	346f, 389m
[18] 2010 Thevissen, Fieuws et al.	Marker M_k: 3rd molar development stage	910, 16 y–22 y
[18] 2010 Thevissen, Fieuws et al.	Bayesian ordinary regress. vs. regression analysis, uniform prior	910, 16 y–22 y
[19] 2010 Langley-Shirley, Jantz	Marker M_k: ephyseal union of medial clavicula	1289
[19] 2010 Langley-Shirley, Jantz	Test of scoring system and transition analysis, prior: fitted Gompertz hazard model	1289
[20] 2011 Bassed, Briggs et al.	Marker M_k: 3rd molar development, clavicular epiphysis, spheno-occipital synchondrosis	184f, 421m
[20] 2011 Bassed, Briggs et al.	Multiple linear regression with scoring system	184f, 421m
[21] 2015 Tangmose, Thevissen et al.	Marker M_k: 3rd molar mineralization	854
[21] 2015 Tangmose, Thevissen et al.	Transition analysis with continuation ratio, with Eq. 8 and ad hoc dependency corrected prediction intervals, uniform prior	854
[22] 2016 Fieuws, Willems et al.	Marker M_k: 3rd molar development (K = 4)	3200m
[22] 2016 Fieuws, Willems et al.	Transition analysis with Eq. 8 and dependency corrected prediction intervals, uniform prior	3200m
[5] 2016 Schmidt, Schramm et al.	Marker M_k: left hand bones, 3rd molar	307, 10 y–29 y
[5] 2016 Schmidt, Schramm et al.	Multiple linear regression model	307, 10 y–29 y
[23] 2016 Cameriere, Pacifici et al.	Marker M_k: sum of open apices of 7 left permanent mandibular teeth	2630, 4 y–17 y
[23] 2016 Cameriere, Pacifici et al.	Linear regression vs. normal Bayesian calibration with uniform prior	2630, 4 y–17 y
[24] 2016 Sironi, Pinchi et al.	Marker M_k: clavicular epiphysis development	380
[24] 2016 Sironi, Pinchi et al.	Bayesian network, transition analysis with continuation ratio, skew-normal distributions as priors	380
[25] 2018 Sironi, Taroni et al.	Marker M_k: 3rd molar mineralization	1143
[25] 2018 Sironi, Taroni et al.	Bayesian network, transition analysis with continuation ratio, skew-normal distributions as priors	1143
[26] 2018 Sironi, Pinchi et al.	Marker M_k: pulp chamber volume	286
[26] 2018 Sironi, Pinchi et al.	Bayesian network, transition analysis with continuation ratio, normal and uniform distributions as priors	286
[6] 2018 Kumagai, Willems et al.	Marker M_k: development teeth, cervical vertebrae, craniofacial growth, hand bones, wrist bones	135f, 121m
[6] 2018 Kumagai, Willems et al.	Transition analysis with continuation ratio and Eq. 8	135f, 121m
[27] 2019 Konigsberg, Frankenberg et al.	Marker M_k: 3rd molar root apex completion	1184m
[27] 2019 Konigsberg, Frankenberg et al.	Transition analysis with probit regression, Laplace‑, normal‑, uniform distribution as prior	1184m
[28] 2020 De Tobel, Fieuws et al.	Marker M_k: 3rd molar development, left wrist, clavicula development Transition analysis with continuation ratio and with Eq. 8 and dependency corrected prediction intervals	160f, 138m

Methods

Terminology for forensic age estimation

Let t be a point in time of interest (e.g. of a crime) and let a* be the true age of a person at t. Let further M_k be the k‑th anatomical feature (e.g. 3rd molar) (k = 1,…, K) and let m_k,j be the j‑th development level of feature M_k (j = 1,…, J(k)). A is the age estimator (estimated age at time of interest) with values a (e.g. A = a = 17 y). The term CSI(M₁,…, M_K|A) stands for the stochastic independence of M₁,…, M_K conditional on age A. The symbol P(A) connotes the prior distribution of the age estimator A with density f(x) := z / (a₀ − a_L) for x in [a_L,a₀], f(x) := (1 − z) / (a_U − a₀) for x in [a₀,a_U] with z in [0,1]. The symbols a₀, a_L, a_U stand for the age threshold of interest (e.g. a₀ = 18 y), a first guess lower limit and a first guess upper limit for the age A. The constant z := P(A < a₀) is determined by the examiner via visual inspection prior to the examination.

