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    beta coefficient for a logistic regression




    I consider a simple model with binary variable Y (diseased vs non-diseased), variable X normally distributed among diseased and non-diseased with means MuX1 (diseased) and MuX0 (non-diseased), and common variance Var(X).

    If we use a simple logistic regression model of Y on X, the regression coefficient would be given by:
    Beta = (MuX1 - MuX0) / Var(X),

    the reference I am using for this is Am J Epidemiol. 1986 Nov;124(5):826-35.
    My interest would be if someone knows another source that explain how to find this equation for Beta, the paper from Am J Epidemiol explains this in its appendix, but very briefly. Thank you in advance.

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    Re: beta coefficient for a logistic regression

    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.
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    Re: beta coefficient for a logistic regression

    The article is not available free online sadly, I don't know if I am allowed to post a part of it, the part I am referring to takes less than half a page. It uses the Bayes theorem to define the odds of disease, and then uses the odds of disease to define the odds ratio as exp[(MuX1 - MuX0) / Var(X)].

    My main question would be this: for linear regression we know that the equation for the regression coefficient is Beta = Cov(Y,X)/Var(X); there are tons of references available showing this. For logistic regression however, references for this equation for the regression coefficient Beta = (MuX1 - MuX0) / Var(X) are harder to find, most references for logistic regression coefficients are about the MLE estimates. I was wondering if someone was familiar with another reference for this equation for logistic regression coefficient.

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    Re: beta coefficient for a logistic regression

    Hmm. Well I was just being lazy, I am a Society for Epidemiologic Research member, so I guess it is my duty to track this thing down. I am pretty busy right now but will see if I can get to this in the PM.
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    Re: beta coefficient for a logistic regression

    I am skimming the article right now, what do you plan to do with this equation when? Since you can just use logistic regression based on MLE like the majority of people and it would be easily understood by your readers?


    Also, have you looked to see if anyone else has cited this article -> then look at all of the references in those article for other and possibly older papers?


    P.S., I am guessing there may be other sensitivity analysis approaches if you are trying to quantify measurement error.
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    Re: beta coefficient for a logistic regression

    I am mainly using the book "Principles of Exposure Measurement in Epidemiology: Collecting, Evaluating, and Improving Measures of Disease Risk Factors", where they use this result (Beta = (MuX1 - MuX0)/VarX) and the similar result for a variable measured with error (they then define more models than the Am J Epidemiol. 1986 Nov;124(5):826-35, but they all depend on this equation of Beta).

    My interest would be to estimate if the equation is valid when the variance of X0 (non-diseased) is different from X1 (diseased), unlike what is assumed in the paper and the book. The trouble I have with the article is in the appendix and how they get from equation A.6 to equation A.7 (from equation A.7 it is easy to get to the final result).

    I am also surprised that this equation for a beta coefficient for a logistic regression is not used more often elsewhere, considering its simplicity.

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    Re: beta coefficient for a logistic regression

    Hmm, I may look at the article again tomorrow if I get time, but it seems like this whole thing could be accomplished using Monte Carlo simulations to get power values. I think simulations are much more commonly used in contemporary papers due to is easy to run compared to in 1986. In addition the authors mention that they are not controlling for exogenous variables.


    I would recommend the following book for a gentle introduction to such topics:


    Applying Quantitative Bias Analysis to Epidemiologic Data, http://www.springer.com/us/book/9780387879604
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    Re: beta coefficient for a logistic regression


    For me personally it was more of theoretical question rather than a practical (maybe another forum would have been more appropriated for me to post this question), this question came about by reading the theory in the book by E. White and al that I cited earlier where they refer to the paper from 1986. This paper explained how they obtain their equation for the beta coefficient which is used a lot in the book by White, but I didn't understand all the steps on how they got to the result. I was then surprised to not find an explanation on this particular equation in books or papers focusing on logistic regression.

    The book by Lash that you recommend is also one that I have used.

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