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![Example illustrating the limitations of using polynomial functions to approximate a nonlinear growth trajectory. Coloured lines represent predicted trajectories from LME models with age as (a) linear term and as (b) quadratic polynomial and (c) cubic polynomial. Points display weight measurements taken from 70 females in the Berkeley Child Guidance Study. Dataset was originally provided as an appendix to the book by Tuddenham and Snyder (1954). The data for this example were taken from the freely accessible ‘Berkeley’ dataset provided with the ‘sitar’ package [15]](publication/359251544/figure/fig1/AS:1146404084948992@1650335441077/Example-illustrating-the-limitations-of-using-polynomial-functions-to-approximate-a.png)
Example illustrating the limitations of using polynomial functions to approximate a nonlinear growth trajectory. Coloured lines represent predicted trajectories from LME models with age as (a) linear term and as (b) quadratic polynomial and (c) cubic polynomial. Points display weight measurements taken from 70 females in the Berkeley Child Guidance Study. Dataset was originally provided as an appendix to the book by Tuddenham and Snyder (1954). The data for this example were taken from the freely accessible ‘Berkeley’ dataset provided with the ‘sitar’ package [15]
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Background
Longitudinal data analysis can improve our understanding of the influences on health trajectories across the life-course. There are a variety of statistical models which can be used, and their fitting and interpretation can be complex, particularly where there is a nonlinear trajectory. Our aim was to provide an accessible guide along wi...
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Introduction
Coagulation is involved in fibroproliferative responses following acute myocardial infarction (AMI). Left ventricular (LV) remodeling following AMI is closely associated with progression to heart failure. This study aims to assess the association between plasma tissue factor activity and LV remodeling in post-AMI patients.
Methods
We...
Citations
... [22] introduced a longitudinal data analysis method using the natural spline regression, modeling time as a continuous variable while accounting for testing version effects to capture the mean trajectory over time. Similarly, [23] provided a practical guide for summarizing nonlinear growth patterns of measured continuous outcomes using linear or natural spline regression. ...
This study evaluates and compares various spline techniques in the nonparametric regression analysis, specifically focusing on the smoothing spline regression, the natural spline regression, the B-spline regression, and the penalized spline regression. The dependent variable in this analysis is time series data generated by a random walk process, while the independent variable is represented as sequential data. The simulation data, derived from a random walk process with diverse variances and sample sizes, ensures an absence of fixed patterns in the variable's changes. In addition, real-world data from the monthly trading volume of the SET (Stock Exchange of Thailand) index is used for practical application. The criterion for model efficiency estimation is based on minimizing the average mean square error for the simulation and SET index data. At the same time, predictive performance for future values is assessed through the minimum of average mean absolute percentage error. Among the models tested, the natural spline regression achieved the minimum average mean square error in all simulations due to SET index data estimation, excelling in model fit. However, the B-spline regression proved highly effective for forecasting future values.
... This large variability, especially in growth outcomes among preterm infants, calls for robust growth models to accurately capture and understand these patterns over time. Growth models (Beath 2007;Cole et al. 2010;Johnson et al. 2013;Elhakeem et al. 2022) are essential for analysing longitudinal data, as they capture overall growth patterns in the population while including and accounting for individual characteristics of growth trajectories. However, traditional growth models, such as linear or nonlinear regression, often struggle to accommodate the unique growth trajectories observed in preterm infants. ...
Background
Children born preterm grow differently from those born at term.
Aim
To compare growth in length/height, weight, and BMI of preterm- and term-born children, grouped by birth weight (BW) and gestational age (GA).
Subjects and Methods
Longitudinal data of 950 children (birth to 12 years) were collected retrospectively. Growth trajectories were modelled using SITAR (Superimposition by Translation and Rotation) by sex, with three groups each for GA and BW.
Results
SITAR summarised growth patterns from birth to 12 years and explained 76–79% of height variance, 90–92% for weight, and 72–75% for BMI. Early preterm and low BW groups were shorter, lighter and thinner on average than their term or normal BW peers, with late preterm and low-normal BW groups intermediate. Effects were larger for BW than GA, e.g. early preterm girls/boys were 0.3/0.8 kg lighter, 0.9/0.9 cm shorter and 0.8/0.8 kg/m² thinner, while low BW girls/boys were 0.5/1.0 kg lighter, 1.5/1.4 cm shorter and 0.8/0.9 kg/m² thinner. Moreover, faster growth rates were associated with lower BW.
Conclusion
Both BW and GA significantly impacted growth, but low BW more so than early preterm birth. This underscores the need for targeted interventions for low BW children to address potential long-term growth challenges.
... We used restricted cubic splines with 2 knots to allow for non-linear changes in air pollutants with time. 30 For each model, a grid of 50 initial values was tested. The same individuals included in the analyses using average levels of air pollution were included in this analysis under the missing at random (MAR) assumption. ...
