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. Author manuscript; available in PMC: 2021 Aug 1.
Published in final edited form as: Ann Hum Biol. 2020 Jun 16;47(5):434–445. doi: 10.1080/03014460.2020.1763458

Estimating peak height velocity in individuals: a comparison of statistical methods

Melanie E Boeyer a,b,c, Kevin M Middleton b, Dana L Duren a,b,c, Emily V Leary a,c
PMCID: PMC7590904  NIHMSID: NIHMS1638510  PMID: 32543236

Abstract

Background:

Estimates pertaining to the timing of the adolescent growth spurt (e.g. peak height velocity; PHV), including age at peak height velocity (aPHV), play a critical role in the diagnosis, treatment, and management of skeletal growth and/or developmental disorders. Yet, distinct statistical methodologies often result in large estimate discrepancies.

Aim:

The aim of the present study was to assess the advantages and disadvantages of three modelling methodologies for height as well as to determine how estimates derived from these methodologies may differ, particularly those that may be useful in paediatric clinical practice.

Subjects and methods:

Height data from 686 individuals of the Fels Longitudinal Study were modelled using 5th order polynomials, natural cubic splines, and SuperImposition by Translation and Rotation (SITAR) to determine aPHV and PHV for all individuals together (i.e. population average) by sex and separately for each individual. Estimates within and between methodologies were calculated and compared.

Results:

In general, mean aPHV was earlier, and PHV was greater for individuals when compared to estimates from population average models. Significant differences between mean aPHV and PHV for individuals were observed in all three methodologies, with SITAR exhibiting the latest aPHV and largest PHV estimates.

Conclusion:

Each statistical methodology has a number of advantages when used for specific purposes. For modelling growth in individuals, as one would in paediatric clinical practice, we recommend the use of the 5th order polynomial methodology due to its parameter flexibility.

Keywords: Growth spurt, growth trajectory, polynomial, natural cubic splin, SITAR

Introduction

The adolescent growth spurt is characterised by one of the most rapid periods of post-natal growth, which is followed closely by epiphyseal fusion and the attainment of final adult height (Bogin 1988; Bogin et al. 2018; Eveleth and Tanner 1990). Over the last several decades, significant progress has been made in the development of statistical methodologies for assessing individual and population average growth in height leading up to and throughout adolescence (Cole 2012; Cole et al. 2010; Preece and Baines 1978). These methods have allowed human biologists and paediatric practitioners to estimate the chronological age at which peak height velocity (aPHV) is attained as well as the rate of growth occurring during peak height velocity (PHV) (e.g. Sanders et al. 2017). Estimates of adolescent ontogenetic parameters are of critical importance, particularly for paediatric practitioners treating children with skeletal growth and/or developmental disorders, including constitutional growth delay (Poyrazoğlu et al. 2005), adolescent idiopathic scoliosis (Busscher et al. 2012; Chazono et al. 2015; Little et al. 2000), or leg length inequality (Green and Anderson 1960; Moseley 1977, 1987). However, the most commonly employed methodologies for predicting aPHV and/or PHV result in large differences in estimates, even when using identical data (Preece and Baines 1978; Simpkin et al. 2017).

Frequently used statistical methodologies to assess longitudinal growth in height during adolescence include cubic splines (Cole and Green 1992; Largo et al. 1978), mixed effect multi-level polynomials (Chirwa et al. 2014; Goldstein 1989; Smith and Buschang 2005), fractional polynomials (Royston and Altman 1994; Tilling et al. 2014), double and triple logistic regressions (Bock et al. 1973; Guo et al. 1992; Lozy 1978; Malina et al. 2016), and Preece-Baines Growth Models (Byard et al. 1993; Guo et al. 1992; Preece and Baines 1978). Natural cubic splines are arguably the most common statistical method employed, in part, because multiple smoothing parameters can be included in model selection (e.g. parametric and non-parametric) (Pan and Goldstein 1998; WHO Multicentre Growth Reference Study Group and de Onis 2007). Additionally, binned and smoothed age distributions (Borghi et al. 2006) can also be employed in natural cubic spline models (see Hamill 1977 for a more detailed discussion). Preece-Baines Growth Models were originally developed to provide a more parsimonious approach (i.e. fewer model parameters) to biological growth than previous methodologies (Preece and Baines 1978). Multi-level polynomials are frequently selected because they simultaneously fit both individual and population average curves, with random effects representing an individual’s deviation from the population average. These models are, therefore, easier to fit with multiple random parameters when compared to Preece-Baines or double logistic methodologies, which often result in a lack of model convergence when parameter complexity is increased (Hauspie and Molinari 2004; Nahhas et al. 2014). More recently, Cole et al. (2010) expanded upon a non-parametric shape-invariant technique (Beath 2007), now known as SuperImposition by Translation And Rotation (SITAR), to model longitudinal changes in height throughout childhood and adolescence. Similar to multi-level polynomials, SITAR also simultaneously fits both individual and population average growth trajectories. Utilisation of this methodology for modelling childhood and/or adolescent growth has recently increased (Blackwell et al. 2017; Cao et al. 2018a; Cole 2018; Cole and Mori 2018; Kuh et al. 2016; Mahmoud et al. 2018; Martin and Valeggia 2018; Simpkin et al. 2017; Spencer et al. 2018), likely due to the high coefficients of determination and ease of model fit.

