Consider model logit(p)=a+bx. I would like to get a analytic formula of a and b like in linear regression. In linear regression, we can get a formula of estimates of a and b.

I tried using MLE to estimate it. But it is too complicated for me.

I am doing research about the statistical genetics. Here p is penetrance and x is genotype. In linear regression, we can use the coefficient b to represent the covariance matrix of x and trait y (both of them should be standardized). I am thinking whether we can use the b in logistic model to recover the information about the covariance matrix of x and y (binary trait) in meta-analysis (they do not provide individual level data). But some cohort will prove the odds ratio and coefficient t-statistic. I do not know how to use it to recover the covariance matrix of x and y.

But some summary statistics will be provided like estimated odds ratio, Coefficient t-statistic.