Consider model $logit(p)=a+bx$. I would like to get an 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. 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.