Given the range of statistical methodologies that can be employed on childhood and/or adolescent growth, it is not surprising that biologically meaningful discrepancies in parameter estimates, such as aPHV and PHV, have been reported (Iuliano-Burns et al. 2001; Largo et al. 1978; Portella et al. 2017). To assess the biological importance of different mathematical models, Simpkin et al. (2017) investigated estimates of aPHV and PHV using four unique modelling techniques: Preece-Baines, multi-level fractional polynomials, SITAR, and principal analysis by conditional expectation. Their results suggest that Preece-Baines and SITAR provide estimates for aPHV with the smallest degree of bias when estimated from simulated data; however, when applied to data that mimic clinical assessments, including unbalanced, biennial assessments with large measurement error, a large number of the tested models failed to converge (11% Preece-Baines; 61.3% SITAR). Moreover, population average aPHV estimates derived from the SITAR methodology showed a difference of more than one year between all four methods, given identical data (Simpkin et al. 2017). This observed difference has the potential to influence clinical interpretations, particularly in intervention outcomes and patient quality of life.

Uniform recommendations for modelling longitudinal changes during childhood and/or adolescence are needed, both for a single individual and for a population. Based primarily on simulated data, the analysis by Simpkin et al. (2017) laid the theoretical ground work for testing the applicability of the available methodologies, which was further assessed using discrete data intervals by Cole (2018). The next step towards assessing the generalisability of these studies and developing recommendations for statistical models that estimate aPHV and/or PHV requires dense longitudinal data (i.e. 10 or more observations over a 10-year period) representative of real-life growth trajectories in both boys and girls. Therefore, we aimed to: (1) elucidate the advantages and disadvantages of three commonly employed growth curve methodologies: 5th order polynomials, natural cubic splines, and SITAR, using a dense longitudinal data set of both healthy boys and girls; and (2) determine how estimates derived from these methodologies differ, particularly those that may be useful in paediatric clinical practice.

Subjects and methods

Study sample and inclusion criteria

The study sample includes individuals from the Fels Longitudinal Study (Roche 1992), all of whom originate from southwest Ohio and are primarily of European descent (2.2% other). Individuals of the Fels Longitudinal Study were examined at routine intervals beginning at birth and continuing into adulthood, with assessments of height every three months for the first year of life, and every six months until adulthood (Roche 1992). Dense longitudinal data of this type are ideal for assessing large-scale statistical inquiries related to growth modelling and the development of growth standards, as pointed out by Cole (2018) because they “consist of frequent regular spaced measurements.” As such, many previous studies have used individuals from the Fels Longitudinal Study (Byard et al. 1991, 1993; Guo et al. 1992; Johnson et al. 2012; Malina et al. 2016; Siervogel et al. 1991). This use of these dense longitudinal data represents an initial approach aimed at bridging the gap between population-based research studies and paediatric clinical practice.

Inclusion in the present analysis required 10 or more assessments between the chronological ages of 7 and 20 years, each with documented sex, chronological age, and standing height. Standing height for each individual was measured using a stadiometer following standard anthropometric protocols (Lohman et al. 1988). The sample includes 686 individuals (12,896 total observations; 19.60 mean observations per individual), 348 boys and 338 girls. The University of Missouri Institutional Review Board approved this study.

Statistical methodology

All statistical modelling was performed separately for both boys and girls using the lm() function or the “sitar” (Version 1.1.0; Cole 2017) and “iapvbs” (Version 0.0.2; Cao et al. 2018b) packages in R (Version 3.4.0). Identical data were used for each methodology. The statistical methodologies employed were selected based on both the existing literature and the ability to be applied to paediatric clinical practice, including: (1) fixed effect 5th order polynomials – prevalent in the literature and impose minimal parameter constraints (e.g. Chirwa et al. 2014; Leung et al. 1999); (2) natural cubic splines – the basis for the Centers for Disease Control and Prevention (CDC) growth standards in children two years of age and older (Flegal and Cole 2013; Kuczmarski 2000); and (3) SITAR, which is gaining traction in the recent literature (e.g. Cole 2018; Cole and Mori 2018; Kuh et al. 2016; Mahmoud et al. 2018).

