Log link
Topic of Underdispersion, positive responses, and quasi-likelihood, part of Generalised linear models · Also in Standard errors, and models for counts
Log link
| Type | Topic |
| Taught in | Underdispersion, positive responses, and quasi-likelihood · Standard errors, and models for counts |
| Part | Generalised linear models |
| Questions using it | 9 |
g(\mu)=\log\mu. It keeps the mean positive for any linear predictor and makes coefficients multiplicative: e^{\beta_j} is the factor by which the mean changes per unit of x_j. It is the canonical link for Poisson counts and the usual choice for gamma responses.
In Underdispersion, positive responses, and quasi-likelihood
A clotting time, or an insurance claim amount, is continuous and cannot be negative. Three links are in ordinary use: \begin{align*} \log\mu_i&={\boldsymbol{X}}_i^\top\boldsymbol{\beta}, & \mu_i&=\exp({\boldsymbol{X}}_i^\top\boldsymbol{\beta}),\\ \mu_i^{-1}&={\boldsymbol{X}}_i^\top\boldsymbol{\beta}, & \mu_i&=1/({\boldsymbol{X}}_i^\top\boldsymbol{\beta}),\\ \mu_i^{-2}&={\boldsymbol{X}}_i^\top\boldsymbol{\beta}, & \mu_i&=1/\sqrt{{\boldsymbol{X}}_i^\top\boldsymbol{\beta}}. \end{align*} The log link is the one with a direct reading, and it keeps \mu positive for every finite predictor. It is not the canonical link for the gamma or the inverse Gaussian, so R will not choose it unless you name it. The canonical link for the gamma is the inverse link. The canonical link for the inverse Gaussian is the inverse quadratic. Both of those are only positive when {\boldsymbol{X}}^\top\boldsymbol{\beta}>0. If the linear predictor changes sign, the inverse link produces a negative mean, and the square root in the inverse quadratic leaves the reals. It is sometimes said that the inverse quadratic “ensures” non-negativity because a complex mean is not a mean. That is a restriction on the domain of \boldsymbol{\beta}, not a guarantee for every coefficient vector. You still have to keep the linear predictor positive.
In Standard errors, and models for counts
Two covariates are enough: \log\mu=\beta_0+\beta_1 x_1+\beta_2 x_2. Increase x_1 by one unit and hold x_2 fixed. The mean is multiplied by e^{\beta_1}. The same sentence with the labels swapped is \beta_2. The intercept is the mean when every covariate is at its baseline, \exp(\beta_0), and it is meaningful only when that baseline is a real situation. Do not judge a slope by whether the coefficient looks small. A small coefficient may be a large effect written in a small unit, and the comparison that matters is with the standard error.
Used in
- Taught in Underdispersion, positive responses, and quasi-likelihood · Standard errors, and models for counts
- Also used in Introduction to generalised linear models · Standard errors, and models for counts · Smooth means · Marginal models, and two responses at once
- Questions that use it (9) — Q30 (answer) · Q31 (answer) · Q45 (answer) · Q54 (answer) · Q55 (answer) · Q56 · Q57 (answer) · Question 19 (answer) · C21
- All 6 questions on Underdispersion, positive responses, and quasi-likelihood
See also
Logit link · Probit link · Complementary log-log link · Identity link · Inverse link · Canonical link · Linear predictor · Gamma regression · Lognormal distribution · Poisson regression