Why doesn't adding additional explanatory variables in a logistic regression model decrease our primary explanatory variables variance? Imagine a clinical trial setting where we have binary outcome Y and we are interested in the effects of treatment X.  Lets say we also have additional explanatory covariates Z and W.  Thus our logistic regression model is:
$$
p_i = Logistic( \mu + X_i + W_i + Z_i ) 
$$
What's confusing me is that in simulations where you assume Z and W are completely independent of each other (and X) and are randomly distributed their inclusion into the model seems to have no meaningful impact on our variance estimate for the beta coefficient for X.  That is the model
$$
p_i = Logistic( \mu + X_i) 
$$
Produces the same point estimate and SE for said estimate.  My expectation was that including additional explanatory terms into the model would have reduced the SE as we would be more confident in the value for X as we have controlled for other sources of variability.  At least this is what happens in a standard linear model.
Simulation code for reference:
https://gist.github.com/gowerc/4191a8f7237f1e08c932f4a3ed4a1231
I am curious as to why this is happening ?
 A: Standard errors get smaller upon addition of new variables with explanatory power only when there is an error variance in the model.  The logistic model does not have an error variance.  So any lack of fit by wrongly omitting variables is reflected by unwanted changes in the regression coefficients of the remaining variables.  This is related to the non-collapsibility of the odds ratio.  As detailed here and papers references therein, this is not to be confused with power reduction.  Adding variables generally increases power for tests on the initial variables, because the model being more correct makes their regression coefficients increase in absolute value.  This increase more than offsets the increase in standard errors.
In linear models, omitting important variables is absorbed by an increase in the residual variance, and if the omitted variables are orthogonal to the included variables, the initial coefficients can be unbiased even when important variables are omitted.  Not so in non-linear non-loglinear models.
