18)with

weights.

 

₀

^diag

₁

 

₀

^,

₂

 

₀

h t   h t h t 

K K K

Where

 

0

diag 

1 0

 , 

2 0

 ,..., 

0



hr t  Kh t t Kh t t Kh tn t 

K andK_{h }. is kernel function

with optimum bandwidth h.

Based on equation (12), nonparametric hat matric, ,for estimate regression function about t₀ is :

 

0

  

t0 ¹

   

t0 t0



¹

 

t0 ¹

 

t0

h t  ^ _h ^{ } _h

A e Z^T V K Z Z^T V K (13)

where

1 0 0 0

0 0 1 0

e  

  

 

So, estimation of f (t) based on local linear estimator can be written as:

 

ˆ

_h

( )

f t A t y^ (14)

Based on equation (5) and (14), local linear estimator for f t

ˆ ( )

can be expressed as :

   

ˆ _h( ) (y )

f t A t X (15)

So, function sum of squared error in equation (4) can be written as :

     ^{ } 

y

_h

(y )

^T

y

_h

(y )

Q^ X



 A t X



^{ }  X



 A t X



^ (16)

The solution of

b ˆ

is gotten by minimizing of Q and we get:

       

¹

       

ˆ

^T

I

_h

t

^T

I

_h

t

^T

I

_h

t

^T

I

_h

t y

  ^  X  A  A X ^ 

^

X  A  A

(17)

By substituting estimator

b ˆ

in equation (17) to equation (15), so equation (15) can expressed as :

         

¹

       

ˆ

_h

( ) (

^T

I

_h ^T

I

_h ^T

I

_h ^T

I

_h

f t t y t t t t y

  





 A    X X   A  A X  X  A  A  

   



¹



( )

^{T T}

(5.21)

h t  t ^ t y

A I X X S  X X S

 

~

  

~

   

~



(18) with

T

h h

S t  IA t I A t

Based on the solution of estimation in equation (17) and equation (18), so we obtained

 

ˆ ˆ

y

ˆ

X



 f t

56

Chamidah. N., Rifada. M., “Estimation of Median Growth Charts for Children Based on Biresponse Semiparametric Regression Model by using Local Linear Estimator”

 

~ ¹

 

~

 

~

 

~ ¹

 

~

ˆ ^{T T} _h ^{T T}

y S t S t y A t I S t S t y

 

       

X X X X    X X X X 

   

~

ˆ

_{par nonpar}

=

_h

y A A y A t y

with

h



par



nonpar

A A A

   

     

1

~ ~

1

~ ~ ~

and

T T

par

T T

nonpar h

S t S t

A t I S t S t





 

  

   

      A X X X X

A X X X X

In this study, we used local linear regression approach. So, there is a parameter i.e. bandwidth (h) that control the smoothness of the fit and also affect the bias-variance trade-off. The optimal bandwidth value is obtained based on generalized cross validation (GCV) method by minimizing the GCV function as follows:

   

   

²

1

ˆ ˆ

GCV( ) (5.24)

( ) tr _h

y f t y f t h

np ^ t

     

   

  I A 

4. Results and Discussion

Theresult of analysis P50 data obtained Pearson's correlation coefficient between weightand length is 0.99. We estimated median of weight and length based on bi-response semi-parametric regression by using local linear estimator. We get optimal bandwidth value, i.e., 1.94, minimum value of GCV i.e., 53.62, coefficient of determination, i.e., 0.997 andthe mean squared error value i.e., 0.21.

Then, we assessed the median growth chart of weight-for-age and length-for-age for boys and girls.

The median growth chart of weight-for-age represents that 50% of the children of a given age and gender haveweight higherthan the fiftieth percentile (P50) value, and 50% of the children of a given age and gender have weight less than P50 value. The children who have weight below P50 value means that their weight less than the average. Hence, the children who have weight higher than P50 value means that their weight above the average. This does not mean thatoverweight or underweight.The median growth chart of weight-for-age and length-for-age for boys and girls are compared by observation median are showed by the following four Figures (Figure 1 -Figure4)

Age (month)

Weight (Kg)

0 5 10 15 20

46810

Age (month)

Weight (Kg)

0 5 10 15 20

46810

The median growth chart of weight-for-age for boys

Points = Observed median , Lines = Smoothed median

Figure 1. Comparison between smoothed median (observed median of weight-for-age for boys)

