Inverse link
Topic of Underdispersion, positive responses, and quasi-likelihood, part of Generalised linear models
Inverse link
| Type | Topic |
| Taught in | Underdispersion, positive responses, and quasi-likelihood |
| Part | Generalised linear models |
| Questions using it | 5 |
g(\mu)=1/\mu, the canonical link for gamma responses, up to sign. It needs the linear predictor to stay positive, which the data do not guarantee, so it can fail where the log link does not.
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.
Used in
- Taught in Underdispersion, positive responses, and quasi-likelihood
- Also used in Introduction to generalised linear models · Logistic regression
- Questions that use it (5) — Q36 (answer) · Q38 (answer) · Q57 (answer) · Question 21 (answer) · C17
- All 6 questions on Underdispersion, positive responses, and quasi-likelihood
See also
Logit link · Probit link · Complementary log-log link · Identity link · Log link · Canonical link · Linear predictor · Gamma regression · Quasi-score and quasi-likelihood · Bids for a bankrupt firm