Combining Bayesian Calibration and Copula Models for Age Estimation
Abstract
:1. Introduction
2. Materials and Methods
2.1. Sample
2.2. Measurements
2.3. Statistical Analysis
- and ;
- and ;
- and ;
- and .
- Mean Absolute Error (MAE);
- Root Mean Squared Error (RMSE);
- the Inter-Quartile Range of error distribution (IQRERR);
- the mean of the quantile-based 95% Bayesian confidence interval (MCI95%) of the calibrating distribution.
3. Results
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
Appendix A.1. Bayesian Calibration
Appendix A.2. Copula Theory
Types of Copula
Appendix A.3. Simulation Study
- Multivariate normal distribution (MVN)
- Multivariate skewed normal distribution (MVSN)
Model | MAE | RMSE | IQRERR | MCI95% |
---|---|---|---|---|
Panel A: S~N; W~N | ||||
Segmented (S single predictor) | 0.92 | 1.39 | (−0.84; 0.40) | 2.77 |
Linear (W single predictor) | 1.01 | 1.51 | (−1.32; 0.53) | 4.22 |
Independent | 0.83 | 1.22 | (−0.65;0.37) | 2.39 |
Gaussian C. | 0.79 | 1.31 | (−0.61;0.21) | 2.31 |
R-Gumbel C. (270°) | 0.83 | 1.23 | (−0.65;0.38) | 2.38 |
R-Clayton C. (90°) | 0.69 | 1.15 | (−0.54;0.26) | 2.35 |
Panel B: S~ALD; W~N | ||||
Segmented (S single predictor) | 0.88 | 1.13 | (−1.01; 0.35) | 3.91 |
Linear (W single predictor) | 1.01 | 1.51 | (−1.32; 0.53) | 4.22 |
Independent | 0.73 | 1.02 | (−0.64;0.39) | 3.79 |
Gaussian C. | 0.5 | 0.76 | (−0.48;0.20) | 3.54 |
R-Gumbel C. (270°) | 0.74 | 1.02 | (−0.63;0.38) | 3.79 |
R-Clayton C. (90°) | 0.53 | 0.78 | (−0.39;0.29) | 3.51 |
Panel C: S~N; W~ALD | ||||
Segmented (S single predictor) | 0.92 | 1.39 | (−0.84; 0.40) | 2.77 |
Linear (W single predictor) | 1.03 | 1.93 | (−0.93; 0.81) | 5.51 |
Independent | 0.86 | 1.31 | (−0.61;0.32) | 2.58 |
Gaussian C. | 1.01 | 1.59 | (−0.63;0.31) | 2.17 |
R-Gumbel C. (270°) | 0.81 | 1.31 | (−0.62;0.31) | 2.13 |
R-Clayton C. (90°) | 0.78 | 1.21 | (−0.56;0.35) | 2.25 |
Panel D: S~ALD; W~ALD | ||||
Segmented (S single predictor) | 0.88 | 1.13 | (−1.01; 0.35) | 3.91 |
Linear (W single predictor) | 1.03 | 1.93 | (−0.93; 0.81) | 5.51 |
Independent | 0.7 | 1.06 | (−0.56;0.31) | 4.84 |
Gaussian C. | 0.75 | 1.16 | (−0.72;0.22) | 3.01 |
R-Gumbel C. (270°) | 0.69 | 1.03 | (−0.73;0.12) | 3.33 |
R-Clayton C. (90°) | 0.68 | 1.03 | (−0.49;0.34) | 3.33 |
Model | MAE | RMSE | IQRERR | MCI95% |
---|---|---|---|---|
Panel A: S~N; W~N | ||||
Segmented (S single predictor) | 0.95 | 1.48 | (−0.96; 0.51) | 4.11 |
Linear (W single predictor) | 1.02 | 1.50 | (−1.31; 0.55) | 4.24 |
Independent | 0.90 | 1.22 | (−0.69; 0.70) | 3.17 |
Gaussian C. | 1.18 | 1.62 | (−0.98; 0.91) | 2.44 |
R-Gumbel C. (270°) | 0.88 | 1.15 | (−0.64; 0.68) | 3.19 |
R-Clayton C. (90°) | 0.86 | 1.14 | (−0.62; 0.65) | 3.17 |
Panel B: S~ALD; W~N | ||||
Segmented (S single predictor) | 0.99 | 1.30 | (−1.11; 0.37) | 3.98 |
Linear (W single predictor) | 1.02 | 1.50 | (−1.31; 0.55) | 4.24 |
Independent | 0.92 | 1.34 | (−0.80; 0.53) | 3.95 |
Gaussian C. | 1.15 | 1.72 | (−0.83;0.71) | 3.77 |
R-Gumbel C. (270°) | 0.91 | 1.31 | (−0.79; 0.54) | 3.89 |
R-Clayton C. (90°) | 0.88 | 1.29 | (−0.73; 0.56) | 3.85 |
Panel C: S~N; W~ALD | ||||
Segmented (S single predictor) | 0.95 | 1.48 | (−0.96; 0.51) | 4.11 |
Linear (W single predictor) | 1.04 | 1.91 | (−0.96; 0.79) | 5.37 |
Independent | 1.04 | 1.42 | (−0.71; 0.79) | 4.20 |
Gaussian C. | 1.35 | 2.02 | (−0.82; 0.85) | 4.05 |
R-Gumbel C. (270°) | 0.94 | 1.32 | (−0.66;0.76) | 4.12 |
R-Clayton C. (90°) | 0.93 | 1.31 | (−0.64; 0.74) | 4.05 |
Panel D: S~ALD; W~ALD | ||||