The primary outcome measures of this study include: (1) PHV and (2) the chronological age at which PHV occurs (aPHV). We focussed on assessing ontogenetic parameters that occur during adolescence; all statistical modelling was performed between the chronological ages of 7 and 20 years, with all growth occurring after seven years of age in both boys and girls being considered adolescent growth (Berkey 1982; Goldstein 1986; Roche and Sun 2005). Estimates of PHV were only considered between the chronological ages of 8 and 17 to avoid undue influence of delayed childhood growth spurts or modelling effects where data are sparse (i.e. at the bounds of the observation data). Adolescent ontogenetic parameter estimates were calculated separately for each individual as well as collectively to obtain sex-specific population averages. Individual estimates of aPHV and PHV were defined by a single individual’s longitudinal data, whereas population average estimates were defined by modelling all available longitudinal data for each sex. Although the population average is known to underestimate PHV and subsequently aPHV (Cao et al. 2018a; Cole et al. 2008; Merrell 1931), we chose to calculate these estimates for two primary reasons: (1) to provide additional evidence that population average estimates are not representative of individual growth trajectories and (2) to assess the extent to which each statistical methodology influenced population average estimates of aPHV and/or PHV.

5th order polynomial modelling

A polynomial model is a special case of multiple regression where the independent variables are composed of higher order terms of a single variable. An important aspect of this methodological approach is that the mean of the dependent variable is a function of a single independent variable. We fit 5th order fixed effect polynomial models to each individual and the model parameters were as follows:

Yij=βi0+βi1(ageij)+βi2(ageij)2+βi3(ageij)3+βi4(ageij)4+βi5(ageij)5+ϵij (1)

where Yij is the height of the ith child at the jth time, βi0 represents the polynomial coefficients where m = 1, …, 5, and ϵij represents error. A multi-level methodological approach was not employed, as it results in a single random effect parameter identifying the upward or downward shift in individual-specific growth trajectories with identical main effects for each individual, thus, constraining the underlying shape of each growth trajectory, which minimises the main advantage of this methodology. Estimates of PHV were calculated using the maximum change in height (i.e. the first derivative; Equation (2)) across the inclusive chronological age range of eight to 17. aPHV is defined as the corresponding age at which PHV was attained (Equation (3)).

PHVi=max{Y(ageij)−Y(ageij−1)(ageij)−(ageij−1)} (2)
aPHVi=argmax{Y(ageij)−Y(ageij−1)(ageij)−(ageij−1)} (3)

Natural cubic spline modelling

The natural cubic spline methodology allows for flexibility in model form and its derivative is continuous beyond the external boundary knots, an important feature for interpreting the cessation of longitudinal growth. Natural cubic splines were also fit to each individual’s data using the lm() function with ns() in R. We specified a three degree piecewise polynomial with two external knots set at the bounds of the chronological age range, which represents the default position within the ns() function (see Hardin et al. 2020; Howe et al. 2016 for additional discussion). Three internal knots were set at the 25th, 50th, and 75th quartiles of the observational age range. The model parameters were as follows:

Yij=∑l=13(βl+βil)ageijl+∑k=15bk(ageij−κk)3 (4)

where Yij is the height of the ith child at the jth time, κ1… κk are the knots, and ℓ is the index used for the natural cubic splines. There are five knots where bk represents the corresponding spline. Estimates of PHV and aPHV were calculated as previously defined (see Equations (2) and (3)).

SITAR modelling

SuperImposition by Translation And Rotation is a shape invariant curvilinear model based on a natural cubic spline, which fits an averaged population curve to an individual’s data. Estimates for an individual’s growth are returned as deviations from the population average through three random effects, which capture differences in the size, tempo, and velocity of the growth trajectory. The model parameters are as follows:

Yij=αi+h(ageij−βiexp(−γi)) (5)

where Yij is the height of the ith child at the jth time, h () represents the population average natural cubic spline curve, and αi, βi, and γi represent an individual’s size, tempo, and velocity, respectively. As recommended in Cole (2017), additional logarithmic data transformations for the SITAR methodology were assessed: log(age), log(height), and log(age and height). Final model selection was based on the Bayesian information criterion (BIC; Schwarz 1978), with only the best-fitting model, represented by the smallest BIC value, for both boys and girls, being used in subsequent statistical analyses. The size, tempo, and velocity parameters derived from SITAR describe each individual’s deviation from the population average (i.e. fixed effects) but do not provide individual estimates of PHV and aPHV that can be assessed without additional data extraction and calculation. Therefore, individual estimates of PHV and aPHV were simultaneously calculated using the “getapv” numeric function developed by Cao et al. (2018a) in the “iapvbs” R package (Cao et al. 2018b). This function uses standard output from predict.sitar(), without reported covariates, to obtain fitted values for each individual.

