廣義線性模式

where Xβ is the linear predictor, a linear combination (depending on X, the known "independent" variables of the experiment) of unknown parameters β, and g is called the link function.

In this framework, the variance is typically a function V of the mean:

It is convenient if the variance follows from the exponential family distribution, but it may simply be that the variance is a function of the predicted value.

The unknown parameters β are typically estimated with maximum likelihood, quasi-maximum likelihood, or Bayesian techniques.

[编辑] Components of the model

The GLM consists of three elements.

1. A distribution function f, from the exponential family.

2. A linear predictor η = X β.

3. A link function g such that E(y) = μ = g^-1(η).

[编辑] Exponential family

Generally speaking, the exponential family of distributions are those probability distributions, parameterized by θ and τ, whose density functions or probability mass functions (depending on whether the distribution is continuous or discrete) can be expressed in the form

τ, called the dispersion parameter, typically is known and is usually related to the variance of the distribution. The functions a, b, c, d, and h are known. Many, although not all, common distributions are in this family.

If a is the identity function, then the distribution is said to be in canonical form. If in addition b is the identity, then θ is called the canonical parameter. θ is related to the mean of the distribution.

[编辑] Linear predictor

The linear predictor is a quantity relating to the expected value of the data (thus, "predictor") through the link function. The symbol η ("eta") is typically used to denote a linear predictor.

η is expressed as linear combinations (thus, "linear") of unknown parameters β. The coefficients of the linear combination are represented as the matrix X; its elements are either fully known by the experimenters or stipulated by them in the modeling process.

Thus η can be expressed as

[编辑] Link function

The link function provides the relationship between the linear predictor and the mean of the distribution function. There are many commonly used link functions, and their choice can be somewhat arbitrary. It can be convenient to match the domain of the link function to the range of the distribution function's mean.

Following is a table of some common link functions and their inverses (sometimes referred to as the mean function, as done here) used for several distributions in the exponential family.

Link Functions

Distribution	Name	Link Function	Mean Function
Normal	Identity
Exponential	Inverse
Gamma
Poisson	Log
Binomial	Logit
Multinomial

In the cases of the exponential and gamma distributions, the domain of the link function is not the same as the permitted range of the mean. In particular, the linear predictor may be negative, which would give an illegal negative mean.

[编辑] Examples

[编辑] General linear models

A possible point of confusion has to do with the distinction between generalized linear models and the general linear model, two broad statistical models. The general linear model may be viewed as a case of the generalized linear model with identity link. As most exact results of interest are obtained only for the general linear model, the development of the general linear model has undergone a somewhat longer historical development. Results for the generalized linear model with non-identity link are asymptotic (tending to work well with large samples).

[编辑] Linear regression

A simple, very important example of a generalized linear model (also an example of a general linear model) is linear regression. Here the distribution function is the normal distribution with constant variance and the link function is the identity.

[编辑] Binomial data

When the response data (Y) are binary (taking on only values 0 and 1), the distribution function is generally chosen to be the binomial distribution and the interpretation of μ_i is then the probability of Y_i taking on the value one. There are several popular link functions for binomial functions; the most typical is the logit:

GLMs with this setup are logistic regression models.

In addition, any inverse cumulative density function (CDF) can be used for the link since the CDF's range is [0, 1], the range of the binomial mean. The normal CDF Φ is a popular choice and yields the probit model. Its link is

The identity link is also sometimes used for binomial data (this is equivalent to using the uniform distribution CDF in the above instead of the Gaussian CDF) but this can be problematic as the predicted probabilities can be greater than one or less than zero. In implementation it is possible to fix the nonsensical probabilities outside of [0,1] but interpreting the coefficients can be difficult in this model. The model's primary merit is that near p = 0.5 it is approximately a linear transformation of the probit and logit — econometricians sometimes call this the Harvard model.

The variance function for binomial data is given by:

where the dispersion parameter τ is typically exactly one. When it is not, the model is often described as binomial with overdispersion or quasibinomial.

[编辑] Count data

Another example of generalized linear models includes Poisson regression which models count data using the Poisson distribution. The link is typically the logarithm.

The variance function is proportional to the mean

where the dispersion parameter τ is typically exactly one. When it is not, the model is often described as poisson with overdispersion or quasipoisson.