Aims: We assessed the association between air pollution from pregnancy (in utero) to 18 years and cardiovascular health markers in early adulthood.
Methods: Data from 3,767 individuals from a UK birth cohort were used. We explored the associations between modelled fine particulate matter (PM2.5), nitrogen dioxide (NO2) and black carbon (BC) across an 18-year period and eight cardiovascular health markers measured at 18 year of age. Long-term exposure to air pollution was assessed by averaging the air pollutants over time and by creating air pollutant trajectories. Linear regressions were used to assess the associations between air pollutants and cardiovascular health markers. Possible sensitive periods of exposure and sex differences in these associations were also explored.
Results: Higher average levels of PM2.5 and NO2 were associated with higher peripheral (pDBP) and central diastolic blood pressure (cDBP); e.g., an interquartile range increase in PM2.5 was associated with 0.46 mmHg (95%CI 0.14, 0.78) higher pDBP and 0.50 mmHg (95%CI 0.17, 0.83) higher cDBP. Higher average PM2.5 levels were also associated with lower carotid intima-media thickness and higher BC levels were associated with higher heart rate (HR). Latent classes showed the same overall patterns of association, with the trajectory classes with the highest levels of air pollution exposure tending to have higher pDBP, cDBP and HR. There was little evidence of sensitive periods of exposure and sex differences in the associations.
Conclusions: Higher lifetime exposure to air pollution up to 18 years was associated with markers of poorer cardiovascular health in early adulthood.
... Fetal growth trajectories were derived using natural cubic spline multilevel models 22 with two levels, clustering measurement occasions (Level 1) within individuals (Level 2). Through this approach, withinperson correlations in repeat measures are accounted for. ...
Background
Obesity during pregnancy is related to fetal overgrowth. Effective interventions that can mitigate this risk are needed.
Objectives
This study aimed to investigate the effect of a lifestyle intervention for pregnant women with obesity on fetal growth trajectories.
Methods
In the DALI trial, pregnant women with a body mass index ≥29.0 kg/m² and without gestational diabetes at baseline were randomized to counselling on physical activity (PA), healthy eating (HE) or a combination (PA + HE), or to usual care (UC). Fetal growth trajectories were modelled based on a combination of estimated fetal weight (EFW) from repeated ultrasound scans and weight measured at birth. Differences in fetal growth trajectories between groups were assessed.
Results
Three hundred eighty‐four women were included. Those in the PA + HE intervention had slower EFW gain from 32 weeks onwards, with differences (PA + HE vs. UC) at 32, 36 and 40 weeks of −54.1 g (−146.7 to 38.9 g), −84.9 g (−194.0 to 24.7 g), and −99.8 g (−227.1 to 28.1 g), respectively. Effects appeared stronger in males, with a difference at 40 weeks of −185.8 g (−362.5 g to −9.2 g) versus −23.4 g (−190.4 g to 143.5 g) in females.
Conclusions
A lifestyle intervention for pregnant women with obesity resulted in attenuated fetal growth, which only reached significance in male offspring. Future larger trials are needed to confirm these findings and elucidate underlying pathways.
... The presence of nonlinearity in a variable is assessed using the p-value of the smooth function; a p-value of 0.05 or below indicates a nonlinearity issue. For the management of nonlinearity, natural splines (NS) have the flexibility to capture nonlinearity while preventing overfitting with linear constraints at the boundaries, making them well-suited for representing stable nonlinearity [31]. The NS applies polynomial functions within specific intervals and enforces linear constraints at the boundaries, which reduces unnecessary variability and allows for smooth, stable nonlinear transitions across the entire observation period. ...
As the operational years of power facilities increase, the importance of accurately assessing the remaining lifespan or risk of failures of distribution transformers for the stable power supply has also grown. However, unlike the typical transformer failures, it is not quantified that the failures are also influenced by weather conditions such as temperature, humidity, precipitation, etc. In this regard, we propose the method for investigating the impact of the weather conditions on the failures of distribution transformers to improve the asset management. By employing survival analysis approach, frameworks are employed to quantify the spatial influence of weather conditions on distribution transformer failures, and the weather risk factors are integrated into lifespan indicators to improve maintenance strategies and reliability assessments. Distribution transformer failure cases of over 11 years in South Korea and the corresponding historical weather conditions are configured to enable the survival analysis and derive the meaningful real-world analysis results. Specifically, weather data are quantified and interpolated to configure the spatial data. The notable results underline quantifying significant impact of high summer temperature and humidity in term of hazard ratios. These findings indicate that incorporating historical weather data into asset management indices can significantly enhance the accuracy of transformer lifespan predictions and inform maintenance planning. This study represents the first long-term survival analysis correlating historical weather conditions with transformer failures and provides a guideline for improving power-facility management in the context of environmental changes.
