[version 1; peer review: 2 approved with reservations]
Assessing and analysing growth is a key activity in paediatric epidemiology, building on centuries of research ^{ 1 }. Anthropometrics are easy to measure with basic equipment and the results are both immediate and meaningful with standardised reference measurements representative of unconstrained growth available from the World Health Organization (WHO) ^{ 2 }. This makes observations of weight, length, and weightforlength attractive as measures of a child’s long and shortterm health ^{ 3, 4 }. One of the main barriers to analysing child growth data is that individual growth trajectories display highly variable and complicated dynamic behaviour, differing markedly between children, even from the same geographic and socioeconomic group. As such, developing growth models from which actionable insights can be extracted – such as identification of interventions at the population level or predictive risk assessments at individual child level – is both methodologically and practically challenging. Here we introduce a new methodology, stochastic differential equation (SDE) ^{ 5 } models, into child growth research.
SDEs describe highly flexible dynamic processes comprising of two components: drift – gradual smooth changes, which could reflect developmental biological aspects such as physiology or gut microbiome ^{ 6 }; and diffusion – sudden shortterm perturbations or shocks – like illness ^{ 7 } or infection ^{ 8 }. This stochastic behaviour could potentially help explain the large variability seen in growth trajectories. SDEs are extensively used in certain specialised applications, most notably in financial modelling ^{ 9, 10 }, to cope with the complicated dynamics of stock price movements. Some case studies utilizing SDEs exist in medicine and biology ^{ 11 }, but they are not yet a part of a typical epidemiologist’s or statistician’s modelling toolbox.
To date, a wide range of different statistical curvefitting methodologies have been applied to child growth trajectories, from common classical approaches such as hierarchical linear mixed models ^{ 12 }, through to methods such as linear spline multilevel/broken stick ^{ 13 } models, SITAR ^{ 14 } growth curves, dynamic regression models ^{ 15 } and functional principle component models ^{ 16 }. SDEs are not curve fitting models but continuous time stochastic processes capable of rich dynamic behaviour.
In the Methods section we provide a brief overview of SDE models. We present a minimum of theory, using instead two empirical case studies to introduce the key features of SDE modelling and how it can be readily applied in practice. In the Results section we present a more complex case study, including quantifying the impact of covariates on growth, using data from the two African sites of the MALED study
^{
17
}. We conclude with a brief discussion of the opportunities for the application of SDEs in child growth research and outline some existing challenges. The computer code required to repeat the modelling results presented are provided as
We use individual child data from the MALED study, whose protocols, methodology and aggregate growth results have been presented previously ^{ 8, 17 }. MALED was initiated as a multicountry cohort study located across eight low and middleincome sites with historically high incidence of diarrhoeal disease and undernutrition, with a research focus on investigating determinants of development in children from birth through early years. Ethical approval was obtained from the National Institute of Medical Research for Tanzania (NIMR/HQ/R.8a/Vol.IX/858 and NIMR/HQ/R.8c/Vol.II/1034) and University of Venda Ethics Committee for South Afirca (SMNS/09/MBY/004). Approval was additionally given by the institutional review board of the University of Virginia, USA and all methods used in this study followed the relevant guidelines and regulations. Informed consent was taken from parents of all children prior to enrolment.
