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Analisis Data Kategorik - STK654 (Materi UAS)
Dr. Kusman Sadik, M.Si
Program Studi Magister Statistika Terapan
Departemen Statistika IPB, Semester Ganjil 2019/2020
IPB University─ Bogor Indonesia ─ Inspiring Innovation with Integrity
Model Log-Linear untuk
Data Respon Poisson(Bagian II)
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When fitting log-linear models to higher-way tables it is
typical to only consider models that are hierarchical in
nature.
These are models that include all lower-order terms for
variables involved in higher-order terms in the model.
For a three-way contingency table, with variables X, Y, and
Z, the saturated model includes all main effects, all two-
way interactions and the three-way interaction, and is
represented symbolically as follows:
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Exp( ) merupakan nilai odds antara kategori pada
baris ke-i dengan kategori baris terakhir.
Exp( ) merupakan nilai rasio odds antara odds
kategori pada baris ke-i dengan kategori baris terakhir
dengan odds kategori pada kolom ke-j dengan
kategori kolom terakhir.
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Lihat : Azen, hlm. 145
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## Model Log-Linear untuk Data Tabel 7.4 (Azen, hlm.145) ##
## relevel --> Memilih Kategori Referensi ##
## Model 1 : Interaksi XY XZ YZ ##
z.gen <- factor(rep(c("1Mal","2Fem"),each=3, times=3))
x.rel <- factor(rep(c("1Lib","2Mod","3Con"),each=1, times=6))
y.god <- factor(rep(c("1Y","2N","3U"),each=6,times=1))
count <- c(26,82,202,44,113,180,121,128,75,221,
204,124,24,52,74,32,49,43)
z.gen <- relevel(z.gen, ref="2Fem")
x.rel <- relevel(x.rel, ref="3Con")
y.god <- relevel(y.god, ref="3U")
data.frame(z.gen, x.rel, y.god, count)
model <- glm(count ~ x.rel + y.god + z.gen +
x.rel*y.god + x.rel*z.gen + y.god*z.gen,
family=poisson("link"=log))
summary(model)
dugaan <- round(fitted(model),2)
data.frame(z.gen, x.rel, y.god, count, dugaan)
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z.gen x.rel y.god count
1 1Mal 1Lib 1Y 26
2 1Mal 2Mod 1Y 82
3 1Mal 3Con 1Y 202
4 2Fem 1Lib 1Y 44
5 2Fem 2Mod 1Y 113
6 2Fem 3Con 1Y 180
7 1Mal 1Lib 2N 121
8 1Mal 2Mod 2N 128
9 1Mal 3Con 2N 75
10 2Fem 1Lib 2N 221
11 2Fem 2Mod 2N 204
12 2Fem 3Con 2N 124
13 1Mal 1Lib 3U 24
14 1Mal 2Mod 3U 52
15 1Mal 3Con 3U 74
16 2Fem 1Lib 3U 32
17 2Fem 2Mod 3U 49
18 2Fem 3Con 3U 43
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Call:
glm(formula = count ~ x.rel + y.god + z.gen + x.rel * y.god +
x.rel * z.gen + y.god * z.gen, family = poisson(link = log))
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 3.862595 0.122713 31.477 < 2e-16 ***
x.rel1Lib -0.511037 0.177163 -2.885 0.003920 **
x.rel2Mod 0.005771 0.150379 0.038 0.969388
y.god1Y 1.367890 0.134994 10.133 < 2e-16 ***
y.god2N 0.860606 0.140582 6.122 9.26e-10 ***
z.gen1Mal 0.377454 0.136017 2.775 0.005519 **
x.rel1Lib:y.god1Y -0.994715 0.209026 -4.759 1.95e-06 ***
x.rel2Mod:y.god1Y -0.548065 0.162483 -3.373 0.000743 ***
x.rel1Lib:y.god2N 1.212754 0.186792 6.493 8.44e-11 ***
x.rel2Mod:y.god2N 0.615654 0.163938 3.755 0.000173 ***
x.rel1Lib:z.gen1Mal -0.416543 0.131545 -3.167 0.001543 **
x.rel2Mod:z.gen1Mal -0.272968 0.114568 -2.383 0.017191 *
y.god1Y:z.gen1Mal -0.334342 0.146372 -2.284 0.022360 *
