# Proof that the one-step estimator from logistic regression is the Mantel Haenszel estimator?

I was rereading some sections from the Breslow and Day IARC publication and noted the following:

Clayton (1982) has shown that [the Mantel Haenszel] estimate arises at the first stage of iteration of one of the computational methods used to find the maximum likelihood estimate.

I cannot find this citation and am curious how one shows this to be the case.

As far as I know, Mantel Haenszel estimation requires that the statistician designate a primary exposure and a (set of) stratifying variable(s) and these cannot be exchanged whereas logistic regression arrives at the same estimates regardless of covariate order in the model matrix. Is that not necessarily the case during the intermediate iterations of Newton Raphson?

The citation is D.G. Clayton (1982) The analysis of prospective studies of disease aetiology, Communications in Statistics - Theory and Methods, 11:19, 2129-2155, DOI:10.1080/03610928208828377

Breslow & Day (and Clayton) aren't talking about the sort of Newton-Raphson algorithm you'd use now. They're talking about an algorithm that alternates between estimating the adjustment coefficients and the parameter of interest. This makes sense when the adjustment variables are a set of indicator variables: the first iteration estimates these coefficients at a zero value for the the parameter of interest and then estimates the parameter of interest based on the estimated adjustment coefficients.