Modelling subject-specific childhood growth using linear mixed-effect models with cubic regression splines

Abstract

BackgroundChildhood growth is a cornerstone of pediatric research. Statistical models need to consider individual trajectories to adequately describe growth outcomes. Specifically, well-defined longitudinal models are essential to characterize both population and subject-specific growth. Linear mixed-effect models with cubic regression splines can account for the nonlinearity of growth curves and provide reasonable estimators of population and subject-specific growth, velocity and acceleration.MethodsWe provide a stepwise approach that builds from simple to complex models, and account for the intrinsic complexity of the data. We start with standard cubic splines regression models and build up to a model that includes subject-specific random intercepts and slopes and residual autocorrelation. We then compared cubic regression splines vis-à-vis linear piecewise splines, and with varying number of knots and positions. Statistical code is provided to ensure reproducibility and improve dissemination of methods. Models are applied to longitudinal height measurements in a cohort of 215 Peruvian children followed from birth until their fourth year of life.ResultsUnexplained variability, as measured by the variance of the regression model, was reduced from 7.34 when using ordinary least squares to 0.81 (p < 0.001) when using a linear mixed-effect models with random slopes and a first order continuous autoregressive error term. There was substantial heterogeneity in both the intercept (p < 0.001) and slopes (p < 0.001) of the individual growth trajectories. We also identified important serial correlation within the structure of the data (ρ = 0.66; 95 % CI 0.64 to 0.68; p < 0.001), which we modeled with a first order continuous autoregressive error term as evidenced by the variogram of the residuals and by a lack of association among residuals. The final model provides a parametric linear regression equation for both estimation and prediction of population- and individual-level growth in height. We show that cubic regression splines are superior to linear regression splines for the case of a small number of knots in both estimation and prediction with the full linear mixed effect model (AIC 19,352 vs. 19,598, respectively). While the regression parameters are more complex to interpret in the former, we argue that inference for any problem depends more on the estimated curve or differences in curves rather than the coefficients. Moreover, use of cubic regression splines provides biological meaningful growth velocity and acceleration curves despite increased complexity in coefficient interpretation.ConclusionsThrough this stepwise approach, we provide a set of tools to model longitudinal childhood data for non-statisticians using linear mixed-effect models.Electronic supplementary materialThe online version of this article (doi:10.1186/s12982-015-0038-3) contains supplementary material, which is available to authorized users.

Measuring mortality due to HIV-associated tuberculosis among adults in South Africa: Comparing verbal autopsy, minimally-invasive autopsy, and research data

“…This study had limitations: the median time from death to VA was slightly longer than the ideal three months that some recommend, but was well within the maximum 12 months recommended by WHO and was therefore considered unlikely to have had a substantial effect on VA-generated estimates [78,79]; physicians who reviewed clinical and VA data were aware that most decedents were likely HIV-positive and had been enrolled into TB-focused studies, which may have led to greater assignment of HIV- and TB-related CoD; missing operational and research data may have affected consistency of the reference standard; pathological autopsy data were available for a small number of decedents; and, although the reference CoD assigned represent our best estimates using the data available, the true CoD may still differ. Questions on ART and TB treatment, added to the VA instrument by the study team, may have led to changes in how events were reported in the free narrative section; the answers to the questions themselves, however, were not provided to reviewing physicians or to either software.…”

Bayesian Hierarchical Modelling of Growth Curve Derivatives via Sequences of Quotient Differences

“…(), Grajeda et al . () and Song ()), local polynomial regression (Gasser and Müller, ; Fan and Gijbels, ) or weighted sequences or quotient differences of the observed data (Müller et al ., ; Härdle, ; De Brabanter et al ., ; Wang and Lin, ; Dai et al ., ; Charnigo et al ., ; De Brabanter and Liu, ). Thus, one approach to estimate derivative curves in a hierarchical setting would be to apply one of the methods that were previously listed independently for each subject.…”

“…One possible reason for this is that f .1/ is rarely (if ever) directly measured, introducing complexities that are associated with modelling and estimating f .1/ that do not exist for f (Ramsay and Silverman, 2005). Although sparse, the literature contains methods for estimating a single f .1/ that are based on splines (see, for example, Ramsay and Silverman (2005), Sangalli et al (2009), Grajeda et al (2016 and Song (2016)), local polynomial regression (Gasser and Müller, 1984;Fan and Gijbels, 1996) or weighted sequences or quotient differences of the observed data (Müller et al, 1987;Härdle, 1999;De Brabanter et al, 2011Wang and Lin, 2015;Dai et al, 2016;Charnigo et al, 2011;De Brabanter and Liu, 2015). Thus, one approach to estimate derivative curves in a hierarchical setting would be to apply one of the methods that were previously listed independently for each subject.…”

Dynamic prediction in functional concurrent regression with an application to child growth

“…() Both approaches are parametric and are not designed to account for subtle or strong departures from the assumed parametric trends. This problem can manifest in a number of ways in longitudinal data, including autocorrelation in the residuals of random intercept/slope models . To address such challenges, one may consider using methods for functional data, which allow more flexible modeling of subject‐specific random curves.…”

“…This problem can manifest in a number of ways in longitudinal data, including autocorrelation in the residuals of random intercept/slope models. 5 To address such challenges, one may consider using methods for functional data, which allow more flexible modeling of subject-specific random curves. A random functional intercept model can be understood as a special case of a broader class of models referred to as functional mixed effects models.…”