In our case studies we use data from Haydom, Tanzania
^{
19
} (n = 224) and Venda, South Africa
^{
20
} (n = 236). Our focus here is on anthropometric data from ages 0–24 months where each child included in these analyses had between 20 and 25 monthly observations (within a window of ±14 days), with 83% of children in Haydom and 86% in Venda having at least 24 observations. Weight and length, collected by trained fieldworkers and with minimal measurement error
^{
8
}, were converted to age and sex standardised zscores using the WHO reference standards
^{
21
}. Here we focus on weightforlength data, reflective of the relative weight of a child given their stature and therefore a child’s current nutritional status
^{
22
}, and one of the growth zscores recommended by the WHO for diagnosing acute malnutrition
^{
23
}.
Large within and between child variability is clear, with three random children highlighted in each site.
High within and betweenchild variability is the predominant feature of the raw trajectory data, which holds from birth through 24 months and for both sites. Three trajectories are highlighted in each site, and these illustrate the dynamic complexity of each child’s growth.
The standard introductory text for SDEs is by Øksendal ^{ 5 }, which contains a detailed mathematical exposition of SDEs. We focus on application and SDE models are introduced through examples with technical details largely omitted. We begin with a wellstudied special case SDE model which we fit to data from three individual children (separately) and compare results with a linear regression model. We then introduce a more general SDE model and fit this to data from all children from the Haydom site.
which models ZWfL at a subsequent age as a linear function of the current value and the elapsed age. In this model, growth velocity – rate of change per unit time (age) – is described by
Consider the same example as above but now where we have
This model now has three parameters, α, β and σ, and (
where α(β − X
_{t}) is called the drift, σ is the diffusion and W
_{t} is a Wiener (Brownian motion) process. The drift can be thought of as the slowmoving trend in growth velocity, while the diffusion is the continual perturbation of the system giving rise to volatility in velocity and therefore growth. This SDE has correlated movements through time, for example, in an OU process that commenced at time t
_{0} the covariance between any two points in time, t and s, is (e
^{−(s+t)α}(−e
^{2t
0α} + e
^{2α Min[s,t]})σ
^{2})/(2α).
To demonstrate the practical application of an SDE model to real data, we fitted OU models to three different ZWfL trajectories from the Haydom data. We compared these with the fit of a classical linear regression (LR) model, e.g. ZWfL(t) ~ N(a
_{0} + a
_{1}t, σ
^{2}), with both models having three parameters.
The difference between a curve fitting approach (fit globally) and time series approach (based on modelling the change over time) is clearly evident.
where X
_{t} is our growth outcome variable of interest (e.g. ZWfL). Functions μ(t,X
_{t},θ) and g(t,X
_{t},θ) generalise the drift and diffusion terms from (
which usefully emphasises that these are models of the evolution of a continuous time stochastic process – here growth of a child.
For model fitting we need to compute the likelihood function given the slice density. If we consider first the likelihood function for trajectory data from a single child, and where we have N observations over time, then the negative loglikelihood for a single child can be written as (see Hurn
^{
10
})
where f
_{0}(X
_{0}θ) is the probability density of the growth outcome variable at the first available data point, f(X
_{k+1}X
_{k},θ)≡f((X
_{k+1},t
_{k+1})(X
_{k},t
_{k}),θ) is the value of the slice density function for a stochastic process starting at (X
_{k},t
_{k}) and evolving to (X
_{k+1},t
_{k+1}).
In summary, the key steps for working with SDE models focussed on model fitting are: (i) choose a form of μ(t,X
_{t},θ) and g(t,X
_{t},θ) in (
Model  No.

