Model Log-Linear (Bagian 2) Dr. Kusman Sadik, M.Si Program Studi Pascasarjana Departemen Statistika IPB, 2018/2019
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1 Model Log-Linear (Bagian 2) Dr. Kusman Sadik, M.Si Program Studi Pascasarjana Departemen Statistika IPB, 2018/2019
2 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: 2
3 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 3
4 Lihat : Azen, hlm
5 ** 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 x.rel y.god <- relevel(z.gen, ref="2fem") <- relevel(x.rel, ref="3con") <- 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) 5
6 z.gen x.rel y.god count 1 1Mal 1Lib 1Y Mal 2Mod 1Y Mal 3Con 1Y Fem 1Lib 1Y Fem 2Mod 1Y Fem 3Con 1Y Mal 1Lib 2N Mal 2Mod 2N Mal 3Con 2N Fem 1Lib 2N Fem 2Mod 2N Fem 3Con 2N Mal 1Lib 3U Mal 2Mod 3U Mal 3Con 3U Fem 1Lib 3U Fem 2Mod 3U Fem 3Con 3U 43 6
7 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) < 2e-16 *** x.rel1lib ** x.rel2mod y.god1y < 2e-16 *** y.god2n e-10 *** z.gen1mal ** x.rel1lib:y.god1y e-06 *** x.rel2mod:y.god1y *** x.rel1lib:y.god2n e-11 *** x.rel2mod:y.god2n *** x.rel1lib:z.gen1mal ** x.rel2mod:z.gen1mal * y.god1y:z.gen1mal * y.god2n:z.gen1mal e-06 *** --- Signif. codes: 0 *** ** 0.01 * Null deviance: on 17 degrees of freedom Residual deviance: on 4 degrees of freedom AIC:
8 z.gen x.rel y.god count dugaan 1 1Mal 1Lib 1Y Mal 2Mod 1Y Mal 3Con 1Y Fem 1Lib 1Y Fem 2Mod 1Y Fem 3Con 1Y Mal 1Lib 2N Mal 2Mod 2N Mal 3Con 2N Fem 1Lib 2N Fem 2Mod 2N Fem 3Con 2N Mal 1Lib 3U Mal 2Mod 3U Mal 3Con 3U Fem 1Lib 3U Fem 2Mod 3U Fem 3Con 3U
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11 Odds X=1 Y=1 = P(X=1 Y=1)/P(X=3 Y=1) Odds X=1 Y=3 P(X=1 Y=3)/P(X=3 Y=3) 11
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13 To determine whether the three-way interaction term in the saturated model H 1 : 0 13
14 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: e+02 on 17 degrees of freedom Residual deviance: e-14 on 0 degrees of freedom AIC:
15 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) < 2e-16 *** x.rel1lib ** x.rel2mod y.god1y < 2e-16 *** y.god2n e-10 *** z.gen1mal ** x.rel1lib:y.god1y e-06 *** x.rel2mod:y.god1y *** x.rel1lib:y.god2n e-11 *** x.rel2mod:y.god2n *** x.rel1lib:z.gen1mal ** x.rel2mod:z.gen1mal * y.god1y:z.gen1mal * y.god2n:z.gen1mal e-06 *** --- Signif. codes: 0 *** ** 0.01 * Null deviance: on 17 degrees of freedom Residual deviance: on 4 degrees of freedom AIC:
16 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) < 2e-16 *** x.rel1lib * x.rel2mod y.god1y < 2e-16 *** y.god2n < 2e-16 *** z.gen1mal *** x.rel1lib:z.gen1mal e-06 *** x.rel2mod:z.gen1mal ** y.god1y:z.gen1mal y.god2n:z.gen1mal e-07 *** --- Null deviance: on 17 degrees of freedom Residual deviance: on 8 degrees of freedom AIC:
17 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 antara X dengan Y, dapat dilakukan dengan membandingkan nilai deviance model (X, Y, Z, XY, XZ, YZ) dengan nilai deviance model (X, Y, Z, XZ, YZ). Jika dari uji hipotesis tersebut menerima model (X, Y, Z, XY, XZ, YZ) maka dapat disimpulkan bahwa ada hubungan yang signifikan antara X dan Y. 17
18 Model 1 : Deviance: on 4 degrees of freedom AIC : Model 2 : Deviance: on 8 degrees of freedom AIC : Apa kesimpulan dari uji deviance tersebut? 18
19 χ 2 (α, db) : qchisq(α, db, lower.tail=false) > qchisq(0.05, 4, lower.tail=false) [1] Jadi χ 2 (α = 0.05, db = 4) =
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26 1. Gunakan Program R untuk menyelesaikan Problem 7.1 (Azen, hlm.177). 26
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28 2. Gunakan Program R untuk menyelesaikan Problem 8.1 (Agresti, hlm. 347). 28
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30 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 2 nd. New York: Wiley. 3. Pustaka lain yang relevan. 30
31 Bisa di-download di kusmansadik.wordpress.com 31
32 Terima Kasih 32
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