Generalized linear model

The generalized linear model (GLM) is a statistical, linear model that generalizes the General linear model in the following ways:

• Error distributions from the Exponential family, besides the normal distribution are permitted.
• The variance may depend on a known function of the mean. (For example, for the binomial distribution, ${\displaystyle \mu =np\,}$ and ${\displaystyle \sigma ^{2}=npq\,}$, and thus ${\displaystyle \sigma ^{2}=\mu (n-\mu )/n\,}$.)
• A non-linear relationship between ${\displaystyle E(Y)\,}$ and ${\displaystyle X\beta \,}$ is allowed, with the aid of a link function.

The GLM may be written as

${\displaystyle g(Y)=X\beta +\epsilon ,\,}$

where g is a monotone, twice-differentiable function, called the link function and Y comes from a multivariate normal distribution with mean E(Y) and variance ${\displaystyle \epsilon }$. It is often assumed that the distribution of y is a member of an exponential family. Each specific choice of the link function and the distribution for the dependent variable yields a different generalized linear model. As in the notation of other regression models such as the General linear model, X is the design matrix, and B is a matrix containing parameters that must be estimated. The residual, U is usually assumed to follow a multivariate normal distribution.

Generalized linear models include, as special cases, ordinary linear regression, logistic regression, Poisson regression, and several other interesting models.

References

• P. McCullagh and J.A. Nelder. Generalized Linear Models. London: Chapman and Hall, 1989.

External links

• MSc Generalised Linear Models
• Module 9: Generalized Linear Models