Methodology In most cases,before fitting the cubic splines,an age transformation was needed to stretch the age scale for values close to zero.Despite their complexity in terms of shape,even the flexible cubic splines fail to adequately fit early infancy growth with reasonable degrees of freedom.When the degrees of freedom are increased excessively,the function can fit well in infancy but it under- smoothes at older ages.The solution is to expand the age scale when growth velocity is high and to compress it when it is low (Cole et al.,1998).A power transformation applied to age,i.e.f)=age", was a good solution for the considered cases.Therefore,prior to determining the best degrees of freedom for the parameter curves,a search was conducted for the best for the age power transformation.For this,an arbitrary starting model was used to search for the best age-transformation power()based only on the global deviance values over a preset grid of values,since the degrees of freedom remained unchanged.The grid of values ranged from 0.05 to 1 in 0.05 intervals,with the exception of the BMI-for-age standards for children younger than 24 months,for which the value 0.01 also was considered.No length/height transformation was necessary for weight-for-length/height. (a)Selecting the best model within a class of models Models were grouped in classes according to the parameters to be modelled.The alternative to modelling parameters was to fix them,e.g.v=1 or r=2.The criteria used to choose among models within the same class were the AIC and the generalized version of it with penalty equal to 3(GAIC(3)) as defined in Rigby and Stasinopoulos(2004a): GAIC(3)=-2L+3p, where L is the maximized likelihood and p is the number of parameters (or the total number of degrees of freedom).While the use of the AlC enhances the fitting of local trends,smoother curves are obtained when the model's choice is based on the GA/C(3)criterion.Consistency in the use of these two criteria was attempted across all indicators.For selecting the best combination of df(u)and df(o), both criteria were used in parallel.In cases of disagreement,AlC was used to select df(u)and GA/C(3) to select df(c),overall favouring the options which offered a good compromise between keeping estimates close to the empirical values and producing smooth curves.Only GAIC(3)values were examined to select df(v)and,whenever needed,df(t).In rare cases,other age-specific diagnostic tools were considered for selecting the model with an adequate number of degrees of freedom for the cubic splines fitting the parameter curves.Worm plots(van Buuren and Fredriks,2001)and Q-test(Royston and Wright,2000)were used conjointly for this purpose. Group-specific Q-test statistics resulting in absolute values of z1,z2,z3 or z4 that were larger than 2 were interpreted to indicate a misfit of,respectively,mean,variance,skewness or kurtosis.The overall Q-test statistics combining all groups were based on a Chi-square distribution,which assumes that observations from different groups are independent.In this case,however,given the repeated measurements in the longitudinal study component,the resulting test's p-values could be distorted slightly.To minimize this potential problem,age groups were designed to avoid repeated measurements of the same child within the same age group.The age groups were formed in time intervals(days)to achieve an approximately even sample size distribution across the entire age range of interest,especially in the cross-sectional component,where sample sizes are smaller than in the longitudinal data. For the longitudinal component,i.e.the first 24 months,time intervals were selected to preserve the longitudinal follow-up structure and avoid having multiple measurements of a given child within one age group.Note that for the longitudinal sample,age ranges were defined to correspond to specific visits,although visits did not always take place at the exact targeted age.For this reason,the constructed age group sample sizes were sometimes slightly different from the designed follow-up
Methodology 9 In most cases, before fitting the cubic splines, an age transformation was needed to stretch the age scale for values close to zero. Despite their complexity in terms of shape, even the flexible cubic splines fail to adequately fit early infancy growth with reasonable degrees of freedom. When the degrees of freedom are increased excessively, the function can fit well in infancy but it undersmoothes at older ages. The solution is to expand the age scale when growth velocity is high and to compress it when it is low (Cole et al., 1998). A power transformation applied to age, i.e. f(λ)=ageλ , was a good solution for the considered cases. Therefore, prior to determining the best degrees of freedom for the parameter curves, a search was conducted for the best λ for the age power transformation. For this, an arbitrary starting model was used to search for the best age-transformation power (λ) based only on the global deviance values over a preset grid of