Cross-sectional study of height and weight in the population of Andalusia from age 3 to adulthood

Target population and considerations about the sample

Once the objectives were clear, it was necessary to make decisions about the sample that would be the object of our study, bearing in mind that a transversal study had been chosen.

In order to achieve the greatest possible representation of the Andalusian population from the age of 3 until they reach final height two samples of individuals were obtained using different sampling systems to assess their weight and height.

Given that the aim of the first sample was to describe the growth in height and weight of the Andalusian child and youth population from the age of 3 to 18 including both sexes and create standards of reference for weight, height and BMI for the Andalusian population within this age range, this first sample needed to represent the whole Andalusian population from the age of 3 to 18. The ideal method, from a methodological viewpoint, for taking a sample of that population would be through simple random sampling taken from the population census; however, this would not be viable, economically speaking. We could obtain a less expensive sample by going to places that are attended by children of these ages, to weigh and measure them.

These places are, of course, education centres, where the population between the ages of 3 and 16 is obliged to attend. Moreover, apart from the age range, if the schooling rate is high enough, a fairly reasonable degree of representation could be achieved.

Data from the Education, Science, and Sports Ministry, show that in 2002–2003 the schooling rate in Andalusia was 96.1% for High school and Professional Education (16–17 years of age) and 17.4% for higher grades of Professional Education (18–19 years of age). Thus, up to the age of 17, the schooling rate in Andalusia is high enough to avoid significant skewness when it comes to taking samples of children and young people from education centres. At the age of 18, the number of students in non-university education is very low and makes it very difficult to sample this age group using only non-university education institutions.

On the other hand, the data from the 2001 census by the National Statistics Institute (Instituto Nacional de Estadística) about rates of schooling (at any level, including university) indicate that, at that time, the schooling rate for 16-year olds was 81.9%, for 17-year olds 71.8% and for 18-year olds 62.2%. Therefore, population aged 16 and 17 would be sufficiently covered by going to education centres, while for 18-year olds it would be necessary to obtain data from other sources in order to cover such deficiencies.

Despite everything, the problem was not serious given that the age where we might encounter problems is one where the variance is very large. This variance is present in all population strata so that the results obtained would not be far from reality in any case.

It was decided to take a random sample per school and within such sample select classes from each year, from which a sample of pupils in the age range being considered would be taken. This sample (despite, as explained later, the risk being reduced to a minimum) could introduce skewness in the assessment due to the resemblance of the pupils in each class. This skewness could artificially reduce the variance of the parameter studied at a certain age, thus strongly affecting the assessment of extreme percentiles. It will be corrected by using adjustments with random effects models to take this fact into account.

For the second sample, concerning the young population between the ages of 18 and 23, it was not viable to use the same sampling method. For this group, 6 non-university education centres were used, where different professional modules were taught and where data were collected from classes in a similar way to the sample of under 18-year olds. In addition, to increase the size of the sample, different university departments in Granada were selected: The School of Health Sciences, The Medicine Faculty, The Civil Engineering School, The Pharmacy Faculty, The Science Faculty and the Computer Engineering School. In all these centres, students in their first, second and third years were measured.

Given the economic determinants, although the statistical quality of this second sample was worse due to the fact that it was not random, this type of sampling was chosen assuming that the strong variance of the different measures would assure a representation of extreme values that would not be reduced even though the sample was not random. On the other hand, if, as was expected, from the age of 18 or 19 the behaviour of the measurements was asyntotic, ages could be put in one group where extreme measures are more probable thus correcting a possible skewness, stemming from underestimating the population's variance.

Types of education centres and sample size

According to data from the Andalusian Board of Education for the academic year 2003–2004, education centres were classified as Private Teaching Centres, Infant and Primary Schools and Institutes for Secondary Education. Moreover, centres are not evenly distributed across the region, but rather they are placed according to the size of the population in a given area. Therefore, there were two already identified criteria to stratify: the type of centre and the size of the population.

The global sample was distributed into 12 strata, in principle, in proportion to the number of pupils in each stratum. Although knowing the distribution of the population in the centres and the pupils in each stratum, it is always possible to calculate the probability of one unit being selected for sampling, and correct this using inverse probability weighting, in case an imbalance in the sample by strata is considered necessary. As explained below, this was the strategy used for sampling.

