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→Intuition: Fixed the order of the log odds ratios Tags: Reverted Mobile edit Mobile web edit |
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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 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
Generalized linear models cover all these situations by allowing for response variables that have arbitrary distributions (rather than simply [[normal distribution]]s), 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.
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