Using Bayes’ theorem for combining information

Bayes’ theorem for the case of K > 1 anatomical age predictors M₁,…, M_K can be written yielding the final formula of information joining for forensic age estimation in the living:

$$P\left(A=a| M_{1}=m_{1}{,}\ldots {,}M_{K}=m_{K}\right)=\frac{P\left(M_{1}=m_{1}{,}\ldots {,}M_{K}=m_{K}| A=a\right)}{\left\{\begin{array}{c} \sum _{b\in \underline{A}}P\left(M_{1}=m_{1}{,}\ldots {,}M_{K}=m_{K}| A=b\right)\\ \cdot P\left(A=b\right)\end{array}\right\}}\cdot P\left(A=a\right)$$

(1)

Using the definition of conditional stochastic independence (CSI) another simplification is possible:

$$P\left(A=a| M_{1}=m_{1}{,}\ldots {,}M_{K}=m_{K}\right)=\frac{\prod _{k=1{,}\ldots {,}K}P\left(M_{k}=m_{k}| A=a\right)}{\left\{\begin{array}{c}\sum _{b\in \underline{A}}\prod _{k=1{,}\ldots {,}K}P\left(M_{k}=m_{k}| A=b\right)\\ \cdot P\left(A=b\right)\end{array}\right\}}\cdot P\left(A=a\right)$$

(2)

The age distribution conditional on a single feature level

If the distribution density f_j of P(A | M = m_j) of the age A under the condition of a feature level value m_j of the feature M is assumed to be normal with mean E_j and standard deviation S_j we write:

$$\forall j=1{,}\ldots {,}J: f_{j}\left(x\right):= \frac{1}{\sqrt{2\cdot \pi \cdot S_{j}}}\cdot exp\left(-\frac{\left(x-E_{j}\right)^{2}}{2\cdot {S_{j}}^{2}}\right)$$

(3)

This is the case in the study of Olze, Peschke et al. [4], which is our source for dental development data.

To yield the distributions P(M | A = a), we apply Bayes’ theorem again:

$$P\left(M=m_{j}|A=a\right)=\frac{P(A=a|M=m_{j})\cdot P(M=m_{j})}{P\left(A=a\right)}=\frac{P\left(A=a|M=m_{j}\right)\cdot P\left(M=m_{j}\right)}{\sum _{m\epsilon \underline{M}}P\left(A=a|M=m\right)\cdot P\left(M=m\right)}$$

(4)

If the joint probability distributions’ absolute histogram N(A, M) is given as in the study of Kreitner, Schweden et al. [2], which we used as our clavicular related data source, we estimate P(M | A = a):

$$\forall j=1{,}\ldots {,}J: \ P\left(M=m_{j}| A=a\right):= \frac{N\left(a{,}m_{j}\right)}{\sum _{m\in \underline{M}}N\left(a{,}m\right)}$$

(5)

Results

Age distribution conditional on tooth 18 eruption state

The third molar tooth eruption status for tooth 18 was investigated by Olze [4]. Table 1 in [4] provides the prior age distribution P(A) by:

$$P\left(A=a\right)={\#}\{\textit{Males}\,ofageA=a\}/{\#}\{\textit{Males}\}$$

(6)

Table 2 in [4] gives the means E_1,j and the standard deviations S_1,j of P(A | M₁ = m_1,j) conditional on the eruption status M₁ with the levels m_1,j = A, B, C, D. Note that the age variable A should not be confused with the eruption status “A” of tooth 18. Figure 1 shows the age distribution P(A | M₁) conditional on the value of the eruption status M₁ = m for all four development level values m = A, m = B, m = C, and m = D. As our approach in Eq. 2 needs the distributions P(M₁ | A = a) for all ages a, we computed the latter from the distributions P(A | M₁ = m_1,j) via the conditional probability definition in the first equality of (Eq. 4) using the intra-study [4]—prior Eq. 6. This approach should not be confused with Bayes theorem’s main application (Eq. 2) with the purpose of joining different data sources.

