RISK OF DENGUE HAEMORRHAGIC FEVER IN BEKASI MUNICIPALITY WITH SMALL AREA APPROACH

RISK OF DENGUE HAEMORRHAGIC FEVERIN BEKASI MUNICIPALITY WITH SMALL AREA APPROACH1

Kismiant ini

Depart ment of Mat hemat ics, Yogyakart a St at e Universit y Anang Kurnia and Khairil A. Not odiput ro

Depart ment of St at ist ics, Bogor Agricult ural Inst it ut e Abst r act

Smal l Ar ea Est i mat i on (SAE) i s a st at i st i cal t echni que t o est i mat e par amet er s of subpopul at i on cont ai ni ng smal l si ze of sampl es. For count dat a, poi sson-gamma model s can be used t o est i mat e par amet er s of t he smal l ar ea. Thi s paper di scusses Bayes est i mat or s and empi r i cal Bayes est i mat or s i n poi sson-gamma model s i n smal l ar ea est i mat i on. We pr ovi de i l l ust r at i on of dengue haemor r hagi c f ever r i sk i n Bekasi Muni ci pal i t y usi ng r eal dat a f r om “ Di nas Kesehat an” and PODES 2003.

Keywords: small area est imat ion, poisson-gamma model, Bayes est imat or, empirical Bayes est imat or.

1. INTRODUCTION

Small Area Est imat ion (SAE) is a st at ist ical t echnique t o est imat e paramet ers of subpopulat ion (domain) cont aining small size of samples. The t erm “ small area” is commonly used t o denot e a small geographical area such as a count y, a municipalit y or census division. Direct est imat ors, based only on t he domain-specif ic sample dat a, are t ypically used t o est imat e paramet ers f or large domains. However, sample sizes in small domains, part icularly small geographical areas, are rarely large enough t o provide accurat e direct est imat es f or specif ic small domains. Theref ore, it is necessary t o f ind indirect est imat ors t o increase t he ef f ect ive sample size and t hus decrease t he st andard error.

There are t hree met hods of est imat ion t o obt ain indirect est imat ions: synt het ic, composit e, and James-St ein met hods. A synt het ic est imat or can brief ly be described as f ollows: “ An unbiased est imat or is obt ained f rom a sample survey f or a large area; and is used t o derive est imat or f or a small area under assumpt ion t hat t he small areas have t he same charact erist ics as t he large area. ” In order t o balance t he pot ent ial bias of a synt het ic est imat or against t he inst abilit y of a direct est imat or, a weight ed average of t he t wo est imat ors need t o be int roduced. This is called composit e est imat ion. James-St ein est imat ion is a composit e approach using common weight .

Indirect est imat ors f rom synt het ic or composit e or James-St ein est imat ion are based on implicit model s. Indirect est imat ors on t he ot her hand are based on explicit models incorporat ing area-specif ic ef f ect s. This includes empirical best linear unbiased predict ion (EBLUP), empirical Bayes (EB), and hierarchical Bayes (HB) est imat ors.

EBLUP met hods are suit able f or cont inuous variables and t hese are not suit able f or binary or count dat a. EB and HB met hods are applicable bot h f or binary and count dat a.

This paper describes a model appropriat e f or analysis of count dat a using small area approach. Bayes est imat or and empirical Bayes (EB) est imat or consider t he est imat ion of small area paramet ers. We use risk of dengue haemorrhagic f ever in Bekasi Municipalit y t o illust rat e t he met hod.

1

Unisba-2. MODEL-BASED INFERENCE FOR COUNT DATA

Empirical Bayes (EB) met hod is more generally valid f or model-based small area est imat ion, especially in handling count dat a.

2. 1 Poisson-Gamma Model

The Poisson model is st andard probabilit y model f or count s

y

_i such as number of risk of dengue haemorrhagic f ever. The Poisson will get it s limit in mean and variance when it is used f or est imat ion of single paramet er. Commonly, count (dengue haemorrhagic f ever) dat a exhibit over dispersion (variance is larger t han t he mean) and t hus a more f lexible Poisson f ormulat ion would include an addit ional paramet er t o accommodat e t he ext ra variabilit y observed in t he sample. Based on above explanat ion, t wo-st age model will int roduce f or count dat a, known as a Poisson-Gamma model:

( )

e

i

m

2. 3 Empirical Bayes (EB) Est imat or

In t he EB approach may be summarized as f ollows: (i) Obt ain t he post erior densit y of t he small area paramet ers of int erest , (ii) Est imat e t he model paramet ers f rom t he marginal densit y, (iii) Use t he est imat ed post erior densit y f or making inf erences.

Marshall (1991) in Rao (2003) obt ained simple moment est imat ors by equat ing t he weight ed sample mean

where

e

_⋅

=

∑

_i

(

e

i

m

)

and



3. AN ILLUSTRATION USING REAL DATA FROM “DINAS KESEHATAN” AND PODES 2003 We are illust rat ing t he poisson-gamma model in small area est imat ion f or risk of dengue haemorrhagic f ever in Bekasi Municipalit y using real dat a f rom “ Dinas Kesehat an” and PODES 2003.

Bekasi Municipalit y is divided int o m non-overlapping small areas. A small area in here means village, so we get 52 villages. Then, let yi is number of risk of dengue

Table one shows est imat ion f or risk of dengue haemorrhagic f ever in villages in Bekasi Municipalit y based on “Di nas Kesehat an” and PODES dat a in 2003. Generally, t he est imat or f rom Empirical Bayes gives result s of Mean Square Error (MSE) t ends t o small values compare wit h t he ot her est imat or. This can give descript ion of a bet t er accurat eness rat e f or Empirical Bayes est imat or. We can t ake anot her int erpret at ion t hat Bekasi Timur, Bekasi Selat an, and Bekasi Barat sub Municipalit ies have a relat ively high risk of dengue haemorrhagic f ever.

Table 1. The est imat ion f or risk of dengue haemorrhagic f ever based on direct , Bayes, and empirical Bayes est imat or.

4. CONCLUSION

Empirical Bayes met hod is used t o est imat e risk of dengue haemorrhagic f ever, which have a bet t er accuracy rat e wit h relat ively smaller MSE when compare wit h direct or Bayes met hods in t his paper.

An illust rat ion in t his paper used poisson-gamma model wit hout covariat es. The model development include covariat es which relat ed wit h response can be done t o improve est imat ion.

5. REFERENCES

Fay, R. E. and Herriot , R. A. (1979). Est imat es of income f or small places: an applicat ion of James-St ein procedures t o census dat a. Jour nal of t he Amer i can St at i st i cal Associ at i on, Vol. 74, p. 269-277.

Kurnia, A. and Not odiput ro, K. A. (2005). Aplikasi Met ode Bayes Pada Smal l Ar ea Est i mat i on. This paper present ed in St at ist ics Nat ional Seminar 7t h at Surabaya Technology Inst it ut e. [ November 26, 2005] .

Ramsini, B. et . all. (2001). Unisured est imat es by count y: a review of opt ions and issues. <www. odh. ohio. gov/ Dat a/ OFHSurv/ of hsrf q7. pdf >. [ August 23, 2005] .

Rao, J. N. K. and Gosh, M. (1994). Small area est imat ion: an appraisal. St at i st i cal Sci ence, Vol. 9, No. 1, pp. 55-76.

Rao, J. N. K. (1999). Some recent advances in model-based small area est imat ion. Sur vey Met hodol ogy, Vol. 25, No. 2, pp. 175-186.