Height Growth

C. Distance plot against reference models

All cases for a given sex of the Brno Growth Study were modeled using Functional Data Analysis (FDA),
which is set as the primary (default), or another combination of growth model and reference data (selected when MODEL AND SAMPLE is entered). From the models, values of stature height for the values you specified for the age of the case being evaluated are extrapolated and these extrapolations were plotted using a boxplot. The evaluated individual is always modelled using the FPCA model. The color of each point of the evaluated case then corresponds to the percentile value of that point relative to the reference boxplot at that age. The points of this plot connected by dotted lines represent the specified values of the evaluated case (raw measurements), and the points connected by dashed lines represent the values of the height interpolation from the model for a given calendar age.

Table of estimates against the BGS longitudinal reference

This table numerically represents the values shown graphically in Plot A and C. The first column (No.) is the measurement number of the case being evaluated, and the second column (Measured percent.) is the raw measurement value expressed as a percentile relative to the BGS reference distribution for that age (points in Plot C connected by a dotted line). The next column (Track percent.) is the value of stature height, extrapolated from the nearest empirical growth trajectory (Plot A – green curve) and again expressed as a percentile of the BGS reference distribution. The last column (FPCA percent.) is the value of stature height extrapolated from the FPCA model (Plot A – red curve, Plot C – points connected by dashed lines) and again expressed as a percentile of the BGS reference distribution. he last row of bolded numbers indicates for all three columns the maximum percentile range over the entire measured trajectory of the individual being assessed. Values derived by interpolation from the models (closest curve, FPCA model) can be expected to have a lower percentile range because they represent smoothed trajectory values with limited influence of measurement noise.

D. Plot of growth velocity against the reference model – calendar age

This plot represents a comparison of the growth velocity of the case being evaluated with the growth velocity of the Brno Growth Study reference sample for the calendar age values of the case under evaluation. It is calculated and displayed only if at least 2 measurements of the evaluated case are given. For each pair of measurements (range of two calendar age values) the program calculates the difference in height, relates it to the range of age and converts it to the growth velocity in centimeters per year. For a given age range, the program calculates an extrapolation of the body height of the reference BGS sample and the growth rate in the same way. The growth rate of the reference file is then displayed as a boxplot and the evaluated case as a point. The points of this plot connected by dotted lines represent the growth velocities calculated from the raw values of the evaluated case, the points connected by dashed lines represent the growth velocities calculated from the interpolations of the height from the FPCA model of the evaluated case.

Table of growth velocity estimates against the BGS longitudinal reference

The table of growth velocity estimates expresses the values shown graphically in Plot D. The first column (No.) is the range of measurement numbers of the evaluated case for which the velocity is calculated. The second column (Age (years)) is the value of the range of ages for which the velocity is calculated. The next column (Measured percentile.) represents the value of the growth velocity of the evaluated case calculated from the raw measurements (dotted line connected points in plot D) as a percentile of the BGS growth velocity reference values for the age range (boxplots in Plot D). The last column (FPCA percentile) represents the value of the growth velocity of the evaluated case calculated from the FPCA model (points of plot D connected by dashed lines) in the form of percentile of the BGS growth velocity reference values for the given age range (boxplots in plot D). For both columns, the last row of bolded numbers indicates the maximum percentile range over the entire measured trajectory of the assessed individual. Values derived by interpolation from the model (FPCA model) can be expected to have a lower percentile range because they represent smoothed trajectory values with limited influence of measurement noise.

E. Plot comparing the child’s growth with the assumption from the parents’ height in adulthood

This diagnostic plot compares the child’s growth to the assumption made from knowing the height of both biological parents in adulthood. From the height of the father and mother of each reference case, the program calculates an estimate of a target height based on some modification of midparental height; currently it uses the calculation of Gray (1948), but we will gradually add other methods of estimating target height. The current estimate method is as follows:

For boys:
target height = (height of father + (height of mother*1.083333))/2
For girls:
target height = ((height of father *0.9230769) + height of mother)/2

