Generalized linear model: Difference between revisions

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{{Short description|Class of statistical models}}

{{Distinguish|generalGeneral linear model|generalizedGeneralized least squares}}

{{Regression bar}}

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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 differently-sized beaches. More specifically, the problem is that if you use the model is used to predict the new attendance with a temperature drop of 10 for a beach that regularly receives 50 beachgoers, youit 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 distribution|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 ratio|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 regression|logistic]] model''.

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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).

{| class="wikitable" style="background:white;"

|+ Common distributions with typical uses and canonical link functions

! Distribution !! Support of distribution !! Typical uses !! Link name !! Link function, <math>\mathbf{X}\boldsymbol{\beta}=g(\mu)\,\!</math> !! Mean function