RISK OF DENGUE HAEMORRHAGIC FEVER IN BEKASI MUNICIPALITY WITH SMALL AREA APPROACH
RISK OF DENGUE HAEMORRHAGIC FEVER IN BEKASI MUNICIPALITY
1 WITH SMALL AREA APPROACH
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 1 Bekasi Municipalit y t o illust rat e t he met hod.
This paper was present ed in The First Int ernat ional Conf erence on Mat hemat ics and St at ist ics, Unisba- Bandung, June 19-21, 2006
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 such as number of i 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: ind
K ~ , 1 , , y Poisson e i m i i θ i = iid ( )
~ , gamma
(2. 1. 1)
θ i ν α ( )
2. 2 Bayes Est imat or ind In t he f irst st age, we assume t hat y ~ Poisson e , and in t he second st age i i θ i iid ( ) assumed t hat ~ gamma , . The probabilit y densit y f unct ion f or is
θ i ν α θ i ( )
ν ν − 1 α − αθ i
, f e
(2. 2. 1)
θ α ν = θ i i ( )
Γ ν ( )
and 2
/ / E ,
V
(2. 2. 2)
θ = ν α = µ θ = ν α ( ) i ( ) i
Not ing t hat ind
y gamma y e , , ~ ( , ) , t he Bayes of and t he post erior variance of are
θ α ν ν α θ θ i i i i i i + +
obt ained f rom (2. 2. 2) by changing t o e and t o y : B α i α ν i ν + +
E y y e (2. 2. 3) θ + α ν = θ α ν = ν α i i i ( ) ( i i )
+
ˆ ( , ) , ,
( )
and 2 V y , , g , , y y e (2. 2. 4)
θ i i i i i i α ν = 1 α ν = ν α + + ( ) ( ) ( )
( )
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
1 ˆ ˆ e e
(2. 3. 1)
θ = θ e ⋅ l ⋅ l l ( ) ∑ m
and t he weight ed sample variance 2 2
1 ˆ ˆ s e e
(2. 3. 2) e ( i ⋅ ) i e ⋅ = θ − θ i ( )
∑ m
t o t heir expect ed values and t hen solving t he result ing moment equat ions f or and ,
α ν
ˆ ˆ
where and . The moment est imat ors, and ,
e e m e n y n = i i = i i i α ν
⋅ ( ) i ∑ ∑ ∑ i i
given by
ˆ ν
ˆ
(2. 3. 3)
= θ e ⋅
ˆ α
and
ˆ ˆ 2 θ ν e ⋅ s
(2. 3. 4) 2 = − e
ˆ e
α ⋅
ˆ ˆ
Then subst it ut e t he moment est imat ors and int o (2. 2. 3) t o get an EB est imat or of
α ν
as
θ i EB B + ˆ ˆ ˆ ˆ ˆ , ˆ ˆ
1 ˆ
(2. 3. 3)
θ = θ α ν = γ θ − γ θ i i ( ) i i ( i ) e ⋅
γ = α i i i ( )
- ˆ ˆ where e e .
ˆ
Not e y e is direct est imat or of , where y and e respect ively denot e t he
θ = θ i i i i i i ˆ observed and t he expect ed number of cases, is synt het ic est imat or.
θ e ⋅
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 y i is number of risk of dengue haemorrhagic f ever in t he i-t h area (f rom “ Di nas Kesehat an” ), n i is number of people in t he i-t h area (f rom PODES 2003), and e i is expect ed number of risk of dengue haemorrhagic f ever in t he i-t h area.
ˆ A direct est imat or of is given by y e , where e n y n .
θ θ = = i i i i i i i i ∑ ∑ i i
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.
