With a complementary-log-log link function, it's not logistic regression -- the term "logistic" implies a logit link. It's still a binomial regression of course.
the estimate of time is 0.015. