# 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.