Int J Legal Med (2015) 129:347–355 DOI 10.1007/s00414-014-1112-z
ORIGINAL ARTICLE
Age estimation in children and young adolescents for forensic purposes using fourth cervical vertebra (C4) R. Cameriere & A. Giuliodori & M. Zampi & I. Galić & M. Cingolani & F. Pagliara & L. Ferrante
Received: 28 July 2014 / Accepted: 3 November 2014 / Published online: 11 November 2014 # Springer-Verlag Berlin Heidelberg 2014
Abstract The aim of this study was to evaluate the applicability of using the growth of the body of C4 vertebra for the estimation of age in children and young adolescents. We used the fact that the proportions between the radiologic projections of the posterior and anterior sides of the C4 vertebral body, which forms a trapezoidal shape, differ with age: in younger individuals, the posterior side is higher, whereas in older individuals, the projections of the sides of the vertebral body form a rectangular shape with the two sides equal or with the anterior side slightly higher. Cephalograms of 444 Italian subjects (214 female and 230 male individuals) aged between 5 and 15 years and with no obvious development abnormalities were analyzed. The projections of the anterior side (a) and of the posterior side (b) of each C4 body were measured, and their ratio (Vba), as a value of the C4 body development, was used for age estimation. Distribution of the Vba suggested that it does not change after 13 years in female and 14 years in male subjects. Consequently, we restricted our analysis of the Vba growing model until 14 years in both sexes. We used a Bayesian calibration method to estimate chronological age as R. Cameriere : A. Giuliodori : F. Pagliara AgEstimation project, Institute of Legal Medicine, University of Macerata, Macerata, Italy M. Zampi : M. Cingolani Institute of Legal Medicine, Macerata, Italy I. Galić (*) Departments of Research in Biomedicine and Health and Dental Medicine, University of Split School of Medicine, Split, Croatia e-mail: [email protected] I. Galić University Hospital Centre Split, Split, Croatia L. Ferrante Department of Biomedical Science and Public Health, Faculty of Medicine, Polytechnic University of Marche, Ancona, Italy
function of Vba as a predicting variable. The intra- and interobserver agreement was satisfactory, using intra-class correlation coefficient of Vba on 30 randomly selected cephalograms. The mean absolute errors were 1.34 years (standard deviation 0.95) and 1.01 years (standard deviation 0.71), and the mean inter-quartile ranges of the calibrating distribution were 2.32 years (standard deviation 0.25) in male and 1.72 years (standard deviation 0.39) in female individuals, respectively. The slopes of the regression of the estimated age error to chronological age were 0.02 in male and 0.06 in female individuals, where both values did not result significantly different from 0 (p>0.12). In conclusion, although our Bayesian calibration method might not really outperform the classical regression models in the precision of its estimates, it appears to be more robust, to greatly reduce the typical bias inherent in the regression model approach, and to have the ability to incorporate multiple predictors. Keywords Age estimation . Forensic science . Fourth cervical vertebra . Children . Young adolescents
Introduction There is a continuing active discussion on penal responsibility and culpability in the judicial systems of Europe and elsewhere in the world. Most countries have a mechanism that presumes responsibility beginning from a given age (chronological criterion), and the criminal code of each country identifies a minimum age for criminal responsibility (MACR) (Table 1) [1, 2]. In response to the increasing number of cases of individuals whose documents are of unreliable origin, it is becoming increasingly necessary to have tools capable to estimate their age as accurate as possible [3–6]. Most systems within the human body change in correlation with the growth and development of the person. This
348 Table 1 Minimum age for criminal responsibility (MACR) in different countries
Int J Legal Med (2015) 129:347–355
10 years
11 years
12 years
13 years
14 years
Australia Cameroon Cook Islands Côte d’Ivoire England
Barbados Turkey
Bolivia Brazil Canada Colombia Costa Rica
Algeria Benin Burkina Faso Burundi Comoros
Bosnia and Herzegovina Bulgaria Central African Republic Croatia Democratic People’s Republic of Korea
Dominica Dominican Republic East Timor Ecuador El Salvador Eritrea Ghana Greece Honduras Ireland Israel Jamaica Netherlands Peru San Marino Uganda Venezuela
Djibouti France Gabon Guinea Haiti Madagascar Mali Monaco Nicaragua Niger Togo Tunisia
Germany Hungary Italy Japan Libyan Arab Jamahiriya Liechtenstein Macedonia Marshall Islands Mauritania New Zealand Panama Paraguay Republic of Korea Romania Rwanda Slovenia Somalia Spain