Within- and between-methodology aPHV and PHV comparisons

Differences between aPHV and PHV for an individual and the population average, within each methodology, were compared using a one-sample, two-sided t-test. Differences in mean aPHV and PHV estimates across methodologies were compared using a one-way analysis of variance and post hoc testing was performed using Tukey’s honestly significant difference (HSD) test. Results that exhibited an alpha (α) of 0.05 or less were considered statistically significant. All p values are reported based on the criteria outlined in Cole (2015).

Leave-one-out cross-validation

To further assess the relationship between methodologies that rely solely on growth trajectories for an individual, such as fixed effect polynomials or natural cubic splines, to SITAR, which relies on population average trajectories to fit individual growth data, a leave-one-out (LOO) cross-validation analysis was performed. LOO cross-validated population average models were generated for all three methodologies employed. Each population average model was used to predict the height of the individual left out, indicating the out-of-sample predictive ability of each model. This methodological approach highlights the number of and degree to which a given individual-specific growth trajectory, and thus estimates of aPHV and/or PHV, would be poorly predicted. The root-mean-squared difference (RMSd) as well as aPHV and PHV was calculated. Differences between aPHV and PHV estimates for an individual and those generated from the LOO population average models were calculated.

Results

Model convergence, for both population average and individual models, occurred for all methodologies (Supplementary Table 1), with the exception of the logarithmic transformation of both chronological age and height in SITAR (e.g. log(age and height)). A number of aPHV and PHV estimates were not biologically meaningful (i.e. within a year of the defined observational age range endpoints; Table 1). Figures 1 and 2 illustrate the population average (solid black) and individual (multi-colour) growth models for height (A) and rate of change in height (B) for boys and girls, respectively.

Table 1.

Estimates of aPHV and PHV for individual and population average models by statistical methodology in boys and girls.

Sex Variable Estimate 5th order polynomial (N = 267 and 306) Natural cubic spline (N = 330 and 309) SITAR (N = 347 and 336)
Boys aPHV Population 12.88<0.01 13.420.2 13.650.6
Individual (CI) 13.17 (12.97, 13.35) 13.34 (13.21, 13.46) 13.68 (13.58, 13.78)
Difference (CI) 0.28 (0.09, 0.47) −0.08 (−2.20, 0.04) 0.03 (−0.07, 0.13)
PHV Population 7.00<0.001 7.07<0.001 9.050.2
Individual (CI) 7.92 (7.81, 8.03) 8.20 (8.09, 8.32) 9.13 (9.00, 9.25)
Difference (CI) 0.91 (0.81, 1.03) 1.13 (1.02, 1.24) 0.08 (−0.05, 0.20)
Girls aPHV Population 10.06<0.001 10.55<0.001 11.700.5
Individual (CI) 10.89 (10.75, 11.02) 11.24 (11.14, 11.35) 11.73 (11.64, 11.82)
Difference (CI) 0.83 (0.69, 0.96) 0.69 (0.59, 0.80) 0.03 (−0.06, 0.12)
PHV Population 6.52<0.01 6.87<0.001 7.940.2
Individual (CI) 7.73 (6.95, 8.51) 7.35 (7.24, 7.46) 8.01 (7.90, 8.12)
Difference (CI) 1.21 (0.43, 1.99) 0.48 (0.37, 0.59) 0.07 (−0.04, 0.18)
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N: number of individuals that exhibited both aPHV and PHV estimates as described in boys and girls; CI: 95% confidence interval; superscript p values were derived from each one-sample, two-sided t-test; difference calculated as individual estimates minus the population estimate.

Estimates for aPHV and PHV between models for each individual and population average were compared using a one-sample two-sided t-test.

Figure 1.

Figure 1.

