KESIMPULAN DAN SARAN

5.1. Kesimpulan

Berdasarkan analisis dan pembahasan pada bab sebelumnya, dapat disimpulkan bahwa :

1. Estimator Parameter untuk Regresi Poisson menggunakan Generalized

Estimating Equation adalah

dengan estimasi untuk Working Correlation Structurenya :

2. Rata-rata Jumlah Kejadian banjir di desa-desa di jawa timur dipengaruhi secara signifikan oleh karakteristik desa sebagai berikut : Topografi, dan Jumlah Pemukiman di Bantaran Sungai. model yang terbentuk berdasarkan nilai QIC maka WCS yang terbaik adalah “Unstructured” dan

“Independent” 5.2. Saran

Berdasakan penelitian yang dilakukan, saran yang dapat diberikan adalah sebagai berikut :

1. Penelitian ini menggunakan Metode Generalized Estimating Equation sebagai estimasi parameter untuk model regresi count poisson, untuk berikutnya bisa dipertimbankan penggunaan metode estimasi parameter yang lain, GEE2, Pendekatan Bayesian dan lain lain

2. Regresi Count untuk data yang mengandung dispersi dan Zero-Inflated diatasi dengan model Zero Inflated Generalized Poisson dan dapat

dipertimbangkan penggunaan metode lain seperti Zero Inflated Negative Binomial, Regresi Hurdle dan lain lain

3. Penelitian terhadap variabel-variabel lain yang mempengaruhi jumlah kejadian banjir

4. Penggunaan jenis Working Correlation Structure lainya untuk penelitian menggunakan Generalized Estimating Equation selanjutnya.

5. Pengujian dan inferensia statistik dengan pengujian hipotesis dapat dilakukan dalam penelitian selanjutnya.

DAFTAR PUSTAKA

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the Mean and Dipersion Level. Lehrstuhl fur Mathematische Statistik.

Technische Universitat Munchen. Garching

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Regression Model with an Application to Accident Data. Journal of Data

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Multilevel. Universitas Brawijaya. Malang

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Fisik dan Frekuensi Banjir Pada 15 DAS di Jawa Timur, Universitas

Jember.

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Pengendaliannya, Dalam Perspektif Lingkungan. Yogyakarta. Pustaka

Pelajar

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Defects in Manufacturing,Technometrics, 34, 1–14.

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Generalized Linear Models. Biometrika 73, 13–22.

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43 Lampiran 1

Pearson's Chi-squared test

data: dtt3$banjir and dtt3$jumlah

X-squared = 19.918, df = 18, p-value = 0.3374 Overdispersion test

data: m.glm

z = 18.858, p-value < 2.2e-16

alternative hypothesis: true dispersion is greater than 1 sample estimates:

dispersion 3.262677

#GEEPOISSON tanpa 0#

geeex<-geeglm(formula = jumlah ~ topo + buangs + jmlbansurt, family = "poisson", data = BANJIR, id = ID, corstr =

"exchangeable")

geear1<-geeglm(formula = jumlah ~ topo + buangs +

jmlbansurt, family = "poisson", data = BANJIR, id = ID, corstr = "ar1")

geein<-geeglm(formula = jumlah ~ topo + buangs + jmlbansurt, family = "poisson", data = BANJIR, id = ID, corstr =

"independence")

geeun<-geeglm(formula = jumlah ~ topo + buangs + jmlbansurt, family = "poisson", data = BANJIR, id = ID, corstr =

"unstructured") B.glm<-glm(jumlah~topo+buangs+jmlbansu,data=BANJIR,family=poisson) summary(geeex) summary(geear1) summary(geein) summary(geeun) summary(B.glm) #GLM MLE POISSON# B.glm2<-glm(jumlah~topo+jmlbansu,data=BANJIR,family=poisson) summary(B.glm2) #QIC QIC = function(model.R) { library(MASS)

model.indep = update(model.R, corstr = "independence") # Quasilikelihood

mu.R = model.R$fitted.values y = model.R$y

44 type = family(model.R)$family quasi.R = switch(type,

poisson = sum((y*log(mu.R)) - mu.R), gaussian = sum(((y - mu.R)^2)/-2), binomial = sum(y*log(mu.R/(1 - mu.R)) +

log(1 - mu.R)),

Gamma = sum(-y/(mu.R - log(mu.R))), stop("Error: distribution not

recognized"))