Fiji Guyana Kiribati Malaysia Nepal Niue Palau Sierra Leone Suriname Tuvalu Vanuatu Wales
primarily refers to the skeleton, which can be observed and evaluated easily using different radiological methods. Depending on the purpose of the body development assessment—whether it is about determining the age, culpability, or personal responsibility—professionals should use all available methods that can give reliable results with a specific range of confidence [7]. Teeth in development are first used to estimate the age in children and adolescents, while their regressive changes are evaluated in adults [8–10]. Permanent teeth, except third molars, usually end their development between 12 and 14 years of age, while the third molars, which appear relatively late, usually around the age of 8 years, have uneven timing of development and can mature until the age of 22 years [11–16]. However, in studying the third molars, the confidence interval of the estimated age is significantly wider when compared with the remaining permanent teeth [17, 18]. Other bone structures, especially the hand and the wrist, radius and ulna, fingers, medial clavicle, first rib, knee, or combination of some of these, were also tested to estimate the age of a person who is still developing [19–30]. Radiological imaging of the hand and the wrist was for decades considered a standard diagnostic procedure in planning and during clinical orthodontic treatments [31]. Methods for the assessment of the skeletal development are based on the comparison of radiographs using the atlases by Greulich and Pyle
[32] or Tanner et al. [33] or using Fishman’s 11 skeletal maturational indicators (SMIs) [34]. The analysis of skeletal development was aimed at assessing the moment of the pubertal development, evaluation of growth rate, and assessment of the remaining development [35]. In addition to the high validity of the method of the hand and the wrist, recognizable indicators on selected anatomical sites (the width of epiphysis, ossification, capping of epiphysis, and fusion) are better predictors of peak and end of puberty, while the onset of puberty is critical in orthodontic practice. Because of the concern about the additional radiation exposure resulting from using supplementary radiographs for the hand and wrist, other complementary methods and indicators have been evaluated [31, 36]. Lateral cephalogram films were used as a standard pretreatment record in planning for orthodontic treatment. In addition to the many important data and reference points on the teeth, bones, and soft tissues of the head needed for cephalometric analysis, projections of the cervical vertebrae are also visible on lateral cephalograms. The use of standard lateral cephalograms instead of hand and wrist X-ray images for diagnosis and monitoring of orthodontic treatment reduces radiation exposure. Methods for cervical vertebrae maturation (CVM) were introduced by Lamparski [37], who defined six stages of cervical vertebral morphology which can be applied to the second through the sixth vertebrae (C2–C6). Hassel and
Int J Legal Med (2015) 129:347–355
Farman [38] showed that Lamparski’s six-stage method for scoring CVM by evaluating the inferior border of the C2–C4 vertebra was as reliable as the hand and wrist method. Baccetti et al. [39] also used six-stage scoring by quantitative analysis of the morphologic characteristic of the C2–C4 vertebra bodies, using the presence or absence of the concavity at the lower border of C2–C4 and the shape of the body of C2–C4. Some modifications of the original Lampinski’s method were also introduced by Mito et al. [40], based on the ratios of measurements in the C3 and C4 vertebral bodies, and by San Roman et al. [41], based on studying the lower border of C2–C6 shape and body height of the C3 and C4 vertebra. De Caldas et al. [42] evaluated the maturation of C3 and C4 by measurements of the anterior, medial, and posterior body heights and anteroposterior body length of the vertebra. However, to the best of our knowledge, there are currently no studies of age assessment based on an analysis of the body of only C4 vertebra. The aim of this study was to assess the relationship between the growth of the vertebra, particularly the body of C4 vertebra, and the age of children and young adolescents for forensic purposes. We focused on the proportion between the radiologic projections of the posterior and the anterior sides of C4 vertebral body, which is clearly seen in lateral cephalograms. The body forms a trapezoidal shape, where the posterior side is higher in younger individuals. In adults, the projections of the vertebral body sides turn into a rectangular shape with either the two sides of the body being equal or the anterior side being slightly higher.
