Designing of Child Growth Chart Based on Multi-Response Local Polynomial Modeling

Problem statements: Anthropometry measures used to measure physical ch ildren growth are not onlyweight but also height and head circumference. In this study we develop the estimation of multi-response localpolynomial regres sion and apply it to design growth chart for children up to five years old based on three respon evariables i.e., weight, height and head circumference. Approach: Based on local polynomial estimator, wedescribe the estimation of multi-response nonparametric regression model by us ing weighted least squared. Themodel is applied to design health card of children up to fiv e years old by using children data in Surabaya, Indonesia. Generalized Cross Validation (GCV) metho d is used to determine the order of local polynomial fit and the bandwidthfor each response v ariable. Results: We formulate the multiresponse local polynomial modeling and give a desi gn of health card of children up to five years old in Surabayacity, Indonesia. Conclusion: The child growth chart based on multi-response loca l polynomial modeling showsincreasing of children nut rition in Surabaya 2010.Because of the strong correlations among all three response variables,the imultaneosly approach for model estimationis better than partly single response approach. The re sult of simultaneosly model estimation based on multi-response local polynomial modeling satisfies goodness of fit criterion i.e., mean squared error value tend to zero and determination coefficient va lue tend to one.


INTRODUCTION
There are many cases that involve the regressionmodel has more than one response variables thatcorrelate each others. In that case, the multiresponsenonparametric regression model provides powerfultools to model the functions which draw association ofthese variables. Local polynomial estimation is widelyused for estimating regression function because it issimple and easy to understand. Localpolynomial estimator is obtained by locally fitting a dth degree polynomial to data by weighted least squares. The local polynomial estimator depends on twoparameters, which must be specified i.e., the order oflocal polynomial fit (d) and the smoothing parameternamed bandwidth (h). These two parameters have asimilar effect, in that a higher order fit or smallerbandwidth reduces bias but increases variance; while alower order or larger bandwidth increases biasbut reduces variance.
Many authors studied multi-responsenonparametric regression model. Wang et al. (2000) proposed spline smoothing for estimatingnonparametric functions from bivariate data with thesame variances of errors for each same response (i.e., 2 j , j 1, 2 σ = ) and applied it to hormone data. Welsh and Yee (2006) considered biresponse local linearregression and applied to blood pressure data which hasresponse variables i.e., systolic and diastolic andpredictor variable i.e., Body Mass Index (BMI). Lestari et al. (2010) studied the estimating of multi-responsenonparametric regression based on spline estimator. In case of heteroscedasticity, Chamidah (2012) studied estimation of biresponses local polynomialregression model and applied the model to estimategrowth curve of children up to 5 years of age basedon their weight and height.
According to pediatrician Roumeliotis (2012) the growthof childrenduring the first 18 months grows rapidly andthen it decreases parallel with increasing of age. It means locally model approach moreappropriate to this data.
Also, Roumeliotis (2012) stated anthropometrymeasures which is used to measure physical child growth are weight,height and head circumference of children. It meansthat physical child growth is more realistic if it ismodeled by multiresponse nonparametricregression approach.
In this study, we discuss the estimation ofmultiresponse local polynomial modeling whichhas bandwidth and polynomial degree of eachresponse must not be equal. For determining thesesmoothing parameters, we use GCV method andthen apply the model to children growth data inSurabaya, Indonesia 2010. It is necessary fordesigning health card that in Indonesia is called as KartuMenuju Sehat (KMS) based on children condition inIndonesia. Currently, the KMS is used formonitoring health and growth children in Indonesiabased on National Center Health Statistics (NCHS)chart, USA. The chart may not appropriate to thecondition of Indonesian children.
Estimation of the function j î f (t ) for each responseis: ( j) j i ĵ f (t ) A(h )y , j 1,2,...,r = = ɶ where,A (h j )represents matrix as follows: To obtain optimal h j and d j based on GCVmethod given by Wu and Zhang (2006), we minimize: In terms of statistical modeling locally around t 0 ,we model (3) as follows Eq. 4: and V −1 is invert of variance-covariance matrix of errorswhich is estimated from sample data. The solution of Eq. 5 is: Eq. 7: elsewhere. The data used for applying the modelcontainsof 1700 children obtained from community healthcenter in Surabaya 2010. The data that describes childrengrowth in Surabaya consists of 3 response variables.These are y 1 : weight (kg), y 2: height (cm) and y 3 :head circumference. (cm). While a predictor variable isage (month). In every month of children age i.e., from 0 until 60, we determine 5 th percentile, 15 th percentile, 25 th percentile, 50th percentile, 75 th percentile, 85 th percentile and 95 th percentile.
Plotting of percentiles of weight, height and headcircumference versus age are shown in Fig. 1-3, respectively.
Based on GCV method, we create R-code forchoosing bandwidth and optimal order of polynomial for each response. These results are given in Table 1.
These results given in Table 1 are used for designing child growth chart in KMS based on the three responses local polynomial estimation. The chart is shown in Fig.  4 as follows. Based on Table 1, the results of the 50 th percentiles estimation of weight, height and head circumference versus age give the mean squared error value 0.0514 and coefficient determination 99%. Plots the estimation of 50 th percentiles of weight, height and head circumference versus age are shown in Fig. 5.

DISCUSSION
In Fig. 4, green area indicates good health, lower yellow area indicates warning for underweight and upper yellow area indicates warning for overweight. Lower red area indicates underweight and upper red area indicates overweight. Based on correlation Pearson formula, we get correlation coefficient between weight and height of children 0.996; correlation coefficient between weight and head circumference of children 0.953; and correlation between height and circumferenceof children 0.946. It means that there are strong correlations among all three response variables. Design of the KMS of child growth in Surabaya 2010 as given in Fig. 4 is quite higher than that currently used to control children health in Surabaya. The simultaneously estimationgives mean squared errorvalue tend to zeroand determinationcoefficient value tend to one. These facts mean that the simultaneously model has satisfied goodness of fit criterion.