Logit link
Topic of Introduction to generalised linear models, part of Generalised linear models
Logit link
| Type | Topic |
| Taught in | Introduction to generalised linear models |
| Part | Generalised linear models |
| Questions using it | 20 |
g(p)=\log\{p/(1-p)\}, the log-odds. It maps (0,1) onto the whole real line, it is the canonical link for binomial responses, and it makes each coefficient a log odds ratio.
In Introduction to generalised linear models
The mean of a binary response is the success probability, p=\mathrm{E}(Y\mid {\boldsymbol{X}})=\mathrm{P}(Y=1\mid {\boldsymbol{X}}). A straight line cannot represent it. The line will eventually leave (0,1), and a probability cannot. The exponential \exp({\boldsymbol{X}}^\top\boldsymbol{\beta}) stays positive, but it is not bounded by 1. Dividing by one plus itself fixes that: p=\frac{\exp({\boldsymbol{X}}^\top\boldsymbol{\beta})}{1+\exp({\boldsymbol{X}}^\top\boldsymbol{\beta})} \qquad\text{equivalently}\qquad {\boldsymbol{X}}^\top\boldsymbol{\beta}=\log\frac{p}{1-p}. The right-hand equation is the logistic link, also called the logit: the log of the odds p/(1-p). The left-hand equation is the inverse link, sometimes called the expit. Log-odds run over the whole real line, which is what a linear predictor needs. The probability stays in (0,1).
Used in
- Taught in Introduction to generalised linear models
- Also used in Linear regression · Standard errors, and models for counts · Underdispersion, positive responses, and quasi-likelihood · Categorical responses · Paired comparisons, and how far a fit is from the data · Smooth means · Additive smooths · Statistical classifiers and discriminant analysis · Logistic regression · Ensembles, random forests, and boosting · Neural units · Backpropagation · Networks in practice · Random slopes, and a binary mixed model · Three species on one hillside
- Questions that use it (20) — A logit-normal prior for a coin (answer) · Odds ratios (answer) · Multinomial probabilities · A logistic rule in an unfamiliar setting (answer) · The exponential-loss minimiser (answer) · Softmax gradient (answer) · Q35 (answer) · Q36 (answer) · Q37 · Q38 · Q39 (answer) · Q40 (answer) · Q41 (answer) · Q42 (answer) · Q46 (answer) · Question 20 (answer) · Question 21 · C20 · C24 · C37 (answer)
- All 9 questions on Introduction to generalised linear models
See also
Binary logistic regression · Interpreting a logistic regression · Odds ratio · Probit link · Complementary log-log link · Identity link · Log link · Inverse link · Canonical link · Baseline-category logit model · Link function · Natural parameter · Odds