In
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, 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
In statistics, a logistic model (or logit model) is a statistical model that models the logit, log-odds of an event as a linear function (calculus), linear combination of one or more independent variables. In regression analysis, logistic regres ...
and
Poisson regression
In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. Poisson regression assumes the response variable ''Y'' has a Poisson distribution, and assumes the lo ...
. 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
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of a given unknown quantity (the ''response variable'', a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
) as a
linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
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 differently-sized beaches. More specifically, the problem is that if the model is used to predict the new attendance with a temperature drop of 10 for a beach that regularly receives 50 beachgoers, it 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
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
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
In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
, depending on exactly how the problem is phrased) and a log-odds (or ''
logit
In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in Data transformation (statistics), data transformations.
Ma ...
'') link function.
Overview
In a generalized linear model (GLM), each outcome Y of the
dependent variable
A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical functio ...
s 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:
:
where E(Y , X) is the
expected value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of Y
conditional on X; 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 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
from among those which are considered exponential families of probability distributions,
: 2. A linear predictor
, and
: 3. A link function
such that
.
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
and
, whose density functions ''f'' (or
probability mass function, for the case of a
discrete distribution) can be expressed in the form
:
The ''dispersion parameter'',
, typically is known and is usually related to the variance of the distribution. The functions
,
,
,
, and
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
and
(denoted
and
in this case), this reduces to
:
is related to the mean of the distribution. If
is the identity function, then the distribution is said to be in
canonical form (or ''natural form''). Note that any distribution can be converted to canonical form by rewriting
as
and then applying the transformation
. It is always possible to convert
in terms of the new parametrization, even if
is not a
one-to-one function; see comments in the page on
exponential families.
If, in addition,
and
are the identity, then
is called the ''canonical parameter'' (or ''natural parameter'') and is related to the mean through
:
For scalar
and
, this reduces to
:
Under this scenario, the variance of the distribution can be shown to be
:
For scalar
and
, this reduces to
:
Linear predictor
The linear predictor is the quantity which incorporates the information about the independent variables into the model. The symbol ''η'' (
Greek "
eta") denotes a linear predictor. It is related to the
expected value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of the data 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
:
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 is informed by several considerations. There is always a well-defined ''canonical'' link function which is derived from the exponential of the response's
density function. However, in some cases it makes sense to try to match the
domain of the link function to the
range of the distribution function's mean, or use a non-canonical link function for algorithmic purposes, for example
Bayesian probit regression.
When using a distribution function with a canonical parameter
the canonical link function is the function that expresses
in terms of
i.e.
For the most common distributions, the mean
is one of the parameters in the standard form of the distribution's
density function, and then
is the function as defined above that maps the density function into its canonical form. When using the canonical link function,
which allows
to be a
sufficient statistic for
.
Following is a table of several exponential-family distributions in common use and the data they are typically used for, along with the canonical link functions and their inverses (sometimes referred to as the mean function, as done here).
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 positive, 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.
In the case of the Bernoulli, binomial, categorical and multinomial distributions, the support of the distributions is not the same type of data as the parameter being predicted. In all of these cases, the predicted parameter is one or more probabilities, i.e. real numbers in the range
logistic regression
In statistics, a logistic model (or logit model) is a statistical model that models the logit, log-odds of an event as a linear function (calculus), linear combination of one or more independent variables. In regression analysis, logistic regres ...
'' (or ''
multinomial logistic regression'' in the case that ''K''-way rather than binary values are being predicted).
For the Bernoulli and binomial distributions, the parameter is a single probability, indicating the likelihood of occurrence of a single event. The Bernoulli still satisfies the basic condition of the generalized linear model in that, even though a single outcome will always be either 0 or 1, the ''
expected value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
'' will nonetheless be a real-valued probability, i.e. the probability of occurrence of a "yes" (or 1) outcome. Similarly, in a binomial distribution, the expected value is ''Np'', i.e. the expected proportion of "yes" outcomes will be the probability to be predicted.
For categorical and multinomial distributions, the parameter to be predicted is a ''K''-vector of probabilities, with the further restriction that all probabilities must add up to 1. Each probability indicates the likelihood of occurrence of one of the ''K'' possible values. For the multinomial distribution, and for the vector form of the categorical distribution, the expected values of the elements of the vector can be related to the predicted probabilities similarly to the binomial and Bernoulli distributions.
Fitting
Maximum likelihood
The
maximum likelihood estimates can be found using an
iteratively reweighted least squares algorithm or a
Newton's method with updates of the form:
:
where
is the
observed information matrix (the negative of the
Hessian matrix) and
is the
score function; or a
Fisher's scoring method:
:
where
is the
Fisher information matrix. Note that if the canonical link function is used, then they are the same.
Bayesian methods
In general, the
posterior distribution cannot be found in
closed form and so must be approximated, usually using
Laplace approximations or some type of
Markov chain Monte Carlo method such as
Gibbs sampling.
Examples
General linear models
A possible point of confusion has to do with the distinction between generalized linear models and
general linear models, two broad statistical models. Co-originator
John Nelder has expressed regret over this terminology.
The general linear model may be viewed as a special case of the generalized linear model with identity link and responses normally distributed. 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
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...
(tending to work well with large samples).
Linear regression
A simple, very important example of a generalized linear model (also an example of a general linear model) is
linear regression. In linear regression, the use of the
least-squares estimator is justified by the
Gauss–Markov theorem, which does not assume that the distribution is normal.
