# Reporting exponentiated coefficients in a logistic regression, t-value and confidence intervals

I recently encountered a situation regarding the reporting of two predictor effects in a logistic regression, which I plan to report their exponentiated coefficients in the text with their SEs, t- & p-values, as well as provide a plot of their predicted effects with 95% confidence bands. I have no issues with computation of the logistic regression and exponentiated coefficients using the following in R

model <- glm(y ~ x1 + x2, family=binomial) or <- exp(coef(model))

My issues concerns more the reporting as follows:

1. Suppose the estimated coefficient of my x1, or β1 in this logistic regression is −1.49 (SE = 0.466). Computing the t-value (i.e. -1.49/0.466) on the basis of these values gives me t=-3.19, which is significant. However, if I were to generate the exponentiated coefficient i.e. exp(β1) = 0.225, and if I were to compute the SE using the delta method, I get SE = 0.154 (in R: deltamethod(~exp(x1), x1, x1se)) and the t-value is 1.465 which is non-significant. Is there something missing in my computation that is causing the discrepancy?

2. A related issue is with regards to computing the 95% confidence intervals for exp(β1) which I initially planned as a prediction plot with different values of x1, however, given the issue in (1), I'm stuck as to how to proceed with this.

## 1 Answer

1. In your second test you test the hypothesis that he Odds Ratio (exponentiated coefficient) is 0, which is impossible, so a useless test. Instead you probably wanted to test the hypothesis that the Odds Ratio is 1 (which is equivalent to the test that the log odds ratio is 0). Now you get a significant effect: (.225-1)/.154= -5.03.

2. I don't understand what the problem is here. Could you clarrify?

• Thanks Maarten for the great answer on #1. My issue #2 concerns the computation of the 95% confidence interval on the basis of exp(β1) and I get 95% CI = [-.077 .527] which indicates non-significance. If I were to work on β1 (unexponentiated), the 95% CI would be [-2.403 -0.577] which makes sense but not quite what I want since I plan to work with exp(β1) – hbox Apr 11 '15 at 12:09
• Your first CI also indicates that the OR is significant (remember, the null hypothesis of no effect is that the OR is 1). However, you typically get CIs with somewhat better properties if you exponentiate the bounds of the CI for the log odds ratio, i.e. [exp(-2.403), exp(-0.577)]. – Maarten Buis Apr 11 '15 at 18:35
• Thanks Maarten. I guess my problem lies in remembering that the null hypothesis for OR is 1 not 0. Another issue I have with exponentiated coefficient is that if β2 (unexponentiated) for x2 is -0.527 (SE=0.152), I get t-value: -0.527/0.152 = 3.46, CI: [-0.82, -0.23]. Working with the exp(β2) = 0.589 (se=0.230) and using the method you suggested, my t-value: (0.589 - 1) / .230 = -1.786, CI: [0.44, 0.79] (if I were to exponentiate the bounds as suggested) - which seemed like a discrepancy between the t and CI. Am I missing something? – hbox Apr 13 '15 at 6:00
• The delta method is an approximation, so you would not expect them to be exactly equal. In general, if you want to test whether there is no effect you do better to report the odds ratio, but test whether the log(odds ratio) = 0, as the sampling distribution of the test statistic is likely to better approximated by a standard normal for the log(odds ratio) than the odds ratio, see e.g. here – Maarten Buis Apr 13 '15 at 7:40
• Many thanks once again for your excellent answer and the link to Stata's FAQ which answers another boggling issue of mine in which I performed a little exercise of reverse coding x1 (say x1 is dichotomous). This yields β1 (unexp) = 1.49 (SE=.466) and exp(β1) = 4.437 (dm SE=3.028), and the t-value I get is (4.437 - 1) / 3.028 = 1.133. Lesson learned here is never to run the test statistic on OR. Thus, answering the question I initially posed about reporting: the test statistic reported should be based on the log(OR) and CI should be based on the exponentiated bounds of the CI of the log(OR). – hbox Apr 13 '15 at 8:32