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Boys: individual (thin line) and population average (thick line) fitted models for growth in height (A) and rate of growth in height (B) for the 5th order polynomial, natural cubic spline, and SITAR (left to right) methodologies. The visualisation of SITAR growth trajectories is based on back-transformed chronological age, as per Cole (2017), despite modelling being performed on the logarithm of chronological age. This image is the property of the authors.

Figure 2.

Figure 2.

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Girls: individual (thin line) and population average (thick line) fitted models for growth in height (A) and rate of growth in height (B) for the 5th order polynomial, natural cubic spline, and SITAR (left to right) methodologies. The visualisation of SITAR growth trajectories is based on back-transformed chronological age, as per Cole (2017), despite modelling being performed on the logarithm of chronological age. This image is the property of the authors.

Within methodology estimates of aPHV and PHV

Polynomial modelling

In boys, the population average for aPHV and PHV were significantly reduced by 3.48 months (0.29 years; 95% CI: 0.09–0.47 years; p < .01) and 0.92 cm/year (95% CI: 0.81–1.03; p < .001), respectively, when compared with the mean of the estimates for each individual (Table 1). In girls, the population average for aPHV and PHV were also significantly reduced by 9.96 months (0.83 years; 95% CI: 0.69–0.96 years; p < .001) and 1.21 cm/year (95% CI: 0.43–1.99; p < .01), respectively, when compared with the mean estimates for each individual (Table 1). The RMSd in height (cm) across the observational age range from the LOO cross-validation analysis was, on average, 5.69 cm in boys and 5.30 cm in girls (Supplementary Figures 1(A), 2(A)). Population average LOO cross-validated estimates for aPHV and PHV tended to be earlier and smaller, respectively, when compared with individual estimates in both boys and girls, represented by a rightward shift in the distribution (Supplementary Figures 1(B,C), 2(B,C)). The magnitude of the difference was larger in girls for both aPHV and PHV (0.82 years and 1.22 cm/year) than when compared to boys (0.28 years and 0.92 cm/year).

Natural cubic spline modelling

In boys, the population average estimate for aPHV (13.42 years) was nearly identical when compared with the mean aPHV for each individual (13.34 years), whereas PHV was significantly reduced by 1.13 cm/year (95% CI: 1.02–1.24; p <.001; Table 1). In girls, the population average estimate for aPHV was reduced by 8.28 months when compared with the mean aPHV estimate for each individual (0.69 years; 95% CI: 0.59–0.80; p <.001; Table 1). The RMSd across the entire length of the observational age ranges from the LOO cross-validated estimates were, on average, 5.67 cm in boys and 5.29 cm in girls (Supplementary Figures 1(D), 2(D)). Population average estimates of PHV from the LOO cross-validation analysis were smaller in both boys and girls by approximately 1.13 cm/year and 0.49 cm/year, respectively. aPHV was also earlier in girls by approximately 0.70 cm/year, but in boys, earlier and later estimates were observed (–0.07 cm/year) (Supplementary Figures 1(E,F), 2(E,F)).

SITAR modelling

Individual and population average estimates for aPHV and PHV in both boys and girls were similar, particularly estimates of aPHV, which differed by less than a month (Table 1). Although not statistically significant, the largest difference observed across individual or population average estimates occurs in PHV for both boys and girls. In boys, the population average estimate for PHV is 0.08 cm/year smaller than the mean estimate for individuals (Table 1); in girls, the population average estimate is 0.07 cm/year smaller (Table 1). Following the LOO cross-validation analysis, the average RMSd for height was 5.69 cm and 5.29 cm for boys and girls, respectively (Supplementary Figures 1(H), 2(H)). The difference between the population average and LOO cross-validated estimates of an individual for aPHV and PHV was almost identical, with a mean of 0.03 years and 0.07 cm/year, respectively, for both boys and girls (Supplementary Figures 1(I,J), 2(I,J)).

Individual aPHV and PHV estimates across methodologies

In boys, all estimates of aPHV and PHV were significantly different between methodologies with the exception of aPHV between 5th order polynomials and natural cubic splines (Supplementary Table 2). In girls, estimates of aPHV were significantly different between methodologies, while no such differences were observed for PHV (Supplementary Table 2). In both boys and girls, mean estimates for aPHV and PHV were largest in SITAR models by at least 4.08 months and 0.28 cm/year, respectively (Supplementary Table 2). The smallest significant difference between mean estimates of aPHV was observed between natural cubic spline and SITAR methodologies in girls, with natural cubic spline estimates being 4.08 months younger than those derived from SITAR; estimates of PHV were smallest between the 5th order polynomial and natural cubic spline methodologies in boys, with a difference of 0.28 cm/year (Supplementary Table 2).

