East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, Issue. 2, 1-12 Mixed Effects Analysis of Height Growth in Ethiopian Children Aged 1-12 Years: A Cohort Study Dereje Danbe Debeko*1 and Ayele Taye Goshu2 1 Department of Statistics, Hawassa University, P.O. Box 05, Hawassa, Ethiopia 2 Department of statistics, Cotebe Teaching University, Addis Ababa, Ethiopia KEYWORDS: Height growth; Mixed effects Models; Children; Ethiopia ABSTRACT Modelling physical growth is a key component to examine and identify defining characteristics in the growth process. The goal of this study was to model and capture known features of height growth in Ethiopian children aged 1–12 years. Height measurements of 1760 children followed from 1 to 12 years at Young Lives Ethiopia, a younger cohort, used in the study. The mixed effects method was used to estimate the rate of change within and between subjects over time and to identify defining covariates. Adult height and rate of change over time were individual-specific resulting individual-level growth differences. There was a negative relationship between individual-specific adult height and rate of change over time. The decelerated rate of change was observed from childhood to the onset of puberty in both sexes. Boys were taller than girls between the ages of 3 and 7 years. Mother’s educational status, access to quality drinking water, age, and sex had a significant effect on height growth. Children who had a decelerated rate of growth change during the childhood period become taller later in life. Adult height could be determined by an individual-specific rate of change over time. INTRODUCTION Growth is a continuous and dynamic process influenced by different unknown factors. Modelling this dynamic process to understand, estimate, and capture the defining characteristics such as initial level, rates of change, periods of acceleration and deceleration when the process enters and leaves different developmental phases (Grimm et al., 2011; Howa et al., 2016). A common practice of child growth is to measure the increase in body mass, to control and modify the external conditions that affect growth gain (Oliveira et al., 2000; Gómez et al., 2008; Aggrey, 2009). There have been different modeling approaches applied to the growth measurement to identify defining characteristics in growth process. An example is the construction of the curve-fitting models that relate age with height and estimate age at which an individual attain maximum growing by associating features in various growth phases (Laird, 1965; Grossman et al., 1985; Grossman and Koops, 1988; Galeano- Vasco et al., 2014). Studying growth in one phase may have an important influence or East African Journal of Biophysical and Computational Sciences Journal homepage : https://journals.hu.edu.et/hu-journals/index.php/eajbcs Hawassa University College of Natural & Computational Sciences Year 2021 Volume xx No xx *Corresponding author: Email: dalbrayii@gmail.com +251913927596 https://dx.doi.org/10.4314/eajbcs.v5i2.1S Research article 2 association with subsequent phases since individual growth is monitored as a sentinel indicator of overall well-being (Tanner, 1981; Gold stein et al., 2002; Richard et al., 2014). Many growth modelling approaches are used to obtain descriptions of change in growth processes accounting individual specific effects observed over time, average change, between- individual differences in change and to identify determinants factors (Grimm et al., 2011). However, modelling growth trajectory is difficult process due to the model parameters that could not be possible to elaborate from biological perspectives (Aggrey, 2002; Aggrey, 2009; Galeano-Vasco et al., 2014). Many scholars have been used cross-sectional data to model features in growth process. Despite of cost effectiveness and easy access to data, cross- sectional centiles for example, only offer a cross-sectional coverage, and hence the growth path of an individual monitored longitudinally in time is unknown since these types of modelling do not describe the dynamic aspect of growth process over time (Grajeda et al., 2016). The mixed effects modeling approaches are among the commonly used methods to capture growth process. This modelling approach is capable of incorporating subject specific rate of change over time and difference across the subjects in the linear predictor expression form (Bates et al., 2015). Mixed effects models have been applied to the longitudinal data in different settings (Devidian