Considering our objective, an additional source for stratification would be each of the age groups from 3 to 18, which would give us 16 strata. In fact, for the sake of accuracy, we would consider not 16 strata, but 32, given that we would choose children of an age calculated in years and "half years"; this way strata would reflect the following ages: 3, 3.5, 4, 4.5, 5, 5.5, .....17, 17.5, 18, 18.5.

Besides, another obvious source of stratification would be the child's sex, due to the fact that the variables measured are different from a very early age in boys and girls.

In terms of the calculation of the sample size we should comment on the normality of the variables involved in the study. For each age, height can be considered to follow a normal distribution, but this is not the case for weight or BMI, calculated based on both of these. Despite this, many authors have managed to normalize these variables using transformations; this way, we consider that if the children's weight for each age group is not a normal variable, once transformed it would become normal, which means that we would not have to re-calculate using non-parametric methods for the size of the sample.

Having discussed the prior considerations, we can concentrate on the calculation of the sample size necessary for the study. Following the methodology set out by Linnet [19] for the calculation of sample size, the key would be the ratio between the width of reference range and the width of the confidence interval for the extremes of the said reference range. Accordingly, for a reasonable quotient between both widths, such as 20%, we would need 126 children per each age group and sex.

However, these calculations do not take into account that the percentiles in the height and weight curves for age are obtained from the ratio of such variables with age using a regression model. Royston [20] addresses this problem, and using his equations the required sample size is reduced due to the variance reduction that occurs when using regression. In figures, such reduction means that if we want a ratio of 0.20 for each group of age and sex, we would only need 74 children. Based on this fact and bearing in mind that it would be applicable to simple random sampling, the size of our global sample would be at least 2,368 males and 2,368 females.

We had opted for a multistage sample. The first stage included a random selection of the education centre, while the second stage included a random selection of the children within the class of their year in the education centre. This leads to a cluster sampling that requires an increase in the size of the sample. Such increase, which is called the design effect, depends on the size of the sample in each centre as well as the degree of resemblance between children in each centre (intraclass correlation coefficient). The larger the effect of the intraclass coefficient, and the larger the sample group in each centre, the larger the design effect. This design effect shows us a coefficient by which we should multiply the size of the sample calculated as a simple random sample so that when we obtain it in clusters we have the same capacity as a simple random sample.

We would choose our sample for each age group (3, 3.5, 4, 4.5 years...) so that in fact our sample would not have to be too big; it would normally be about 4 or 5 pupils in each age group per school. So, supposing we chose 4 children of each age from all the children of the school and that the intraclass correlation coefficient was 5%, the value of the design effect would be 1.15. Therefore, our total sample would be approximately 1.15 × 4608 ≈ 5300 pupils of whom half would be female and half male. This means around 166 pupils per age group of whom half would be female and half male.

The choice of 0.05 as the intra-class correlation coefficient was made on the assumption that although the children in class would resemble each other, variance increases with age. Therefore, in classes of older children variance between observations is guaranteed, while in the case of younger children an important variance is also guaranteed due to the fact that the age range within a class, although small, is strong enough to mean that the intra-class correlation is not very large.

In the case of the sample of students between the ages of 18 and 23, the size of the sample calculated following the same steps included 2000 people.

Sample distribution in education centres

As discussed earlier, knowing the distribution of the population in the centres and that of pupils in each stratum, it is always possible to calculate the probability of a unit being selected for sampling and to correct using inverse probability weighting, if an imbalance in the sample strata is considered necessary.

We made use of this methodological resource, with the exact intention of under-representing samples from centres with a smaller population, since otherwise it would be more costly in resources and time to reach those centres, which would then lead to a different weighting of results to take this additional imbalance into account. Bearing this in mind, the distribution shown in the column "Definitive Sample" in Table 2 was obtained, this distribution was used for the study.

