Introduction 1 13 15 3 6 9 10 17 18 4 13 5 15 16 In this study, we aim to assess from a large sample whether or not variables such as sex, socioeconomic status, and ethnicity affect the probability of having completely fused clavicles at a certain age. We further aim to compare results from X-ray and CT scan examinations with those from studies based on dry bone specimens. To do so, we have combined data from different studies and performed a binomial logistic regression analysis. We then illustrate how the resulting model can be applied to estimate the probability of completely fused clavicles at a given age. Data 3 6 8 10 13 15 18 19 6 18 1 1 Fig. 1 Age distribution of individuals in the sample  Table 1 Number of individuals per category Category Values Total number of individuals 3,552 Sex Males 2,133 Females 1,123 Unspecified 296 a b 1,316 Afro-American descent 287 Asian descent 695 Unspecified 1,254 Method Dry bone specimen 1,374 X-ray 1,326 CT scan 852 a 6 b Statistical analysis Analysis of the data was performed using the SPSS (version 10.0) software package for statistical analysis. We carried out a binomial logistic regression analysis to assess the effect of various variables on the probability of mature clavicles. In logistic regression, the dependent variable is transformed into a logit variable, i.e., the natural log of the odds of the dependent occurring or not. This transformation ensures that the estimated probabilities are between 0 and 1. A logit model is a form of the generalized linear model. \documentclass[12pt]{minimal}
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\begin{document}$${\text{ln}}{\left[ {p \mathord{\left/ {\vphantom {p {{\left( {{\text{1 - }}p} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {{\text{1 - }}p} \right)}}} \right]} = \alpha {\text{ }} + {\text{ }}\beta _{{{\text{age}}}} {\text{ $   \times  $ age }} + {\text{ }}\beta _{{{\text{HDI}}}} {\text{ $   \times  $ HDI }} + {\text{ }}\gamma _{{{\text{sex}}}} {\text{ }} + {\text{ }}\gamma _{{{\text{ethnicity}}}} {\text{ }} + {\text{ }}\gamma _{{{\text{method}}}} {\text{ }} + {\text{ }}\beta _{{{\text{sex $   \times  $ age}}}} {\text{ $   \times  $ age}} + \beta _{{{\text{ethnicity}} * {\text{age}}}} {\text{ $   \times  $ age}} + \beta _{{{\text{method}} * {\text{age}}}} {\text{ $   \times  $ }}age$$\end{document} p α β γ p 15 5 R 2 Results 2 p Table 2 Estimated parameters of logistic regression models for the logit of the probability of mature clavicles Model Independent variables B Standard error p N a   0.355 N Age 0.629 0.034 0.000 Sex (ref: male)   0.026  Female 0.368 0.166 Human development index 2.746 0.590 0.000 Method (ref: dry bone specimen)   0.000  X-ray −21.046 3.055  CT scan −7.624 2.346 Method × age   0.000  X-ray 1.029 0.135  CT scan 0.423 0.104 Constant −18.638 1.002 0.000 N Age 0.630 0.034 0.000 Human development index 3.012 0.584 0.000 Method (ref: dry bone specimen)   0.000  X-ray −20.729 3.060  CT scan −7.526 1.928 Method × age   0.000  X-ray 1.023 0.135  CT scan 0.401 0.083 Constant −18.803 0.998 0.000 a p p R 2 Model M3, which could be fitted to all available data on clavicle fusion, resulted in comparable model parameters for the dummy variable for CT scans. Confounding by an observer effect is therefore reduced. Application of the model \documentclass[12pt]{minimal}
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\begin{document}$${\text{ln}}{\left[ {p \mathord{\left/ {\vphantom {p {{\left( {1 - p} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 - p} \right)}}} \right]} = {\text{ }} - {\text{18}}{\text{.638 }} + {\text{ 0}}{\text{.629 $   \times  $ age }} + {\text{ 2}}{\text{.746 $   \times  $ HDI }} + {\text{ }}\gamma _{{{\text{sex}}}} {\text{ }} + {\text{ }}\gamma _{{{\text{method}}}} {\text{ }} + {\text{ }}\beta _{{{\text{method $   \times  $ age}}}} {\text{ $   \times  $ age}}{\text{.}}$$\end{document} 2 \documentclass[12pt]{minimal}