and Giltinan, 1995; Pan and Goldstein, 1998; Grimm et al., 2011; Richard et al., 2014; Chirwaet al., 2014). In this approach, mixed effects refer to the population mean of the parameter and random effects that indicate the differences between the mean value of the parameter and the adjusted value for each East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12 subject (Littell et al., 2000; Wang and Zuidhof, 2004). The mixed effects methods quantify variability between and within individuals letting a flexible covariance structure (Pinheiro and Bates, 1995; Aggrey, 2009) to accommodate time dependent and time independent covariates within individual residual terms (Pan and Goldstein, 1998). Many studies have been used the mixed effects models in linear and nonlinear approaches (Pan and Goldstein, 1998; Craig and Schinckel, 2001; Schinckel et al., 2005; Aggrey, 2009; Grimm et al., 2011). However, both modelling approaches have not been applied to the longitudinally collected data from low income countries’ settings. Thus, the main objective of the current study was to model height growth in Ethiopian children aged 1-12 years and to identify determinant factors using mixed effects modelling approaches in linear and nonlinear forms. MATERIALS AND METHODS The study design and source of data Data on growth measurement gathered over time by Young Lives Ethiopia, a younger cohort, was used in this study. Young Lives is an international collaborative research project supported and coordinated by a team based at Oxford University, UK. The cohort has been studying the lives of children in Ethiopia aiming to reduce childhood poverty. This project built up lives of 3,000 children living in 20 sites across Addis Ababa (the capital) and four other regions (Amhara, Oromia, Former Southern Nations Nationalities and Peoples (SNNPs), and Tigrai). The Young Lives Ethiopia cohorts have been aimed to follow children in two age groups: a younger cohort following of 2,000 East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12 3 children who were 0.5 to 1.5 years old and an older cohort following 1,000 children aged 7.5 to 8.4 years at the baseline (first round) in 2002. The rest three rounds of surveys were carried out in 2006, 2009, and 2013, respectively, for both cohorts. Details of the cohort studies have been referred via the official website of the project (www.younglivesethiopia.org). Based on the inclusion criteria, a total of 1760 children were included in the study. Height growth measurements observed from each subject in four survey rounds were used as outcome variable (see Figure 1). Age and other socio- economic, demographic and health related covariates were included in the study. Statistical Methods Two modeling approaches were used in the data analysis as described in the next section. The linear mixed effects modeling approach was used to identify determinant factors associated with height growth difference between and within subjects. The nonlinear mixed effects modeling approach was used to capture growth trajectories and to estimate relationship between rate of change over time and maximum (adult) height. Linear Mixed Effects Models Linear mixed models (LMMs) may be expressed in different but equivalent forms. It is common to express such a model in hierarchical form, or just as a mixed model, including additional random-effect terms and associating variance and covariance components (John and Sanford, 2015). When the levels that we observed represent a random sample from the set of all possible values, the random effects can be incorporated in the model (Bates, 2010). This approach decomposes the outcome of an observations as fixed effect (population mean) and random effect (subject specific change over time), and it account for the correlation structure of variations among subjects. Figure 1: Individual height growth measurements plotted on measurement time by sex (male: left and females right). East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12 4 Model description LMMs used in this study have only two-levels, across and within individual variations. For ijy is height measurements of ith subject taken on jth measurement occasion, where j = 4; i = 1,2,…,n. The extended form of linear model with random components is described as follows: ijqijiqijipijpijij zbzbxxy   ...... 