Materials and methods Sample This cross-sectional study included 444 cephalograms from 444 healthy living Italian subjects, aged between 5 and 15 years and with no obvious development abnormalities. Table 2 presents the age and gender distribution of each age category. All subjects were Italian descendents who lived in Italy. The following data were recorded for each subject: (1) dates of birth and of the radiograph, (2) gender, as reported on the radiograph, (3) ancestry, reported by either self or family. The observer was blinded to the information on the subject’s gender and age. The study was conducted in accordance also to the ethical standards laid down by the Declaration of Helsinki [43]. The chronological age of each individual was calculated by subtracting the date of the image and the date of birth. All radiographs were in a digital format and were processed by computer-aided drawing program ImageJ [44]. When necessary, digital images were submitted to contrast enhancement for easier interpretation. We decided to start with a single vertebra and chose C4 because it was most clearly visible in the radiographs included in the study. The exclusion criteria
349
for cephalograms were missing C4 in the image or difficulty of readings because of poor projection of the vertebrae. The projections of the posterior (b) and the anterior side (a) of each C4 body were measured, and their ratio (Vba) was used for age estimation. The measurement of the anterior side of the C4 body was made before the anterior side curvature toward the top of the body so that the anterior side was measured only on the straight part and not on the curved part (Fig. 1). Statistical analysis To assess intra-observer reliability, 30 cephalograms were reanalyzed after 2 weeks by the same reader. Inter-observer variability was assessed based on a random subsample of 30 cephalograms, which were separately and independently evaluated by two authors. Intra-class correlation coefficient, ICC, was used to evaluate intra- and inter-observer agreement. Although several regression models could be used to analyze the relationship between age and Vba, serious limitations concern the assumptions about the shape (normal) and variance (usually constant) of residual distributions and the unavoidable bias in age estimation when regression models are used [45]. To overcome these problems, we developed a full Bayesian calibration method for age estimation, with the asymmetric Laplace distribution as the probability model [46, 59]. In our asymmetric Laplace Bayesian calibration (ALBC) method, the term “calibration” is used to define a two-step process which 1. Used training data to establish a statistical model between the age and a related predictor, i.e., C4 body development (Vba). 2. This information was then used to estimate the unknown value of the age from the related measured value of C4 body development (Vba). Let us consider a sample of n individuals (training sample) whose ages, t1,…,tn, are known and assume that we can observe the values of the age predictor, Vba, x1,…,xn. If we have a measure y of Vba in a new individual of unknown age u, we are interested in making inferences about age u, which corresponds to the additional observation y. The Bayesian approach to this problem consists of determining the a posteriori distribution of age u, conditioned to the value of Vba of the new individual, y, and the vectors of ages t=(t1,…, tn) and the values of Vba, x=(x1,…,xn): f u y; t; x
This conditional distribution of unknown age u is called “calibrating distribution” (see Appendix).