# Trace Term (penalty for model complexity)

omegaI = ginv(model.indep$geese$vbeta.naiv) # Omega-hat(I) via Moore-Penrose generalized inverse of a matrix in MASS package

#AIinverse = solve(model.indep$geese$vbeta.naiv) # solve via indenity

Vr = model.R$geese$vbeta

trace.R = sum(diag(omegaI %*% Vr))

px = length(mu.R) # number non-redunant columns in design matrix

# QIC

QIC = 2*(trace.R - quasi.R)

#QICu = (-2)*quasi.R + 2*px # Approximation assuming model structured correctly

output = c(QIC, quasi.R, trace.R, px)

names(output) = c('QIC', 'Quasi Lik', 'Trace', 'px') return(output)

}

sapply(list(geeex, geeun, geein, geear1), QIC)

#ZIP ZIP<-zeroinfl(formula=jumlah~topo+buangs+jmlbansu,dist="poisson" ,data=dtt3) summary(ZIP) #GEEPOISSON dengan 0 #

gee0ex<-geeglm(formula = jumlah ~ topo + buangs + jmlbansurt, family = "poisson", data = dtt3, id = ID, corstr = "exchangeable")

gee0ar1<-geeglm(formula = jumlah ~ topo + buangs + jmlbansurt, family = "poisson", data = dtt3, id = ID, corstr = "ar1")

gee0ind<-geeglm(formula = jumlah ~ topo + buangs + jmlbansurt, family = "poisson", data = dtt3, id = ID, corstr = "independence")

gee0un<-geeglm(formula = jumlah ~ topo + buangs + jmlbansurt, family = "poisson", data = dtt3, id = ID, corstr = "unstructured")

45 summary(gee0ex)

summary(gee0ar1) summary(gee0ind) summary(gee0un)

#GEEPOISSON dengan 0 variabel signifikan#

gee0ex2<-geeglm(formula = jumlah ~ topo + jmlbansurt, family = "poisson", data = dtt3, id = ID, corstr = "exchangeable") gee0ar12<-geeglm(formula = jumlah ~ topo + jmlbansurt,

family = "poisson", data = dtt3, id = ID, corstr = "ar1") gee0ind2<-geeglm(formula = jumlah ~ topo + jmlbansurt,

family = "poisson", data = dtt3, id = ID, corstr = "independence")

gee0un2<-geeglm(formula = jumlah ~ topo + jmlbansurt, family = "poisson", data = dtt3, id = ID, corstr = "unstructured") summary(gee0ex2) summary(gee0ar12) summary(gee0ind2) summary(gee0un2) #QIC QIC = function(model.R) { library(MASS)

model.indep = update(model.R, corstr = "independence") # Quasilikelihood

mu.R = model.R$fitted.values y = model.R$y

type = family(model.R)$family quasi.R = switch(type,

poisson = sum((y*log(mu.R)) - mu.R), gaussian = sum(((y - mu.R)^2)/-2), binomial = sum(y*log(mu.R/(1 - mu.R)) +

log(1 - mu.R)),

Gamma = sum(-y/(mu.R - log(mu.R))), stop("Error: distribution not

recognized"))

# Trace Term (penalty for model complexity)

omegaI = ginv(model.indep$geese$vbeta.naiv) # Omega-hat(I) via Moore-Penrose generalized inverse of a matrix in MASS package

#AIinverse = solve(model.indep$geese$vbeta.naiv) # solve via indenity

Vr = model.R$geese$vbeta

trace.R = sum(diag(omegaI %*% Vr))

px = length(mu.R) # number non-redunant columns in design matrix

# QIC

QIC = 2*(trace.R - quasi.R)

#QICu = (-2)*quasi.R + 2*px # Approximation assuming model structured correctly

names(output) = c('QIC', 'Quasi Lik', 'Trace', 'px') return(output)

}

sapply(list(gee0ex2, gee0un2, gee0ind2, gee0ar12), QIC)

OUTPUT Call:

geeglm(formula = jumlah ~ factor(topo) + buangs + jmlbansurt,

family = "poisson", data = BANJIR, id = ID, corstr = "exchangeable")

Coefficients:

Estimate Std.err Wald Pr(>|W|) (Intercept) 0.6573973 0.1684969 15.222 9.56e-05 *** factor(topo)2 0.0531249 0.2212208 0.058 0.810 factor(topo)3 -0.1722873 0.1332154 1.673 0.196 buangs -0.0240812 0.0548999 0.192 0.661 jmlbansurt -0.0004632 0.0003788 1.496 0.221 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Estimated Scale Parameters: Estimate Std.err (Intercept) 1.897 0.1012