Chamidah. N., Rifada. M., “Estimation of Median Growth Charts for Children Based on Biresponse Semiparametric Regression Model by using Local Linear Estimator”

ISBN : 978-602-99849-2-7

Copyright © 2015by Advances on Science and Technology

Age (month)

Weight (Kg)

0 5 10 15 20

46810

Age (month)

Weight (Kg)

0 5 10 15 20

46810

The median growth chart of weight-for-age for girls

Points = Observed median , Lines = Smoothed median

Figure 2. Comparison between smoothed median (observed median of weight-for-age for girls)

Age (month)

Length (cm)

0 5 10 15 20

50607080

Age (month)

Length (cm)

0 5 10 15 20

50607080

The median growth chart of length-for-age for boys

Points = Observed median , Lines = Smoothed median

Figure 3. Comparison between smoothed median (observed median of length-for-age for boys)

Age (month)

Length (cm)

0 5 10 15 20

50607080

Age (month)

Length (cm)

0 5 10 15 20

50607080

The median growth chart of length-for-age for girls

Points = Observed median , Lines = Smoothed median

Figure 4.Comparison between smoothed median (observed median of length-for-age for girls)

Overall, the comparison between smoothed median and observed or empirical median growth chart of weight-for-age and length-for-age has been fit.The average absolute difference of smoothed and observed median was small i.e., 0.15 kg for weight-for-age for boys (Figure 1) and 0.14 kg for girls (Figure 2).Meanwhile,the average absolute difference of smoothed and observed medianfor length-for-age i.e. 0.62cm for boys (Figure 3) and 0.47cm for girls (Figure 4). Next, plotting of comparison median growth chart between boys and girls for weight-for-age and length-for-age are showed in Figure 5 and Figure 6, respectively.

58

The First International Conference on Statistical Methods in Engineering, Science, Economy, and Education 1^st SESEE-2015Department of Statistics, Yogyakarta September 19-20 2015

Age (month)

Weight (Kg)

0 5 10 15 20

46810

Age (month)

Weight (Kg)

0 5 10 15 20

46810

The median growth chart of weight-for-age for boys and girls

Blue = boys, Red = girls

Figure 5. Comparison between median growth (chart of weight-for-age boys and girls)

Age (month)

Length (cm)

0 5 10 15 20

50607080

Age (month)

Length (cm)

0 5 10 15 20

50607080

The median growth chart of length-for-age for boys and girls

Blue = boys, Red = girls

Figure 6.Comparison between median growth (chart of length-for-age boys and girls)

We can showed from Figure 5 and Figure 6 that the median growth chart of weight-for-age and length-for-age for boys were higher than girls.The average of weight and length growth of boys from birth up to two years old in Surabaya is higher than girls i.e., 0.24 kg and 1.78 cm, respectively.

5. Conclusion

The estimation median of weight and length children in Surabaya, Indonesia 2015, based on bi- response semiparametric regression model by using local linear estimator has been appropriate to goodness of fit criterions, i.e.,determination coefficient tend to one and mean squared error tend to zero. There is difference pattern of growth chart between boys and girls. The median growth chart of weight-for-age and length-for-age for boys were higher than girls.

Acknowledgement.

Many thanks to Directorate General of Higher Education of Indonesia for financial support of this research through FeaturedResearch University Grant 2015.

References

[1] WHO-Multicenter Growth Reference Study Group, “WHO Child Growth Standards based on length/height, weight and age”, ActaPædiatrica, Suppl 450 : 76 - 85 (2006)

[2] M.B. Narendra, T.S. Sularyo, Soetjiningsih, H. Suyitno, and I.G.N.G Ranuh, “ TumbuhKembangAnakdanRemaja “, Jakarta: CV. SagungSeto, 2002

[3] N.Chamidah and Eridani, “Designing of Growth Reference Chart by Using Birespon Semiparametric Regression Approach Based on P-Spline Estimator”, International Journal of Applied Mathematics and Statistics53 (3),150-158 (2015).

[4] P.Roumeliotis, Children’s health and wellness. Growth and Development, 2012

[5] Eubank, R., Spline Smoothing and Nonparametric Regression,Marcel Dekker, New York, 1998.

The First International Conference on Statistical Methods in Engineering, Science, Economy, and Education 1^st SESEE-2015Department of Statistics, Yogyakarta September 19-20 2015

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