In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value.

Generalized linear models were formulated by John Nelder and Robert Wedderburn as a way of unifying various other statistical models, including linear regression, logistic regression and Poisson regression.^[1] They proposed an iteratively reweighted least squares method for maximum likelihood estimation (MLE) of the model parameters. MLE remains popular and is the default method on many statistical computing packages. Other approaches, including Bayesian regression and least squares fitting to variance stabilized responses, have been developed.

Intuition

Ordinary linear regression predicts the expected value of a given unknown quantity (the response variable, a random variable) as a linear combination of a set of observed values (predictors). This implies that a constant change in a predictor leads to a constant change in the response variable (i.e. a linear-response model). This is appropriate when the response variable can vary, to a good approximation, indefinitely in either direction, or more generally for any quantity that only varies by a relatively small amount compared to the variation in the predictive variables, e.g. human heights.

However, these assumptions are inappropriate for some types of response variables. For example, in cases where the response variable is expected to be always positive and varying over a wide range, constant input changes lead to geometrically (i.e. exponentially) varying, rather than constantly varying, output changes. As an example, suppose a linear prediction model learns from some data (perhaps primarily drawn from large beaches) that a 10 degree temperature decrease would lead to 1,000 fewer people visiting the beach. This model is unlikely to generalize well over different sized beaches. More specifically, the problem is that if you use the model to predict the new attendance with a temperature drop of 10 for a beach that regularly receives 50 beachgoers, you would predict an impossible attendance value of −950. Logically, a more realistic model would instead predict a constant rate of increased beach attendance (e.g. an increase of 10 degrees leads to a doubling in beach attendance, and a drop of 10 degrees leads to a halving in attendance). Such a model is termed an exponential-response model (or log-linear model, since the logarithm of the response is predicted to vary linearly).

Similarly, a model that predicts a probability of making a yes/no choice (a Bernoulli variable) is even less suitable as a linear-response model, since probabilities are bounded on both ends (they must be between 0 and 1). Imagine, for example, a model that predicts the likelihood of a given person going to the beach as a function of temperature. A reasonable model might predict, for example, that a change in 10 degrees makes a person two times more or less likely to go to the beach. But what does "twice as likely" mean in terms of a probability? It cannot literally mean to double the probability value (e.g. 50% becomes 100%, 75% becomes 150%, etc.). Rather, it is the odds that are doubling: from 2:1 odds, to 4:1 odds, to 8:1 odds, etc. Such a model is a log-odds or logistic model.

Generalized linear models cover all these situations by allowing for response variables that have arbitrary distributions (rather than simply normal distributions), and for an arbitrary function of the response variable (the link function) to vary linearly with the predictors (rather than assuming that the response itself must vary linearly). For example, the case above of predicted number of beach attendees would typically be modeled with a Poisson distribution and a log link, while the case of predicted probability of beach attendance would typically be modelled with a Bernoulli distribution (or binomial distribution, depending on exactly how the problem is phrased) and a log-odds (or logit) link function.

Overview

In a generalized linear model (GLM), each outcome Y of the dependent variables is assumed to be generated from a particular distribution in an exponential family, a large class of probability distributions that includes the normal, binomial, Poisson and gamma distributions, among others. The conditional mean μ of the distribution depends on the independent variables X through:

${\displaystyle \operatorname {E} (\mathbf {Y} \mid \mathbf {X} )={\boldsymbol {\mu }}=g^{-1}(\mathbf {X} {\boldsymbol {\beta }}),}$

where E(Y | X) is the expected value of Y conditional on X; Xβ 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:

${\displaystyle \operatorname {Var} (\mathbf {Y} \mid \mathbf {X} )=\operatorname {V} (g^{-1}(\mathbf {X} {\boldsymbol {\beta }})).}$

It is convenient if V follows from an exponential family of distributions, 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 Bayesian techniques.

Model components

The GLM consists of three elements:

1. A particular distribution for modeling ${\displaystyle Y}$ from among those which are considered exponential families of probability distributions,

2. A linear predictor ${\displaystyle \eta =X\beta }$, and

3. A link function ${\displaystyle g}$ such that ${\displaystyle \operatorname {E} (Y\mid X)=\mu =g^{-1}(\eta )}$.

Probability distribution

An overdispersed exponential family of distributions is a generalization of an exponential family and the exponential dispersion model of distributions and includes those families of probability distributions, parameterized by ${\displaystyle {\boldsymbol {\theta }}}$ and ${\displaystyle \tau }$, whose density functions f (or probability mass function, for the case of a discrete distribution) can be expressed in the form

${\displaystyle f_{Y}(\mathbf {y} \mid {\boldsymbol {\theta }},\tau )=h(\mathbf {y} ,\tau )\exp \left({\frac {\mathbf {b} ({\boldsymbol {\theta }})^{\rm {T}}\mathbf {T} (\mathbf {y} )-A({\boldsymbol {\theta }})}{d(\tau )}}\right).\,\!}$

The dispersion parameter, ${\displaystyle \tau }$, typically is known and is usually related to the variance of the distribution. The functions ${\displaystyle h(\mathbf {y} ,\tau )}$, ${\displaystyle \mathbf {b} ({\boldsymbol {\theta }})}$, ${\displaystyle \mathbf {T} (\mathbf {y} )}$, ${\displaystyle A({\boldsymbol {\theta }})}$, and ${\displaystyle d(\tau )}$ are known. Many common distributions are in this family, including the normal, exponential, gamma, Poisson, Bernoulli, and (for fixed number of trials) binomial, multinomial, and negative binomial.

For scalar ${\displaystyle \mathbf {y} }$ and ${\displaystyle {\boldsymbol {\theta }}}$ (denoted ${\displaystyle y}$ and ${\displaystyle \theta }$ in this case), this reduces to

${\displaystyle f_{Y}(y\mid \theta ,\tau )=h(y,\tau )\exp \left({\frac {b(\theta )T(y)-A(\theta )}{d(\tau )}}\right).\,\!}$

${\displaystyle {\boldsymbol {\theta }}}$ is related to the mean of the distribution. If ${\displaystyle \mathbf{b}(\mathit{\theta <!--\; \theta \; -->})}$Zdroj:https://en.wikipedia.org?pojem=Generalized_linear_model
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