If coefficient variance is incorrect (for a regression parameter), does that mean the model's log-likelihood is incorrect? I am using logistic regression to estimate ~probability of a sample unit being used by an animal.
Due to my sampling design it is unavoidable that there is overlap between 'used' sample units and 'available' sample units. (Overlap refers to the situation where 'used' units also occur in the sample of 'available' units. That is, sample units with identical characteristics are coded as both '0' and '1' in the data. So the situation is not analogous to a case-control application of logistic regression.)
According to Johnson et al 2006 "Resource Selection Functions Based on Use–Availability
Data: Theoretical Motivation and Evaluation Methods";
"When overlap occurs, the variance estimates reported by the
logistic regression procedure for coefficients are not correct, even
though coefficients estimates are. If no overlap occurs, variance
estimates are correct. Coefficient variances in the case of overlap
should be estimated by bootstrap methods that resample distinct
units."
My question is, do incorrect coefficient variances imply incorrect log-likelihoods?  I ask because I want to compare about 10 models using AIC, but it seems I'd have to bootstrap the variances for each model's coefficients separately and then calculate the likelihood, and then use my model selection procedure.
Thank you for any thoughts
 A: Assume that you have $m$ observations for which there is no overlap, and $m_o$ observations that each appear twice in the sample. Your total sample size is therefore $N = m+2m_o$.
If observation $y_k$ is overlapped, then in the sample there exists another observation $\tilde y_k = 1-y_k$, but for which all explanatory variables are identical, both as a set, and as numerical realizations, with those associated with $y_k$. We assume that some $y_k$ have the value $1$ while others the values $0$ (so in the subset with the overlaps, both ones and zeros appear).  
If I understand correctly, overlap is akin to sample contamination of some sort. If this is the case, then the correctly specified model and likelihood should be one that includes only the $m+m_o$ observations, i.e. discarding the subset including the $\tilde y_k$'s.
Therefore the correct log-likelihood is (written separately for later purposes)
$$\ln L=\sum_{i=1}^m\left[y_i \ln p_i+(1-y_i)\ln (1-p_i)\right] + \sum_{k=m+1}^{m+m_o}\left[y_k \ln p_k+(1-y_k)\ln (1-p_k)\right]$$
where $p_i$ and $p_k$ are logistic functions of the explanatory variables and the unknown coefficients in the usual way.
The first order conditions that the maximum likelihood estimator should satisfy are (taking the gradient with respect to the unknown coefficients)
$$\sum_{i=1}^m\left[y_i - p_i\right]\mathbf x_i + \sum_{k=m+1}^{m+m_o}\left[y_k - p_k\right]\mathbf x_k=0 \tag{1}$$
Suppose now that we ignore the issue of overlap and we go on and specify a  log-likelihood over all $m+2m_o$ observations
$$\ln L_o=\sum_{i=1}^{m+2m_o}\left[y_i \ln p_i+(1-y_i)\ln (1-p_i)\right]$$
which will give us the first-order conditions that the estimator must satisfy
$$\sum_{i=1}^{m+2m_o}\left[y_i - p_i\right]\mathbf x_i =0$$
which we can decompose due to overlap (that exists irrespective of whether we dealt with it or not), into
$$\sum_{i=1}^{m}\left[y_i - p_i\right]\mathbf x_i +\sum_{k=m+1}^{m+m_o}\left[y_k - p_k\right]\mathbf x_k +\sum_{k=m+1}^{m+m_o}\left[\tilde y_k - p_k\right]\mathbf x_k =0$$
$$\Rightarrow \sum_{i=1}^{m}\left[y_i - p_i\right]\mathbf x_i +\sum_{k=m+1}^{m+m_o}\left[y_k - p_k\right]\mathbf x_k+\sum_{k=m+1}^{m+m_o}\left[1- y_k - p_k\right]\mathbf x_k =0$$
$$\Rightarrow \sum_{i=1}^{m}\left[y_i - p_i\right]\mathbf x_i +\sum_{k=m+1}^{m+m_o}\left[y_k - p_k +1 - y_k-p_k\right]\mathbf x_k =0$$
$$\Rightarrow \sum_{i=1}^{m}\left[y_i - p_i\right]\mathbf x_i +\sum_{k=m+1}^{m+m_o}\left[1-2p_k \right]\mathbf x_k =0 \tag{2}$$
Compare $(1)$ and $(2)$. I don't see how the coefficient estimates that will satisfy $(2)$, will be the same as those that satisfy $(1)$. So I cannot understand the claim that "coefficient estimates are correct even in the presence of overlap", let alone the variance issue.
