# How do I calculate confidence intervals for a linear combination of coefficients of a logistic regression?

I want to test a series of hypotheses on linear combinations of coefficients for logistic regression on categorical count data implemented as a GLM model. I know how to do linear combinations under normality, and I know how to do basic confidence intervals and significance tests for logistic models. But I just realized with some embarrassment that knowing these two things does not add up to knowing how to construct significance tests and confidence intervals on linear combinations of logistic regression coefficients.

How do you construct significance tests and confidence intervals on linear combinations of logistic regression coefficients?

• The test is done using likelihood ratio tests - the change in deviance between the null model and the full model is asymptotically chisquare with df equal to the number of independent constraints imposed by the null hypothesis (the column dimension of the matrix $C$ in $C\beta=d_0$). For a single linear combination, this is just 1 df. As for a CI, one way to do it is to find the values that cause the chisquare to reach the critical values, though you can work on the scale of the linear predictor ... (ctd) – Glen_b Aug 5 '13 at 8:56
• (ctd) ... (which the scale your combination is on anyway) and treat it as asymptotically normal, like a Wald test. (You should be able to back out an interval from that approach.) If you use R, one way to do the Wald test is with the linear.hypothesis() function in the car package (which aslo suggests Fox's Companion to Applied Regression as a reference); you can do the LR by fitting the two models and use the anova function. – Glen_b Aug 5 '13 at 8:58