So you are using maximum likelihood estimates from a simple logistic regression model and you want to know how they are calculating the beta coefficient for the one explanatory variable that is a continuous, correct?

I can probably look up some better definition, but it is my understanding that they use MLE for the estimate and the coefficient actually represents a log odds ratio, in particular the standard formula you would see for a categorical variable but its ratio for a continuous variable is the variable + 1 unit over the variable. So it is a log odds ratio for a relative 1 unit increase, with the estimates coming from a maximization to make the coefficient best fit data. I apologized for my bastardized interpretation, I am not theoretical but very much applied. Is the AMJE article available free, if so please post a link so we can quickly check it out.