The final procedure for sampling that was used as an alternative to simple random sampling based on census, requires two precautionary measures in the analysis phase. In the first place, as has already been mentioned, it is necessary to carry out an inverse probability weighting of the selection of each of the samples in each stratum; in the second place, a random effects model should be used, unless it is proven to be unnecessary, to adjust the sampling by schools, and within them by class and within them by pupils. Such precautions were taken into account as explained in the results section.

Statistical methods for adjusting Reference Curves

We used Cole's LMS method [21], which models the relationship between percentiles and age using a regression technique and assumes a normal distribution of the transformed variable.

This is a widely used method, especially in Europe, for the creation of reference tables depending on age; moreover, computer software in standard statistical packages is also available.

The method assumes that, in each age group, the anthropometric data can be adjusted to a normal distribution after having been adequately transformed, taking into account the degree of asymmetry (L), central tendency (M) and dispersion (S).

Using the original data, for each moment of time (t) the following quantities are obtained:

• L(t) value of the parameter λ of the transformation of Box-Cox to obtain the normality of the variable.

• M(t) median of the original data in the t instant.

• S(t), coefficient of variation of the original data in the t instant.

Obtained for the different t values, within the time frame considered, they are adjusted, using a penalized likelihood, a method that connects them with age.

Using the formula that appears below (and that is simply the result of undoing the change applied in order to normalize the variable) the α percentile is calculated for the t instant, which is given by this expression:

C _α (t) = M(t)(1+L(t)S(t)z _α )^1/L(t)

where z _α is the value of the function of an N (0.1) distribution that leaves to its left a α probability.

What Cole's method does is to model the skewness and the kurtosis of the variable (through the transformation that has to be made to convert the original variable into a Normal one), the central position of the variable (through the median) and the variance, and also the kurtosis in an indirect way (through the coefficient of variation of data in the t instant). Once these coefficients have been modelled in relation to time, thus obtaining the desired percentiles.

An assessment using the method of penalized likelihood requires specific statistical software. The STATA 8.1 and Splus S 6.0 packages were used.

Measuring the "goodness of fit"

The quality of the fit was assessed with tests proposed by Royston [22] that evaluate whether the model residuals follow a normal distribution based on their average, symmetry and kurtosis. Royston proposes a series of tests called Q-tests that characterize certain properties of the model residuals so that if the test yields a significant result it implies that the residuals do not adjust well to the normal random variable and the model does not adjust well either. They are based on the assumption that model residuals should be distributed according to an N (0.1) regardless time, if the model adjusts well.

The tests applied in the assessment of the goodness of fit were as follows:

• Q₁test: If the model adjusts well, the sum of the squares of the mean model residuals in each group, weighed by the size of the group, follows an χ² distribution with G-1 g.l.

• Q₂test: If the model adjusts well, the sum of a variances function in each model residuals group, follows an χ² distribution with G-1 g.l.

• Q₃test: If the model adjusts well, the sum of the squares of the experimental quantities from D'Agostino's Normality test for skewness, in each model residuals group, follows an χ² distribution with G g.l.

• Q₄test: If the model adjusts well, the sum of a function of the significance levels, P values, Shapiro-Wilks' Normality test, combined tests for skewness and Kurtosis, in each model residuals group follows an χ² distribution with 2 G g.l.

Each test responds to the imbalance in normality that the model can create. The first test concerns the differences between the residuals and the median of the distribution they should present, meaning, the N (0.1); the second test is related to a model residuals variance which is either too high or too low, that would suggest very or little sharp and therefore abnormal distributions;; the third test characterizes the skewness of the distributions of the model residuals that would indicate a way of non-normality of such distributions and therefore an imbalance; lastly, the fourth test refers at the same time to the skewness and kurtosis, which are two characteristics that mark non-normality. The latter could be significant for this study more easily, given that Cole's method does not directly model the kurtosis of the base distribution.

Royston suggests declaring the imbalance of the model when any of the tests proves to be significant at 5% error, a situation that rejects completely the adjusted model, given the great number of tests to be performed this is easy to happen simply thanks to the accumulation of errors in each one of them.

In our case, these tests were used to determine the number of edf necessary for each model's fit (Cole's LMS method), and we kept those models which achieved tests with more than 10% significance. When significant results were obtained in some tests, a study of the original observations was performed in case there were extreme data possibly contaminating the model to a great extent. The type of analysis applied is shown in the next section.