Correlation: Structure = exchangeable Link = identity Estimated Correlation Parameters:

Estimate Std.err alpha 0.8074 0.01577

Number of clusters: 1218 Maximum cluster size: 3 > summary(geear1)

geeglm(formula = jumlah ~ factor(topo) + buangs + jmlbansurt,

family = "poisson", data = BANJIR, id = ID, corstr = "ar1")

Coefficients:

Estimate Std.err Wald Pr(>|W|) (Intercept) 0.625459 0.171297 13.33 0.00026 *** factor(topo)2 0.054520 0.221304 0.06 0.80541 factor(topo)3 -0.133372 0.135788 0.96 0.32599 buangs -0.023844 0.054833 0.19 0.66368 jmlbansurt -0.000396 0.000367 1.16 0.28063 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Estimated Scale Parameters: Estimate Std.err (Intercept) 1.89 0.101

Correlation: Structure = ar1 Link = identity Estimated Correlation Parameters:

Estimate Std.err alpha 0.859 0.0118

Number of clusters: 1218 Maximum cluster size: 3 > summary(geein)

Call:

geeglm(formula = jumlah ~ factor(topo) + buangs + jmlbansurt,

family = "poisson", data = BANJIR, id = ID, corstr = "independence")

Coefficients:

Estimate Std.err Wald Pr(>|W|) (Intercept) 0.657397 0.168497 15.22 9.6e-05 *** factor(topo)2 0.053125 0.221221 0.06 0.81 factor(topo)3 -0.172287 0.133215 1.67 0.20 buangs -0.024081 0.054900 0.19 0.66 jmlbansurt -0.000463 0.000379 1.50 0.22 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Estimated Scale Parameters: Estimate Std.err (Intercept) 1.9 0.101

Correlation: Structure = independenceNumber of clusters: 1218 Maximum cluster size: 3

48 Call:

geeglm(formula = jumlah ~ factor(topo) + buangs + jmlbansurt,

family = "poisson", data = BANJIR, id = ID, corstr = "unstructured")

Coefficients:

Estimate Std.err Wald Pr(>|W|) (Intercept) 0.652736 0.168608 14.99 0.00011 *** factor(topo)2 0.053369 0.220685 0.06 0.80891 factor(topo)3 -0.166623 0.133355 1.56 0.21149 buangs -0.024043 0.054849 0.19 0.66114 jmlbansurt -0.000453 0.000376 1.45 0.22852 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Estimated Scale Parameters: Estimate Std.err (Intercept) 1.9 0.101

Correlation: Structure = unstructured Link = identity Estimated Correlation Parameters:

Estimate Std.err alpha.1:2 0.817 0.0216 alpha.1:3 0.787 0.0213 alpha.2:3 0.818 0.0210

Number of clusters: 1218 Maximum cluster size: 3 > summary(B.glm)

Call:

glm(formula = jumlah ~ factor(topo) + buangs + jmlbansu, family = poisson,

data = BANJIR) Deviance Residuals:

Min 1Q Median 3Q Max -1.990 -0.619 -0.451 0.370 4.173 Coefficients:

Estimate Std. Error z value Pr(>|z|) (Intercept) 0.63849 0.06891 9.27 < 2e-16 *** factor(topo)2 0.05440 0.11072 0.49 0.62316 factor(topo)3 -0.16853 0.04835 -3.49 0.00049 *** buangs -0.02505 0.02080 -1.20 0.22852 jmlbansu 0.01017 0.00451 2.25 0.02420 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1) Null deviance: 5978.5 on 3653 degrees of freedom Residual deviance: 5958.4 on 3649 degrees of freedom AIC: 12830

Number of Fisher Scoring iterations: 5 > > #GLM MLE POISSON# > B.glm2<-glm(jumlah~factor(topo)+jmlbansu,data=BANJIR,family=poisson ) > summary(B.glm2) Call:

glm(formula = jumlah ~ factor(topo) + jmlbansu, family = poisson,

data = BANJIR) Deviance Residuals:

Min 1Q Median 3Q Max -1.997 -0.636 -0.454 0.367 4.123 Coefficients:

