# Coefficients in a Logistic Regression with Multiple Predictors

i was wondering about the beta (predictor coefficients) in a logistic regression; i know when it's is with one variable, the $\beta$ (ie coefficient) is definitely a logodds ratio, so $e^{\beta_1}$ would be the odds ratio for that predictor, does this remain true for multiple variables with no interactions set? I am having trouble finding answers online and have been noticing that the logistic regression results in changes in the coefficient with addition of the predictors.

Yes, it is still a log of the odds ratio, only now, it is controlling for the other variables in the model (in the same sense that this is true in "regular" regression).

• So how would the odds ratio be interpreted in the setting of other variables? I had thought that it would be increased/decreased odds relative to not having a certain feature? Clearly the interpretation is different from a two-way table. (and thank you for your help!) – Cenoc Dec 24 '14 at 16:53
• The odds ratio is how much the odds change when that variable changes, while holding the other variables constant. – Peter Flom - Reinstate Monica Dec 24 '14 at 17:00
• But how would you compare that to the univariate case? In articles it seems only the odds ratio in the context of the logistic regression are reported, but not one that can be obtained from the univariate/unipredictor case? (as in gender, you can say that this is increased/decreased odds relative to not being male, if female = 1). Is this relative to a new baseline? Why does the baseline change? I guess I'm trying to understand how to understand the changes in OR when you have one predictor versus when you add more predictors. – Cenoc Dec 24 '14 at 17:13
• Is there a proof somewhere showing that the coefficents are OR in the multiple predictor case? – Cenoc Dec 24 '14 at 18:04
• Or what probabilities is the OR a ratio of in that case? – Cenoc Dec 24 '14 at 18:30

Logit is defined as:

ln(p/(1-p))=X*B

If you have two variables, then it is:

ln(p/(1-p))=a+bx+cz

p/q = exp(a+bx+cz)

where q=1-p

OR is a ratio of two odds.

Suppose that both x and z are dichotomous (1/0 values) and we are interested about effect of z, controlling for x=0

OR=p(z=1)/(1-p(z=1))/(p(z=0)/(1-p(z=0))
OR=exp(a+c)/exp(a)
OR=exp(c)

which means that it is simply exp(c) which is estimated when controlling other variables in the model.

Of course your results change if the second variable is capturing some of the effect which in univariate case is wrongly attributed just to the one variable present in the model.