Model Log-Linear
(Bagian 1)
Dr. Kusman Sadik, M.Si
Program Studi Magister (S2)
Using a log-linear modeling approach is advantageous to
conducting inferential tests of the associations in contingency tables because the models can handle more complicated
situations.
For example, the Breslow–Day statistic is limited to 2x2xK tables and estimates of common odds ratios cannot be obtained for tables larger than 2x2.
Conversely, a log-linear modeling approach is not restricted to two- or three-way tables so it can be used for testing
3
Log-linear models are used to model the cell counts in contingency tables.
The ultimate goal of fitting a log-linear model is to estimate parameters that describe the relationships between
categorical variables.
Specifically, for a set of categorical variables, log-linear models do not really distinguish between explanatory and response variables but rather treat all variables as response variables by modeling the cell counts for all combinations of the levels of the categorical variables included in the model.
In general, the
number of parameters
in a log-linear
model depends on the number of categories of the
variables of interest.
More specifically, in any log-linear model the effect of a
categorical variable with a total of C categories requires
(C – 1) unique parameters
.
For example, if variable X is gender (with two
categories), then C = 2 and only one predictor, thus one
parameter, is needed to model the effect of X.
5
When dummy coding is used, the last category of the variable is used as a reference category.
Therefore, the parameter associated with the last category is set to zero, and each of the remaining parameters of the model is interpreted relative to the last category.
For example, if male is the last category of the gender variable, then the one gender parameter in the log-linear model will be interpreted as the difference between females and males because the parameter reflects the odds for
7
Instead of representing the parameter associated with the ith variable (Xi) as βi, in log-linear models this parameter is
represented by the Greek letter lambda, λ, with the variable indicated in the superscript and the (dummy-coded) indicator of the variable in the subscript.
For example, if the variable X has a total of I categories (i = 1, 2, …, I), λix is the parameter associated with the ith
indicator (dummy variable) for X.
Similarly, if the variable Y has a total of J categories (j = 1, 2, …, J), then λjy is the parameter associated with the jth
For two categorical variables, the expected cell counts,
denoted by μij for the cell in the ith row and jth column, are the outcome values from a log-linear model.
In general, main effects in log-linear models are interpreted as odds.
The (exponentiated) parameter values associated with X, λix , can be interpreted as the odds of being in the ith row versus being in the last row of the table regardless of the value of the other variable, Y.
Likewise, the (exponentiated) parameter values associated with Y, λjy , can be interpreted as the odds of being in the jth column versus being in the last column of the table
11
i = 1, 2, ..., k : kategori terakhir (i = k) sebagai referensi j = 1, 2, ..., m : kategori terakhir (j = m) sebagai referensi
𝜆 𝑖𝑋 = log 𝑜𝑑𝑑𝑠 𝑖 = log 𝑃 𝑋=𝑖 𝑃 𝑋=𝑘 = log 𝑛𝑖./𝑛.. 𝑛𝑘./𝑛.. 𝜆 𝑗𝑌 = log 𝑜𝑑𝑑𝑠 𝑗 = log 𝑃 𝑌=𝑗 𝑃 𝑌=𝑚 = log 𝑛.𝑗/𝑛.. 𝑛.𝑚/𝑛..
13 Data : Azen, Table.7.2
𝜆 𝑖𝑋 = log 𝑜𝑑𝑑𝑠 𝑖 = log 𝑃 𝑋=𝑖 𝑃 𝑋=𝑘 = log 𝑛𝑖./𝑛.. 𝑛𝑘./𝑛.. 𝜆 1𝑋 = log 𝑃 𝑋=1 𝑃 𝑋=3 = log 𝑛1./𝑛.. 𝑛3./𝑛.. = log 450/1776 698/1776 = −0.43897 𝜆 2𝑋 = log 𝑃 𝑋=2 𝑃 𝑋=3 = log 𝑛2./𝑛.. 𝑛3./𝑛.. = log 628/1776 698/1776 = −0.10568 𝜆 3𝑋 = log 𝑃 𝑋=3 𝑃 𝑋=3 = log 𝑛3./𝑛.. 𝑛3./𝑛.. = log 1 = 0
𝜆 𝑗𝑌 = log 𝑜𝑑𝑑𝑠 𝑗 = log 𝑃 𝑌=𝑗 𝑃 𝑌=𝑚 = log 𝑛.𝑗/𝑛.. 𝑛.𝑚/𝑛.. 𝜆 1𝑌 = log 𝑃 𝑌=1 𝑃 𝑌=3 = log 𝑛.1/𝑛.. 𝑛.3/𝑛.. = log 647/1776 274/1776 = 0.859218 𝜆 𝑌2 = log 𝑃 𝑌=2 𝑃 𝑌=3 = log 𝑛.2/𝑛.. 𝑛.3/𝑛.. = log 855/1776 274/1776 = 1.137973 𝜆 𝑌3 = log 𝑃 𝑌=3 𝑃 𝑌=3 = log 𝑛.3/𝑛.. 𝑛.3/𝑛.. = log 1 = 0
15
Bagaimana menduga λ ?