Estimate Std. Error z value Pr(>|z|) (Intercept) 0.57485 0.04468 12.87 <2e-16 *** factor(topo)2 0.06520 0.11037 0.59 0.5547 factor(topo)3 -0.15266 0.04661 -3.28 0.0011 ** jmlbansu 0.01001 0.00452 2.21 0.0269 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1) Null deviance: 5978.5 on 3653 degrees of freedom Residual deviance: 5959.9 on 3650 degrees of freedom AIC: 12829

Number of Fisher Scoring iterations: 5 >

> sapply(list(geeex, geeun, geein, geear1), QIC) [,1] [,2] [,3] [,4] QIC 6330.0 6330.0 6330.0 6331 Quasi Lik -3151.1 -3151.1 -3151.1 -3152 Trace 13.9 13.9 13.9 14 px 3654.0 3654.0 3654.0 3654 > Call:

geeglm(formula = jumlah ~ factor(topo) + buangs + jmlbansurt,

family = "poisson", data = dtt3, id = ID, corstr = "exchangeable")

Coefficients:

Estimate Std.err Wald Pr(>|W|) (Intercept) -1.940156 0.183519 111.77 <2e-16 *** factor(topo)2 0.753250 0.329919 5.21 0.0224 * factor(topo)3 0.433299 0.163960 6.98 0.0082 ** buangs 0.004450 0.069169 0.00 0.9487 jmlbansurt 0.003613 0.000391 85.21 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Estimated Scale Parameters: Estimate Std.err (Intercept) 3.29 0.183

Correlation: Structure = exchangeable Link = identity Estimated Correlation Parameters:

Estimate Std.err alpha 0.887 0.0598

Number of clusters: 8502 Maximum cluster size: 3 > summary(gee0ar1)

Call:

geeglm(formula = jumlah ~ factor(topo) + buangs + jmlbansurt,

family = "poisson", data = dtt3, id = ID, corstr = "ar1")

Coefficients:

Estimate Std.err Wald Pr(>|W|) (Intercept) -1.972654 0.187012 111.27 <2e-16 *** factor(topo)2 0.755930 0.330345 5.24 0.0221 * factor(topo)3 0.476734 0.166811 8.17 0.0043 ** buangs 0.003362 0.069128 0.00 0.9612 jmlbansurt 0.003658 0.000392 86.98 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Estimated Scale Parameters: Estimate Std.err (Intercept) 3.29 0.197

Correlation: Structure = ar1 Link = identity Estimated Correlation Parameters:

51 Estimate Std.err

alpha 0.92 0.045

Number of clusters: 8502 Maximum cluster size: 3 > summary(gee0ind)

Call:

geeglm(formula = jumlah ~ factor(topo) + buangs + jmlbansurt,

family = "poisson", data = dtt3, id = ID, corstr = "independence")

Coefficients:

Estimated Scale Parameters: Estimate Std.err (Intercept) 3.29 0.183

Correlation: Structure = independenceNumber of clusters: 8502 Maximum cluster size: 3

> summary(gee0un) Call:

geeglm(formula = jumlah ~ factor(topo) + buangs + jmlbansurt,

family = "poisson", data = dtt3, id = ID, corstr = "unstructured")

Coefficients:

Estimate Std.err Wald Pr(>|W|) (Intercept) -1.95934 0.18455 112.72 <2e-16 *** factor(topo)2 0.76670 0.33161 5.35 0.0208 * factor(topo)3 0.44851 0.16532 7.36 0.0067 ** buangs 0.00522 0.06920 0.01 0.9399 jmlbansurt 0.00362 0.00039 86.16 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Estimated Scale Parameters: Estimate Std.err (Intercept) 3.31 0.189

52 Estimated Correlation Parameters: Estimate Std.err

alpha.1:2 0.873 0.0613 alpha.1:3 0.877 0.0617 alpha.2:3 0.913 0.0629

Number of clusters: 8502 Maximum cluster size: 3 >

> #GEEPOISSON dengan 0 variabel signifikan#

> gee0ex2<-geeglm(formula = jumlah ~ factor(topo) + jmlbansurt, family = "poisson", data = dtt3, id = ID, corstr = "exchangeable")

> gee0ar12<-geeglm(formula = jumlah ~ factor(topo) + jmlbansurt, family = "poisson", data = dtt3, id = ID, corstr = "ar1")

> gee0ind2<-geeglm(formula = jumlah ~ factor(topo) + jmlbansurt, family = "poisson", data = dtt3, id = ID, corstr = "independence")

> gee0un2<-geeglm(formula = jumlah ~ factor(topo) + jmlbansurt, family = "poisson", data = dtt3, id = ID, corstr = "unstructured")

