Q38. Changing from a logit to a probit link
[3 marks]
Another researcher fits:
gam(
disease ~ s(cholesterol),
family = binomial(link = "probit")
)Write down explicitly the model statement, defining any additional mathematical function you use.
Identify precisely which component changes relative to the corresponding model with a logit link.
Definitions. Let Y_i indicate disease, c_i denote fixed cholesterol, and p_i=\mathbb P(Y_i=1\mid c_i). For Z\sim N(0,1), \Phi(z)=\mathbb P(Z\le z); the probit link is \Phi^{-1}(p). Tildes on the intercept and smooth distinguish a refitted probit model from its logit counterpart. The Bernoulli responses are independent conditional on their covariates.
The model is Y_i\mid c_i\overset{\mathrm{ind}}{\sim}\operatorname{Bernoulli}(p_i), \boxed{\Phi^{-1}(p_i)=\widetilde\beta_0+\widetilde s(c_i).} Equivalently, \boxed{p_i=\Phi\{\widetilde\beta_0+\widetilde s(c_i)\}.} Use a centring constraint such as \sum_i\widetilde s(c_i)=0 to separate the smooth from the intercept.
The comparison is with the corresponding intercept-plus-smooth logit model:
| Component | Logit GAM | Probit GAM |
|---|---|---|
| Response distribution | Bernoulli | Bernoulli |
| Additive structure | Intercept plus smooth | Intercept plus smooth |
| Link | \log\{p/(1-p)\} | \Phi^{-1}(p) |
| Inverse link | e^\eta/(1+e^\eta) | \Phi(\eta) |
The component that changes is the link function, and therefore the inverse link converting the predictor into a probability. The tildes emphasise that the intercept and smooth are fitted anew: their numerical values need not equal those obtained using the logit link.
Key result. A probit GAM models \Phi^{-1}(p), not log odds. Changing the link does not change the Bernoulli response distribution.