λ values, since the degrees of freedom remained unchanged. The grid of λ values ranged from 0.05 to 1 in 0.05 intervals, with the exception of the BMI-for-age standards for children younger than 24 months, for which the value 0.01 also was considered. No length/height transformation was necessary for weight-for-length/height. (a) Selecting the best model within a class of models Models were grouped in classes according to the parameters to be modelled. The alternative to modelling parameters was to fix them, e.g. ν=1 or τ=2. The criteria used to choose among models within the same class were the AIC and the generalized version of it with penalty equal to 3 (GAIC(3)) as defined in Rigby and Stasinopoulos (2004a): GAIC(3) = −2L + 3p, where L is the maximized likelihood and p is the number of parameters (or the total number of degrees of freedom). While the use of the AIC enhances the fitting of local trends, smoother curves are obtained when the model's choice is based on the GAIC(3) criterion. Consistency in the use of these two criteria was attempted across all indicators. For selecting the best combination of df(µ) and df(σ), both criteria were used in parallel. In cases of disagreement, AIC was used to select df(µ) and GAIC(3) to select df(σ), overall favouring the options which offered a good compromise between keeping estimates close to the empirical values and producing smooth curves. Only GAIC(3) values were examined to select df(ν) and, whenever needed, df(τ). In rare cases, other age-specific diagnostic tools were considered for selecting the model with an adequate number of degrees of freedom for the cubic splines fitting the parameter curves. Worm plots (van Buuren and Fredriks, 2001) and Q-test (Royston and Wright, 2000) were used conjointly for this purpose. Group-specific Q-test statistics resulting in absolute values of z1, z2, z3 or z4 that were larger than 2 were interpreted to indicate a misfit of, respectively, mean, variance, skewness or kurtosis. The overall Q-test statistics combining all groups were based on a Chi-square distribution, which assumes that observations from different groups are independent. In this case, however, given the repeated measurements in the longitudinal study component, the resulting test's p-values could be distorted slightly. To minimize this potential problem, age groups were designed to avoid repeated measurements of the same child within the same age group. The age groups were formed in time intervals (days) to achieve an approximately even sample size distribution across the entire age range of interest, especially in the cross-sectional component, where sample sizes are smaller than in the longitudinal data. For the longitudinal component, i.e. the first 24 months, time intervals were selected to preserve the longitudinal follow-up structure and avoid having multiple measurements of a given child within one age group. Note that for the longitudinal sample, age ranges were defined to correspond to specific visits, although visits did not always take place at the exact targeted age. For this reason, the constructed age group sample sizes were sometimes slightly different from the designed follow-up
10 Methodology visit sample sizes.Moreover,cross-sectional observations were added to the longitudinal sample between 18 and 24 months.In the cross-sectional data,it is possible that in a few cases more than one measurement from the same child occurs because of the multiple visits in Brazil and the USA, combined with the lower data density in this component.Similarly,it was impossible to break the sample into independent groups for the weight-for-length/height indicators.For this reason,the Q-test results required a conservative interpretation. Overall,Q-test results were interpreted with caution and considered simultaneously with results of worm plots(van Buuren and Fredriks,2001)which do not require any assumption and still offer very specific information about the goodness of fit for each group.The same age grouping was used as defined for the Q-test.Interpretation of results requires careful review of the shapes of the worms formed by a cubic polynomial (the red line in all worm plots)fitted to the points of the detrended Q-Q plots based on z-score values derived from the model being evaluated.A detrended Q-Q plot is presented for each age group.Confidence intervals(95%)are displayed for each of the worms(dotted curves in all worm plots).Table 7 summarizes the interpretation of various worm plot patterns.Flat worms indicate an adequate fit.The Q-test combined with the worm plot patterns provide a robust assessment of a model's goodness of fit,especially in terms of evaluating local fit. Table 7 Interpretation of various patterns in the worm plot" Shape Moment If the worm Then the Intercept Mean passes above the origin, fitted mean is too small passes below the origin, fitted mean is too large. Slope Variance has a positive slope, fitted variance is too small. has a negative slope, fitted variance is too large. Parabola Skewness