# Interpreting odds ratios for logistic regression with intercept removed

So I ran a binomial glm to look at the effect of minimum temperature (continuous data) and moon phase (categorical data with 3 categories) on lion incidents. I removed the intercept to look at all 3 categories of moon by using + 0 in my glm. These were the results: I then calculated the odd ratios and confidence intervals using the code:

exp(cbind(OR = coef(GLMoon), confint(GLMoon)))

which gave me this: I'm not fully clear on how to interpret these as most of the odds ratios I've seen have been above 1, but from what I've read I guess it would mean that the odds for incidents during the 'new.moonthe rest' phase are 45% lower than during other phases? And that holding moon phases at a fixed value, we will see a 1.14% increase in the odds of an incident, for a one-unit decrease in temperature?

I'd prefer to calculate probabilities of attack, but I'm unsure how to do this from the coefficients or odds ratios when the intercept is removed. I'd really appreciate any help!

Cheers

For the various levels of New.moon, these are not odds ratios, but odds. So the odds of an "incident" is $0.55$ during the 'new.moonthe rest' phase when TMIN (minimun temperature) is at 0. You could also back-translate this into the chances of an incident (i.e., $0.55 / (1 + 0.55) \approx 0.35$).
If you want an odds ratio, you have to compare two odds against each other. So, for example, the odds are $0.55$ for the 'the rest' phase and $0.36$ for the 'pre' phase. So, the odds ratio is $0.55 / 0.36 \approx 1.53$, or in other words, the odds are $1.53$ times higher during 'the rest' phase compared to the 'pre' phase.
For TMIN, the value is an odds ratio, comparing the odds of an incident for a one-unit increase in minimum temperature (so the odds ratio of $x+1$ versus $x$, where $x$ is the minimum temperature value).
• Yes, the 35% applies when temperature is at 0. But then it is an extrapolation and you probably shouldn't interpret it. Of course you can easily estimate the odds (and hence chances) for other temperature values based on the model. For TMIN, the odds ratio is .986, so as temperature increases, the odds decrease (and vice-versa). If you want to phrase it in terms of a one-unit decrease in temperature, you should say that the odds increase by $(1-.986)*100 = 1.4$%. – Wolfgang Aug 21 '16 at 19:15