y.god2N:z.gen1Mal -0.640833 0.142437 -4.499 6.83e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Null deviance: 724.8365 on 17 degrees of freedom
Residual deviance: 6.5154 on 4 degrees of freedom
AIC: 146.38
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z.gen x.rel y.god count dugaan
1 1Mal 1Lib 1Y 26 28.54
2 1Mal 2Mod 1Y 82 86.34
3 1Mal 3Con 1Y 202 195.12
4 2Fem 1Lib 1Y 44 41.46
5 2Fem 2Mod 1Y 113 108.66
6 2Fem 3Con 1Y 180 186.88
7 1Mal 1Lib 2N 121 115.01
8 1Mal 2Mod 2N 128 122.52
9 1Mal 3Con 2N 75 86.47
10 2Fem 1Lib 2N 221 226.99
11 2Fem 2Mod 2N 204 209.48
12 2Fem 3Con 2N 124 112.53
13 1Mal 1Lib 3U 24 27.45
14 1Mal 2Mod 3U 52 53.14
15 1Mal 3Con 3U 74 69.41
16 2Fem 1Lib 3U 32 28.55
17 2Fem 2Mod 3U 49 47.86
18 2Fem 3Con 3U 43 47.59
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𝑂𝑑𝑑𝑠 𝑋=1 |𝑌=1
𝑂𝑑𝑑𝑠 𝑋=1 |𝑌=3=
𝑃(𝑋=1|𝑌=1)/𝑃(𝑋=3|𝑌=1)
𝑃(𝑋=1|𝑌=3)/𝑃(𝑋=3|𝑌=3)
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Nilai deviance dapat digunakan untuk menguji hipotesis
ada tidaknya hubungan suatu variabel dengan variabel
lainnya.
Misalkan ada tiga variabel X, Y, dan Z. Untuk menguji
apakah ada hubungan sekaligus antara X, Y, dan Z, dapat
dilakukan dengan membandingkan nilai deviance model
(X, Y, Z, XY, XZ, YZ, XYZ) dengan nilai deviance model (X,
Y, Z, XY, XZ, YZ).
Jika dari uji hipotesis tersebut menerima model (X, Y, Z,
XY, XZ, YZ) maka dapat disimpulkan bahwa tidak ada
interaksi XYZ, demikian pula sebaliknya.
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To determine whether the three-way interaction term in the
saturated model
H1: ≠ 0
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Call:
glm(formula = count ~ x.rel + y.god + z.gen + x.rel * y.god
+ x.rel * z.gen + y.god * z.gen
+ x.rel * y.god * z.gen,
family = poisson(link = log))
Null deviance: 7.2484e+02 on 17 degrees of freedom
Residual deviance: -4.6851e-14 on 0 degrees of freedom
AIC: 147.86
H1: ≠ 0
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Call:
glm(formula = count ~ x.rel + y.god + z.gen + x.rel * y.god +
x.rel * z.gen + y.god * z.gen, family = poisson(link = log))
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 3.862595 0.122713 31.477 < 2e-16 ***
x.rel1Lib -0.511037 0.177163 -2.885 0.003920 **
x.rel2Mod 0.005771 0.150379 0.038 0.969388
y.god1Y 1.367890 0.134994 10.133 < 2e-16 ***
y.god2N 0.860606 0.140582 6.122 9.26e-10 ***
z.gen1Mal 0.377454 0.136017 2.775 0.005519 **
x.rel1Lib:y.god1Y -0.994715 0.209026 -4.759 1.95e-06 ***
x.rel2Mod:y.god1Y -0.548065 0.162483 -3.373 0.000743 ***
x.rel1Lib:y.god2N 1.212754 0.186792 6.493 8.44e-11 ***
x.rel2Mod:y.god2N 0.615654 0.163938 3.755 0.000173 ***
x.rel1Lib:z.gen1Mal -0.416543 0.131545 -3.167 0.001543 **
x.rel2Mod:z.gen1Mal -0.272968 0.114568 -2.383 0.017191 *
y.god1Y:z.gen1Mal -0.334342 0.146372 -2.284 0.022360 *
y.god2N:z.gen1Mal -0.640833 0.142437 -4.499 6.83e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Null deviance: 724.8365 on 17 degrees of freedom
Residual deviance: 6.5154 on 4 degrees of freedom
AIC: 146.38
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𝑣𝑠
Deviance: 6.5154 on 4 degrees of freedom
AIC : 146.38
Model H0:
Deviance: 0.0000 on 0 degrees of freedom
AIC : 147.86
Model H1:
H1: ≠ 0
Menolak ataukah menerima H0 pada α = 0.05? Apa kesimpulan dari hasil uji deviance tersebut?