... in the field of human growth and development the modelling of time trends in growth patterns allows missing data to be input within known measurement errors (e.g. anderson et al. 2019;Elhakeem et al. 2022;Howe et al. 2016) resulting in useful data covering the entire pre-adult period. in psychological and health research which relies on recalled events and potential exposures during childhood and adolescence, there is unreliability of recall in terms of timing, duration, and severity of exposure increases over time. in such research designs, methods that link historical factual data with age at time of exposure provide more reliable data but are subject to the limitations of dated exposure measurement techniques. ...
... ; supports trajectory modelling using fixed linear, quadratic, cubic and quartic polynomial terms, and random intercept, linear and quadratic terms (thereby allowing random slopes), as per previous research. 1,7,15 To guide the choice of modelling, a line plot of the mean scores at each occasion is presented, alongside descriptive information (N, mean, standard deviation, median and interquartile range of the outcome variable at each occasion). Users need to select a model to initiate the descriptive information and plot, so we recommend running a linear model to begin exploring the data. ...
Motivation:
Growth curve modelling is one method used to model trajectories of traits and behaviours over time. However, accessing, analysing and interpreting trajectories requires statistical expertise, thereby creating potential barriers for users to implement and understand longitudinal traits. TIDAL is a user-friendly research tool designed to facilitate trajectory modelling by improving access, analysis and interpretation of trajectory and longitudinal data.
Implementation:
TIDAL is available in two formats: an R package and an online Shiny application. The R package can be used offline, negating the need to upload potentially sensitive data.
General features:
TIDAL includes all the main steps of trajectory analysis including: 1) data preparation, (converting data from wide to long format); 2) data exploration, via basic plots and descriptive information; 3) analysis of trajectories using mixed effects modelling, interpretation of results, visualisation of trajectories, and extraction of key features (scores at different ages; area under the curve); and 4) interactions to derive population specific trajectories, combined with all the above. TIDAL is built with a simple graphical interface to guide users through each step. R syntax accompanies each step.
Availability:
Both versions of TIDAL can be found here: [https://tidal-modelling.github.io/].
... Age at menarche was calculated as the first reported age at onset of menstruation. Pubertal age for all other measures was derived using the SITAR (Super Imposition by Translation And Rotation) method of growth curve analysis 28,29 . SITAR is a shape invariant nonlinear mixed effects model that fits a single (mean) natural spline growth curve in the study sample and tailors it (using random effects) to define how individual growth curves differ from the mean curve. ...
Background
Pubertal timing is heritable, varies between individuals, and has implications for life-course health. There are many different indicators of pubertal timing, and how they relate to each other is unclear. Our aim was to quantitatively compare nine indicators of pubertal timing.
Methods
We used data from questionnaires and height, weight, and bone measurements from ages 7–17 y in a population-based cohort of 4267 females and 4251 males to compare nine growth and development-based indicators of pubertal timing. We summarise age of each indicator, their phenotypic and genetic correlations, and how they relate to established genetic risk score (GRS) for puberty timing, and phenotypic childhood body composition measures.
Results
We show that pubic hair in males (mean: 12.6 y) and breasts in females (11.5 y) are early indicators of puberty, and voice breaking (14.2 y) and menarche (12.7 y) are late indicators however, there is substantial variation between individuals in pubertal age. All indicators show evidence of positive phenotypic intercorrelations (e.g., r = 0.49: male genitalia and pubic hair ages), and positive genetic intercorrelations. An age at menarche GRS positively associates with all other pubertal age indicators (e.g., difference in female age at peak height velocity per SD higher GRS: 0.24 y, 95%CI: 0.21 to 0.26), as does an age at voice breaking GRS (e.g., difference in age at male axillary hair: 0.11 y, 0.07 to 0.15). Higher childhood fat mass and lean mass associated with earlier puberty timing.
Conclusions
Our findings provide insights into the measurements of the timing of pubertal growth and development and illustrate value of various pubertal timing indicators in life-course research.
... The primary explanatory factors included in the models are time, partner status, age, the number of active modules, and the specific module associated with an outcome (eg, the nutrition module for the PDR score). To appropriately model time as a continuous variable, we will use linear functions, polynomial terms, or natural cubic splines [37]. ...
Background
Many couples undergoing fertility treatment face multiple lifestyle risk factors that lower their chances of achieving pregnancy. The MyFertiCoach (MFC) app was designed as an integrated lifestyle program featuring modules on healthy weight management, nutrition, exercise, quitting smoking, reducing alcohol and drug use, and managing stress. We hypothesized that supplementing standard care with the MFC app would improve lifestyle outcomes.