Segmented (S single predictor) | 0.99 | 1.30 | (−1.11; 0.37) | 3.98 |
Linear (W single predictor) | 1.04 | 1.91 | (−0.96; 0.79) | 5.37 |
Independent | 0.92 | 1.28 | (−0.56; 0.31) | 6.34 |
Gaussian C. | 1.15 | 1.65 | (−0.65; 0.40) | 7.07 |
R-Gumbel C. (270°) | 0.87 | 1.17 | (−0.51; 0.33) | 6.29 |
R-Clayton C. (90°) | 0.85 | 1.12 | (−0.46; 0.38) | 6.15 |
References
- Cameriere, R.; Ferrante, L.; Ermenc, B.; Mirtella, D.; Strus, K. Age estimation using carpals: Study of a Slovenian sample to test Cameriere’s method. Forensic Sci. Int. 2007, 174, 178–181. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Cameriere, R.; Ferrante, L.; Cingolani, M. Age estimation in children by measurement of open apices in teeth. Int. J. Leg. Med. 2006, 120, 49–52. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Schulz, R.; Mühler, M.; Reisinger, W.; Schmidt, S.; Schmeling, A. Radiographic staging of ossification of the medial clavicular epiphysis. Int. J. Leg. Med. 2008, 122, 55–58. [Google Scholar] [CrossRef] [PubMed]
- Aykroyd, R.G.; Lucy, D.; Pollard, A.M.; Solheim, T. Technical note: Regression analysis in adult age estimation. Am. J. Phys. Anthr. 1997, 104, 259–265. [Google Scholar] [CrossRef]
- Lucy, D.; Pollard, A.M. Further comments on the estimation of error associated with the Gustafson dental age estimation method. J. Forensic Sci. 1995, 40, 222–227. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Masset, C. Age Estimation on the basis of cranial sutures. In Age Markers in the Human Skeleton; Iscan, M.Y., Ed.; Springfield: Hong Kong, China, 1989. [Google Scholar]
- Ferrante, L.; Skrami, E.; Gesuita, R.; Cameriere, R. Bayesian calibration for forensic age estimation. Stat. Med. 2015, 34, 1779–1990. [Google Scholar] [CrossRef] [PubMed]
- Bucci, A.; Skrami, E.; Faragalli, A.; Gesuita, R.; Cameriere, R.; Carle, F.; Ferrante, L. Segmented Bayesian Calibration Approach for Estimating Age in Forensic Science. Biom. J. 2019, 61, 1575–1594. [Google Scholar] [CrossRef] [PubMed]
- Kumagai, A.; Willems, G.; Franco, A.; Thevissen, P. Age estimation combining radiographic information of two dental and four skeletal predictors in children and subadults. Int. J. Leg. Med. 2018, 132, 1769–1777. [Google Scholar] [CrossRef] [PubMed]
- De Tobel, J.; Ottow, C.; Widek, T.; Klasinc, I.; Mörnstad, H.; Thevissen, P.W.; Verstraete, K.L. Dental and Skeletal Imaging in Forensic Age Estimation: Disparities in Current Approaches and the Continuing Search for Optimization. Semin. Musculoskelet. Radiol. 2020, 24, 510–522. [Google Scholar] [CrossRef] [PubMed]
- Sklar, A. Fonctions de répartition à n dimensions et leurs marges. Inst. Statist. Univ. Paris 1959, 8, 229–231. [Google Scholar]
- Anderson, M.J.; de Valpine, P.; Punnett, A.; Miller, A.E. A pathway for multivariate analysis of ecological communities using copulas. Ecol. Evol. 2019, 9, 3276–3294. [Google Scholar] [CrossRef] [PubMed]
- Nikoloulopoulos, A.K.; Karlis, D. Multivariate logit copula model with an application to dental data. Stat. Med. 2008, 27, 6393–6406. [Google Scholar] [CrossRef] [PubMed]
- Patton, A. Estimation of multivariate copula models for time series of possibly different lengths. J. Appl. Econom. 2006, 21, 147–173. [Google Scholar] [CrossRef]
- Cameriere, R.; De Luca, S.; Cingolani, M.; Ferrante, L. Measurements of developing teeth, and carpals and epiphyses of the ulna and radius for assessing new cut-offs at the age thresholds of 10, 11, 12, 13 and 14 years. J. Forensic Leg. Med. 2015, 34, 50–54. [Google Scholar] [CrossRef] [PubMed]
