Clarifications about probit and logit models

Question

I know that there is a very good explanation of the technical differences of probit and logit model in this question. However, I would appreciate some common sense clarifications which can be very helpful when deciding which model to use.

So let's consider this model: $$ Prob(y=1|x) = G(\beta_0 + x\beta) $$

where $G(z)$ is the respective link function.

1. Do I get it right that when the inner linear model yields z=0 then both the probit and logit predict 50% probability that y=1 and 50% that y=0?

2. I know that there isn't any general rule which model to choose, however one of the differences which should be considered are the slightly different shapes of the curves (the tails). So again, am I right if I say that the logit model "spreads" probability across wider range of values of z? That is, the probit will predict lower probabilities for negative values and higher probabilities for positive values of z than logit?

Answer 1

Let me start with a couple of persnickety details: We usually refer to the link function as being applied to the LHS, and the inverse of the link function being applied to the RHS. Thus, it would be better to write: $Prob(y=1|x)=G^{-1}(β0+xβ)$. Second, if the probability that y=1 is 50%, then the probability y=0 must also be 50%, so it's best to leave that out.

1. Yes, in both the case of the logit and probit link functions, when the linear predictor, z sums to 0, the predicted probability that y=1 is $0$. However, this is a little bit tricky. People usually talk about what happens when x=0, in which case you have the predicted probability of y=1 being $g^{-1}(\beta_0)$, which is only $50\%$ if $\hat\beta_0=0$.

2. It isn't quite right that the probabilities are spread over a wider range with the logit than the probit. They both range $(0,\ 1)$. Instead, they differ in the rate of change in predicted probabilities as they approach the bounds and 'turn the corner'. I think the main issue that may be causing you difficulty is that the fitted values of $\hat\beta_1$ will differ depending on whether you use the logit or the probit. The slope with the logit link will be larger than the slope with the probit link. Thus, what looks like a large difference in your plot will mostly disappear.