1111 The matrix form of this model is equivalent and considerably simpler to write as: ,0 ijiiijij bZXY   where ijY is the × 1 vector of response observations in the thi subject at thj measurement occasion, ijX is the pni  model matrix of fixed-effect regressors,  is the 1p vector of fixed-effect coefficients which is invariant across groups, iZ is the qni  matrix of regressors for the random effects of observations in subject i , ib is the q × 1 vector of random effects for group i , potentially different in different groups and ij is the × 1 vector of errors for thj measurement in thi subject. Model assumptions The linear mixed effects analysis was performed with some assumptions: random effects are different across the subject and normally distributed with mean zero and variance co- variance structure, covariates are uncorrelated with each other (no multicollinearity), time variant covariates are a subset of the time invariant covariates, error terms are assumed to have a multivariate normal distribution and within subject measurement errors are auto correlated. Under certain conditions, physical growth does not follow linear pattern over time. Modeling growth spurts using linear form of models could lead us into wrong conclusion and may have weak prediction power. Thus, the nonlinear models are alternative ways of modeling growth spurts. This modelling approach uses mixed or fixed effects form based on the objectives of underlying study. Non-Linear Mixed Effects Models Nonlinear mixed effects models refer to the population mean of the parameter and random effects that indicates the differences between the mean value of the parameter and the adjusted value for each individual growth over time (Wang and Zuidhof, 2004).The nonlinear mixed effects growth curve models used in this study are Logistic and Gompertz which are most commonly used growth curve models due to their mathematical tractability with biologically meaningful parameters. Logistic Model (Nelder, 1961) ij ijii i ij tbb b y        ))(exp()(1 3322 11 5 Gompertz Model (Winsor, 1932) ijijiiiij tbbby   ))(exp()exp(()( 332211 where, ijy is height of thi subject at thj measurement occasion, ijt is age of thi subject at thj measurement time, 1 is asymptotic or maximum height (adult height), 2 is scaling parameter and 3 is growth rate (between subjects); ib1 is random effects for 1 , ib2 is random effects for i2 , ib3 is random effect for i3 and ij is error term. kik Nb ,0(~ ), ),0(~ 2 Nij for 3,2,1k ,               333231 232221 131211 bbb bbb bbb k Since our sample data is repeated measurements taken on the same subject, the expression for the within-subject variance-covariance matrix can be formed in the following way: inii IR  2 , inI = An identity matrix ( ii nn  ), i = Correlation structures and 2 = Residual variance of the model. East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12 Methods of parameter estimation Maximum likelihood estimation method was implemented to estimate underlying model parameters using R-package “lme4” and “nlme”. Statistical test was done at 5% level of significance. Further details of mixed effects model parameter estimation has been described by Lindstrom and Bates (1990), Lindstrom and Bates (1995), Davidian and Giltinan (1995) and, Sedigheh and Debasis (2012). Model adequacy checking The model assumptions were checked using residual plots versus fitted values, QQ and P-P plots. The goodness of fit was tested based on the Bayesian Information Criteria (BIC) and Akaikie Information Criteria (AIC). However, some scholars argue that several measures of model fit, even the likelihood ratio chi-square, that often appear to reflect relatively poor fit – when a model fits data very closely – that is, if residual variances are quite small (Browne et al., 2002). RESULTS Descriptive statistics Descriptive statistics of height measurements taken on each survey visit presented below in Table 1. More height growth variation was observed in girls than in boys after period I. Growth variation became higher in the fourth period for both sexes East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12 6 Table 1: Summary statistics of height in each measurement periods by sex. Measurement Period in years Female Male N Min Max Mean Std. D N Min Max Mean Std. D Period I (age 1) 841 56.00 89.50 70.425 5.498 947 55.30 89.50 71.45 5.227 Period II (age 5) 843 86.50 124.00 103.563 5.512 947 80.90 124.90 103.95 5.291 Period III (age 8) 843 99.10 157.00 120.628 6.469 947 102.0 146.00 120.66 6.129 Period IV (age 12) 843 118.2 178.00 142.205 7.880 947 120.0 161.50 139.80 6.628 Figure 2: Smooth line curve of height growth over measurement time by sex Boys and girls had