- Muggeo, V.M.R. Estimating regression models with unknown break-points. Stat. Med. 2003, 22, 3055–3071. [Google Scholar] [CrossRef] [PubMed]
- Joe, H. Multivariate Models and Dependence Concepts; CRC Press: Boca Raton, FL, USA, 1997. [Google Scholar]
- Nelsen, R.B. An Introduction to Copulas; Springer: New York, NY, USA, 2006. [Google Scholar] [CrossRef]
- Rynkiewicz, A.; Bar-Yosef, O.; Smith, P. Age estimation using wrist radiography: A Bayesian approach. J. Forensic Sci. 2011, 56, 1614–1620. [Google Scholar]
- Chen, X.; Liu, J.; Wang, Y. A Bayesian approach to age estimation using dental and wrist bone maturation. J. Forensic Sci. 2018, 63, 200–206. [Google Scholar]
Copula | BIC |
---|---|
Clayton: | |
Not rotated Clayton | −40.879 |
R-Clayton C. (90°) | −119.814 |
R-Clayton C. (180°) | −40.992 |
R-Clayton C. (270°) | −56.342 |
Gumbel: | |
Not rotated Gumbel | −42.033 |
R-Gumbel C. (90°) | −79.181 |
R-Gumbel C. (180°) | −42.168 |
R-Gumbel C. (270°) | −124.289 |
Model | MAE | RMSE | IQRERR | MCI95% |
---|---|---|---|---|
Panel A: S~N; W~N | ||||
Segmented (S single predictor) | 1.13 | 1.41 | (−1.46; 0.35) | 2.62 |
Linear (W single predictor) | 1.21 | 1.68 | (−1.44; 0.67) | 2.81 |
Independent | 1.11 | 1.40 | (−1.42; 0.28) | 2.17 |
Gaussian C. | 1.03 | 1.32 | (−1.18; 0.65) | 1.92 |
R-Gumbel C. (270°) | 1.02 | 1.32 | (−1.16; 0.63) | 1.94 |
R-Clayton C. (90°) | 1.01 | 1.31 | (−1.16; 0.63) | 1.96 |
Panel B: S~ALD (τ = 0:43); W~N | ||||
Segmented (S single predictor) | 1.09 | 1.39 | (−1.46; 0.35) | 2.33 |
Linear (W single predictor) | 1.21 | 1.68 | (−1.44; 0.67) | 2.81 |
Independent | 1.01 | 1.34 | (−1.17; 0.59) | 2.07 |
Gaussian C. | 1.02 | 1.32 | (−1.16; 0.61) | 1.99 |
R-Gumbel C. (270°) | 1.03 | 1.31 | (−1.13; 0.65) | 2.41 |
R-Clayton C. (90°) | 1.01 | 1.30 | (−1.14; 0.62) | 2.57 |
Panel C: S~N; W~ALD (τ = 0:51) | ||||
Segmented (S single predictor) | 1.13 | 1.41 | (−1.46; 0.35) | 2.62 |
Linear (W single predictor) | 1.20 | 1.65 | (−1.41; 0.66) | 2.78 |
Independent | 1.28 | 1.55 | (−0.78; 1.16) | 8.51 |
Gaussian C. | 1.03 | 1.32 | (−1.10; 0.74) | 2.26 |
R-Gumbel C. (270°) | 1.02 | 1.32 | (−1.15; 0.61) | 1.98 |
R-Clayton C. (90°) | 1.01 | 1.31 | (−1.23; 0.57) | 2.06 |
Panel D: S~ALD (τ = 0:43); W~ALD (τ = 0:51) | ||||
Segmented (S single predictor) | 1.09 | 1.39 | (−1.46; 0.35) | 2.33 |
Linear (W single predictor) | 1.20 | 1.65 | (−1.41; 0.66) | 2.78 |
Independent | 1.12 | 1.41 | (−1.51; 0.18) | 3.12 |
Gaussian C. | 1.03 | 1.32 | (−1.10; 0.74) | 1.91 |
R-Gumbel C. (270°) | 1.02 | 1.30 | (−1.10; 0.74) | 1.93 |
R-Clayton C. (90°) | 1.03 | 1.31 | (−1.11; 0.73) | 1.95 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Faragalli, A.; Skrami, E.; Bucci, A.; Gesuita, R.; Cameriere, R.; Carle, F.; Ferrante, L. Combining Bayesian Calibration and Copula Models for Age Estimation. Int. J. Environ. Res. Public Health 2023, 20, 1201. https://doi.org/10.3390/ijerph20021201
Faragalli A, Skrami E, Bucci A, Gesuita R, Cameriere R, Carle F, Ferrante L. Combining Bayesian Calibration and Copula Models for Age Estimation. International Journal of Environmental Research and Public Health. 2023; 20(2):1201. https://doi.org/10.3390/ijerph20021201
Chicago/Turabian StyleFaragalli, Andrea, Edlira Skrami, Andrea Bucci, Rosaria Gesuita, Roberto Cameriere, Flavia Carle, and Luigi Ferrante. 2023. "Combining Bayesian Calibration and Copula Models for Age Estimation" International Journal of Environmental Research and Public Health 20, no. 2: 1201. https://doi.org/10.3390/ijerph20021201