has a U-shape, fitted distribution is too skew to the left. has an inverted U-shape, fitted distribution is too skew to the right. S-curve Kurtosis has an S-shape on the left tails of the fitted distribution are bent down, too light. has an S-shape on the left tails of the fitted distribution are bent up, too heavy a Reproduced from van Buuren and Fredriks(2001)with permission from John Wiley Sons Limited. Pan and Cole(2004)proposed using a new tool to guide the choice of degrees of freedom for cubic splines fitting each of the parameter curves.They suggested plotting standardized Q-statistics against the number of age groups minus the corresponding degrees of freedom,for each of the L,M and S curves of the LMS method (Cole and Green,1992).If fitting is adequate,the Q-statistics should be normally distributed with values within the range -2 to 2.This tool provides a global rather than a local test of significance and gives an accurate impression of the underlying goodness of fit because it does not depend on the precise choice of the number of groups.The proposed test is very useful for cross-sectional data where the choice of the number of groups can affect the Q-test results considerably.For example,points that are close in age but in opposite tails of the distribution generate opposing skewness when they fall into separate groups but cancel each other out when they are in the same group.This test was not implemented for the MGRS sample for two reasons.First,the largest number of observations was obtained in the study's longitudinal component,i.e.data were collected frequently at relatively well-defined ages from birth to 24 months.Second,splitting age intervals in a manner that failed to follow the study design,e.g.from birth to one month (which includes
10 Methodology visit sample sizes. Moreover, cross-sectional observations were added to the longitudinal sample between 18 and 24 months. In the cross-sectional data, it is possible that in a few cases more than one measurement from the same child occurs because of the multiple visits in Brazil and the USA, combined with the lower data density in this component. Similarly, it was impossible to break the sample into independent groups for the weight-for-length/height indicators. For this reason, the Q-test results required a conservative interpretation. Overall, Q-test results were interpreted with caution and considered simultaneously with results of worm plots (van Buuren and Fredriks, 2001) which do not require any assumption and still offer very specific information about the goodness of fit for each group. The same age grouping was used as defined for the Q-test. Interpretation of results requires careful review of the shapes of the worms formed by a cubic polynomial (the red line in all worm plots) fitted to the points of the detrended Q-Q plots based on z-score values derived from the model being evaluated. A detrended Q-Q plot is presented for each age group. Confidence intervals (95%) are displayed for each of the worms (dotted curves in all worm plots). Table 7 summarizes the interpretation of various worm plot patterns. Flat worms indicate an adequate fit. The Q-test combined with the worm plot patterns provide a robust assessment of a model's goodness of fit, especially in terms of evaluating local fit. Table 7 Interpretation of various patterns in the worm plota Shape Moment If the worm Then the Intercept Mean passes above the origin, fitted mean is too small. passes below the origin, fitted mean is too large. Slope Variance has a positive slope, fitted variance is too small. has a negative slope, fitted variance is too large. Parabola Skewness has a U-shape, fitted distribution is too skew to the left. has an inverted U-shape, fitted distribution is too skew to the right. S-curve Kurtosis has an S-shape on the left bent down, tails of the fitted distribution are too light. has an S-shape on the left bent up, tails of the fitted distribution are too heavy. a Reproduced from van Buuren and Fredriks (2001) with permission from © John Wiley & Sons Limited. Pan and Cole (2004) proposed using a new tool to guide the choice of degrees of freedom for cubic splines fitting each of the parameter curves. They suggested plotting standardized Q-statistics against the number of age groups minus the corresponding degrees of freedom, for each of the L, M and S curves of the LMS method (Cole and Green, 1992). If fitting is adequate, the Q-statistics should be normally distributed with values within the range -2 to 2. This tool provides a global rather than a local test of significance and gives an accurate impression of the underlying goodness of fit because it does not depend on the precise choice of the number of groups. The proposed test is very useful for cross-sectional data where the choice of the number of groups can affect the Q-test results considerably. For example, points that are close in age but in opposite tails of the distribution generate opposing skewness when they fall into separate groups but cancel each other out when they are in the same group. This test was not implemented for the MGRS sample for two reasons. First, the largest number of observations was obtained in the study's longitudinal component, i.e. data were collected frequently at relatively well-defined ages from birth to 24 months. Second, splitting age intervals in a manner that failed to follow the study design, e.g. from birth to one month (which includes
