MULTIRESPONSE SEMIPARAMETRIC REGRESSION MODEL APPROACH TO STANDARD GROWTH CHARTS DESIGN FOR ASSESSING NUTRITIONAL STATUS OF EAST JAVA TODDLERS

: A nutritional status of toddlers characterized by a lack of weight based on anthropometric index weight-for-age (W/A) is called underweight. In Indonesia, the anthropometric index is recorded on a Card Towards Health called KMS ( Kartu Menuju Sehat ) which refers to WHO–2005 Standard Growth Charts (WHO–2005 SGC). Samples used to design the WHO–2005 SGC were toddlers taken from Brazil, Ghana, India, Norway, Oman


INTRODUCTION
An underweight status of toddler is a nutritional status of toddler which is characterized by a lack of weight based on anthropometric index weight-for-age (W/A). In terms of malnutrition, Indonesia ranked fifth in the world that is about 3.8 per cent of 87 millions of total Indonesian toddlers. Nationally, the second position after East Nusa Tenggara province for cases of malnutrition in toddlers aged under five years old is East Java province. According to Health Department of East Java province, Indonesia there had been an increase in cases of malnutrition by 31.36 per cent that is from 4.716 cases to 6.195 cases in 2017 [1].
A Card Towards Health (KMS) is an instrument containing normal growth curves for toddlers based on the anthropometric index weight-for-age (W/A). As we know that up to now, Indonesia uses KMS based on WHO-2005 anthropometric standards. The use of KMS in Indonesia is based on the Z-Score standard growth charts of weight-for-age. Designing WHO-2005 standard growth charts used samples of toddlers aged 0-60 months who took from Brazil, Ghana, India, Norway, Oman, and USA. These samples are considered to represent regions of the world that are recommended as an assessment of global nutritional status [2]. However, there are different physical characteristics between Indonesian toddlers and those toddlers who used by WHO-2005. This, of course, makes a difference in chart patterns between the WHO-2005 standard growth charts and Indonesian standard growth charts that we propose in this study. Therefore, an effort that can be done to overcome the discrepancy is to design a KMS chart locally using data on toddlers aged under five years old whose physical condition is in accordance with Indonesian 3 MULTIRESPONSE SEMIPARAMETRIC REGRESSION MODEL APPROACH toddlers. Additionally, the growth charts for toddlers every age show different patterns at each stage [3]. In usual, the pattern does not form a linear curve or a particular shape, so that the appropriate model approach for this case is nonparametric regression model [4]. However, in several cases, some parts of toddlers growth charts form a linear pattern while other parts form nonlinear pattern, and there is correlation between responses, then for these cases, a multiresponse semiparametric regression (MSR) model approach is very suitable to use for designing the standard growth charts of toddlers [5]. Furthermore, to estimate these nonparametric regression models and semiparametric regression models, we use smoothing techniques. There are several smoothing techniques in nonparametric regression and semiparametric regression, for examples local linear [6][7][8][9][10][11], local polynomial [12][13][14][15][16][17], spline [4,5,[18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34], and kernel [24,25,30,[35][36][37][38], and two smoothing techniques of which are the LS-spline and local linear with the advantage of being able to overcome data patterns that show a sharp rise or fall with the help of knots, the resulting curve is relatively smooth [29,34], and able to determine the localness properties of data [6,7,10].
A study on designing standard growth charts of toddlers locally has been carried out by [39] in Padang City, but the underweight samples in this study were not differentiated by sex. Further, researches using local linear estimators have been done by [8][9][10] that have differentiated the sex of toddlers and found that the results of the East Java toddlers growth chart design curves lie in lower position than the WHO-2005 standard. In common, the standard growth charts of toddlers are designed based on percentile values. The advantage of using the percentile is that the calculation results are more accurate and able to be compared for each age group and anthropometric index. Calculating the percentile can indicate the problem of malnutrition and stunting which is more suitable compared to conventional systems [40].
For creating a standard growth chart of weight-for-age (W/A) for baby under five years old by using the nonparametric regression model approach, there are two variables used, namely toddler weight (kg) and toddler age (month) for each sex. While, if use gender as a variable in constructing the model then the appropriate model approach is semiparametric regression model.
Semiparametric regression model is formed if in a regression model there are components of the 4 CHAMIDAH, ZAMAN, MUNIROH, LESTARI model in which some components can be estimated parametrically and remain components are estimated non-parametrically. The use of the semiparametric regression models based on the LSspline estimator were also carried out by [29,41] that were to design the standard growth charts for East Java toddlers aged under five years old as a determinant of wasting nutrition status. The growth of toddlers that are different for each age will be appropriate if they are approached by semiparametric regression model where the toddlers gender variable is a parametric component which is as a dummy variable. Based on these facts, in this study we proposed an approach, namely multiresponse semiparametric regression (MSR) model, to design Percentile Standard Growth Charts (Percentile SGC) of weight-for-age (W/A) for East Java toddlers based on both LS-spline and local linear estimators. Next, the obtained these Percentile SGC designs will be compared with WHO-2005 Standard Growth Charts (WHO-2005 SGC) which can then be used as a reference for determining underweight or malnutrition status of East Java toddlers aged under five years old which is one of the indicators of stunting.