350 Table 2 Frequency distribution by gender and age cohorts
Int J Legal Med (2015) 129:347–355
Age (years)
5
6
7
8
9
10
11
12
13
14
15
Total
Females Males Total
7 14 21
8 8 16
20 12 32
29 39 68
36 38 74
24 30 54
27 38 65
28 25 53
13 12 25
11 10 21
11 4 15
214 230 444
Mean, û, of the calibrating distribution was used for point estimate, and the confidence interval was evaluated by the quantile-based 95 % credible interval (CI95) of the calibrating distribution. To measure model performance, taking into account the several age cohorts for each gender, we did not divide the data into separate training and test sets. Hence, all the estimates were obtained using all the available data, but the performance of the model was evaluated by bootstrap resampling strategy [47]. The accuracy and precision of the models used in age estimation were evaluated by N
1. The mean absolute error (MAE), M AE ¼ N1 ∑ jt i −b ui j; i¼1
2. The slope (βERR) of the regression of the estimated age error to chronological age, the inter-quartile range (IQRERR) of the distribution of the error, 3. The mean inter-quartile range (MIQR) of the calibrating distribution, and
Fig. 1 Example of the posterior (b) and anterior (a) sides of the fourth cervical vertebral body. Anterior side of the body is measured up to the point where anterior side (a) curves (C1) toward the superior side (C2) of the vertebral body
4. The mean width of the quantile-based 95 % credible interval (MCI95) of the calibrating distribution. Statistical analyses were performed with the R statistical program [48].
Results The Pearson correlations between age and Vba are 0.89 and 0.78 for female and male individuals, respectively. Distribution of the Vba suggested that it does not change after 13 years in female and 14 years in male individuals (Fig. 2). We restricted our analysis of the Vba growing model until 14 years in both sex (203 female and 226 male individuals). Age distribution gradually increased as Vba increased in both male and female individuals (Fig. 2). There were no statistically significant intra-observer differences between paired sets of measurements carried out on re-examined cephalograms (intra-class correlation coefficient (ICC)=0.93). Similar result was obtained considering the agreement between the two observers (ICC=0.92). Wilcoxon–Mann–Whitney test revealed a statistical difference of Vba distribution between male and female individuals (p0 and a skewness parameter 0
ORIGINAL ARTICLE
Age estimation in children and young adolescents for forensic purposes using fourth cervical vertebra (C4) R. Cameriere & A. Giuliodori & M. Zampi & I. Galić & M. Cingolani & F. Pagliara & L. Ferrante
Received: 28 July 2014 / Accepted: 3 November 2014 / Published online: 11 November 2014 # Springer-Verlag Berlin Heidelberg 2014
Abstract The aim of this study was to evaluate the applicability of using the growth of the body of C4 vertebra for the estimation of age in children and young adolescents. We used the fact that the proportions between the radiologic projections of the posterior and anterior sides of the C4 vertebral body, which forms a trapezoidal shape, differ with age: in younger individuals, the posterior side is higher, whereas in older individuals, the projections of the sides of the vertebral body form a rectangular shape with the two sides equal or with the anterior side slightly higher. Cephalograms of 444 Italian subjects (214 female and 230 male individuals) aged between 5 and 15 years and with no obvious development abnormalities were analyzed. The projections of the anterior side (a) and of the posterior side (b) of each C4 body were measured, and their ratio (Vba), as a value of the C4 body development, was used for age estimation. Distribution of the Vba suggested that it does not change after 13 years in female and 14 years in male subjects. Consequently, we restricted our analysis of the Vba growing model until 14 years in both sexes. We used a Bayesian calibration method to estimate chronological age as R. Cameriere : A. Giuliodori : F. Pagliara AgEstimation project, Institute of Legal Medicine, University of Macerata, Macerata, Italy M. Zampi : M. Cingolani Institute of Legal Medicine, Macerata, Italy I. Galić (*) Departments of Research in Biomedicine and Health and Dental Medicine, University of Split School of Medicine, Split, Croatia e-mail: [email protected] I. Galić University Hospital Centre Split, Split, Croatia L. Ferrante Department of Biomedical Science and Public Health, Faculty of Medicine, Polytechnic University of Marche, Ancona, Italy