Remarks  AIC

BIC


OrnsteinUhlenbeck

3  No random effects 


Linear regression  15454  15473  
OrnsteinUhlenbeck

6  Random speed of reversion and



Linear mixed

Random intercept and slope with

11607  11627 
AIC, Akaike information criterion; BIC, Bayesian information criterion.
For the same number of parameters, the OU process gives substantially better Akaike information criterion (AIC) and Bayesian information criterion (BIC) metrics, and fitting these SDE models including random effects is straightforward, requiring only with a few lines of code in SAS's proc nlmixed. These mixed models can also be implemented in the Stan ^{ 27 } language, with an OU specific example using Stan provided by Goodman (2018) ^{ 28 }.
Our main results comprise of an illustrative case study where we considered the combined data from Haydom and Venda. The general model formulations considered, and model search process are detailed below. To keep the analysis as clear as possible we considered only one covariate, (in addition to age) in the modelling, a categorical variable indicating site.
where X
_{t} is ZWfL at age
Comparing
The main realworld application area of SDEs is in predictive modelling (e.g. Iversen
Predictions are calculated using a 10fold random sampling approach, where we draw from all the parameters estimated (fixed and random) in the best fitting SDE model – one set for each child. Which parameter sets are chosen to generate predictions depends on how likely trajectories generated from each set are to have visited each given starting point across an ageZWfL grid. This adds an important element of “locality” to our predictions, combined with 10fold sampling to provide an indication of robustness of our predictions. A more detailed description of the prediction algorithm is given below, with full R code provided in SI.5 (see
For each point across an ageZWfL grid we compute the likelihood of observing this point for each set of parameters, using the relevant slice density, where the initial starting point is the first age available for each trajectory. Predictions from each grid point progress through increasing ages using the new slice distribution at each next point in time. For example, for predictions in Venda we have n=236 likelihood values for each ageZWfL grid point. The most likely parameter set is then used for the prediction of the next ZWfL at the next age, with 10fold sampling used to indicate how robust this prediction is. The 10fold sampling splits these n=236 parameter sets into 10 random groups, and within each group we choose the parameter set with the highest likelihood value as the one to be used for the next prediction. In summary, from a fixed starting point in an ageZWfL grid we have a main prediction at each future age up to 24 months, plus 10 additional predictions at each age as a sensitivity analysis (incrementing age in small steps).
Using individual child trajectory data over 0–24 months for ZWfL from all n=460 children from the two sites, the best fitting LMM model was a cubic polynomial with a single interaction term between age squared and site (with no separate term for site), which gave AIC=21913 and BIC=21983. The best fitting SDE model had corresponding values of AIC=21718 and BIC=21772, where this model had random effects in four of the six model parameters, a diagonal covariance matrix and site included in two of the drift parameters and the diffusion parameter. Full modelling descriptions can be found in SI.4a and results, including parameter estimates, can be found in the SI4.b (see
SDE model has narrower confidence intervals in each site, entirely contained within the wider LMM confidence interval.
In summary, our results so far suggest that our SDE model is at least as good, and appears superior in some respects, to a reasonable choice of classical LMM.
The bestfitting predictions are shown in red and 10fold cross validation in grey. The distributions of observed weightforlength at both zero and 24 months in each site is shown to the right of the respective plot. The predictions appear to cluster together into a smaller set of “paths”, which also differ between sites.
We have presented a novel approach for analysing child growth trajectories, using a modelling methodology, stochastic differential equations, widely used in other fields but not yet in child growth research. The use of a continuous time stochastic process approach, such as SDEs, to model child growth trajectories is conceptually appealing as it explicitly acknowledges  through drift and diffusion processes  the highly complicated dynamic environment into which a new born child is delivered and exposed, particularly in resourcelimited settings. Our results show that SDEs also have practical appeal as they offer very different (highly nonlinear) formulations from the usual additive linear models, which gave good results with our case study data. This suggests that SDEs may be an attractive alternative to other established methods, at least as supporting analyses, moreover because SDEs can also be readily fitted using standard software such as SAS or open source alternatives such as Stan. Our supplementary information contains modelling code that can adapted to other study data ^{ 18 }.
While only an initial exploration of a subset of the MALED data using SDE modelling, our predictive results presented in
We restricted our presentation to a narrow selection of simple SDE models, many more parameterizations are available with explicit expressions for the slice density (e.g. using software like Mathematica). More complex formulations, particularly for the diffusion function g(t,X _{t},θ), may add considerable richness to an SDE model’s dynamic behaviour; however, these would require numerical methods to compute the likelihood function. Initial explorations suggest this is far from straightforward, both in terms of computational feasibility and also numerical stability, and is another area ripe for future development.
Data from the MALED study are available from
Zenodo: Introducing a drift and diffusion framework for childhood growth research.
This project contains the following extended data:
SI1.pdf (Model code for OU and linear regression models, in SAS)
SI2.pdf (Illustrative reference slide densities, using Mathematica)
SI3a.pdf (SAS code to compare OU and LMM models for
SI3b.pdf (SAS model output comparing OU and LMM models in
