- Generalized linear model
statistics, the generalized linear model (GLM) is a flexible generalization of ordinary least squares regression. It relates the random distribution of the measured variable of the experiment (the "distribution function") to the systematic (non-random) portion of the experiment (the "linear predictor") through a function called the link function.
Generalized linear models were formulated by
John Nelderand Robert Wedderburn as a way of unifying various other statistical models, including linear regression, logistic regressionand Poisson regression, under one framework. [ cite book | last = McCullagh | first = Peter | coauthors = Nelder, John | title = Generalized Linear Models | publisher = London: Chapman and Hall | date = 1989 | isbn = 0-412-31760-5 Chapter 1. ] This allowed them to develop a general algorithm for maximum likelihoodestimation in all these models. It extends naturally to encompass many other models as well.
In a GLM, each outcome of the
dependent variables, Y, is assumed to be generated from a particular distribution functionin the exponential family, a large range of probability distributionsthat includes the normal, binomial and poisson distributions, among others. The mean, "μ", of the distribution depends on the independent variables, X, through:
where E(Y) is the
expected valueof Y; X"β" is the "linear predictor", a linear combination of unknown parameters, "β"; "g" is the link function.
In this framework, the variance is typically a function, V, of the mean:
It is convenient if V 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, maximum quasi-likelihood, or Bayesiantechniques.
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").
exponential familyof distributions are those probability distributions, parameterized by "θ" and "τ", whose density functions "f" (or probability mass function, for the case of a discrete distribution) 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.
"θ" is related to the mean of the distribution. If "a" is the identity function, then the distribution is said to be in
canonical form. If, in addition, "b" is the identity and "τ" is known, then "θ" is called the "canonical parameter" and is related to the mean through
Under this scenario, the variance of the distribution can be shown to be [ cite book | last = McCullagh | first = Peter | coauthors = Nelder, John | title = Generalized Linear Models | publisher = London: Chapman and Hall | date = 1989 | isbn = 0-412-31760-5 Chapter 2. ]
The linear predictor is the quantity which incorporates the information about the independent variables into the model. The symbol "η" (Greek "eta") is typically used to denote a linear predictor. It is related to the
expected valueof the data (thus, "predictor") through the link function.
"η" is expressed as linear combinations (thus, "linear") of unknown parameters "β". The coefficients of the linear combination are represented as the matrix of independent variables X. "η" can thus be expressed as
The elements of X are either measured by the experimenters or stipulated by them in the modeling design process.
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.
When using a distribution function with a canonical parameter "θ", a link function exists which allows for XTY to be a sufficient statistic for "β". This occurs when the link function equates "θ" and the linear predictor. Following is a table of canonical link functions and their inverses (sometimes referred to as the mean function, as done here) used for several distributions in the exponential family.
In the cases of the exponential and gamma distributions, the domain of the canonical 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 impossible negative mean. When maximizing the likelihood, precautions must be taken to avoid this. An alternative is to use a noncanonical link function.
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 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).
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, which is the canonical link if the variance is known. Unlike most other GLMs, there is a closed form solution for the maximum likelihood parameter estimates.
When the response data, "Y", are binary (taking on only values 0 and 1), the distribution function is generally chosen to be the
binomial distributionand the interpretation of "μ"i is then the probability, "p", of "Y"i taking on the value one.
There are several popular link functions for binomial functions; the most typical is the canonical
GLMs with this setup are
In addition, the inverse of any continuous
cumulative distribution 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 is a popular choice and yields the probit model. Its link is
The identity link is also sometimes used for binomial data, but a drawback of doing this is that 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 fixed at exactly one. When it is not, the resulting
quasi-likelihoodmodel often described as binomial with overdispersionor "quasibinomial".
Another example of generalized linear models includes
Poisson regressionwhich models count datausing the Poisson distribution. The link is typically the logarithm, the canonical link.
The variance function is proportional to the mean
where the dispersion parameter "τ" is typically fixed at exactly one. When it is not, the resulting
quasi-likelihoodmodel is often described as poisson with overdispersionor "quasipoisson".
