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.

enter image description here

  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?


1 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.

  • $\begingroup$ Thank you for a nice answer I think I get it now, but could you maybe try to give some example of an implication of the differences in the marginal probabilities. I mean is this difference of big importance for practical matters? In other words, what would you answer is you said: "they differ in the rate of change in predicted probabilities" and someone asked "So what?" :) $\endgroup$
    – m3div0
    Dec 21, 2014 at 17:16
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    $\begingroup$ @m3d1v0, for the logit vs probit links (but not necessarily other links), the difference is miniscule. It is hard to see that the difference will be of big or practical importance. There is more in my answer at the linked thread, but I really wouldn't fuss over l vs p. If someone asked "so what", I would probably just acknowledge that they have a point. $\endgroup$ Dec 21, 2014 at 17:23
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    $\begingroup$ @gung: this is a major point. The estimates of the $\beta_i$'s will vary between models because of the difference in the shape of the two link functions. $\endgroup$
    – Xi'an
    Dec 21, 2014 at 20:29
  • $\begingroup$ I think that the reason that probit is preferred in some economics context is that it falls out of certain theoretical models -- or so I've heard. Logit has the advantage in that it can be interpreted in terms of odds ratios. (Google this if you haven't heard of it). These have the advantage of being constant with respect to other factors. If you're really worried about the link function, there are methods for nonparametically estimating it -- but they are difficult, and not canned (at least I'm not familiar with any). $\endgroup$ Dec 21, 2014 at 22:18

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