Objective
This study aims to assess the impact of the MFC app on changing multiple lifestyle habits in women seeking fertility treatment. The primary outcome is the change in the total risk score (TRS) at 3- and six-month follow-ups. The TRS is calculated for each individual as the sum of all risk scores per behavior (eg, vegetable/fruit/folic acid intake, smoking, and alcohol use) at 3 and 6 months. A higher TRS indicates unhealthier nutrition and lifestyle habits and a lower likelihood of achieving pregnancy. The secondary endpoints include changes in BMI, activity score, preconception dietary risk score, distress score (eg, perceived burden), smoking habits, alcohol intake, and program adherence.
Methods
This retrospective, observational, single-center evaluation included patients between January 1, 2022, and December 31, 2023. Subfertile female patients aged 18-43 years and their partners, who were referred to a gynecologist, were invited to participate in online lifestyle coaching via the MFC app. The gynecologist selected relevant lifestyle modules based on the results of integrated screening questionnaires. We used (hierarchical) linear mixed models (LMMs) to estimate changes in outcomes. For missing data patterns deemed missing not at random, joint modeling was applied. Statistical significance was set at P≤.05, with methods in place to maintain the same false-positive rate.
Results
A total of 1805 patients were invited to participate in the evaluation, with an average of 737 (40.83%) completing the screening questionnaire at baseline. For the TRS, 798 (44.21%) patients were included at baseline, of whom 517 (64.8%) involved their partner. On average, 282 of 744 (37.9%) patients submitted at least one follow-up questionnaire. Patients rated the app above average (n=137, median score of 7 on a 1-10 scale) on days 7 and 14. The TRS decreased by an average of 1.5 points (P<.001) at T3 and T6 compared with baseline, a clinically meaningful improvement. All secondary outcomes showed statistically significant positive changes for patients who used a relevant lifestyle module (P<.001). Most improvements were achieved by 3 months and remained significant at 6 months (P<.001), except for alcohol intake (P<.53). These findings were consistent across both LMMs and joint models.
Conclusions
Our evaluation of a mobile health app integrated into standard care demonstrates immediate and clinically meaningful improvements in key lifestyle parameters among women seeking to become pregnant. Additional scientific research is needed to identify the causal pathways leading to sustained effectiveness. To maintain and enhance these outcomes, further tailoring of patient-specific programs is essential.
... Additionally, SITAR random effects are easily interpretable in the original units of the modelled parameter and estimates are specific values along a continuous scale that can be used in subsequent analyses of associations with exposures or outcomes (Cole et al., 2016;Crozier et al., 2019). Detailed information about the differences between statistical methods that are commonly used to characterize nonlinear longitudinal change trajectories have been describe elsewhere (Crozier et al., 2019;Elhakeem et al., 2022). In the second stage, we used Bayesian Kernel Machine Regression (BKMR) to examine the associations between first trimester plasma PFAS concentrations and BP trajectory parameters. ...
Background: Evidence suggests that exposure to per- and polyfluoroalkyl substances (PFAS) increases risk of high blood pressure (BP) during pregnancy. Prior studies did not examine associations with BP trajectory parameters (i.e., overall magnitude and velocity) during pregnancy, which is linked to adverse pregnancy outcomes.
Objectives: To estimate associations of multiple plasma PFAS in early pregnancy with BP trajectory parameters across the second and third trimesters. To assess potential effect modification by maternal age and parity.
Methods: In 1297 individuals, we quantified six PFAS in plasma collected during early pregnancy (median gestational age: 9.4 weeks). We abstracted from medical records systolic BP (SBP) and diastolic BP (DBP) measurements, recorded from 12 weeks gestation until delivery. BP trajectory parameters were estimated via Super Imposition by Translation and Rotation modeling. Subsequently, Bayesian Kernel Machine Regression (BKMR) was employed to estimate individual and joint associations of PFAS concentrations with trajectory parameters – adjusting for maternal age, race/ethnicity, pre-pregnancy body mass index, income, parity, smoking status, and seafood intake. We evaluated effect modification by age at enrollment and parity.
Results: We collected a median of 13 BP measurements per participant. In BKMR, higher concentration of perfluorooctane sulfonate (PFOS) was independently associated with higher magnitude of overall SBP and DBP trajectories (i.e., upward shift of trajectories) and faster SBP trajectory velocity, holding all other PFAS at their medians. In stratified BKMR analyses, participants with ≥ 1 live birth had more pronounced positive associations between PFOS and SBP velocity, DBP magnitude, and DBP velocity – compared to nulliparous participants. We did not observe significant associations between concentrations of the overall PFAS mixture and either magnitude or velocity of the BP trajectories.
Conclusion: Early pregnancy plasma PFOS concentrations were associated with altered BP trajectory in pregnancy, which may impact future cardiovascular health of the mother.