function of Vba as a predicting variable. The intra- and interobserver agreement was satisfactory, using intra-class correlation coefficient of Vba on 30 randomly selected cephalograms. The mean absolute errors were 1.34 years (standard deviation 0.95) and 1.01 years (standard deviation 0.71), and the mean inter-quartile ranges of the calibrating distribution were 2.32 years (standard deviation 0.25) in male and 1.72 years (standard deviation 0.39) in female individuals, respectively. The slopes of the regression of the estimated age error to chronological age were 0.02 in male and 0.06 in female individuals, where both values did not result significantly different from 0 (p>0.12). In conclusion, although our Bayesian calibration method might not really outperform the classical regression models in the precision of its estimates, it appears to be more robust, to greatly reduce the typical bias inherent in the regression model approach, and to have the ability to incorporate multiple predictors. Keywords Age estimation . Forensic science . Fourth cervical vertebra . Children . Young adolescents
Introduction There is a continuing active discussion on penal responsibility and culpability in the judicial systems of Europe and elsewhere in the world. Most countries have a mechanism that presumes responsibility beginning from a given age (chronological criterion), and the criminal code of each country identifies a minimum age for criminal responsibility (MACR) (Table 1) [1, 2]. In response to the increasing number of cases of individuals whose documents are of unreliable origin, it is becoming increasingly necessary to have tools capable to estimate their age as accurate as possible [3–6]. Most systems within the human body change in correlation with the growth and development of the person. This
348 Table 1 Minimum age for criminal responsibility (MACR) in different countries
Int J Legal Med (2015) 129:347–355
10 years
11 years
12 years
13 years
14 years
Australia Cameroon Cook Islands Côte d’Ivoire England
Barbados Turkey
Bolivia Brazil Canada Colombia Costa Rica
Algeria Benin Burkina Faso Burundi Comoros
Bosnia and Herzegovina Bulgaria Central African Republic Croatia Democratic People’s Republic of Korea
Dominica Dominican Republic East Timor Ecuador El Salvador Eritrea Ghana Greece Honduras Ireland Israel Jamaica Netherlands Peru San Marino Uganda Venezuela
Djibouti France Gabon Guinea Haiti Madagascar Mali Monaco Nicaragua Niger Togo Tunisia
Germany Hungary Italy Japan Libyan Arab Jamahiriya Liechtenstein Macedonia Marshall Islands Mauritania New Zealand Panama Paraguay Republic of Korea Romania Rwanda Slovenia Somalia Spain
Fiji Guyana Kiribati Malaysia Nepal Niue Palau Sierra Leone Suriname Tuvalu Vanuatu Wales
primarily refers to the skeleton, which can be observed and evaluated easily using different radiological methods. Depending on the purpose of the body development assessment—whether it is about determining the age, culpability, or personal responsibility—professionals should use all available methods that can give reliable results with a specific range of confidence [7]. Teeth in development are first used to estimate the age in children and adolescents, while their regressive changes are evaluated in adults [8–10]. Permanent teeth, except third molars, usually end their development between 12 and 14 years of age, while the third molars, which appear relatively late, usually around the age of 8 years, have uneven timing of development and can mature until the age of 22 years [11–16]. However, in studying the third molars, the confidence interval of the estimated age is significantly wider when compared with the remaining permanent teeth [17, 18]. Other bone structures, especially the hand and the wrist, radius and ulna, fingers, medial clavicle, first rib, knee, or combination of some of these, were also tested to estimate the age of a person who is still developing [19–30]. Radiological imaging of the hand and the wrist was for decades considered a standard diagnostic procedure in planning and during clinical orthodontic treatments [31]. Methods for the assessment of the skeletal development are based on the comparison of radiographs using the atlases by Greulich and Pyle