When comparing each individual estimate of both aPHV and PHV between methodologies, several trends become apparent (Figures 3 and 4). In boys, estimates of aPHV between 5th order polynomials and natural cubic splines exhibit both younger and older estimates depending on the age at which PHV was achieved (Figure 3(A)). For boys that reach PHV at both younger and older ages – representing the tails of the distribution – we see consistently younger estimates of aPHV from the 5th order polynomial methodology when compared with natural cubic splines (Figure 3(A)). In girls, a similar, but opposite trend is observed, with tail estimates frequently being larger by the natural cubic spline methodology when compared with the 5th order polynomial methodology (Figure 3(B)). The observed trend in estimates of aPHV is less pronounced when comparing estimates derived from either the natural cubic spline or 5th order polynomial methodology to the SITAR methodology in both boys and girls (Figure 3(B)). Here, we observe that estimates of aPHV derived from the SITAR methodology are almost always later than those derived from either 5th order polynomial or natural cubic spline methodologies (Figure 3(A,B)), except for in boys where the latest aPHV estimates were derived from the 5th order polynomial methodology. Estimates of PHV exhibit a unique trend to that observed in aPHV estimates (Figure 4). In general, estimates of PHV between all three methodologies appear more diffuse with no clear pattern across the distribution of estimates. Estimates of PHV for boys and girls derived from the 5th order polynomial and natural cubic spline methodologies appear similar to one another with no systemic directional bias. However, estimates of PHV derived from the SITAR methodology appear to be consistently larger, in both boys and girls, when compared with 5th order polynomial or natural cubic spline methodologies (Figure 4(A,B)).

Figure 3.

Figure 3.

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aPHV: all pairwise estimate comparisons based on methodology for boys (A) and girls (B), where the trend line indicates a 1:1 relationship between methodology estimates. This image is the property of the authors.

Figure 4.

Figure 4.

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PHV: all pairwise estimate comparisons based on methodology for boys (A) and girls (B), where the trend line indicates a 1:1 relationship between methodology estimates. This image is the property of the authors.

The observable differences in estimates of PHV and aPHV between methodologies are driven by model fit. It is often difficult to detect the impact of these differences (see Figure 5(A)); but, differences in individual-specific velocity curves between methodologies are readily apparent (Figure 5(B)). The implications of this finding are critical for methodology selection, as all estimates of the adolescent growth spurt, like PHV and by extension aPHV, rely on mathematical assessments of both the velocity curve and the growth curve. To illustrate this nuanced but critical point, we have selected a single representative boy and girl from the Fels Longitudinal Study (Figure 5). Substantial overlap in the growth curves between methodologies can be observed in Figure 5(A) and is reflected in nearly identical RMSd across all methodologies for both boys and girls (Supplementary Figures 1 and 2). The largest visual differences in each methodology-specific growth curve occurs near PHV, represented by different slopes. These differences are magnified when considering the corresponding velocity curves, where estimates for both PHV and aPHV vary substantially. Estimates of PHV are largest in the SITAR methodology than when compared to either the 5th order polynomial or natural cubic spline methodology, with a maximum of 2.06 cm/year and 2.03 cm/year for the representative boy and girl, respectively. For the representative boy, aPHV ranged from 12.20 to 12.83 years of age, with the polynomial estimate being the earliest and the SITAR estimate being the latest. This was also observed for the representative girl, with an aPHV that ranged from 12.05 to 13.26 years of age. In general, estimates derived from the polynomial and natural cubic spline methodologies were most similar, with aPHV and PHV estimate differences being less than 9.84 months and 1.15 cm/year, respectively.

Figure 5.

Figure 5.

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Comparison of growth in height (A) and rate of growth in height (B) trajectories for a single representative boy (left panel) and girl (right panel) for each of the following methodologies: 5th order polynomial (Poly), natural cubic splines (NCS), and SITAR where the black dots represent individual height observations. This image is the property of the authors.

Discussion

Assessments of adolescent growth trajectories for individuals play a critical role in the diagnosis, treatment, and management of many skeletal growth and/or developmental disorders. Although a number of statistical methodologies exist for characterising this period of growth, methodological recommendations for their potential application to individual assessments of longitudinal growth are needed. The present study outlines the advantages and disadvantages of three commonly employed statistical methodologies to assess adolescent growth (e.g. 5th order polynomials, natural cubic splines, and SITAR; Table 2). Significant differences in estimates of aPHV and PHV were observed both within and between these methodologies. As expected, models for an individual more accurately represent growth trajectories that may be observed in paediatric clinical practice than when compared to population average models. Growth trajectories are highly variable (see Figures 1 and 2) requiring model flexibility in order to more accurately reflect individual growth; estimates of aPHV and PHV from polynomial methodologies impose the fewest structural restrictions to growth trajectories, a critical component for future application in paediatric clinical practice.