> summary(gee0ex2) Call:

geeglm(formula = jumlah ~ factor(topo) + jmlbansurt, family = "poisson",

data = dtt3, id = ID, corstr = "exchangeable") Coefficients:

Estimate Std.err Wald Pr(>|W|) (Intercept) -1.929929 0.167468 132.81 <2e-16 *** factor(topo)2 0.753414 0.329277 5.24 0.022 * factor(topo)3 0.431713 0.171873 6.31 0.012 * jmlbansurt 0.003610 0.000389 86.09 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Estimated Scale Parameters: Estimate Std.err (Intercept) 3.29 0.17

Correlation: Structure = exchangeable Link = identity Estimated Correlation Parameters:

Estimate Std.err alpha 0.887 0.0579

Number of clusters: 8502 Maximum cluster size: 3 > summary(gee0ar12)

geeglm(formula = jumlah ~ factor(topo) + jmlbansurt, family = "poisson",

data = dtt3, id = ID, corstr = "ar1") Coefficients:

Estimate Std.err Wald Pr(>|W|) (Intercept) -1.96492 0.17018 133.32 <2e-16 *** factor(topo)2 0.75605 0.32975 5.26 0.0219 * factor(topo)3 0.47553 0.17444 7.43 0.0064 ** jmlbansurt 0.00365 0.00039 87.87 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Estimated Scale Parameters: Estimate Std.err (Intercept) 3.29 0.181

Correlation: Structure = ar1 Link = identity Estimated Correlation Parameters:

Estimate Std.err alpha 0.92 0.0433

Number of clusters: 8502 Maximum cluster size: 3 > summary(gee0ind2)

Call:

geeglm(formula = jumlah ~ factor(topo) + jmlbansurt, family = "poisson",

data = dtt3, id = ID, corstr = "independence") Coefficients:

Estimated Scale Parameters: Estimate Std.err (Intercept) 3.29 0.17

Correlation: Structure = independenceNumber of clusters: 8502 Maximum cluster size: 3

> summary(gee0un2) Call:

geeglm(formula = jumlah ~ factor(topo) + jmlbansurt, family = "poisson",

data = dtt3, id = ID, corstr = "unstructured") Coefficients:

Estimate Std.err Wald Pr(>|W|) (Intercept) -1.947364 0.168878 132.97 <2e-16 *** factor(topo)2 0.766886 0.330962 5.37 0.0205 * factor(topo)3 0.446673 0.173233 6.65 0.0099 ** jmlbansurt 0.003617 0.000388 87.00 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Estimated Scale Parameters: Estimate Std.err (Intercept) 3.31 0.175

Correlation: Structure = unstructured Link = identity Estimated Correlation Parameters:

Estimate Std.err alpha.1:2 0.873 0.0594 alpha.1:3 0.877 0.0597 alpha.2:3 0.913 0.0610

Number of clusters: 8502 Maximum cluster size: 3 >

> sapply(list(gee0ex2, gee0un2, gee0ind2, gee0ar12), QIC) [,1] [,2] [,3] [,4]

QIC 28148.4 28148.8 28148.4 28149.9 Quasi Lik -14063.1 -14063.2 -14063.1 -14063.7 Trace 11.1 11.2 11.1 11.2

BIOGRAFI PENULIS

Penulis dilahirkan di Yogyakarta pada tanggal 26 Maret 1983 dan putri pertama dari pasangan suami istri Bapak Nyono Sunarso dan Ibu Sumaryanti. Saat ini penulis telah berkeluarga dengan istri bernama Mariatul Kiftiyah dan telah dikaruniai dengan satu putri Mahya Aretya Diah Garini dan satu orang putra Satrio Pinandito .

Riwayat pendidikan penulis diawali dari SDN Pujokusuman, SMPN 2 Yogyakarta SMUN 6 Yogyakarta dan Sekolah Tinggi Ilmu Statistik (STIS) Jakarta jurusan Statistik Ekonomi. Setelah menamatkan pendidikan D-IV STIS, Pembaca yang ingin memberikan kritik, saran dan pertanyaan mengenai penelitian ini dapat menghubungi penulis melalui email [email protected]

Dalam dokumen Regresi Poisson Menggunakan Generalized Estimating Equation (Case Study: Flood Occurance Longitudinal Data In East Java Province Year 2011-2013) - ITS Repository (Halaman 48-66)