# Interpreting odds ratio of multiple comparisons from a logistic regression model (using R)

I recently ran a logistic regression on categorical data and ran a Tukey multiple comparisons post hoc analysis using the glht function in multcomp package.

id.glm3<-glm(detec~apptreat+marker+exp+inter_MarEx, family=binomial, data=indiv_detec3)

Where inter_MarEx is the interaction between marker type (marker) and time after exposure (exp)

I understand that I must convert the coefficients from log odds ratios to odds ratios via exp(coefficient) and generally understand how to interpret them with mixed continuous. However, I do not quite understand how to interpret the odds ratio resulting from the multiple comparisons.

summary(glht(id.glm4, linfct=mcp(inter_MarEx="Tukey")))

Simultaneous Tests for General Linear Hypotheses

Multiple Comparisons of Means: Tukey Contrasts

Fit: glm(formula = detec ~ apptreat + inter_MarEx, family = binomial,
data = indiv_detec3)

Linear Hypotheses:
Estimate Std. Error z value Pr(>|z|)
EW 24 - EW 0 == 0       -1.5989     0.6196  -2.581   0.0937 .
EW 48 - EW 0 == 0        0.9306     0.6190   1.503   0.6429
EW 48 - EW 24 == 0       2.5295     0.3575   7.076   <0.001 ***
Milk 24 - Milk 0 == 0    0.3010     0.4442   0.678   0.9828
Milk 48 - Milk 0 == 0    4.0374     0.5190   7.779   <0.001 ***
Milk 48 - Milk 24 == 0   3.7364     0.4681   7.983   <0.001 ***
Milk 0 - EW 0 == 0       0.8109     0.6180   1.312   0.7632
Milk 24 - EW 24 == 0     2.7108     0.2716   9.980   <0.001 ***
Milk 48 - EW 48 == 0     3.9177     0.5161   7.592   <0.001 ***

---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Adjusted p values reported -- single-step method)


Now I convert the coefficients to get the odds ratio from log odds ratio.

Linear Hypotheses:

Estimate Std. Error z value Pr(>|z|)
EW 24 - EW 0 == 0        0.20     0.6196  -2.581   0.0937 .
EW 48 - EW 0 == 0        2.54     0.6190   1.503   0.6429
EW 48 - EW 24 == 0       12.55     0.3575   7.076   <0.001 ***
Milk 24 - Milk 0 == 0    1.35     0.4442   0.678   0.9828
Milk 48 - Milk 0 == 0    56.68     0.5190   7.779   <0.001 ***
Milk 48 - Milk 24 == 0   41.95     0.4681   7.983   <0.001 ***
Milk 0 - EW 0 == 0       2.25     0.6180   1.312   0.7632
Milk 24 - EW 24 == 0     15.04     0.2716   9.980   <0.001 ***
Milk 48 - EW 48 == 0     50.28     0.5161   7.592   <0.001 ***


How would I go about interpreting the odds ratios for the multiple comparisons above? For example, when comparing Milk 48 and EW 48 (odds ratio = 50.28), would I interpret the output as saying the odds of detecting EW at 48 hours after exposure are 50:1 greater than the odds of detecting Milk at 48 hrs after exposure? If this is incorrect, how should this be interpreted?