Age distribution conditional on clavicular epiphysis stage

Kreitner [2] published joined histograms N(A, M₂) of patient’s CTs (age < 30 y). The development level M₂ of the epiphyseal union was quantified according to a four-stage system {S1, S2, S3, S4} from [29]. Figure 2 shows the age distribution P(A | M₂) computed by:

$$P\left(A=a| M_{2}=m\right)=N\left(A=a,M_{2}=m\right)/\sum _{a^{\prime}\in \underline{A}}N\left(A=a^{\prime},M_{2}=m\right)$$

(7)

Joining clavicular and tooth state age information

As we assume Eq. 8 for our example, we use Eq. 3 instead of Eq. 2 to compute P(A | M₁,…, M_K). Figure 3a, b, c visualize the result for three distinct combinations (m_1a,m_2a), (m_1b,m_2b), (m_1c,m_2c) of molar 18 eruption and clavicular development statuses M₁ and M₂, respectively, the distributions P(A | M_k = m_k,x) for k = 1,2 and P(A | M₁ = m_1,x, M₂ = m_2,x) resulting from our joining operation.

Application in a hypothetical case example

Let us assume a male individual undergoing a criminal proceeding. He asserts to be younger than 18 years without confirming documents. The prosecution requests a medico-legal investigation (see AGFAD criteria e.g. [9]) whether he is older than 18 years. He matches the selection criteria of [4] and [2].

The persons’ medial clavicular epiphysis status is M₂ = S2 (see [2]), the eruption state of tooth 18 is M₁ = B (see [4]). Table 2 in [4] gives expectation E₁ = 20.8 y and standard deviation S₁ = 2.7 y as well as min₁ = 14.9 y, max₁ = 25.7 y. The normal assumption leads to p₁ = P(A ≥ 18 y | M₁ = B) = 0.85. With Tab. 1 in [2] we yield p₂ = P(A ≥ 18 y | M₂ = S2) = (1 + 1 + 4 + 8) / 51 = 0.27. Our Bayesian approach results in p = P(A ≥ 18 y | M₁ = B, M₂ = S2) = P(A = 18 y | M₁ = B, M₂ = S2) + P(A = 19 y | M₁ = B, M₂ = S2) + P(A = 20 y | M₁ = B, M₂ = S2) + P(A = 22 y | M₁ = B, M₂ = S2) = 0.01 + 0.01 + 0.035 + 0.1 = 0.155 (as all other relevant conditional age probabilities are 0), which can be directly read from Fig. 4.

Discussion

The approach should not be misunderstood as an attempt to overwrite the suggestions and established procedures (see [7‐11, 30]) given by experienced experts or boards like AGFAD. It should merely provide another tool for dealing with the information at hand.

We would like to emphasize that our assumption Eq. 8 of the age marker M₁, …, M_K’s stochastic independence conditional on the age A is a hypothetical auxiliary construct, enabling us to perform example calculations. Its usage in casework should be scrutinized in every application case based on empirical data. A general usage of Eq. 8 is not recommendable, although Eq. 8 is applied in several studies, as it is disputed in the literature.

Joining empirical distributions from different study samples and extending the term Eq. 8 needs an explanation: Per definitionem Eq. 8 is a potential property of the random variables M₁ and M₂: interpreting Eq. 8 for this application, the variables M₁ and M₂ have to be defined in an overall population‘ H which is not experimentally accessible. Here Eq. 8 is realized:

$$P_{H}\left(M_{1}=m_{1}.M_{2}=m_{2}| A=a\right)=P_{H}\left(M_{1}=m_{1}| A=a\right)\cdot P_{H}(M_{2}=m_{2}| A=a)$$

(8)

Two disjoint subsamples H1 taken from Kreitner, Schweden et al. [2] and H2 from Olze, Peschke et al. [4] respectively, were used to compute the probability distributions P_H1(M₁) and P_H2(M₂) respectively. H1 and H2 being representative subsamples in H means:

$$\forall i=1.2\colon P_{H}\left(M_{i}=m_{i}| A=a\right)=P_{Hi}\left(M_{i}=m_{i}| A=a\right)$$

(9)

Note that assumptions like Eq. (9) are the basis of every application of experimental statistics. Thus, we can extend Eq. 8 though none of the individuals in H1 belong to H2:

$$P_{H}\left(M_{1}=m_{1}.M_{2}=m_{2}| A=a\right)=P_{H1}\left(M_{1}=m_{1}| A=a\right)\cdot P_{H2}(M_{2}=m_{2}| A=a)$$

(10)

As in Eq. 2 the number K of anatomical features is not bounded. In principle one distribution P(A | M₁ = m₁,…, M_K = m_K) could contain all information from different features.