MATERIALS AND METHODS
In this section, we provide a brief overview of materials and methods used in this study such as multiresponse semiparametric regression model, curve of percentile, LS-spline and local linear estimators, generalized cross validation and coefficient of determination, and description of data.

Multiresponse Semiparametric Regression Model
In a regression model, if there is a combination of parametric regression and nonparametric regression then it is called semiparametric regression model [29]. Furthermore, if this semiparametric regression model draws a functional relationship between two or more response is vector of unknown parameters of the ℎ -response; is unknown regression function of the ℎ -response; and is a zero mean independent random error with variance 2 of the ℎresponse [5,32].
Next, in the following section we provide a brief overview of Percentile curve that will be used to design standard growth charts of East Java toddlers.

Curve of Percentile
For calculating values of standard deviation ( ( )) on the Percentile curve of an observation, it involves mathematical calculations on normally distributed data from measurements that describe the population. The formula used is as follows [2]: . Next, in the following section we provide a brief overview of LS-spline and local linear estimators that will be used to 6 CHAMIDAH, ZAMAN, MUNIROH, LESTARI design standard growth charts of East Java toddlers.

LS-Spline and Local Linear Estimators
If regression function ( ) in Eq. (2) is approximated by using the LS-spline estimator where p is the order of spline and m is the number of knots, then we can express the model in Eq.(2) as follows [34]: where 0 = 0 + 0 and ( − ) + meets the following equation: Hence, the semiparametric regression model in Eq.(4) based on the LS-spline estimator can be presented into a matrix notation as follows: Therefore, semiparametric regression model presented in (5) can be stated into: where C = ( X T ) and = ( 0   1 ) .
Next, estimation of in model (6) can be obtained by solving the following optimization: Estimation of , namely ̂, is obtained by differentiating Q with respect to parameter that is = . It would give the solution to optimization in (7) that is ̂= (C C) − C . In this step, ̂ consists of estimation values of constant 0 , parameter 1 of parametric component and parameter of nonparametric component, so that we obtain the estimation of response variable 7 MULTIRESPONSE SEMIPARAMETRIC REGRESSION MODEL APPROACH of semiparametric regression model as follows [5,32,34]: Furthermore, if the regression function ( ) in model (2) is estimated by using local linear estimator, then it can be expressed as follows [7]: where ̂( 0 ) is estimator for ( 0 ) that obtained by minimizing the weighted least square function as follows: and we obtain: Hence, by considering Eq.(9) and Eq. (11), we obtain the estimation of regression function based on local linear estimator as follows [7]: Next, in the following section we provide a brief overview of generalized cross validation and coefficient of determination that will be used to design standard growth charts of East Java toddlers.

