I am studying some researches and found that authors have used probit function on some calculated/observed data instead of using inverse CDF and then applied CDF after probit. Also sometimes author applies inverse of probit on predicted/estimated CDF and then apply inverse CDF to get estimated realisations of random variables. I am unable to understand this process why should we use probit funciton.
Hi, logistic has slightly flatter tails. i.e probit curve approaches the axes more quickly than the curve. Logit has better interpretation than probit. Logistic regression can be interpreted as modeling log odds. Usually people start the modeling with logit. You could use likelihood value to decide logit or probit.
A logit link function is a softer way of approximating/implementing a binary step function (so that its derivatives exist, for gradient descent type algorithms for ML/MAP estimation and max/min type objective functions). it is useful when your dependent variable ('y') is categorical/binary or you are dealing with a binary classification problem.
The logit function is ideal if you have a variable x that can take values between 0 and 1 (a frequency) and the effect of this variable on another one, y, is going stronger and stronger when it is approaching either 0 or 1. For example, consider the frequency of a individual factor that protects against a infectious disease, and suppose it is 80% in a population. Then it raises to 90%. If you take just the ratio 0.9 / 0.8 the increase was small, but if you think in terms of "distance to 1", that distance was halved. Thus, na increase from 0.8 to 0.9 is importante, and from 0.9 to 0.95 also important and from 0.95 to 0.99 probably even more important, because the number of people *without* the factor was reduced to just 1%, which is 5 times lower than 5%. The logit function is ideal to capture that effect.
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