almost the same height between 12-84 months (1-7 years). After 84 months girls become taller than boys (Figure 2). However, the smooth curve plot in Figure 2 doesn’t show where the rate of change was accelerated or decelerated. In order to identify where rapid and slow rate of change lies we have used height growth velocity curves presented in Figure 3. Height growth Velocity Height growth velocity was calculated to investigate time of accelerated and decelerated growth periods. To assess these curvatures, the following velocity formula used (Grajeda et al., 2016): )( ijtheight = height change for thi individual at thj measurement period. Between 60 to 84 months, boys and girls had shown very similar rate of growth change. After 84 months (7 years) rate of change in girls became accelerated than boys. The growth difference in this time could be due to various factors. Thus, we have used linear mixed effects models to identify factors associated with rate of change for both sexes. On the other hand, , )()( )( )1( )1()1( where tt theighttheight theight jiij jiji ij      East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12 7 nonlinear mixed effects models were used to estimate height at maximum growth and rate of change over time within and between subjects. Figure 3: Smooth line and growth velocity curves by sex Linear fixed and mixed effects models Table 2 presents the goodness of fit test for the linear models. Linear mixed effects model better fit the data than fixed effects model. Table 2: Assessing the goodness of fit of linear models Fit statistics Linear fixed Linear mixed AIC 44485.57 43010.67 Log.Lik -24546.60 -21446.34 The estimated values of the mixed effects model are presented in Table 3. Within an individual growth variation over time was around 0.03 cm. There was a positive relationship between within-individual rate of change (slope) and height at the baseline (random intercept). This points that a child who was taller at the baseline shown faster growth over time (r = 0.728). There was a positive correlation between two consecutive measurements taken on the same subjects at different measurement occasions (AR (1): = 0.2) (Table 3). Table 3: Estimated values of random effects model Random Components Estimates Serial correlation Continuous AR(1) St. Dev. Correlation (Intercept) 1.893 (Intr) = 0.2 Age in months 0.0299 0.728 Residual = 4.664 Access to quality drinking water, mother’s educational status and sex had a significant effect on height growth. Children who had access to quality drinking water were 3.8 cm East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12 8 taller than children who had no access to quality drinking water (Table 4). Table 4: Estimated values of different covariates on height growth of children aged 1-12 years based on the linear mixed effects model Covariates Categories β Std.Error DF t-value p-value (Intercept) - 66.97115 2.618180 5067 25.57927 0.0000 Age - 0.49914 0.007732 5067 64.55163 0.0000 Sex Male (ref.) Female -1.30927 1.348283 1698 -0.97106 0.3317 Birth order - -0.01189 0.042960 1698 -0.27685 0.7819 BCG status Yes (ref.) No -0.21590 0.244046 1698 -0.88469 0.3765 Had quality drinking water: No (ref.) Yes -3.86616 0.748315 5067 -5.16649 0.0000 Household size - 0.24943 0.132781 5067 1.87850 0.0604 Had ANC visit No (ref.) Yes 0.20970 0.225077 5067 0.93170 0.3515 Breast feeding duration Never fed (ref.) fed for 1-3 months 0.48811 2.516145 1698 0.19399 0.8462 fed for 4-6 months 2.07767 2.539510 1698 0.81814 0.4134 fed for > 6 months 1.48964 2.419669 1698 0.61564 0.5382 Mother’s age at birth - -0.00786 0.019595 1698 -0.40107 0.6884 Father’s education: No education ref. Elementary (1-8) 1.70812 0.210840 5067 8.10153 0.0000 0ther -1.52816 3.234500 5067 -0.47246 0.6366 Mother’s education: No education ref. Elementary (1-8) 0.83562 0.233946 5067 3.57187 0.0004 >= High school 1.93923 0.290056 5067 6.68570 0.0000 Region Addis Ababa ref. Amhara -1.60553 1.437172 5067 -1.11714 0.2640 Oromia -4.24911 1.382365 5067 -3.07380 0.0021 SNNP -1.19146 1.384214 5067 -0.86074 0.3894 Tigrai -2.45187 1.403808 5067 -1.74658 0.0808 Area of residence Urban ref. Rural -0.72144 0.436144 5067 -1.65413 0.0982 Interaction effects Age*sex Male (ref.) Female 0.02708 0.007462 5067 3.62902 0.0003 Quality drinking water*region A.A (No ref.) Amhara 2.17289 0.919798 5067 2.36236 0.0182 Oromia 6.04705 0.928457 5067 6.51301 0.0000 SNNP 2.99661 0.911643 5067 3.28704 0.0010 Tigrai 2.14791 0.907350 5067 2.36724 0.0180 Age*Had quality drinking water (No