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The link function is the cumulative probability function that the error terms follow. For instance, if it is logistic (and we will use that the logistic distribution is symmetric in the fourth equality), P(yi=1)=P(y2217i>0)=P(x2032i03b2+03f5i>0)=P(03f5i>2212x2032i03b2)=P(03f5i<x2032i03b2)=039b(x2032i03b2).

Canonical Link Poisson: log link g(µ) = log(Aµ). 2022 Gamma: negative inverse link g(Aµ) = 22121/Aµ. 2022 Negative binomial: g(Aµ) = logAµ/k(1 + Aµ/k).

Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters. A Poisson regression model is sometimes known as a log-linear model, especially when used to model contingency tables.

A link function in a Generalized Linear Model maps a non-linear relationship to a linear one, which means you can fit a linear model to the data. More specifically, it connects the predictors in a model with the expected value of the response (dependent) variable in a linear way.

The log link function is very commonly used for parameters that are positive. Here, all logarithms are natural logarithms, i.e., to base \\(e\\). Numerical values of theta close to 0 or out of range result in Inf , -Inf , NA or NaN .

A link function transforms the probabilities of the levels of a categorical response variable to a continuous scale that is unbounded. Once the transformation is complete, the relationship between the predictors and the response can be modeled with 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.