Model Linear Terampat (Generalized Linear Model / GLM) Dr. Kusman Sadik, M.Si Departemen Statistika IPB, 2017/2018

Transcription

1 Model Linear Terampat (Generalized Linear Model / GLM) Dr. Kusman Sadik, M.Si Departemen Statistika IPB, 2017/2018

2 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. 2

3 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. 3

4 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. 4

5 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 5

6 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. 6

7 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). 7

8 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. 8

9 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, 9

10 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. 10

11 11

12 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: kuartil, persentil, dsb. e. Gunakan data pada Tabel 1 (terlampir) untuk mengerjakan poin b, c, dan d tersebut. 12

13 Responden J.Kelamin T.Pendidikan T.Pendapatan Responden J.Kelamin T.Pendidikan T.Pendapatan

14 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. 14

15 Bisa di-download di kusmansadik.wordpress.com 15

16 Terima Kasih 16