Matskási István (szerk.): A Magyar Természettudományi Múzeum évkönyve 94. (Budapest 2002)

Radoinova, D., Tenekedjiev, K. ; Yordanov, Y.: Stature estimation according to bone length in Hungarian population

Table 1. Means (x), standard deviations (SD), minimal (min) and maximal (max) values for the measurements of stature, long bones and age used in this study Females (n=83) Males (n = 186) x SD min max X SD min max Age (years) 50.93 12.17 20 66 52.23 11.68 20 66 Stature (cm) 162.8 5.72 148 174.5 171.4 6.27 157 189 Humerus (cm) 32.90 2.36 26.9 38.0 33.76 2.42 28.3 40.1 Fibula (cm) 35.12 2.41 29.6 44.4 36.72 2.55 31.2 43.8 Tibia (cm) 35.03 2.62 29.5 40.9 36.57 2.72 30.9 44.3 four F-tests (RAMSEY, WHITE, GLEJSER & GOLDFELD-QUANDT) and by one x

2-test (BREUSCH & PAGAN) in eleven modifications (linear, squared, root and reciprocal) (MADDALA 1988, GUJARATI 1995). When heteroskedasticity was detected, four regression models were built for the residuals' module as a squared, linear, reciprocal and (if possible) root function of the corresponding bone lengths. Let an adequate model (according to an ANOVA-test) be an acceptable one but with adjusted coefficient of multiple determination (R

2

), not less than the predetermined boundary value (Amin)- The accepted model will be the one of the acceptable models, which has the greatest R

2 (TENEKEDJIEV & RADOINOVA 2001 ). The logic of this approach is that R

2 shows what is the percentage of the initial variance of the residuals of the constant model, which is explained using the newly developed regression dependence (of course taking into account the degree of freedom). On the other hand heteroskedasticity means a regular change of the residuals variance. For that reason if the variance cannot be explained sufficiently enough from the regression dependence of the residuals' module then it meant that heteroskedasticity itself is not that regular and as a consequence is practically negligible, even if it is statistically significant. Let the acceptable model give the expected value of the residual e

( = E[ I residual, I ] for experiment No. i. Then the regression equation required can be built using the method of WLS, where a regression of the stature divided by e^s built on the predicting bones, divided by ei and on 1/ e

( without intercept term. In this way the heteroskedasticity of this model is tested again and if it is eliminated the confidence intervals of the coefficients, the covariance matrix, the characteristics of the standard error, and the coefficients of multiple determination (R

2 and R

2

) are calculated. The adequacy of the model is checked by an ANOVA-test. The confidence intervals of the predicted maximal stature are calculated with respect to the standard error at the point of prediction and the Mahalonobis distance between the point of prediction and the mean point of the learning sample in the space of regressors (MADDALA 1988, PRESS et al. 1992). The applicability of 16 well-known formulae: TROTTER, M. and GLESER, G. for either sex on H, on Fi, on T and on both H and T, BREITINGER, E. for males on H and on T, DUPERTUIS, W. and HADDEN, J. for either sex on H and on T and BACH, H. for females on H and T is assessed. The adequacy of these reference formulae is estimated within the limits of the regression parameters (Fig. 3) as a decreasing function of the Mahalonobis distance between the vectors corresponding to the two models (the one built on the gathered data and the tested reference model), with the presupposition of a multi-dimensional normal distribution of the parameters (PRESS etal. 1992).