[32] or Tanner et al. [33] or using Fishman’s 11 skeletal maturational indicators (SMIs) [34]. The analysis of skeletal development was aimed at assessing the moment of the pubertal development, evaluation of growth rate, and assessment of the remaining development [35]. In addition to the high validity of the method of the hand and the wrist, recognizable indicators on selected anatomical sites (the width of epiphysis, ossification, capping of epiphysis, and fusion) are better predictors of peak and end of puberty, while the onset of puberty is critical in orthodontic practice. Because of the concern about the additional radiation exposure resulting from using supplementary radiographs for the hand and wrist, other complementary methods and indicators have been evaluated [31, 36]. Lateral cephalogram films were used as a standard pretreatment record in planning for orthodontic treatment. In addition to the many important data and reference points on the teeth, bones, and soft tissues of the head needed for cephalometric analysis, projections of the cervical vertebrae are also visible on lateral cephalograms. The use of standard lateral cephalograms instead of hand and wrist X-ray images for diagnosis and monitoring of orthodontic treatment reduces radiation exposure. Methods for cervical vertebrae maturation (CVM) were introduced by Lamparski [37], who defined six stages of cervical vertebral morphology which can be applied to the second through the sixth vertebrae (C2–C6). Hassel and
Int J Legal Med (2015) 129:347–355
Farman [38] showed that Lamparski’s six-stage method for scoring CVM by evaluating the inferior border of the C2–C4 vertebra was as reliable as the hand and wrist method. Baccetti et al. [39] also used six-stage scoring by quantitative analysis of the morphologic characteristic of the C2–C4 vertebra bodies, using the presence or absence of the concavity at the lower border of C2–C4 and the shape of the body of C2–C4. Some modifications of the original Lampinski’s method were also introduced by Mito et al. [40], based on the ratios of measurements in the C3 and C4 vertebral bodies, and by San Roman et al. [41], based on studying the lower border of C2–C6 shape and body height of the C3 and C4 vertebra. De Caldas et al. [42] evaluated the maturation of C3 and C4 by measurements of the anterior, medial, and posterior body heights and anteroposterior body length of the vertebra. However, to the best of our knowledge, there are currently no studies of age assessment based on an analysis of the body of only C4 vertebra. The aim of this study was to assess the relationship between the growth of the vertebra, particularly the body of C4 vertebra, and the age of children and young adolescents for forensic purposes. We focused on the proportion between the radiologic projections of the posterior and the anterior sides of C4 vertebral body, which is clearly seen in lateral cephalograms. The body forms a trapezoidal shape, where the posterior side is higher in younger individuals. In adults, the projections of the vertebral body sides turn into a rectangular shape with either the two sides of the body being equal or the anterior side being slightly higher.
Materials and methods Sample This cross-sectional study included 444 cephalograms from 444 healthy living Italian subjects, aged between 5 and 15 years and with no obvious development abnormalities. Table 2 presents the age and gender distribution of each age category. All subjects were Italian descendents who lived in Italy. The following data were recorded for each subject: (1) dates of birth and of the radiograph, (2) gender, as reported on the radiograph, (3) ancestry, reported by either self or family. The observer was blinded to the information on the subject’s gender and age. The study was conducted in accordance also to the ethical standards laid down by the Declaration of Helsinki [43]. The chronological age of each individual was calculated by subtracting the date of the image and the date of birth. All radiographs were in a digital format and were processed by computer-aided drawing program ImageJ [44]. When necessary, digital images were submitted to contrast enhancement for easier interpretation. We decided to start with a single vertebra and chose C4 because it was most clearly visible in the radiographs included in the study. The exclusion criteria
349