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χ2(α, db) : qchisq(α, db, lower.tail=FALSE)
> qchisq(0.05, 4, lower.tail=FALSE)
[1] 9.487729
Jadi χ2(α = 0.05, db = 4) = 9.497729
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𝐻0: 𝜆𝑖𝑗𝑋𝑌 = 0 𝑣𝑠 𝐻1: 𝜆𝑖𝑗
𝑋𝑌 ≠ 0
Model H0:
Model H1:
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Call:
glm(formula = count ~ x.rel + y.god + z.gen + x.rel * z.gen +
y.god * z.gen, family = poisson(link = log))
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 3.75190 0.09978 37.601 < 2e-16 ***
x.rel1Lib -0.15559 0.07905 -1.968 0.049035 *
x.rel2Mod 0.05331 0.07493 0.711 0.476792
y.god1Y 0.99980 0.10503 9.519 < 2e-16 ***
y.god2N 1.48782 0.09943 14.964 < 2e-16 ***
z.gen1Mal 0.45511 0.13489 3.374 0.000741 ***
x.rel1Lib:z.gen1Mal -0.56353 0.12225 -4.610 4.04e-06 ***
x.rel2Mod:z.gen1Mal -0.34575 0.11081 -3.120 0.001808 **
y.god1Y:z.gen1Mal -0.27386 0.14465 -1.893 0.058324 .
y.god2N:z.gen1Mal -0.71771 0.14014 -5.121 3.03e-07 ***
---
Null deviance: 724.84 on 17 degrees of freedom
Residual deviance: 256.75 on 8 degrees of freedom
AIC: 388.61
𝐻0: 𝜆𝑖𝑗𝑋𝑌 = 0
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Menolak ataukah menerima H0 pada α = 0.05? Apa kesimpulan dari hasil uji deviance tersebut?
𝐻0: 𝜆𝑖𝑗𝑋𝑌 = 0 𝑣𝑠 𝐻1: 𝜆𝑖𝑗
𝑋𝑌 ≠ 0
Deviance: 256.75 on 8 degrees of freedom
AIC : 388.61
Model H0:
Deviance: 6.5154 on 4 degrees of freedom
AIC : 146.38
Model H1:
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1. Gunakan Program R untuk menyelesaikan Problem 7.1 (Azen,
hlm.177) .
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2. Gunakan Program R untuk menyelesaikan Problem 8.1 (Agresti,
hlm. 347) .
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Pustaka
1. Azen, R. dan Walker, C.R. (2011). Categorical Data
Analysis for the Behavioral and Social Sciences.
Routledge, Taylor and Francis Group, New York.
2. Agresti, A. (2002). Categorical Data Analysis 2nd. New
York: Wiley.
3. Pustaka lain yang relevan.
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Bisa di-download di
kusmansadik.wordpress.com
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