From the perspective of generalized linear models, however, it is useful to suppose that 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. Under these assumptions, the least-squares estimator is obtained as the maximum-likelihood parameter estimate.
For the normal distribution, the generalized linear model has a
closed form expression for the maximum-likelihood estimates, which is convenient. Most other GLMs lack
closed form estimates.
Binary data
When the response data, ''Y'', are binary (taking on only values 0 and 1), the distribution function is generally chosen to be the
Bernoulli distribution and 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.
Logit link function
The most typical link function is the canonical
logit
In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in Data transformation (statistics), data transformations.
Ma ...
link:
:
GLMs with this setup are
logistic regression
In statistics, a logistic model (or logit model) is a statistical model that models the logit, log-odds of an event as a linear function (calculus), linear combination of one or more independent variables. In regression analysis, logistic regres ...
models (or ''logit models'').
Probit link function as popular choice of inverse cumulative distribution function
Alternatively, the inverse of any continuous
cumulative distribution function (CDF) can be used for the link since the CDF's range is
Complementary log-log (cloglog)
The complementary log-log function may also be used:
:
g(p) = \log(-\log(1-p)).
This link function is asymmetric and will often produce different results from the logit and probit link functions. The cloglog model corresponds to applications where we observe either zero events (e.g., defects) or one or more, where the number of events is assumed to follow the
Poisson distribution. The Poisson assumption means that
:
\Pr(0) = \exp(-\mu),
where ''μ'' is a positive number denoting the expected number of events. If ''p'' represents the proportion of observations with at least one event, its complement
:
1-p = \Pr(0) = \exp(-\mu),
and then
:
-\log(1-p) = \mu.
A linear model requires the response variable to take values over the entire real line. Since ''μ'' must be positive, we can enforce that by taking the logarithm, and letting log(''μ'') be a linear model. This produces the "cloglog" transformation
:
\log(-\log(1-p)) = \log(\mu).
Identity link
The identity link ''g(p) = p'' is also sometimes used for binomial data to yield a
linear probability model. However, the identity link can predict nonsense "probabilities" less than zero or greater than one. This can be avoided by using a transformation like cloglog, probit or logit (or any inverse cumulative distribution function). A primary merit of the identity link is that it can be estimated using linear math—and other standard link functions are approximately linear matching the identity link near ''p'' = 0.5.
Variance function
The
variance function for "" data is:
:
\operatorname(Y_i)= \tau\mu_i (1-\mu_i)\,\!
where the dispersion parameter ''τ'' is exactly 1 for the binomial distribution. Indeed, the standard binomial likelihood omits ''τ''. When it is present, the model is called "quasibinomial", and the modified likelihood is called a
quasi-likelihood, since it is not generally the likelihood corresponding to any real family of probability distributions. If ''τ'' exceeds 1, the model is said to exhibit
overdispersion.
Multinomial regression
The binomial case may be easily extended to allow for a
multinomial distribution as the response (also, a Generalized Linear Model for counts, with a constrained total). There are two ways in which this is usually done:
Ordered response
If the response variable is
ordinal, then one may fit a model function of the form:
:
g(\mu_m) = \eta_m = \beta_0 + X_1 \beta_1 + \cdots + X_p \beta_p + \gamma_2 + \cdots + \gamma_m = \eta_1 + \gamma_2 + \cdots + \gamma_m \text \mu_m = \operatorname(Y \leq m). \,
for ''m'' > 2. Different links ''g'' lead to
ordinal regression models like
proportional odds models or
ordered probit models.
Unordered response
If the response variable is a
nominal measurement, or the data do not satisfy the assumptions of an ordered model, one may fit a model of the following form:
:
g(\mu_m) = \eta_m = \beta_ + X_1 \beta_ + \cdots + X_p \beta_ \text \mu_m = \mathrm(Y = m \mid Y \in \ ). \,
for ''m'' > 2. Different links ''g'' lead to
multinomial logit or
multinomial probit models. These are more general than the ordered response models, and more parameters are estimated.
Count data
Another example of generalized linear models includes
Poisson regression
In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. Poisson regression assumes the response variable ''Y'' has a Poisson distribution, and assumes the lo ...
which models
count data using the
Poisson distribution. The link is typically the logarithm, the canonical link.
The variance function is proportional to the mean
:
\operatorname(Y_i) = \tau\mu_i,\,
where the dispersion parameter ''τ'' is typically fixed at exactly one. When it is not, the resulting
quasi-likelihood model is often described as Poisson with
overdispersion or ''quasi-Poisson''.
Extensions
Correlated or clustered data
The standard GLM assumes that the observations are
uncorrelated. Extensions have been developed to allow for
correlation between observations, as occurs for example in
longitudinal studies and 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 effects and 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 effects in 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 also referred to as
multilevel models and as
mixed model. In general, fitting GLMMs is more computationally complex and intensive than fitting GEEs.
Generalized additive models
Generalized additive models (GAMs) are another extension to GLMs in which the linear predictor ''η'' is not restricted to be linear in the covariates X but is the sum of
smoothing functions applied to the ''x
i''s:
:
\eta = \beta_0 + f_1(x_1) + f_2(x_2) + \cdots \,\!
The smoothing functions ''f
i'' are estimated from the data. In general this requires a large number of data points and is computationally intensive.
See also
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* (VGLM)
*
Generalized estimating equation
References
Citations
Bibliography
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Further reading
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External links
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{{DEFAULTSORT:Generalized Linear Model
Actuarial science
Regression models