Based on the target height estimate from the parental data, the program calculates the Z-score to which this target height corresponds at age 18 of the reference population (BGS by default). Next, the interpolation of the height for a given age of measurement is calculated according to the model curve (i.e. the height of each child at each age of measurement adjusted for noise). The difference between each child’s height at a given age and the percentile estimate of their height based on the target height percentile (height minus the estimate of height from the target height) is then calculated. Thus, the difference on the scale on both the positive and negative axes corresponds to whether the child is taller or shorter in stature at a given age than would correspond to the percentile of his or her target height estimate (Figure 5 bottom). The distribution of these differences is converted into a standardized distribution (Z-score) using the mean of 0 and the standard deviation of the distribution of these differences of the reference sample of children, and finally, the values of the evaluated case is computed in the same manner and plotted on this standardized scale (Figure 5 top). The evaluated case is then also plotted as a Z-score against this reference. If the evaluated case differs in height from the Z-score of the target height (i.e his or her parents) by more than +/-2SD (more than about 95% of all cases in the reference population), the points are shown in red and indicate a growth problem. The points in this graph connected by a dotted line represent the Z-score calculated from the raw values of the case being evaluated, the points connected by a dashed line represent the Z-score calculated from interpolating the height from the selected growth model of the case being evaluated.

Table comparing standardized values of the differences between height and the assumption from the target height

This table shows the standardized values of the differences between the child’s height and the assumption derived from the parents’ target height (points from the bottom graph). In the first column (No.) is the measurement number of the evaluated case for which the difference is calculated. In the second column (Age (years)) is the value of the age for which the difference is calculated. In the next column (Measured SD) is the value of the standardized difference for the raw measurements and in the last column (FPCA SD) is the value of the standardized difference for the measurements derived by interpolation from the FPCA model.

F. Pubertal growth velocity curve – biological age

The FPCA method is based on two separate FPCAs, one for phase and the other for amplitude, which are summed during model generation. Thus, it allows to analyze both aspects of the growth curve separately. This can be used to synchronize the phase of all growth curves. All the reference curves are registered at point 0, which corresponds to the APV, their phase is neglected and their shape is deformed (using the landmarkreg function from the fda package) so that the overall magnitude matches the average growth curve of the reference sample. The curves then differ only in amplitude. The curve of the individual being evaluated is identically processed. As a result, we can compare the growth of individuals of the same biological age, free from differences in their calendar age (or from differences in the calendar timing of the pubertal growth phase), in other words: below each other in the graph are values that correspond in the timing of their phases for all individuals. In this plots, we use the first derivative of such synchronized curves and compare the growth velocity of the evaluated case with the growth velocity of a reference set at the same growth age. In Plot F we see a comparison of the phase-synchronized velocity curve of the evaluated case with the average velocity curve of the reference sample during the pubertal growth acceleration period. For this procedure, the results of the FPCA method are needed; if we do not select it when selecting methods (Compute FPCA model), the plot will not be displayed.

G. Plot of growth velocity against a reference model of the same growth age

This plot represents a comparison of the phase-synchronized growth velocity curve of the case under evaluation (points connected by a solid line) with the reference, phase-synchronized growth velocities values for all measured ages of the case under evaluation (box plots). We can then assess what the growth rate is relative to other individuals in the same growth phase. For this procedure, the results of the FPCA method are needed; if we do not select it when selecting methods (Compute FPCA model), the plot will not be displayed.

Table of the height growth velocity after phase synchronization

The table shows the growth velocity values of body height for the respective growth age, expressed in relation to the APV of the average reference growth velocity curve (in years). The first column (No.) is the measurement number of the case being evaluated for which the velocity is calculated. The second column (Age (years)) is the value of the age for which the rate is calculated. The next column (FPCA Percentiles) is the value of the growth rate for that measurement after synchronization. The last row is the overall range of the standardized distribution over which the speed of the individual being evaluated varies over the entire set of comparisons (once is the standard deviation of the growth rate).