Method for detecting extreme and/or influential data

In the case in hand, the detection of extreme data should be carefully assessed. With the reference curves depending on time, our objective is to determine extreme values (high or low), thus eliminating extreme data from the sample could affect the curves in an obvious way, "pruning" the distribution of values that are valuable when it comes to determining percentiles. Nevertheless, in this kind of projects, it is not possible to leave out an analysis of extreme data given that this could entail an important risk of working with data which, almost certainly, do not belong to the target population.

There is a classic rule for detecting extreme data, which involves labelling a piece of data x as extreme in the following situations (Q(25) being the 25th percentile of the sample, Q(75) the 75th percentile and IQR the interquartile range):

x < Q(25) - 1,5 × IQR

x > Q(75) + 1,5 × IQR

We used, however, a modification to this rule, which makes it much more conservative:

x < Q(25) - 4 × IQR

x > Q(75) + 4 × IQR

With this modification we only detected data that were unusually extreme and in this way we do not risk "pruning" the distribution of data that could be influencing the tables.

Data are considered influential if, when eliminated from the sample, they generate a significant change in the model that is being adjusted, giving rise to a profoundly different one. Traditionally, influential data have been identified with extreme or very extreme data, although this is not always the case. A conservative, but reasonable, option would be to look for influential data only among the very extreme data in such a way that will not prejudice areas of distribution that are closest to the centre.

Assuming the previous considerations, the process for the detection and, when necessary, the elimination of data was as follows:

1) Extreme values were labelled for each age, using the rule explained above.

2) Taking labelled data as extreme into consideration, the model was adjusted and the tests were evaluated to determine whether they were significant or not. If this was not the case, it was acknowledged that the most extreme data, up to that moment, were not influential and the process was stopped, conserving all sample data. When the test was significant, the most extreme data were eliminated and the process was repeated until the adjustment quality control tests were not proven to be significant. The iterant process was always carried out on the most extreme data used.

We insist that this is a very conservative process, because this is required by the type of study we are carrying out and, as the results will show, it has led us to reject a very small number of observations.

Sample data

It can be appreciated that in the sample of women the approximate height value becomes stable as of the age of 18, while in men, it stabilizes beyond the age of 20. For this reason, comparisons of mean heights were performed for age groups calculated every half year beyond the above-mentioned values. The results obtained in both cases (women F_exp = 1.36 (10; 1049) g.l. p = 0.1938 and men F_exp = 0.44 (6; 655) g.l. p = 0.8496) confirm the impression that the average height does not vary significantly beyond these ages. As a result the values beyond the age of 20 were grouped together, in the sample of women and men, to refer to the Andalusian adult height in each gender.

Extreme values

Following the method already explained in the methodology section, the values considered extreme were categorized. In the women's group, 1 three cases were corrected because they contained automation errors that had not been detected during the first phase of corrections. Another two were kept because they were not proven significant during the quality adjustment tests.

On the contrary, the two cases detected as extreme in the men's group were eliminated because, when they were included, they were proven significant during the quality adjustment tests.

Model adjustment

The adjustment of the model can be considered adequate since none of the tests used (Q1, Q2, Q3, and Q4) yielded significant results. Adjustment was analysed using a random effects model (the classroom was the grouping unit for the 4–6 pupils chosen in it); following the performance of the likelihood ratio test to compare with the fixed effects model the values obtained were χ²_exp = 1.63 (1 g.l.) p = 0.2014 for women and χ²_exp = 1.08 (1 g.l.) p = 0.3102 for men. The use of the fixed effects model adjustment, produced the height table for men and women, in decimals of years from the age of three to 19 concentrating the values beyond this in 20 (Additional data file 1: Table A and B). Figures 1 and 2 present percentiles 3, 5, 10, 25, 50, 75, 90, 95 and 97 for height, of both men and women.

