Link functions, Advanced Statistics

Link functions:

The link function relates the linear predictor ηi to the expected value of the data. In classical linear models the mean and the linear predictor are identical. However, when dealing with counts and the distribution is Poisson, we must have the Poisson distribution parameter satisfy µi > 0 so that the identity link is less attractive, partly because ηi may be negative while µi > 0 must not be. It is advisable to utilize the log link η = log(µ) in this case. Similarly, when dealing with binomial distribution, the parameter p of probability of success in a single trial is restricted to be in (0,1) and the link function serves the purpose to map the interval (0,1) on to R1 . Therefore, links like the following (here µ is replaced by p):

614_Link functions6.png

have been suggested and widely used.

Let us illustrate the most commonly met examples of GLM together with describing the types of response variable, distribution, and the link function:

- Traditional Linear Model:

i) response variable: continuous

ii) distribution: normal

iii) link function: identity : η = µ

- Logistic Regression:

i) response variable: probability ( µ)

ii) distribution: binomial

iii) link function: logit: η = log( µ/1-µ)

- Poisson Regression in Log Linear Model:

i) response variable: count

ii) distribution: Poisson
iii) link function: η = log(µ)
- Gamma model with Log Link:
i) response variable: a positive continuous variable
ii) distribution: Gamma
iii) link function: η = log(µ)

Intermezzo and history. If you read di?erent references, you may get confused about the terminology. You may have already come across the term "general linear model" in your introductory Statistics courses or in some reference books. Note, however, that this term refers to a conventional linear regression model for a continuous response variables given continuous and/or categorical predictors. It includes multiple linear regression, as well as ANOVA and ANCOVA. In SAS, such models are ?t by least squares and weighted least squares using (typically) proc glm. HOWEVER, the "generalized linear model" we are speaking about here, refers to the larger class discussed in this section. The ?rst widely used software package for ?tting these models was called GLIM. Because of this program, "GLIM" became a well-accepted abbreviation for generalized linear models, as opposed to "GLM". Since we clari?ed the confusion though, we will continue using "GLM" for generalized linear models since many recent references use it. Today, generalized linear models are ?t by many packages, notably by the SAS proc genmod. (End of intermezzo).

One of the advantages of the full probabilistic speci?cation of the GLM model is that ML Estimation suggests itself as a natural general estimation method. We have to maximize the log-likelihood

2487_Link functions2.png

where β is linked to θ through the link function. Recall that the main parameter- vector of interest is β, the vector of regression coeffcients in the relation ηi = g(µi) = x0

1971_Link functions3.png

There is nowadays, with the availability of modern computing power, seldom any reason to consider estimators of β that are di?erent from the MLE. By using the chain rule, we get for the components of the score function:

1533_Link functions4.png

The (expected) Fisher information matrix is given then by

687_Link functions5.png

The ML Estimator is de?ned by equating the score function to zero. Numerically, the equation is solved by applying iterative procedures which we discuss next.

Posted Date: 2/27/2013 1:11:48 AM | Location : United States







Related Discussions:- Link functions, Assignment Help, Ask Question on Link functions, Get Answer, Expert's Help, Link functions Discussions

Write discussion on Link functions
Your posts are moderated
Related Questions
It is the art of attempting to exchange something quite small and certain, for something which are large and uncertain. Gambling is big business; in the US, for instance, it is at

The particular projection which an investigator believes is most likely to give an accurate prediction of the future value of some process. Commonly used in the context of the anal

Inliers is the term used for the observations most likely to be subject to error in situations where the dichotomy is developed by making a ‘cut’ on an ordered scale, and where th


Johnson-Neyman technique:  The technique which can be used in the situations where analysis of the covariance is not valid because of the heterogeneity of slopes. With this method

Protopathic bias is the type of bias (also called as reverse-causality) that is a consequence of differential misclassification of the exposure related to timing of occurrence. It

The statistical methods for estimation and inference which are based on a function of sample observations, probability distribution of which does not rely upon a complete speci?cat

Different approaches to the study of early indian history

Recursive models are the statistical models in which the causality flows in one direction, that is models which include only unidirectional effects. Such type of models do not inc

Regression diagnostics is the process designed to investigate the suppositions underlying particular forms of regression examination, for instance, homogeneity of variance, norma