Direct Est. Bayes Est. EB Est. Sub No. Villages Municipalities Pred. MSE Pred. MSE Pred. MSE
1 Perwira 0. 103 0. 011 0. 264 0. 022 0. 452 0. 033
2 Harapan Jaya 1. 174 0. 051 1. 159 0. 046 1. 137 0. 041
3 Kal i Abang Tengah 0. 660 0. 023 0. 684 0. 022 0. 720 0. 021
Bekasi Ut ara
4 Harapan Baru 0. 686 0. 157 0. 789 0. 121 0. 870 0. 088
5 Teluk Pucung 0. 825 0. 036 0. 840 0. 033 0. 862 0. 030
6 Marga Mulya 0. 134 0. 018 0. 327 0. 034 0. 527 0. 044
Medan Sat ria
7 Pej uang 0. 405 0. 016 0. 452 0. 017 0. 524 0. 018
8 Medan Sat ria 0. 491 0. 040 0. 567 0. 039 0. 662 0. 038
Direct Est. Bayes Est. EB Est. No. Sub Municipalities Villages Pred. MSE Pred. MSE Pred. MSE
41 Jat i Rasa 0. 753 0. 063 0. 791 0. 056 0. 837 0. 048
33 Pondok Gede Jat i Makmur 0. 681 0. 036 0. 713 0. 034 0. 759 0. 031
34 Jat i Sampurna 1. 130 0. 128 1. 105 0. 101 1. 077 0. 076
35 Jat i Ranggon 0. 313 0. 049 0. 485 0. 057 0. 651 0. 057
36 Jat i Murni 0. 385 0. 049 0. 517 0. 052 0. 657 0. 051
37 Jat i Rangga 0. 217 0. 047 0. 464 0. 069 0. 665 0. 068
38 PWK Jat i Sampurna Jat i Karya 0. 000 0. 000 0. 390 0. 071 0. 650 0. 077
39 Jat i Asih 2. 155 0. 245 1. 930 0. 176 1. 679 0. 129
40 Jat i Kramat 1. 275 0. 096 1. 237 0. 080 1. 188 0. 065
42 Jat i Mekar 0. 559 0. 045 0. 623 0. 042 0. 705 0. 040
31 Jat i Rahayu 0. 797 0. 035 0. 815 0. 033 0. 841 0. 030
43 Jat i Luhur 0. 730 0. 133 0. 806 0. 106 0. 873 0. 079
44 Jat i Asih Jat i Sari 1. 365 0. 186 1. 282 0. 136 1. 198 0. 095
45 Bant ar Gebang 0. 608 0. 092 0. 704 0. 081 0. 798 0. 066
46 Sumur Bat u 0. 000 0. 000 0. 417 0. 082 0. 675 0. 082
47 Cikiwul 0. 000 0. 000 0. 371 0. 065 0. 631 0. 073
48 Ciket ing 0. 333 0. 111 0. 610 0. 119 0. 782 0. 092
49 Pedurenan 0. 000 0. 000 0. 232 0. 025 0. 467 0. 042
50 Cimuning 1. 549 0. 480 1. 331 0. 248 1. 188 0. 137
51 Must ika Sari 0. 644 0. 138 0. 755 0. 111 0. 847 0. 083
32 Jat i Warna 1. 547 0. 126 1. 466 0. 102 1. 364 0. 080
30 Jat i Bening 1. 055 0. 044 1. 050 0. 041 1. 043 0. 036
9 Kali Baru 0. 899 0. 074 0. 914 0. 064 0. 933 0. 053
18 Pekayon 1. 206 0. 063 1. 186 0. 056 1. 156 0. 048
10 Harapan Mul ya 0. 438 0. 048 0. 544 0. 048 0. 665 0. 047
11 Bint ara Jaya 0. 650 0. 047 0. 696 0. 044 0. 758 0. 040
12 Bint ara 0. 816 0. 033 0. 831 0. 031 0. 853 0. 029
13 Kot a Baru 1. 070 0. 052 1. 063 0. 047 1. 054 0. 041
14 Kranj i 1. 175 0. 060 1. 158 0. 053 1. 133 0. 046
15 Bekasi Barat Jaka Sampurna 1. 021 0. 036 1. 020 0. 033 1. 018 0. 030
16 Kayuringin 1. 144 0. 049 1. 132 0. 044 1. 114 0. 039
17 Marga Jaya 1. 052 0. 138 1. 041 0. 107 1. 029 0. 079
19 Jaka Set ia 1. 222 0. 107 1. 187 0. 087 1. 144 0. 069
29 Jat i Waringin 1. 073 0. 031 1. 069 0. 029 1. 062 0. 027
20 Bekasi Sel at an Jaka Mul ya 1. 088 0. 108 1. 073 0. 088 1. 055 0. 068
21 Margahayu 2. 219 0. 100 2. 111 0. 087 1. 952 0. 078
22 Aren Jaya 1. 356 0. 054 1. 328 0. 049 1. 286 0. 043
23 Duren Jaya 0. 932 0. 036 0. 938 0. 034 0. 945 0. 031
24 Bekasi Timur Bekasi Jaya 1. 635 0. 076 1. 577 0. 067 1. 492 0. 058
25 Sepanj ang Jaya 2. 243 0. 314 1. 957 0. 211 1. 666 0. 146
26 Pengasinan 2. 029 0. 114 1. 919 0. 097 1. 763 0. 083
27 Boj ong Rawal umbu 0. 928 0. 029 0. 932 0. 027 0. 939 0. 025
28 Rawa Lumbu Boj ong Ment eng 0. 624 0. 078 0. 703 0. 069 0. 788 0. 059
52 Bant ar Gebang Must ika Jaya 1. 555 0. 242 1. 417 0. 165 1. 283 0. 111
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.
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