Our prior P(A) represents an age estimation expert’s first guess of the age before examination but on the basis of informal visual inspection. The value of z in the density f(x) of the age estimator A’s prior distribution P(A), as introduced in the Terminology paragraph of the Methods section, can be interpreted as the a priori probability of the persons age A being below the bound a₀ of interest.

Figure 3a, b, c show the dominance in P(A | M₁ = m₁, M₂ = m₂) of clavicular feature M₂ over molar feature M₁ with smaller variance of P(A | M₂) than P(A | M₁) and wider spread maxima on the age scale (14 y–25 y) (see Fig. 1). In Fig. 3a, b, c the graphs of P(A | M₁ = m₁, M₂ = m₂) are more similar to the graphs of P(A | M₂ = m₂) than to the ones of P(A | M₁ = m₁). The molar distribution P(A | M₁ = m₁) smooths the form of the P(A | M₁ = m₁, M₂ = m₂) in comparison to the P(A | M₂ = m₂).

Finally, we encourage all scientists to publish their empirical data on multiple features joint distributions.

Practical conclusion

The study presented demonstrates information fusing from different single predictors, providing probabilities for forensic age estimation. Equation 8 of the single age indicators as presupposition has to be scrutinized.

Declarations

Conflict of interest

M. Hubig, D. Wittschieber, T. Hunold, H. Muggenthaler, S. Schenkl and G. Mall declare that they have no competing interests.

Ethical standards

We herewith confirm the ethical standards: For this article no studies with human participants or animals were performed by any of the authors. All studies mentioned were in accordance with the ethical standards indicated in each case. This article uses only published data of published studies with human participants. The data used are of strictly statistical nature and no identifying individual data can be reconstructed.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Supplementary Information

Code availability: The program code is provided in the electronic supplementary as a Maple-sheet written in Maple 2022.2.

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Titel: Combination of several predictors for forensic age estimation
verfasst von: Michael Hubig
Daniel Wittschieber
Thomas Hunold
Holger Muggenthaler
Sebastian Schenkl
Gita Mall
Publikationsdatum: 03.01.2024
Verlag: Springer Medizin
Erschienen in: Rechtsmedizin / Ausgabe 2/2024
Print ISSN: 0937-9819
Elektronische ISSN: 1434-5196
DOI: https://doi.org/10.1007/s00194-023-00672-7

Die Highlights vom Kongress des American College of Cardiology 2024

Springer Medizin

Combination of several predictors for forensic age estimation

Abstract

Background

Objective

Material and methods

Results

Conclusion

Supplementary Information

Publisher’s Note

Introduction

Methods

Terminology for forensic age estimation

Using Bayes’ theorem for combining information

The age distribution conditional on a single feature level

Results

Age distribution conditional on tooth 18 eruption state

Age distribution conditional on clavicular epiphysis stage

Joining clavicular and tooth state age information

Application in a hypothetical case example

Discussion

Practical conclusion

Declarations

Conflict of interest

Ethical standards

Publisher’s Note

Unsere Produktempfehlungen

Rechtsmedizin

e.Med Interdisziplinär

Supplementary Information

Neu im Fachgebiet Rechtsmedizin

Molekularpathologische Untersuchungen im Wandel der Zeit

Vergleichende Pathologie in der onkologischen Forschung

Gastrointestinale Stromatumoren

Personalisierte Medizin in der Onkologie

Die Highlights vom Kongress des American College of Cardiology 2024

Springer Medizin

Abstract

Background

Objective

Material and methods

Results

Conclusion

Supplementary Information

Publisher’s Note

Introduction

Methods

Terminology for forensic age estimation

Using Bayes’ theorem for combining information

The age distribution conditional on a single feature level

Results

Age distribution conditional on tooth 18 eruption state

Age distribution conditional on clavicular epiphysis stage

Joining clavicular and tooth state age information

Application in a hypothetical case example

Discussion

Practical conclusion

Declarations

Conflict of interest

Ethical standards

Publisher’s Note

Unsere Produktempfehlungen

Rechtsmedizin

e.Med Interdisziplinär

Supplementary Information

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