Methodology L measurements taken at birth,and at 7,14 and 28 days)would group together four measurements per child,thereby reducing the reliability of the Q-test results. (b)Selecting the best model across different classes of models The search for the best model was done in an add-up stepwise form,starting from the simplest class of models comprising the age transformation,if any,and the fitting of the u and o curves,while keeping fixed v=l and t=2 as described in section (a)above.The next step was to fit the v curve,fixing only t=2 and using the df(u)and df(o)selected in the previous step.Once the best model within this class of models was selected,Q-test and worm plot results were evaluated to inform the decision on whether or not to select the more complex model.In a few cases when Q-test and worm plots were not sufficient to assess the improvement offered by the more complex model,comparison of observed and fitted percentiles was used to determine if differences were of clinical significance. The fit of t was considered only when Q-test or worm plots indicated misfit with respect to kurtosis.In this case,a third class of models was considered and comparison of observed against fitted percentiles was done to assess the improvement in the final curves.Among the rare cases where this occurred, fitting the fourth parameter always led to change that was negligible in practical terms.Therefore,all the models fitted had at most 3 non-fixed parameters (u,o and v). With df(v)thus selected (i.e.when y was not fixed to value 1).a new iteration was done to re-search for df(u)and df(o).However,none of the additional iterations indicated any need to change either df(u) or df(o).A further iteration was carried out to investigate if it was necessary to change the age- transformation power.This exercise did not lead to any changes in the selected models. The methodology described above was used for all the indicators.Methodological aspects that are specific to the construction of each of the standards are described hereafter in relevant sections. As part of the internal validation for each indicator,a detailed examination was made of the differences between empirical and the fitted centiles resulting from the selected model.Comparisons were also made between the observed and expected proportions of children with measurements below selected centiles across age (or length/height for weight-for-length/height)groups.For these two diagnostic tools,evidence of systematic patterns indicative of biases and the magnitude of deviations were examined. Length/height-for-age,weight-for-age and BMI-for age curves were constructed using all available data (i.e.from birth to 71 months)but final age-based standards were truncated at 60 completed months to avoid the right-edge effect (Borghi et al.,2006).The weight-for-length standards go from 45 to 110 cm and weight-for-height from 65 to 120 cm
Methodology 11 measurements taken at birth, and at 7, 14 and 28 days) would group together four measurements per child, thereby reducing the reliability of the Q-test results. (b) Selecting the best model across different classes of models The search for the best model was done in an add-up stepwise form, starting from the simplest class of models comprising the age transformation, if any, and the fitting of the µ and σ curves, while keeping fixed ν=1 and τ=2 as described in section (a) above. The next step was to fit the ν curve, fixing only τ=2 and using the df(µ) and df(σ) selected in the previous step. Once the best model within this class of models was selected, Q-test and worm plot results were evaluated to inform the decision on whether or not to select the more complex model. In a few cases when Q-test and worm plots were not sufficient to assess the improvement offered by the more complex model, comparison of observed and fitted percentiles was used to determine if differences were of clinical significance. The fit of τ was considered only when Q-test or worm plots indicated misfit with respect to kurtosis. In this case, a third class of models was considered and comparison of observed against fitted percentiles was done to assess the improvement in the final curves. Among the rare cases where this occurred, fitting the fourth parameter always led to change that was negligible in practical terms. Therefore, all the models fitted had at most 3 non-fixed parameters (µ, σ and ν). With df(ν) thus selected (i.e. when ν was not fixed to value 1), a new iteration was done to re-search for df(µ) and df(σ). However, none of the additional iterations indicated any need to change either df(µ) or