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I am trying to understand some hypothesis testing results I am getting from a logistic regression analysis. Say I have 2 estimates of the log-odds of an event happening (the betas from my GLM). On the logit scale these these are

log.odd.trt1 <- -0.803410
log.odd.trt2 <-  0.772911

On the probability scale these are

prob.trt.1 <- 0.3092966
prob.trt.2 <- 0.6841503

The difference between the two log odds is -1.576321

diff.log.odds <- -0.803410 - 0.772911

If you convert this difference to the probability scale you get 0.1713171

inv.logit(diff.log.odds)

In contrast, The difference between the probabilities is -0.3748537

prob.trt.1 - prob.trt.2

-1.576321 is not -0.3748537. These seems reasonable given how the logit transformation works. However, my concern is whether that these means it is valid to conduct hypothesis tests that ask "is the difference between two log odds 0." This is in contrast to the question "is the log odds different from 0".

Some context: I am using logistic regression to compare 2 treatments implemented over 6 years. I am interested in determining for each year whether there are differences between the 2 treatments. I am using the R package multcomp with a contrast matrix to carry out these pairwise multiple comparisons and also conduct a test for linear trend within each treatment. I am uncertain, however, if I am doing this correctly because the hypothesis that I am testing it essentially "is the difference between to logits zero" and the difference between two logits does not map directly to the difference between two probabilities.

My inverse logit function is inv.logit <- function(x) {1/(1+exp(-x))}

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1 Answer 1

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A difference in log odds maps to a ratio of odds, which is typically called the odds ratio. A test of the hypothesis that the difference in log odds is equal to 0 maps to a test of the hypothesis that odds ratio equals 1, i.e. both odds are equal. When both odds are equal, then both probabilities are also equal.

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