Logit, probit and complementary log-log links
Section of Introduction to generalised linear models, part of Generalised linear models
Logit, probit and complementary log-log links
| Type | Section |
| Taught in | Introduction to generalised linear models |
| Part | Generalised linear models |
| Questions using it | 2 |
The logistic link is Equation (5) in Introduction to generalised linear models. It is the link we will interpret, and it is the canonical link.
The probit link replaces the logistic curve by a standard normal distribution function. If \Phi is the cumulative distribution function of \mathrm{N}(0,1), \Phi^{-1}(p_i)={\boldsymbol{X}}_i^\top\boldsymbol{\beta} \qquad\text{equivalently}\qquad p_i=\Phi({\boldsymbol{X}}_i^\top\boldsymbol{\beta}). Read it as a quantile. Compute {\boldsymbol{X}}_i^\top\boldsymbol{\beta}, find that point on the standard normal axis, and take the probability to its left. That left-tail probability is p_i. Any other continuous distribution function would give a link of the same type. The normal is the default.
The complementary log-log link is \log\bigl(-\log(1-p_i)\bigr)={\boldsymbol{X}}_i^\top\boldsymbol{\beta} \qquad\text{equivalently}\qquad p_i=1-\exp\bigl\{-\exp({\boldsymbol{X}}_i^\top\boldsymbol{\beta})\bigr\}. The exercise asks you to check that the right-hand side lies in (0,1) for every real linear predictor. This link is the one that does not treat the two tails symmetrically. Logit and probit both approach 0 and 1 in the same fashion, one a mirror of the other. Complementary log-log can stay near zero for longer and then rise more sharply towards one, or the reverse, depending on the sign of the slope. Use it when that asymmetry is the scientific point. Do not use it because you want a sentence about odds. The coefficient is a log-hazard, an instantaneous risk, and a hazard is much harder to explain to a medical collaborator than an odds ratio.
In Figure the logit and the probit differ in the tails and agree through the middle. In a fitted model they almost always agree, close enough that the two curves will not separate. A probit essentially never has to be the link reported to an applied scientist. A log-odds can be explained. A computer scientist or an engineer will sometimes prefer the probit, because a normal quantile is already their language. When there is a choice, and the person across the table is not one of those, use the logit. Goodness of fit will not be the reason to switch.
Used in
- Taught in Introduction to generalised linear models
- Also used in Paired comparisons, and how far a fit is from the data
- Questions that use it (2) — Q38 (answer) · Question 21 (answer)
- All 9 questions on Introduction to generalised linear models
See also
Exponential family · Mean and variance of an exponential family · Generalised linear model · Canonical link · What a regression statement is hiding · Three examples the linear model does not fit · Runs in the past week · Clotting-time data