# How does statistical control work in logistic regression?

I want to make sure that I can generally interpret model findings accurately. Is it fair to say that each log-odds associated with a predictor assumes that the others are held constant at 0? Making it similar to standard linear regression. And would this change based on continuous vs. binary predictors?

Is it fair to say that each log-odds associated with a predictor assumes that the others are held constant at 0?

The logit model is

\begin{align*} E(Y_t\vert x_t) &= p_t \\ \log \left( \frac{p_t}{1-p_t}\right) &= \vec{\beta}^\top \vec{x}_t \end{align*}

So $$\beta_j$$ describes a unit move on the log-odds scale of the $$j$$th covariate regardless of whether the other covariates are zero or not as long as the other covariates stays fixed.

And would this change based on continuous vs. binary predictors?

No that does not change anything.

• EdM has some good comments. I assume in the above that you have no interactions and that you familiar with interpreting intercepts when you have factors. – Benjamin Christoffersen Feb 25 '18 at 18:50

Even in standard linear regression, the coefficient for a predictor represents the association of the outcome variable with that particular predictor when all other predictors are held constant. There is no requirement that they be held constant at values of 0; so long as the others are held constant, under the assumptions of the linear model it won't matter what their values are. That's true for both standard and logistic regression based on linear models.

The interpretation of the intercept, however, will differ dramatically depending on how the values of the predictors are coded. With the default treatments contrasts used in R, the intercept will be the value of the outcome variable when continuous predictors have values of 0 and categorical variables are at their reference levels. You have to be careful with intercepts as not all statistical software programs use the same defaults for choosing reference values of categorical variables, and different handling of predictor contrasts might also affect them.

Things do get a bit tricky when you consider interactions, however, with both standard and logistic regression. An interaction term then effectively represents the product of the values of the interacting predictors. For continuous variables, interpretation of interaction coefficients can be simpler if you center the continuous predictors to mean values of 0 first. Then any such interaction term will have a value of 0 when all individual predictors are at their mean values.