df(σ). A further iteration was carried out to investigate if it was necessary to change the agetransformation power λ. This exercise did not lead to any changes in the selected models. The methodology described above was used for all the indicators. Methodological aspects that are specific to the construction of each of the standards are described hereafter in relevant sections. As part of the internal validation for each indicator, a detailed examination was made of the differences between empirical and the fitted centiles resulting from the selected model. Comparisons were also made between the observed and expected proportions of children with measurements below selected centiles across age (or length/height for weight-for-length/height) groups. For these two diagnostic tools, evidence of systematic patterns indicative of biases and the magnitude of deviations were examined. Length/height-for-age, weight-for-age and BMI-for age curves were constructed using all available data (i.e. from birth to 71 months) but final age-based standards were truncated at 60 completed months to avoid the right-edge effect (Borghi et al., 2006). The weight-for-length standards go from 45 to 110 cm and weight-for-height from 65 to 120 cm
3. CONSTRUCTION OF THE LENGTH/HEIGHT-FOR-AGE STANDARDS 3.1 Indicator-specific methodology For the linear growth indicator,the objective was to construct a length-for-age(birth to 2 years)and height-for-age (2 to 5 years)standard using the same model and yet reflect the average difference between recumbent length and standing height.By design,children between 18 and 30 months in the cross-sectional component had both length and height measurements taken.The average difference between the two measurements in this set of 1625 children was 0.73 cm.The results by age group are shown in Table 8. Table 8 Summary of differences between recumbent length and standing height in a sample of children measured both ways Age (months) 18to<21 21to<24 24t0<27 27to≤30 18to≤30 Sample size 334 354 476 461 1625 Mean(cm)" 0.75 0.69 0.72 0.77 0.73 St Deviation(cm)" 0.61 0.67 0.61 0.61 0.62 Recumbent length minus standing height. To fit a single model for the whole age range,0.7 cm was thus added to the cross-sectional height values.After the model was fitted,the median curve was shifted back downwards by 0.7 cm for ages above two years and the coefficient of variation curve adjusted to the new median values to construct the height-for-age growth curves.The adjusted coefficient of variation(S)was calculated as follows: P* StDev MXS M M*, where M and S are,respectively,the fitted median and coefficient of variation values,and M'are the shifted-down median values:StDev is the standard deviation calculated as the median times the coefficient of variation. The curves were derived directly from a model that used cubic spline fitting functions for the median and coefficient of variation curves.The age transformation used to stretch the x-axis resulted in a large gap between the birth and day 14 measurements,and when the centiles were shrunk back to the original age scale,the cubic spline-fitted curves formed an artificial pattern in this interval.Therefore, keeping the cubic spline-fitted points at days 0 and 14,linear interpolation was applied to derive estimates of the median and the coefficient of variation curves from day 1 to 13 for the final standards. Although all available data (birth to 71 months)were used when modelling the curves,to minimize the right-edge effect the length/height-for-age and all the other age-based standards extend up to 60 completed months only. 3.2 Length/height-for-age for boys 3.2.1 Sample size There were 13 551 length/height observations for boys.The longitudinal and cross-sectional sample sizes by visit and age are shown in Tables 9 and 10. -13-
- 13 - 3. CONSTRUCTION OF THE LENGTH/HEIGHT-FOR-AGE STANDARDS 3.1 Indicator-specific methodology For the linear growth indicator, the objective was to construct a length-for-age (birth to 2 years) and height-for-age (2 to 5 years) standard using the same model and yet reflect the average difference between recumbent length and standing height. By design, children between 18 and 30 months in the cross-sectional component had both length and height measurements taken. The average difference between the two measurements in this set of 1625 children was 0.73 cm. The results by age group are shown in Table 8. Table 8 Summary of differences between recumbent length and standing height in a sample of children measured both ways Age (months) 18 to <21 21 to <24 24 to <27 27 to ≤30 18 to ≤30 Sample size 334 354 476 461 1625 Mean (cm)a 0.75 0.69 0.72 0.77 0.73 St Deviation (cm)a 0.61 0.67 0.61 0.61 0.62 a Recumbent length minus standing height. To fit a single model for the whole age range, 0.7 cm was thus added to the cross-sectional height values. After the model was fitted, the median curve was shifted back downwards by 0.7 cm for ages above two years and the coefficient of variation curve adjusted to the new median