ref.) Yes 0.03497 0.008313 5067 4.20725 0.0000 East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12 9 Mother’s educational status also plays a significant role on height growth of children. Children whose mother had elementary (1-8) and high school plus educational status were 0.84cm and 1.9 cm taller than children whose mother had no formal education, respectively. As age increased by a month, girls’ height increased by 0.27 cm compared to boys holding other covariates constant in the model. The interaction effect between age and sex was higher in girls compared to boys. Height in girls increased by 0.035 cm than boys as age increased by one month. On the other hand, access to quality drinking water had a significant effect on height growth of children (Table 4). Children those who had access to quality drinking was had 0.035 cm increased height compared to children those who had no access to quality drinking water. Nonlinear Models After identifying the best model fit to the data, the average rate of change over time and maximum (adult height) estimated height for both sexes separately. Table 5: Goodness of fit test for nonlinear mixed effects models by sex Models Fit statistics Fixed effects models Random effects Male Female Male Female Gompertz AIC 23759.5 21781.9 4243.28 2902.96 BIC 23799.2 21798.4 4280.23 2956.96 Logistic AIC 23784.4 21772.6 4216.93 2921.48 Table 6: Estimated values and fit statistics based on nonlinear mixed effects growth curve models by sex Female Nonlinear mixed models Fixed estimate Random components Fit statistics paramete r Estimated Std.err Ó Ó Ó AIC BIC Gompertz b1 179.44 2.398 -0.889 10.582 0.049 0.374 0.718 2901.48 2934.96 b2 1.054 0.0108 b3 0.0105 0.0003 Logistic b1 166.703 1.716 -0.802 9.750 0.103 0.758 0.752 2922.96 2956.44 b2 1.623 0.0215 b3 0.0156 0.0003 b3 0.0087 0.0003 Male Gompertz b1 171.652 1.3951 -0.879 7.9496 0.039 0.227 0.717 4216.93 4253.88 b2 0.98945 0.0066 b3 0.0111 0.0002 Logistic b1 161.340 1.034 -0.802 7.2219 0.047 0.340 0.995 4243.28 4280.23 b2 1.4912 0.0125 b3 0.01618 0.0002 10 Model comparisons Incorporating an individual-specific rate of change over time in the nonlinear models had dramatically reduced the estimation error and increased the fitting performance of the model. For instance, when adult height, rate of change and scaling (point of growth change) allow to varying across individuals, the fitting performance of the models had improved (Table 6). Random effects parameters were selected based on their capability to map with theoretical and physical meanings. When adult height and rate of change vary across individual over time, Logistic models better fitted the data for both sexes. The mean adult height was estimated to be 166.7 cm and 171.6 cm in girls and boys, respectively. Adult height and rate of change had inverse relationship for both sexes (for girls r13, = -0.802; for boys r13 = -0.879). DISCUSSION This study was aimed to model individual specific growth spurt over time in children aged 1-12 years old. The mixed effects models were applied to the height growth measurements to interpolate growth spurt within and between subjects over time. The effect size of different covariates on height growth was estimated using the linear mixed effects models. Decelerated and accelerated growth periods were identified using growth velocity curves. The fitting performance of the models increased when adult height and rate of change were allowed to as individual specific. Random effects were partitioned into between (σ) and East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12 within (Σ) subject variations. The consistent result by other study (Spyrides et al., 2008) reported that considering random effects in the proposed models increase the precision of estimated parameters since parameters vary from individual to individual that might be demanded using margin of errors in the fixed effects model set-up. However, most of the mixed effects models applied to human physical growth data had no clear identification towards random and fixed effects parameters that should be included in the models. Thus, one of the objects of this study was to identify model that best fits height growth in the defined time points when individual specific effects were taken in to account. In addition, this study was concerned to identify mixed effects parameters that best map on theoretical basis having biologically meaningful interpretations. Of course, here the study was concerned to use some notations suggested by