I know this or a similar question has been addressed more than once on CV, and I've tried to read/understand the responses, but I'm still stuck and hoping for some further clarification.
Results from a mark-recapture analysis that uses a logit link to model survival probability. This is the raw output from the model, so the estimates represent change on the logit(response) (i.e., I haven't back-transformed):
estimate se lcl ucl S:(Intercept) -0.7805226 0.7506744 -2.2518445 0.6907993 S:grass -4.6189001 2.0282830 -8.5943350 -0.6434653 S:shrub 0.0120384 0.0295291 -0.0458387 0.0699155 S:season2014 0.7729132 0.5820680 -0.3679401 1.9137665 S:grass:shrub 0.1835990 0.0894282 0.0083198 0.3588782
I want to get mean, LCL and UCL odds ratios for the interaction, grass*shrub. I understand there isn't one OR that represents the entire interaction, since the OR depends on the values of the covariates. I am primarily interested in holding shrub constant and calculating an OR +/- error for grass.
I attempted to follow this helpful post (How to interpret interaction continuos variables in logistic regression?) and my OR's for the mean make sense given my probability estimates (assuming that's a useful comparison to be making). However, when I apply the same steps to the LCL and UCL, the estimates are far from what I would expect.
1) When shrub = 0, a unit change in grass is associated with a (exp(-4.62)-1)*100 = -99% decrease in odds of survival.
2) For every one unit increase in shrub, the effect of grass increases by (exp(0.18)-1)*100 = 19.72%.
(exp(.18)=1.19, exp(-4.62) = 0.0098)
3) The OR for grass when shrub = 1 is 1.19^1 * 0.0098 = 0.01.
- When shrub = 15: 1.19^15 * 0.0098 = 0.15
- When shrub = 30: 1.19^30 * 0.0098 = 2.18
However, when I apply these same steps to the LCL the OR is always essentially zero.
- When shrub = 1: 1.01^1 * 0.0002 = 0.0002
- When shrub = 15: 1.01^15 * 0.0002 = 0.0002
- When shrub = 30: 1.01^30 * 0.0002 = 0.0003
The OR's are huge when I calculate them for the UCL, e.g.:
When shrub = 30, OR = 25848.
I'm using these probability predictions (+/- 95% CI) from this model to help determine my expectation for the OR's, but maybe that's part of my problem. Based on this plot, I would expect the OR for grass to be at least marginally significantly positive (LCL of OR >=1) at the highest values of shrub cover, but that's far from what the OR's are showing above:
Am I calculating or interpreting the OR's incorrectly?
Does the method I'm using not apply to LCL and UCL? If so, what is the alternative/preferred method?
Am I just really bad at math?