Table 2.

Advantages for each statistical modelling methodology by characteristic including, modelling considerations, bias, assumptions, and recommended use, where ● indicates yes and ○ indicates no.

Characteristics Type Description 5th order polynomial Natural cubic spline SITAR
Modelling considerations Flexibility Growth trajectory shape based on data ● ● ○
Growth trajectory shape constrained by knot placement ○ ● ○
Data type modelled Individuals ● ● ●
Populations ● ● ●
Sample density requirement Many observations for each individual ● ○ ○
Many observations for a population ○ ○ ●
Biological relevance Curve shape with cessation plateau ○ ● ○
Bias aPHV Older than expected relative to LOO population average ● ● ○
No bias ○ ○ ●
PHV Higher than expected relative to LOO population average ● ● ●
Assumptions Shape invariant model Individual curve shape constrained to population average with three random effect deviation parameters ○ ○ ●
Unique model Individual shape not influenced by population average ● ● ○
Recommended use Models for individuals Dense individual data ● ● ○
Dense population data ○ ○ ●
Models for population average Inter-population comparisons with dense data ○ ○ ●
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Population average models often underestimate PHV, which simultaneously impacts estimates of aPHV (Cao et al. 2018a; Cole et al. 2008; Merrell 1931). This occurs due to the inherent nature of the statistical model, which aims to minimise error. This results in a single growth trajectory that may not accurately represent an individual. However, these trajectories may play a critical role in how paediatric practitioners use previously published estimates of aPHV and/or PHV when assessing patients in their clinical practice. More specifically, paediatric practitioners might conclude that PHV has already occurred in their patients – changing their treatment plan – when in reality, it has yet to occur or vice versa. Both polynomial and natural cubic spline methodologies exhibit these differences between aPHV and PHV estimates for individuals and the population average, likely because these models allow for more flexibility in growth trajectory shape, each with differing levels of constraint. The natural cubic splines employed here are grounded at the 25th, 50th, and 75th percentiles of the observational age range for each individual, which provides inherent structure while maintaining trajectory flexibility. Yet, differences in population average and individual estimates for aPHV appear unique among boys and girls (Table 2; Supplementary Figures 1(E,F), 2(E,F)). As suggested by Tanner (1981), this may be due to underlying differences in growth trajectory shape between boys and girls. Individual and population average estimates derived from the SITAR methodology were almost identical. As SITAR requires individual growth trajectories to reflect the structure of the population average trajectory (i.e. shape invariant), traditional estimates and LOO cross-validated estimates are expected to be more similar than when compared to those for polynomial or natural cubic spline methodologies. However, there is no a priori reason to assume that individual growth trajectory shape should be identical across an entire population, a weakness of SITAR acknowledged by Cole et al. (2016). The size, tempo, and velocity parameters derived from SITAR are also estimated together, which may result in additional bias (Cole et al. 2016), particularly when strict differences in underlying curve shape exist. In fact, Cao et al. (2018a) recently discovered that when individual growth trajectory shapes differed from the population average trajectory, estimates of aPHV and PHV derived from the SITAR methodology were significantly biased. Specifically, Cao et al. (2018a) observed that irregular measurement frequencies resulted in generally larger estimates of PHV and earlier estimates of aPHV. Simpkin et al. (2017) also noted significant negative bias in estimates of aPHV derived from the SITAR methodology when irregular biennial measures were fit. Our data reinforce these concerns regarding biased estimates of aPHV and/or PHV derived from the SITAR methodology (see Supplementary Figures 1 and 2). This is particularly pertinent because the data presented herein are representative of dense longitudinal data without irregular measurement frequencies or increased measurement error, indicating that even with ideal data the inherent bias of a shape invariant methodology is not eliminated.