Gunakan kategori terakhir untuk X dan Y, untuk data di atas (i=3 dan j=3), sehingga: log(μ33) = λ, karena 𝜆 3𝑋 = 𝜆 3𝑌 = 0
μ33 = (n3.) (n.3)/ (n..) = (274)(698)/(1776) = 107.6869
19
Program R : Wajib
** Model Log-Linear untuk Data Tabel 7.2 (Azen, hlm.140) **
** relevel --> Memilih Kategori Referensi **
** Model 1 : Tanpa Interaksi **
pol <- factor(rep(c("1Lib","2Mod","3Con"),3))
pre <- factor(rep(c("1Bus","2Cli","3Per"),rep(3,3))) count <- c(70, 195, 382, 324, 332, 199, 56, 101, 117) pol <- relevel(pol, ref="3Con")
pre <- relevel(pre, ref="3Per") data.frame(pol, pre, count)
model1 <- glm(count ~ pol + pre, family=poisson("link"=log)) summary(model1)
dugaan <- round(fitted(model1),2) data.frame(pol,pre, count, dugaan)
21
pol pre count 1 1Lib 1Bus 70 2 2Mod 1Bus 195 3 3Con 1Bus 382 4 1Lib 2Cli 324 5 2Mod 2Cli 332 6 3Con 2Cli 199 7 1Lib 3Per 56 8 2Mod 3Per 101 9 3Con 3Per 117
Coefficients:
Estimate Std. Error z value Pr(>|z|) (Intercept) 4.67923 0.06723 69.605 < 2e-16 *** pol1Lib -0.43897 0.06045 -7.261 3.84e-13 *** pol2Mod -0.10568 0.05500 -1.921 0.0547 . pre1Bus 0.85922 0.07208 11.921 < 2e-16 *** pre2Cli 1.13797 0.06942 16.392 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for poisson family taken to be 1)
Null deviance: 626.32 on 8 degrees of freedom Residual deviance: 247.70 on 4 degrees of freedom AIC: 320
23
pol pre count dugaan 1 1Lib 1Bus 70 163.94 2 2Mod 1Bus 195 228.78 3 3Con 1Bus 382 254.28 4 1Lib 2Cli 324 216.64 5 2Mod 2Cli 332 302.33 6 3Con 2Cli 199 336.03 7 1Lib 3Per 56 69.43 8 2Mod 3Per 101 96.89 9 3Con 3Per 117 107.69
When there is evidence for dependency between the row and column variables of a two-way table, the dependency is modeled using two-way interaction terms in the log-linear modeling framework.
However, fitting a log-linear model with a two-way interaction to a two-way contingency table is analogous to fitting the
saturated model.
27
** Model Log-Linear untuk Data Tabel 7.2 (Azen, hlm.140) **
** relevel --> Memilih Kategori Referensi **
** Model 2 : Ada Interaksi **
pol <- factor(rep(c("1Lib","2Mod","3Con"),3))
pre <- factor(rep(c("1Bus","2Cli","3Per"),rep(3,3))) count <- c(70, 195, 382, 324, 332, 199, 56, 101, 117) pol <- relevel(pol, ref="3Con")
pre <- relevel(pre, ref="3Per") data.frame(pol, pre, count)
Model1 <- glm(count ~ pol + pre + pol*pre,
family=poisson("link"=log)) summary(model1)
dugaan <- round(fitted(model1),2) data.frame(pol,pre, count, dugaan)
Coefficients:
Estimate Std. Error z value Pr(>|z|) (Intercept) 4.76217 0.09245 51.511 < 2e-16 *** pol1Lib -0.73682 0.16249 -4.534 5.77e-06 *** pol2Mod -0.14705 0.13582 -1.083 0.27895 pre1Bus 1.18325 0.10566 11.198 < 2e-16 *** pre2Cli 0.53113 0.11650 4.559 5.14e-06 *** pol1Lib:pre1Bus -0.96010 0.20810 -4.614 3.96e-06 *** pol2Mod:pre1Bus -0.52537 0.16185 -3.246 0.00117 ** pol1Lib:pre2Cli 1.22426 0.18578 6.590 4.41e-11 *** pol2Mod:pre2Cli 0.65888 0.16274 4.049 5.15e-05 *** ---Signif.codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for poisson family taken to be 1)
Null deviance: 6.2632e+02 on 8 degrees of freedom Residual deviance: 9.5701e-14 on 0 degrees of freedom
1. Gunakan Program R untuk menganalisis data yang terdapat pada Tabel 7.2 (Azen, hlm.140) :
a. Lakukan pemodelan log-linear dengan menjadikan Conservative dan Perot sebagai pembanding/referensi. Apa interpretasinya? b. Lakukan pemodelan log-linear dengan menjadikan Liberal dan
Bush sebagai pembanding/referensi. Apa interpretasinya?
c. Berdasarkan dua pendekatan tersebut (a dan b), tentukan penduga bagi ij, untuk i = 1, 2, 3 dan j = 1, 2, 3. Apakah hasilnya berbeda antara (a) dan (b) di atas?
d. Lakukan uji hipotesis untuk mengetahui ada tidaknya hubungan antara afiliasi politik dengan pilihan menggunakan model penuh
31
2. Gunakan Program R untuk melakukan analisis data pada Tabel 2 dibawah ini:
a. Tentukan model log-linear dan dugaan parameternya. Apa interpretasinya?
b. Berdasarkan model tersebut, tentukan penduga bagi ij,
untuk i = 1, 2, 3 dan j = 1, 2, 3, 4.
c. Lakukan uji hipotesis untuk mengetahui ada tidaknya hubungan antara afiliasi politik dengan umur
menggunakan model penuh (saturated model). Apa kesimpulan Anda?
33
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.
Bisa di-download di
35