SI4a.pdf (SAS code to fit mixed effects OU and LMM models)
SI4b.pdf (SAS model output for OU and LMM models)
SI5.pdf (R code to generate figures, including the prediction algorithm to generate
Data are available under the terms of the
Source code available from:
Archived code at time of publication:
License:
We thank the participants and staff of the MALED study for their vital contributions and we thank Prof. Laura Caulfield for her insightful and constructive input. The findings and conclusions in this report are those of the authors and do not necessarily represent the official position of the U.S. National Institutes of Health or Department of Health and Human Services.
This is an interesting and novel methods paper outlining an approach to analyzing child growth through translation of the SDE approach from physics and financial modelling. SDEs extend the concept common to growth modelling methods that individuals vary stochastically about a population curve.
It would be helpful in the introduction to outline which uses the SDE approach is aiming at. For example, descriptive, explanatory or forecasting in populations or realtime use in individuals.
In the introduction, for clarity please also include whether ‘drift’ is also thought to represent exposures such as seasonal food security and dietary intake patterns and feeding practices that may change with age (e.g. breastfeeding) which may be more influential than the described ‘developmental biological aspects’.
Reference 8 suggests that good reliability weight and length can be assumed, however length is more difficult to measure precisely in younger infants and I notice extreme values appear more common close to the time of birth on the figures, including 4 of the 6 individuals presented in figures 1 and 2. Translation to Z scores (a ratio) may compound errors. Was reliability assessed for (computer calculated) ZWfL? If the data are available, I suggest including it in this report, including performance across the age range 024m. It would also be worth commenting that apparent weightforlength may be affected by hydration status as both may provide additional stochastic variability that varies with age. The former may potentially affect the constant term X _{0} and the starting points across the ageZWfL grid.
An issue worth considering is that when examining changes in Z scores is that the standard deviation (the denominator for ZWfL) varies across the range of length (see:
The formulations of the SDE models themselves is outside my expertise.
In Methods, example 2, it may be helpful to the readers to point out if child ID was a random effect term in the nonlinear mixed model variants and explain ‘random speed of reversion’.
The longitudinal clustering is very interesting, with potential genetic and environmental interpretations. However, the finding would be strengthened by validation using another method such as simple probability estimates for ending up in one of the final ‘bins’ identified in the model to reassure readers this is not an artifact of the method (quantization?) or due to different error in the first measurements (as above).
The discussion should expand on future research that the method leads to. Anthropometry is a practical proxy for health and nutritional status and validation of criteria for intervention are typically based on mortality risk in large populations. As a next stage of research, prediction of actual health events such as death, serious illness or impaired neurodevelopment would be valuable to mention. Additionally, it would be helpful to comment on the models’ ability to include timevarying covariates such as seasonality, illness or food security shock, as well as preterm birth.
The narrower CIs suggest the SDE model is a better approximation of the data than the linear mixed models. However, linear models may not be regarded as a current gold standard for growth models, and comparisons with others including latent growth models, SITAR, multilevel fractional polynomial models or penalised splines etc. This is a limitation of the paper and area for future work.
Is the work clearly and accurately presented and does it cite the current literature?
Yes
If applicable, is the statistical analysis and its interpretation appropriate?
I cannot comment. A qualified statistician is required.
Are all the source data underlying the results available to ensure full reproducibility?
Yes
Is the study design appropriate and is the work technically sound?
Yes
Are the conclusions drawn adequately supported by the results?
Partly
Are sufficient details of methods and analysis provided to allow replication by others?
Yes
Reviewer Expertise:
Child survival, infectious diseases, malnutrition, newborn health, clinical trials, anthropometry
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.
We have amended the introduction to note that we believe this approach to be particularly useful in explanatory analysis of growth data (“SDEs can enhance mechanistic interpretation of the drivers of variability”). We note that in the discussion we do discuss the advantage SDEs offer to forecasting because they are processbased models rather than purely descriptive. Forecasts ought, assuming appropriate covariables, better account for variance.
We have added these to the text, but we note that the terms that are included in the drift and diffusion are specific to the conceptual model examined. Certainly, these terms could be included (in either or both components) and it is a matter of the appropriate technical details. This reflects the richness of the method, because such exposures can be explored mechanistically. Diet might conceivably impact both drift (‘usual’ diet) or diffusion (seasonal food insecurity).
Reliability (noted in reference 8) was measured on the basis of 5% remeasurement. Reliability was indeed lower for measurements of younger children, and lower for the combined WFL than either weight or length, albeit with correlation coefficients over a range of 0.92 (1 month) to 0.98 (24 months). We have noted that this could be incorporated into the model.
The Leroy et al. argument is particularly interesting in the context of recovery from stunting, which tends to occur post24 months. For this particular study, WFL was used as an example and the methodology would be equally applicable to other anthropometric metrics.