values to construct the height-for-age growth curves. The adjusted coefficient of variation (S* ) was calculated as follows: * * * M M S M StDev S × = = , where M and S are, respectively, the fitted median and coefficient of variation values, and M* are the shifted-down median values; StDev is the standard deviation calculated as the median times the coefficient of variation. The curves were derived directly from a model that used cubic spline fitting functions for the median and coefficient of variation curves. The age transformation used to stretch the x-axis resulted in a large gap between the birth and day 14 measurements, and when the centiles were shrunk back to the original age scale, the cubic spline-fitted curves formed an artificial pattern in this interval. Therefore, keeping the cubic spline-fitted points at days 0 and 14, linear interpolation was applied to derive estimates of the median and the coefficient of variation curves from day 1 to 13 for the final standards. Although all available data (birth to 71 months) were used when modelling the curves, to minimize the right-edge effect the length/height-for-age and all the other age-based standards extend up to 60 completed months only. 3.2 Length/height-for-age for boys 3.2.1 Sample size There were 13 551 length/height observations for boys. The longitudinal and cross-sectional sample sizes by visit and age are shown in Tables 9 and 10
14 Length/height-for-age,boys Table 9 Longitudinal sample sizes for length/height-for-age for boys Visit Birth 1 2 3 4 5 6 Age 0 2 wk 4 wk 6 wk 2 mo 3mo 4 mo N 893 425 424 424 424 420 419 Visit > P 9 10 11 12 13 Age 5 mo 6mo 7mo 8 mo 9m0 10m0 11mo N 420 424 420 420 416 411 422 Visit 14 15 16 17 18 19 20 Age 12 mo 14m0 16m0 18 mo 20 mo 22 mo 24 mo N 422 419 418 417 422 418 421 Table 10 Cross-sectional sample sizes for length/height-for-age for boys Age (mo) <18 18-20 21-23 24-26 27-29 30-32 33-35 N 3 177 185 238 263 232 259 Age (mo) 36-38 39-41 42-44 45-47 48-50 51-53 54-56 N 273 255 263 244 245 229 234 Age(mo) 57-59 60-62 63-65 66-68 69-71 >71 N 245 236 221 225 221 4 3.2.2 Model selection and results The model BCPE(x=age",df(u)=10,df(o)=6,v=1,t=2)served as a starting point to construct the length-for-age growth curves.Improvement of the model's fit was investigated by studying changes in global deviance at varying levels of the age-transformation power A.Table 11 shows the global deviance for a grid of values.The smallest global deviance corresponds to age-transformation power 入=0.35. Table 11 Global deviance (GD)for models within the class BCPE(x=age",df(u)=10,df()=6, v=1,t=2)for length/height-for-age for boys 0.05 0.100.150.200.250.30 0.350.40 0.45 0.50 GD339.9333.7329.1325.6323.0321.3320.8322.2326.2332.6 0.55 0.600.650.700.750.800.850.900.95 1.00 GD2340.5347.1349.4345.5337.2331.4340.8383.1479.1648.0 a In excess of 65 000 Having chosen the age-transformation power A=0.35,the search for the best df(u)and df(o)followed, comparing models in which the parameters v and t had the fixed values 1 and 2,respectively.For this, all possible combinations of df(u)ranging from 5 to 15 and df(o)from 2 to 10 were considered.Partial results are presented in Table 12
14 Length/height-for-age, boys Table 9 Longitudinal sample sizes for length/height-for-age for boys Visit Birth 1 2 3 4 5 6 Age 0 2 wk 4 wk 6 wk 2 mo 3 mo 4 mo N 893 425 424 424 424 420 419 Visit 7 8 9 10 11 12 13 Age 5 mo 6 mo 7 mo 8 mo 9 mo 10 mo 11 mo N 420 424 420 420 416 411 422 Visit 14 15 16 17 18 19 20 Age 12 mo 14 mo 16 mo 18 mo 20 mo 22 mo 24 mo N 422 419 418 417 422 418 421 Table 10 Cross-sectional sample sizes for length/height-for-age for boys Age (mo) <18 18–20 21–23 24–26 27–29 30–32 33–35 N 3 177 185 238 263 232 259 Age (mo) 36–38 39–41 42–44 45–47 48–50 51–53 54–56 N 273 255 263 244 245 229 234 Age (mo) 57–59 60–62 63–65 66–68 69–71 >71 N 245 236 221 225 221 4 3.2.2 Model selection and results The model BCPE(x=ageλ , df(µ)=10, df(σ)=6, ν=1, τ=2) served as a starting point to construct the length-for-age growth curves. Improvement of the model's fit was investigated by studying changes in global deviance at varying levels of the age-transformation power λ. Table 11 shows the global deviance for a grid of λ values. The smallest global deviance corresponds to age-transformation power λ=0.35. Table 11 Global deviance (GD) for models within the class BCPE(x=ageλ , df(µ)=10, df(σ)=6, ν=1, τ=2) for length/height-for-age for boys λ 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 GDa 339.9 333.7 329.1 325.6 323.0 321.3 320.8 322.2 326.2 332.6 λ 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 GDa 340.5 347.1 349.4 345.5 337.2 331.4 340.8 383.1 479.1 648.0 a In excess of 65 000 Having chosen the age-transformation power λ=0.35, the search for the best df(µ) and df(σ) followed, comparing models in which the parameters ν and τ had the fixed values 1 and 2, respectively. For this, all possible combinations of df(µ) ranging from 5 to 15 and df(σ) from 2 to 10 were considered. Partial results are presented in Table 12