Hello,
I have focused on
- normal distribution,
- linearity
- multicollinearity
on a study based on a logistic regression statistic.
However, it is stated that there is no need for normal distribution for logistic regression:
https://www.statisticssolutions.com/assumptions-of-logistic-regression/
https://statistics.laerd.com/spss-tutorials/binomial-logistic-regression-using-spss-statistics.php
What do you think?
Yes, that's correct.
"no need" is actually a strange statement. The logistic model is about binary data and is based on a binomial distribution. There are no choices to model the distribution of a binary variable - it's neccesarily binomial distributed (actually, a single binary variable that can take one of only two possible values codes as 0 and 1 has a Bernoulli distribution, and the sum of n such variables -all independent and having the same event probability- is binomial distributed).
EDIT: included Emmanuels correct comment that a sum of (n) Bernoullis is binomial ONLY when all the Bernoullis are stochastically independent and have the same event probability (p). In this case the variance of the sum equals its expectation (np). If the event probability in the Bernoullis itself has a beta-distribution, the sum will have a beta-binomial distribution, in which the variance can be larger than the expectation [a.k.a over-dispesion]). Thanks Emmanuel for pointing this out.
Just a small precision: the sum of n Bernoulli variables is a binomial if, and only if, they are independent and identically distributed (at least, conditional to the predictors). Which are two strong hypothesis.
If you are performing logistic regression on aggregated data (that is, on counts instead on the raw binary data), violation of any of these hypotheses, especially the first one (independance), can lead to [most often] overdispersion and the need for more generic models than logistic binomial regression. However, residuals are in that case asymptotically Gaussian, so similar checks than linear regression make sense.
If you are performing on raw binary data, it's more tricky to explain the failure, but basically the (usual) methods are not anymore maximum likelihood methods in case of failure of the independance assumption, hence *all* usual results do not apply anymore. But in that case, residuals are not at all Gaussian (really too far from infinity), so normality checks on them does not make sense.
Tests for these are, somehow, the pendant of the normal assumption checks.
Note that you may also encounter normal assumption in the logistic regression approach if you assume that your binary variable is a binarization of a continuous one, that is Y = 1 if X > 0, 0 if X
Mutlu Gulbay do you mean normality of dependent variable or the predictors?
The dependent variable can't be normal obviously. And the predictors don't need to be normal neither.
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