It is our hope that this introduction may inspire others to consider this approach and that it will make the methods less opaque by demonstrating their application and accessibility in conventional software.
We have now noted that these were childlevel parameters (random effects). The random speed of reversion is the covariance between time points, so the closer time points are the more similar they are. We have added a note to the text.
We apologise that we miswrote the code for the prediction presented in figure 4. The model to evaluate predictions was correct, but the 10fold selection of the best fitting points incorrectly selected observations, not trajectories. We have updated the figure, which more accurately captures the density of observations and have added a ‘rug plot’ to the figure, with marks to indicate the observed WFL at months zero and 24.
It is still the case the trajectories tend to cluster into a smaller number of ‘streams’ as different initial conditions tend to converge, but to a much lesser extent.
This is an interesting idea and perhaps the opposite way to how we were framing the question (looking at how events during childhood manifest in growth outcomes rather than growth as a predictor of later outcomes). Individuallevel coefficients from the model could indeed be used as parameters to predict other health outcomes. We have noted the incorporation of seasonal parameters.
It is not clear that there is a gold standard (linear mixed models are still widely used) as such and as Reviewer 1 notes, there is not much to pick between such models that are fundamentally similar in concept (e.g. fitting to the mean using some variant of least squares). We note that the accessibility of the method in conventional software might mean that this approach can be run in parallel for routine comparisons. That said, we have noted this important future avenue in the discussion.
The authors need to explain why yet another model for growth is required or necessary or advantageous.
Introduction: “One of the main barriers to analysing child growth data is that individual growth trajectories display highly variable and complicated dynamic behaviour, differing markedly between children, even from the same geographic and socioeconomic group. As such, developing growth models from which actionable insights can be extracted – such as identification of interventions at the population level or predictive risk assessments at individual child level – is both methodologically and practically challenging.”
The variability demonstrated between children is not a “barrier” to analysing growth data. The variability is a measure of the variation in normal growth rates and in response to potential constraints on growth. To smooth out or diminish this variation without investigation destroys the very essence of growth studies particularly in clinical scenarios.
“WHO reference standards” – there is no such thing as a “reference standard”. Growth charts are either based on crosssectional large sample data and are “references” or selected longitudinal samples and are “standards”. The WHO charts are standards.
Whilst I am impressed by the comparison between the OU method and LMM in Figure 4, the lack of any apparently significant improvements or insights into the growth trajectory makes me wonder if the new method really does represent a significant advantage over previous methods.
I would like the authors to explore what precisely is gained by applying this method as opposed to previous methods. Does it allow earlier detection of poor growth? Does it provide a more reliable prediction of future growth? If used as an intervention trigger what preceding data are required and prediction accuracy is obtained?
References include only minimal publications relating to growth modelling.
Is the work clearly and accurately presented and does it cite the current literature?
Partly
If applicable, is the statistical analysis and its interpretation appropriate?
I cannot comment. A qualified statistician is required.
Are all the source data underlying the results available to ensure full reproducibility?
Yes
Is the study design appropriate and is the work technically sound?
Yes
Are the conclusions drawn adequately supported by the results?
Partly
Are sufficient details of methods and analysis provided to allow replication by others?
Yes
Reviewer Expertise:
Human growth and development  Auxology
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.
This is an important consideration true of any modelling work. Given that models can serve a multitude of purposes in this case, we believe that this alternative modelling strategy (rather than the model results
We have revised the text to “challenges” and thank the reviewer for their comment. The principle purpose of an SDE model is that it explicitly describes variability in growth rates rather than fitting a population mean and treating variance as a nuisance parameter. SDEs are most often used where a process has noise and variability, as the reviewer notes, is the case with growth modelling. In this sense, the SDE is the perfect match to produce better (more robustly capturing uncertainty) forecasts of growth. We have amended the text to better draw out this point (“more accurately capturing external sources of variability and uncertainty”).
Thank you for this clarification, we have amended the text accordingly.
We illustrate an improved model fit using a relatively simple model akin to Brownian motion. This manuscript lays out an introduction to the approach, but the SDE method is much richer in terms of capturing a dynamic stochastic process.
As with the first question posed by reviewer 2, we have noted that the principle advantage of the SDE approach is the descriptive partitioning of variance (“SDEs can enhance mechanistic interpretation of the drivers of variability (both long and shortterm) in growth”). In that sense, we believe this method can (with appropriate terms) offer superior mechanistic insight: variables can relate to one or both of the drift and shift components. With such insight, one might more appropriately distinguish between factors relating to growth trends and those relating to shortterm fluctuations (albeit these fluctuations can have lasting consequences if negative effects aren’t removed).
Most published growth models take a very similar methodological approach (curvefitting) whereas out intention was to introduce a conceptually different approach rather for contrast.