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\begin{document}$${p{\left( {\operatorname{age} } \right)} = e^{ \wedge } {\left( { - 18.638 + 0.629{\text{ $   \times  $ }}\operatorname{age}  + {\text{ }}2.746{\text{ $   \times  $ }}HDI{\text{ }} + {\text{ }}\gamma _{{sex}}  + {\text{ }}\gamma _{{method}} {\text{ }} + {\text{ }}\beta _{{\operatorname{method} {\text{ $   \times  $ }}\operatorname{age} }} {\text{ $   \times  $ }}age} \right)}} \mathord{\left/ {\vphantom {{p{\left( {\operatorname{age} } \right)} = e^{ \wedge } {\left( { - 18.638 + 0.629{\text{ $   \times  $ }}\operatorname{age}  + {\text{ }}2.746{\text{ $   \times  $ }}HDI{\text{ }} + {\text{ }}\gamma _{{sex}}  + {\text{ }}\gamma _{{method}} {\text{ }} + {\text{ }}\beta _{{\operatorname{method} {\text{ $   \times  $ }}\operatorname{age} }} {\text{ $   \times  $ }}age} \right)}} {\left( {1 + e\,^{ \wedge } } \right.\,{\left( { - 18.638{\text{ }} + {\text{ }}0.629{\text{ $   \times  $ }}age{\text{ }} + {\text{ }}2.746{\text{ $   \times  $ }}HDI{\text{ }} + {\text{ }}\gamma _{{sex}} {\text{ }} + {\text{ }}\gamma _{{method}} {\text{ }} + {\text{ }}\beta _{{method{\text{ $   \times  $ }}age}} {\text{ $   \times  $ }}\operatorname{age} } \right)}}}} \right. \kern-\nulldelimiterspace} {\left( {1 + e\,^{ \wedge } } \right.\,{\left( { - 18.638{\text{ }} + {\text{ }}0.629{\text{ $   \times  $ }}age{\text{ }} + {\text{ }}2.746{\text{ $   \times  $ }}HDI{\text{ }} + {\text{ }}\gamma _{{sex}} {\text{ }} + {\text{ }}\gamma _{{method}} {\text{ }} + {\text{ }}\beta _{{method{\text{ $   \times  $ }}age}} {\text{ $   \times  $ }}\operatorname{age} } \right)}}.$$\end{document} Based on the currently available data on clavicle fusion, we can now predict the probability of having mature clavicles for different individuals. Here, we provide a few examples. The predicted probability of mature clavicles is 0.016 (1.6%) for a contemporary 19-year-old German male (HDI is 0.932). For a 20- or 21-year-old German male, this probability increases to 0.029 (2.9%) and 0.054 (5.4%), respectively. For females, the predicted probability values are increased to 0.023 (2.3%), 0.042 (4.2%), and 0.076 (7.6%), respectively. However, if the individuals have a lower socioeconomic status, these values are decreased. For contemporary 19-, 20-, and 21-year-old males from India (HDI is 0.611), for instance, they are 0.007 (0.7%), 0.012 (1.2%), and 0.023 (2.3%), respectively. When diagnosed from X-ray or CT scan, the values increase. For the 21-year-old Indian male, it would be 0.040 (4%) if diagnosed from an X-ray and 0.076 (7.6%) if diagnosed from a CT scan (composed using a slice thickness that is far from optimal). 2 3 2 3 Fig. 2 Predicted probability of having completely fused clavicles: a comparison between German males, German females and Indian males  Fig. 3 Predicted probability of being diagnosed with mature clavicles: a comparison between examination by means of dry bone specimens, X-rays and CT scans  Discussion 12 6 1 11 12 14 2 The method of clavicle examination appeared to also be significant. The significant interaction term method × age indicates that we should assume a difference in the estimated effect of age depending on the method applied. For the age interval in which mature clavicles have been observed, the predicted probability of being diagnosed with mature clavicles is greater when X-rays or CT scans are used instead of dry bone specimens. It is noteworthy that the difference in the predicted probability is initially very small, but becomes increasingly greater with increasing age during a limited age interval. When keeping in mind that the curve for the predicted probability will be s-shaped, one may image that the difference is again only minimal for the older individuals. A number of theories can be postulated to explain a difference in the predicted probabilities between methods of examination. Inappropriate slice thickness 7 5 15 16 Persistent small grooves or notches between diaphysis and epiphysis 6 9 5 Reduced visibility of the epiphyseal disc on X-rays Interobserver variability Expectation bias 13 15 Statistical artifact shape Conclusions The probability of having mature clavicles at a certain age is affected by the sex and socioeconomic status of the individual. The probability of being diagnosed with mature clavicles further appears affected by the method of clavicle examination. Predicted probabilities from the model may be used to illustrate acceptance or rejection of individual age claims. Please note that the odds on having mature clavicles given a certain age should not be confused with the (posterior) odds of having reached a certain age given that the clavicles have matured. These probabilities are only equal if we assume a prior odds of 1.