Model Linear Terampat(Generalized Linear Model / GLM)

Dr. Kusman Sadik, M.Si

Program Studi Magister (S2)

Departemen Statistika IPB, 2017/2018

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Pada model linear klasik, seperti regresi linear,

memerlukan asumsi bahwa peubah respon y menyebar

Normal.

Pada kenyataanya banyak ditemukan bahwa peubah

respon y tidak menyebar Normal. Misalnya menyebar

Binomial, Poisson, Gamma, Eksponensial, dsb.

Maka dikembangkan Model Linear Terampat (GLM) untuk

mengatasi masalah ini.

Metode GLM bisa digunakan untuk memodelkan peubah

respon y yang mengikuti sebaran keluarga eksponensial

(exponential family).

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Normal

Binomial

Multinomial

Poisson

Gamma

Eksponensial

Negatif Binomial

Dsb.

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Function: The structure of the association between the

variables (e.g., linear or some other function).

Parameters: How a change in a predictor variable, X, is

expected to affect an outcome variable, Y.

Partial parameters: How a change in one of the predictor

variables affects the outcome variable while controlling for

the effects of other predictor variables included in the model.

Smooth prediction: What the expected (or predicted) value of

the outcome variable might be for any given values of the

predictor variables.

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The random component : refers to the distribution of the

outcome variable (Y);

The systematic component : refers to the predictor

variables (X);

The link function : refers to the way in which the outcome

variable (or, more specifically, its expected value) is

transformed so that a linear relationship can be used to

model the association between the predictors (X) and the

transformed outcome.

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The random component of a GLM is the probability

distribution that is assumed to underlie the dependent or

outcome variable.

When the outcome or response variable is continuous, such

as in simple linear regression or analysis of variance

(ANOVA), we typically assume that the normal distribution is

the random component.

When the dependent or outcome variable is categorical it can

no longer be assumed that its values in the population are

normally distributed.

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The systematic component of a GLM consists of the

independent, predictor, or explanatory variables (X)

that a researcher hypothesizes will predict (or explain)

differences in the dependent or outcome variables.

These variables are combined to form the linear

predictor, which is simply a linear combination of the

predictors

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The key to GLMs is to “link” the random and systematic

components of the model with some mathematical

function, call it g(.), such that this function of the

expected value of the outcome can be properly modeled

using the systematic component:

The link function is the mathematical function that is

used to transform the dependent or outcome variable so

that it can be modeled as a linear function of the

predictors.

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In this case, the predicted or expected outcome, E(Y),

does not need to be transformed to be linearly related to

the predictor.

More technically, if g(.) represents the link function, the

transformation of E(Y) by g in this case is g(E(Y)) = E(Y).

This is referred to as the identity link function because

applying the g(.) function of E(Y) in this case results in

the same value, E(Y).

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For example, suppose that the outcome variable was the

probability that a student will pass (as opposed to fail) a

specific test, so the predicted value is E(Y) = π.

Transformation:

This particular link function (or transformation) is called the

logit link function, and the resulting GLM is called the logistic

regression model.

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When the outcome variable is a count variable, and thus the

random component is assumed to follow a Poisson

distribution.

The outcome variable is a count so by definition it cannot be

lower than zero, but if a linear regression model was fit using

the untransformed outcome, nonsensical negative values

could theoretically result as predictions for low values of X.

On the other hand, when the predicted outcome, E(Y), is

transformed using the natural log function,

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This particular transformation is called the log link function

and this model is called the Poisson regression model.

The log function typically works well with outcome variables

that represent counts or a random component that follows a

Poisson distribution.

Another GLM that uses the log link function is the log-linear

model, in which the predictor variables are typically

categorical and the outcome variable, rather than

representing yet another, separate variable, is the count or

frequency obtained in each of the categories of the

predictors.

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1. Komponen Acak (Random Component)

Komponen acaknya adalah peubah respon y.

Pada regresi linear, peubah respon y

diasumsikan menyebar Normal dengan nilai

tengah dan ragam 2.

E(y) = 0 + 1x1 + … kxk = (ixi)

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2. Komponen Sistematik (Systematic Component)

Komponen sistematik dalam regresi linear

adalah kombinasi linear dari kovariat x1, x2, …,

xp. Sehingga dapat dituliskan sebagai berikut:

= (ixi)

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3. Fungsi Hubung (Link Function)

Fungsi hubung pada regresi linear adalah fungsi yang

menghubungkan antara komponen acak (y) dengan

komponen sistematik (x1, x2, …, xp). Misalkan E(y) = ,

selanjutnya dapat dibuat hubungan sebagai berikut :

g((E(Y)) = g() = = (ixi) = X

g(.) pada regresi linear adalah fungsi identitas, yaitu

g() = = E(y).

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Sebaran y Fungsi Hubung

Normal Identitas

Binomial Logit

Gamma Invers

Poisson Log

Multinomial Logit Kumulatif

Negatif Binomial Log

Inverse Gaussian Invers Kuadrat

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Pendugaan Parameter

Metode Fisher Scoring

L(,y) adalah fungsi kemungkinan (likelihood), I

disebut matrik informasi Fisher. Maka penduga

secara iteratif adalah sebagai berikut :

srr

r

yLE

yLU

),( ;

),( 2

I

)1()1()1()()1( ˆˆ kkkkkUβIβI

)1()1()1()( )(ˆˆ kkkkUIββ

-

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Kelayakan model (goodness of fit) pada GLM dapat diukur berdasarkan Deviance (D).

Deviance adalah dua kali perbedaan antara log likelihood nilai aktual dengan log likelihood nilaidugaan.

Nilai deviance dapat digunakan sebagai statistikuji mengenai kelayakan model.

Deviance merupakan peubah acak yang sebarannya mendekati sebaran 2.

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Sebaran asimptotik bagi deviance (D) adalah

2(n-p)

dimana n adalah banyaknya data, sedangkan p adalah banyaknya parameter dalam model.

)ˆ ;ˆ(2) ;(2 iiii yLyLD

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1

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2

23

3

24

4

25

5

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a. Install Program R versi terbaru.

b. Membaca input data dalam format : txt, excel, csv, dsb.

c. Deskripsi data melalui tabel dan grafik untuk data

kategorik: histogram, x-y plot, tabel frekuensi, tabel

kontingensi, dsb.

d. Deskripsi data secara numerik untuk data kategorik

e. Gunakan data pada Tabel 1 (terlampir) untuk mengerjakan

poin b, c, dan d tersebut.

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Responden J.Kelamin T.Pendidikan T.Pendapatan

1 1 1 4

2 1 3 6

3 1 2 4

4 1 2 5

5 1 4 4

6 1 4 1

7 1 3 3

8 0 4 3

9 1 1 5

10 1 2 5

11 1 2 2

12 1 3 5

13 0 4 5

14 1 4 4

15 1 3 3

16 1 3 4

17 0 4 4

18 1 1 6

19 1 2 3

20 0 4 3

21 1 1 6

22 1 1 2

23 1 3 3

24 1 1 5

25 1 2 3

Responden J.Kelamin T.Pendidikan T.Pendapatan

26 1 2 3

27 0 2 5

28 1 3 2

29 1 1 6

30 1 4 2

31 1 2 3

32 1 1 4

33 1 3 2

34 1 1 6

35 1 3 1

36 1 2 4

37 1 1 3

38 1 4 1

39 1 4 5

40 0 4 1

41 1 4 6

42 1 2 4

43 0 2 2

44 1 1 1

45 1 2 4

46 0 4 3

47 0 2 3

48 1 4 5

49 1 1 5

50 1 1 1

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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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