Many of the differences observed in mean estimates for aPHV and PHV within and between methodologies are a result of the underlying structure of each methodology. This includes the minimum number of observations required for each individual and the chronological age range over which those observations are spread. In a clinical scenario, data are often limited to a few years of biennial visits. Polynomial models, although flexible, require dense longitudinal data over extended age ranges to generate biologically meaningful estimates of aPHV and PHV. Although SITAR models for individuals can be fit with as few as four measurements during puberty (Cole 2018), significant bias towards earlier aPHV estimates occurs with irregular biennial measures (Simpkin et al. 2017). Additionally, convergence of the SITAR methodology requires a large population with dense longitudinal data. Natural cubic splines also require this structure for convergence, although data may be sparser than what is required for either polynomial or population average SITAR methodologies. An advantage of spline based modelling techniques, including SITAR, is that they assume a linear or zero asymptotic relationship between growth in height and age beyond the bounds of the observational data. This includes the plateau in height that occurs during the cessation of growth. Conversely, polynomial based methodologies do not have this characteristic, which often results in Runge’s phenomenon (i.e. fluctuations of growth trajectories at the bounds of the observational data; Epperson 1987), as seen in Figures 1 and 2. Estimates of aPHV and PHV can also be influenced by knot placement in spine-based methodologies. The natural cubic splines described here are grounded at the 25th, 50th, and 75th age range quartiles; however, there is no reason to assume that this knot placement is superior to other knot selection methods, such as bracketed age ranges. This method may blunt the slope of the piecewise polynomial near PHV compared to variable knot selection (Hamill 1977), particularly when quartiles are near true PHV estimates.

Although a number of statistical methodologies exist for modelling longitudinal changes in height during adolescence, certain methodologies may be more appropriate for particular questions. The advantages and disadvantages of the three statistical methodologies assessed here are summarised in Table 2. When global, population questions are of interest, SITAR should be used. This methodology results in summary statistics regarding both the general population and its individuals through three random effect estimates. In addition, the shape invariant nature of this methodology produces the smallest overall bias between the population average estimates and estimates for an individual, including those derived from the LOO cross-validated analysis. Yet, this methodology assumes that individual trajectory shape is identical to the population average trajectory (Cole et al. 2016), which is likely not appropriate for modelling individuals. Polynomials and natural cubic spline methodologies used to model individual growth trajectories are not dependent on population average trajectories, which may better reflect patient-based assessments. Although polynomial methodologies can result in biased estimates of aPHV and PHV, they allow for increased flexibility by restricting only the total number of curvilinear slope changes across the observational age range, as opposed to restricting both their number and placement, as in natural cubic splines. The data presented here are likely denser than that encountered in paediatric clinical practice; however, our results are critical to understanding the consequences of aPHV and/or PHV estimates derived from distinct statistical methodologies and their potential impact on clinical interpretations. As such, these data could be used by paediatric practitioners to answer relevant questions related to patient-specific interventions, including: (1) were the reported estimates derived from individual or population average mathematical models; (2) what statistical methodology was employed and how do the underlying assumptions of that methodology impact estimates of aPHV and/or PHV; and (3) is there any inherent bias in estimates derived from the statistical methodology employed? Although these data represent an important first step towards impacting paediatric clinical practice, future efforts should focus on: (1) assessing all applicable longitudinal growth models; (2) better understanding the relationship between childhood and adolescent growth in data derived from paediatric clinical practice; and (3) the development of predictive models that could be used by paediatric practitioners, including irregular biennial measures, as discussed by Cole (2018).

Conclusion

Understanding an adolescent’s current and future growth trajectory has critical implications for the treatment of many skeletal growth and/or developmental disorders. Many population average estimates of aPHV and PHV attenuate the timing and intensity of a single individual growth trajectory by regressing towards the mean (Cao et al. 2018a; Cole et al. 2008; Merrell 1931), which may influence the applicability of these estimates in paediatric clinical practice. Here, we have highlighted the advantages and disadvantages of three commonly employed growth curve methodologies (Table 2). Specifically, we have contributed the following to the existing literature: (1) a mathematical validation of three commonly employed growth curve methodologies, all of which were based on a large sample of normal boys and girls that were independent from those used to develop the methodologies; (2) a detailed assessment of methodological bias through a LOO cross-validated analysis; and (3) a discussion regarding the importance of individual and population average estimates of aPHV and/or PHV, particularly in how these estimates could be used to inform paediatric clinical practice. Taken together, these results suggest the use of the 5th order polynomial and SITAR methodologies when assessing estimates of aPHV and/or PHV in individual and population growth trajectories, respectively.

Supplementary Material

Published Supplementary Materials

Funding

This work was supported by the Eunice Kennedy Shriver National Institute of Child Health and Human Development (F31 HD091939, R01 HD012252) and the National Institute of Arthritis and Musculoskeletal and Skin Diseases (R01 AR055927).

Footnotes

Disclosure statement

The authors have nothing to disclose.

Supplemental data for this article is available online at https://doi.org/10.1080/03014460.2020.1763458.

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