Calculating the target height

Target height is an estimate of the height a child will reach as an adult (or at age 18) based on currently
available information. Since genetic predispositions (“genetic growth potential”) resulting from the combined genetic contribution of both parents have always been considered as a strong determinant, the target height of a child is often derived from the body height of the biological mother and biological father. Different methods of calculating the target height then take into account in different ways the height of the mother and father, the sex of the child and possibly also secular trends (average intergenerational differences) in stature or differences between populations more generally. However, genetic theory suggests that each child (even within siblings of the same sex) of the same parents represents a slightly different combination of the father’s and mother’s traits, so it can be assumed that different children of the same sex of the same parental couple will in fact have different heights in adulthood, even though the target height of the child of the two parents will be only one.
The table of results currently includes these estimates:

Midparent 1 – the method of Tanner et al., represents the average of the father’s and mother’s stature
(midparental height), to which 6.5 cm is added for boys and the same value is subtracted for girls. This method does not take into account a possible secular trend in increasing height in the 20th century.

Midparent 2 – the method of Molinari et al. (1984), follows a similar procedure, but adds 6.4 cm to the
midparental height for boys and subtracts 6.4 cm for girls, but accounts for the secular trend in height increase by adding 3.8 cm to all estimates.

Centile – this method obtains the Z-score of the mother’s height relative to a reference sample of mothers of BGS children, and similarly the Z-score of the father’s height relative to a reference sample of fathers of BGS children, and calculates the average of the two Z-scores. This value is then multiplied by the standard deviation of the interpolations of the BGS children’s growth patterns (of the relevant sex) for age 18, and this multiple is finally added to the average value of the interpolation of the BGS children’s stature height (of respective sex) for age 18, thus getting back to the stature height scale in centimeters and obtaining an estimate of the stature height of the individual being assessed at age 18. This method addresses the intergenerational difference (secular trend) in absolute stature height between the parents’ and children’s generation by transferring it through the relative Zscore, but de facto increases the stature height of children relative to their parents to a degree corresponding to
the difference in their means. It is therefore appropriate in a situation where the difference between the parents’ and children’s generation will be similar to the BGS study.

Linear Regression – our original multiple linear regression method, based on BGS data, where child height is estimated directly from parental height by fitting it to a linear equation. For boys, the equation takes the form:
Son’s height at adulthood (cm) = 21.36057 + 0.52969*mother’s height (cm) + 0.41558*father’s height (cm)
For girls, the equation takes the form:
Daughter’s height at adulthood (cm) = 30.62437 + 0.48017*mother’s height (cm) + 0.33279*father’s height (cm)
This regression model is derived from the BGS and accounts for the specific change between the generation of parents and the children born to them in the 1960s in Brno.

Longitudinal – this estimate of target height is the only one included so far in the Program that does not take into account the height of the parents, but the previously measured height of the child whose target height we are estimating. The measured stature heights to date are fitted with the FPCA model and the target height estimate is an extrapolation of the stature height for this model at age 18. The fewer the number of measurements of the child available and the further the age at the time of measurement from age 18, the less accurate the estimate will be. At the same time, given the behavior of the model, it can be expected that in this case the estimates will tend towards average values, i.e. the stature height of smaller individuals will tend to be overestimated and the stature height of taller individuals will tend to be underestimated.

Calculation of growth milestones at puberty

From the growth curve obtained for the individual under evaluation, the program calculates growth
milestones using the getPeak and getTakeoff functions from the sitar package available for the R
environment. For each individual modeled, the following growth milestones/parameters are determined from the model curves:

APV (Age at Peak Velocity) – age at peak velocity at pubertal growth spurt in body height (in years).
VPV (Velocity at Peak Velocity) – maximum velocity at pubertal height growth spurt (in cm/year).
HPV (Height at Peak Velocity) – body height at the time of peak velocity in puberty (in cm).
ATO (Age at Take Off) – age of onset of pubertal height spurt (in years), corresponds to the point of lowest growth velocity before puberty, the position of the foot at the beginning of the pubertal velocity wave.
VTO (Velocity at Take Off) – velocity at the ATO point, the lowest velocity of the pre-pubertal growth spurt of body height (in cm/year)
HTO (Height at Take Off) – body height at the ATO point, the height reached at the start of the pubertal growth spurt in height (in cm).

In this text field, the person evaluating the body height (physician, anthropologist) can enter his/her textual evaluation of the program outputs, which will then also appear in the report downloaded in PDF format. It is recommended to follow the individual outputs and comment on the message of the corresponding table or plot. The main focus should be on situations where the values of the examined person deviate from the specified reference frame. In some plots the degree of deviation should be assessed using scale units, in others there are defining limits and colored highlighting of deviating values.