for cephalograms were missing C4 in the image or difficulty of readings because of poor projection of the vertebrae. The projections of the posterior (b) and the anterior side (a) of each C4 body were measured, and their ratio (Vba) was used for age estimation. The measurement of the anterior side of the C4 body was made before the anterior side curvature toward the top of the body so that the anterior side was measured only on the straight part and not on the curved part (Fig. 1). Statistical analysis To assess intra-observer reliability, 30 cephalograms were reanalyzed after 2 weeks by the same reader. Inter-observer variability was assessed based on a random subsample of 30 cephalograms, which were separately and independently evaluated by two authors. Intra-class correlation coefficient, ICC, was used to evaluate intra- and inter-observer agreement. Although several regression models could be used to analyze the relationship between age and Vba, serious limitations concern the assumptions about the shape (normal) and variance (usually constant) of residual distributions and the unavoidable bias in age estimation when regression models are used [45]. To overcome these problems, we developed a full Bayesian calibration method for age estimation, with the asymmetric Laplace distribution as the probability model [46, 59]. In our asymmetric Laplace Bayesian calibration (ALBC) method, the term “calibration” is used to define a two-step process which 1. Used training data to establish a statistical model between the age and a related predictor, i.e., C4 body development (Vba). 2. This information was then used to estimate the unknown value of the age from the related measured value of C4 body development (Vba). Let us consider a sample of n individuals (training sample) whose ages, t1,…,tn, are known and assume that we can observe the values of the age predictor, Vba, x1,…,xn. If we have a measure y of Vba in a new individual of unknown age u, we are interested in making inferences about age u, which corresponds to the additional observation y. The Bayesian approach to this problem consists of determining the a posteriori distribution of age u, conditioned to the value of Vba of the new individual, y, and the vectors of ages t=(t1,…, tn) and the values of Vba, x=(x1,…,xn): f u y; t; x
This conditional distribution of unknown age u is called “calibrating distribution” (see Appendix).
350 Table 2 Frequency distribution by gender and age cohorts
Int J Legal Med (2015) 129:347–355
Age (years)
5
6
7
8
9
10
11
12
13
14
15
Total
Females Males Total
7 14 21
8 8 16
20 12 32
29 39 68
36 38 74
24 30 54
27 38 65
28 25 53
13 12 25
11 10 21
11 4 15
214 230 444
Mean, û, of the calibrating distribution was used for point estimate, and the confidence interval was evaluated by the quantile-based 95 % credible interval (CI95) of the calibrating distribution. To measure model performance, taking into account the several age cohorts for each gender, we did not divide the data into separate training and test sets. Hence, all the estimates were obtained using all the available data, but the performance of the model was evaluated by bootstrap resampling strategy [47]. The accuracy and precision of the models used in age estimation were evaluated by N
1. The mean absolute error (MAE), M AE ¼ N1 ∑ jt i −b ui j; i¼1
2. The slope (βERR) of the regression of the estimated age error to chronological age, the inter-quartile range (IQRERR) of the distribution of the error, 3. The mean inter-quartile range (MIQR) of the calibrating distribution, and
Fig. 1 Example of the posterior (b) and anterior (a) sides of the fourth cervical vertebral body. Anterior side of the body is measured up to the point where anterior side (a) curves (C1) toward the superior side (C2) of the vertebral body
4. The mean width of the quantile-based 95 % credible interval (MCI95) of the calibrating distribution. Statistical analyses were performed with the R statistical program [48].
Results The Pearson correlations between age and Vba are 0.89 and 0.78 for female and male individuals, respectively. Distribution of the Vba suggested that it does not change after 13 years in female and 14 years in male individuals (Fig. 2). We restricted our analysis of the Vba growing model until 14 years in both sex (203 female and 226 male individuals). Age distribution gradually increased as Vba increased in both male and female individuals (Fig. 2). There were no statistically significant intra-observer differences between paired sets of measurements carried out on re-examined cephalograms (intra-class correlation coefficient (ICC)=0.93). Similar result was obtained considering the agreement between the two observers (ICC=0.92). Wilcoxon–Mann–Whitney test revealed a statistical difference of Vba distribution between male and female individuals (p0 and a skewness parameter 0