Sensitivity analysis

As explained in the methods section, a sensitivity analysis was performed on the final model using the most unfavourable scenario. To this end, it was assumed that 10% of the individuals measured could not be measured, and that all of them were below the median. Then, when we compared this hypothetical situation to the real model, we found that for estimating the median, in the worst case (around the age of ten) the difference in the tables would be 8 mm in women and 9 mm in men, an amount that seems small. For the most important situation, from a clinical point of view, which would be that of the percentile 3, the maximum difference was half a centimetre, assuming that 10% of the children were not measured and that all of them were below median measured.

Sample data

Extreme values

All values, from both samples that were characterized as extreme (5 in women and 6 in men) were eliminated since the goodness of fit test was proven significant in case they were included, and they had a considerable effect on the kurtosis of the distribution.

Model adjustment

The adjustment is adequate because none of the tests assessing its goodness of fit (Q1, Q2, Q3, and Q4) yielded a significant result. The adjustment was analysed using a random effects model (the classroom was the grouping unit for the 4–6 pupils selected in it); following the performance of the likelihood ratio test to compare with the fixed effects model, the values obtained were χ²_exp = 1.08 (1 g.l.) p = 0.2987 for women and χ²_exp = 1.49 (1 g.l.) p = 0.2222 for men. The use of the fixed effects model adjustment, produced the height table for men and women, in decimals of years from the age of three to 19 concentrating the values beyond this in 20 (Additional data file 1: Tables C and D). Figures 3 and 4 show percentiles 3, 5, 10, 25, 50, 75, 90, 95 and 97 for weight, in both women and men.

Sensitivity analysis

In the same way as with height, a sensitivity analysis of the final models was performed, using the most unfavourable scenario. To this end, it was assumed that 10% of the individuals weighed could not be weighed, and that all of them were below the, median. Then, when we compared this hypothetical situation with the real model, we found that for the estimation both of the median and of the percentile 3, differences in women were between 0.1 and 0.8 kg (around the age of 14). In the case of men, differences in the estimation of the median ranged between 0.1 and 1 kg, while in the case of the percentile 3 which has the greatest clinical significance, it ranged between 0.1 and 0.3 kg.

Sample data

Extreme values

Both in the case of men and women, all values characterised as extreme were eliminated since the tests showed that they significantly affected the quality of adjustment. 13 cases were eliminated in the women's sample (0.29% of the sample) and 8 in the men's group (0.18% of the sample).

Model adjustment

The adjustment of the model is adequate given that none of the tests evaluating the goodness of fit (Q1, Q2, Q3, and Q4) yielded a significant result. The adjustment was analysed using a random effects model (the classroom was the grouping unit for the 4–6 pupils chosen in it); following the performance of the likelihood ratio test to compare with the fixed effects model the values obtained were χ²_exp = 2.37 (1 g.l.) p = 0.1237 for women and χ²_exp = 2.21 (1 g.l.) p = 0.1371 for men. The use of the adjustment of the fixed effects model produced the BMI table for women and men, grouped in ages of half years from the age of three to 19, with all ages over 20 concentrated at 20 (Additional data file 1: Table E). Figures 5 and 6 show percentiles 3, 5, 50, 85, 95 and 97 for BMI, in both women and men. These are the percentiles considered both in the BMI table and graphs since from a clinical point of view the most useful values are those corresponding to percentiles 3, 50, 85 and 95.

With regards to the BMI, the evolution of the percentiles below the median is very different from the evolution of the percentiles that are above such median for both genders, especially that of the most extreme percentiles. In the women's model, percentiles 85, 95 and 97 increase significantly until the age of 14, when they start to decrease. So Andalusian girls from the age of 8 or 9 have a higher probability of being overweight or obese and this probability reaches its maximum at the age of 14, when it starts to diminish. In the case of men, percentiles 95 and 97 show a similar evolution, although they reach their peak around the age of 16.

Sensitivity analysis

In the sensitivity analysis of the final BMI model, it was also assumed that 10% of the individuals measured were not actually measured, and that the BMI values of all of them were below the median. This way, when we compared this hypothetical situation with the real model, we found that for the estimation of the median and of the percentile 85, both in the sample of men and women, differences ranged between 0.1 and 0.3. These differences are noticeably lower in the extreme percentiles, both above and below the median.