# Working out/Derivation of Standard Error of Coefficient in Logistic Regression

Let's keep it simple and just go with Simple Logistic Regression where there is only 1 continuous independent variable.

I.e. Log(P/1-P)=B0 + B1X1

What is the working out to derive the standard error for the B1 coefficient? Or what does the formula look like? I would also like to see the solution for this without getting linear algebra and matrices involved.

I'm guessing it wouldn't look anything like the formula for the standard error for B1 coefficient in simple linear regression i.e. √(s^2 / Sum of (Xi - Xbar)^2) or would it? Thank you

In logistic regression, as with most nonlinear models, you need to rely on asymptotic normality to characterize the sampling distribution of the coefficient. The variance of this distribution is given by the Fisher information of the logit log-likelihood. More formally, $$\sqrt{n}(\hat{\beta}-\beta_0) \to N(0, \mathbf{A}^{-1})$$ where $$\mathbf{A}$$ is the negative Hessian of the log-likelihood, and given by $$\mathbf{A} = \lim_{n \to \infty} - \frac{1}{n} \operatorname{E} \left. \frac{\partial^2 \log L}{\partial \beta \partial \beta^\mathsf{T}} \right|_{\beta_0} \approx \frac{1}{n} \sum_{i=1}^n \Lambda_{i}(1-\Lambda_{i}) x_i x_i^\mathsf{T}$$ where $$\Lambda_{i} = \frac{1}{1 + e^{-x_i\beta}}$$ is the logistic function. For a more rigorous derivation and all the necessary assumptions behind this result, see Gourieroux & Monfort (1981).