Correlated or clustered data
The standard GLM assumes that the observations are
uncorrelated. Extensions have been developed to allow for correlationbetween observations, as occurs for example in longitudinal studiesand clustered designs:
Generalized estimating equations(GEEs) allow for the correlation between observations without the use of an explicit probability model for the origin of the correlations, so there is no explicit likelihood. They are suitable when the random effectsand their variances are not of inherent interest, as they allow for the correlation without explaining its origin. The focus is on estimating the average response over the population ("population-averaged" effects) rather than the regression parameters that would enable prediction of the effect of changing one or more components of X on a given individual. GEEs are usually used in conjunction with Huber-White standard errors.
Generalized linear mixed models (GLMMs) are an extension to GLMs that includes random effectsin the linear predictor, giving an explicit probability model that explains the origin of the correlations. The resulting "subject-specific" parameter estimates are suitable when the focus is on estimating the effect of changing one or more components of X on a given individual. GLMMs are a particular type of multilevel model( mixed model). In general, fitting GLMMs is more computationally complex and intensive than fitting GEEs.
* Hierarchical generalized linear models (HGLMs) are similar to GLMMs apart from two distinctions:
#The random effects can have any distribution in the
exponential family, whereas current GLMMs nearly always have normal random effects;
#They are not as computationally intensive, as instead of integrating out the random effects they are based on a modified form of likelihood known as the "hierarchical likelihood" or "h"-likelihood.The theoretical basis and accuracy of the methods used in HGLMs have been the subject of some debate in the statistical literature. As of 2008, the method is only available in one statistical software package, namely
Genstat. [cite book|author = Youngjo Lee|coauthors = John Nelder and Yudi Pawitan|title = Generalized Linear Models with Random Effects: Unified Analysis via H-likelihood|publisher = Chapman & Hall/CRC|year = 2006| url=http://www.crcpress.com/shopping_cart/products/product_detail.asp?sku=C6315]
Generalized additive models
Generalized additive models (GAMs) [cite book|author = Hastie, T. J. and Tibshirani, R. J.|title = Generalized Additive Models|publisher = Chapman & Hall/CRC|year = 1990|isbn=9780412343902 ] are another extension to GLMs in which the link function "η" is not restricted to be linear in the covariates X but is an additive function of the "xi"s::The smooth functions "fi" are estimated from the data. In general this requires a large number of data points and is computationally intensive.
The binomial case may be easily extended to allow for a
multinomial distributionas the response. There are two ways in which this is usually done:
If the response variable is an ordinal measurement, then one may fit a model function of the form:
: where .
for . See:
Ordered logit(also known as the proportional odds model).
If the response variable nominal measurement, or the data does not satisfy the assumptions of an ordered model, one may fit a model of the following form:
: where .
for . This is much less efficient then the ordered response model, as it needs to estimate more parameters. See:
The term "generalized linear model" and especially its abbreviation GLM can cause confusion with the
general linear model. John Nelderhas expressed regret about this in a conversation with Stephen Senn:
Senn: I must confess to having some confusionwhen I was a young statistician between general linearmodels and generalized linear models. Do you regretthe terminology?
Nelder: I think probably I do. I suspect we shouldhave found some more fancy name for it that wouldhave stuck and not been confused with the generallinear model, although general and generalized are notquite the same. I can see why it might have been betterto have thought of something else. [cite journal |last= Senn|first=Stephen |year=2003 |title=A conversation with John Nelder |journal=Statistical Science |volume=18 |issue=1 |pages=118–131 |doi=10.1214/ss/1056397489 |url=http://projecteuclid.org/euclid.ss/1056397489]
For a more detailed discussion of the more common types of generalised linear models, see:
* [http://www.eng.ox.ac.uk/samp Systems Analysis, Modelling and Prediction (SAMP), University of Oxford] [http://www.eng.ox.ac.uk/samp/glm_soft.html Open-source MATLAB code for GLM fitting.]
* [http://stats.ma.ic.ac.uk/j/jan01/public_html/ John Nelder FRS]
* [http://www.royalsoc.ac.uk/DServe/dserve.exe?dsqIni=Dserve.ini&dsqApp=Archive&dsqCmd=Show.tcl&dsqSearch=RefNo='EC/1981/28'&dsqDb=Catalog Royal Society citation for Nelder]
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