Browne and co-authors (Browne et al., 2002) to identify those parameters and to select best fit to the data. The authors discussed their arguments that many measures of model fit, even the likelihood ratio chi-square, and often appear to reflect relatively poor fit: This suggest a poor fit despite the model closely fitting the data. Therefore, this study had carefully examined and identified that adult height and rate of change over time were individual specific and had random effect on height growth. The present study found that rapid growth was observed in children between 1 to 3 years followed by decelerated rate of change. Other consistent study reported that children of both sexes grow at approximately the same rate until the adolescent growth spurt (Rogol et al., 2000). Adult height and rate of change had inverse relationship for both sexes. The consistent East Afr. J. Biophys. Comput. Sci. (2024), Vol. 5, No. 2, 1-12 11 findings reported that the adolescents with a later APHV tended to have a higher height later in life (Chen et al., 2022). The interaction effect of age on rate of change in growth was higher in girls compared to boys. This may be due to girls experience adolescence earlier than boys. CONCLUSIONS The logistic mixed effects models best fit and captured height growth pattern for both sexes. Boys were taller than females until onset of puberty. The most decelerated rate of growth was observed in early childhood period for both sexes. Mother’s educational status, access to quality drinking water, age and sex could be one of the determinant factors of height growth in children aged 1- 12 years. Rate of change at the base line had no a significant effect on adult height. Child whose growth was accelerated during childhood period attains adult height later in life. Competing interest Authors have no conflict of interest Acknowledgement We, the authors, would like to thank Oxford University and Young Live Ethiopia for the data access. References Aggrey S. E. 2002. Comparison of Three Nonlinear and Spline Regression Models for Describing Chicken Growth Curves. Poult. Sci. 81:1782-1788. Aggrey S. E. 2009. Logiatic Nonlinear Mixed Effects Models for Estimating Growth Parameters. Poult. Sci. Asso. 88:276-280. Agudelo Gómez D. A., Cerón Muñoz M. F. and Restrepo Betancur L. F. 2008. Modeling of growth Functions Applied to Animal Production. Rev. Colomb. Cienc. Pecu. 21:3-8. Bates D., Mächler M., Bolker B. and Walker S. 2015. 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INTRODUCTION MATHEMATICAL MODEL FORMULATION The modified mathematical model Model formulation QUALITATIVE ANALYSIS OF THE MODIFIED MODEL Well-posedness Steady state Basic reproduction number Local stability of disease free equilibrium Global stability of disease free equilibrium Endemic equilibrium point Local stability of endemic equilibrium Bifurcation analysis Sensitivity analysis NUMERICAL SIMULATIONS AND DISCUSSION EXTENSION OF THE MODIFIED MODEL INTO AN OPTIMAL CONTROL Optimal protection and hospitalization using modified model Existence of an optimal control The Hamiltonian and optimality system Numerical simulations of optimal control problem Optimal control comparisons and strategies CONCLUSION INTRODUCTION PRELIMINARIES Intuitionistic Fuzzy Set Operations over Intuitionistic Fuzzy Sets Multi-objective programming problem MATHEMATICAL FORMULATION OF PROBLEM Intuitionistic Fuzzy Multi-objective Linear Decision-Making Problem Intuitionistic Fuzzy Goal Model of IF-MOLDMP PROPOSED SOLUTION METHOD Extended Yager-membership function in the IFDE Develop a new intuitionistic fuzzy (IF) aggregation operator The intuitionistic fuzzy goal programming method Interactive Penalty Function Method (IPFM) ALGORITHM FOR PENALIZED IFGP METHOD NUMERICAL EXAMPLE RESULTS AND DISCUSSION CONCLUSION INTRODUCTION MATHEMATICAL MODEL FORMULATION The modified mathematical model Model formulation QUALITATIVE ANALYSIS OF THE MODIFIED MODEL Well-posedness Steady state Basic reproduction number Local stability of disease free equilibrium Global stability of disease free equilibrium Endemic equilibrium point Local stability of endemic equilibrium Bifurcation analysis Sensitivity analysis NUMERICAL SIMULATIONS AND DISCUSSION EXTENSION OF THE MODIFIED MODEL INTO AN OPTIMAL CONTROL Optimal protection and hospitalization using modified model Existence of an optimal control The Hamiltonian and optimality system Numerical simulations of optimal control problem Optimal control comparisons and strategies CONCLUSION