Generalized Cross Validation and Coefficient of Determination
In LS-spline regression it is very important to calculate the optimum knot point. Based on optimum knots, the best spline function is obtained. There are several methods for calculating the optimum knot point, one of which is the GCV (Generalized Cross Validation) method. The GCV for the MSR model is defined as follows [26,27,30,31]: is order of spline and is the number of knots. Optimal knot values are determined from a minimum GCV value from a combination of observation points which are assumed to begin to change behavior patterns.
Next, the coefficient of determination notated R 2 is a measure of the accuracy of the regression curve [41,42]. The purpose of calculating the R 2 value is to determine the variation of the response and 0 ≤ R 2 ≤ 1 .
Hereinafter, in the next section we give a brief presentation on description of data that will be used to determine standard growth charts design which is suitable for determining the nutritional status of East Java toddlers.

Description of Data
Here we use a secondary data that contains observations on gender and toddlers' body weight of toddlers aged 0-60 months which have been recorded from twenty three districts and cities in East

RESULTS AND DISCUSSION
In this section we provide results and discussions on estimation of Percentile values based on both LS-spline estimator and local linear estimator.

Estimating Percentile Values Using LS-Spline Estimator
The best estimation model for creating a standard growth chart based on the anthropometric index weight-for-age (W/A) is obtained by estimating the percentile values at each age based on the LSspline estimator of the MSR model approach. In the MSR model approach, the optimal order of 9 MULTIRESPONSE SEMIPARAMETRIC REGRESSION MODEL APPROACH splines and the optimal knots for each percentile value, namely 3-rd percentile ( 3 ); 15-th percentile ( 15 ); 50-th percentile ( 50 ); 85-th percentile ( 85 ); and 97-th percentile ( 97 ), are estimated based on GCV criterion namely the minimum GCV value. The estimation results which include order of spline, the optimal knots, minimum GCV values, MSE values, and coefficient of determination (R 2 ) for each percentile are presented in Table 1.  where subscript "(1)" represents the first response, namely weight-for-age (W/A).

Figure 1. Percentile SGC for W/A of Boy Toddlers (a), and for H/A of Boy Toddlers (b).
Hereinafter, we can express Eq.(19) into a truncated function as follows: where subscript "(2)" represents the second response, namely height-for-age (H/A).
Next, we can express Eq.(21) into a truncated function as follows: Based on truncated function presented in Eq. (22), we obtained the highest mean baby height gain in East Java which occurs in the age interval 0 ≤ < 6. This means that, every one month the mean baby's height increases by 2.11 cm while the lowest mean height gain occurs at the interval 18 ≤ ≤ 24, and every one month increase, the mean height gain is 0.709 cm.   yellow area, whereas a light green area indicates a toddler who has a nutritional status with a threshold of more than 85 . Malnutrition status lies in the area below the red line with a Percentile threshold value of less than 3 . Next, comparison between Percentile SGC and WHO-2005 SGC of W/A boys and girls in East Java using LS-spline estimator are presented in Figure 3.  Table 2.  Table 3 and Table 4, respectively. criterion which is the minimum value of CV. In this step we get the optimal bandwidth (h) values and the minimum CV values for every percentile which is shown in Table 5.  Table 6. and 97 and by substituting = 0 (for girl toddlers) and = 1 (for boy toddlers) into Eq. (23) and Eq. (24), we obtain the Percentile SGC of W/A for boy and girl toddlers in East Java province of Indonesia by using local linear estimator of MSR model as shown in Figure 4.   Table 7. and by substituting = 0 (for girl toddlers) and = 1 (for boy toddlers) into Eq.(23) and Eq. (24), we obtain the Percentile SGC of anthropometric index height-for-age (H/A) by using local linear estimator of MSR model for boys and girls in East Java as shown in Figure 6.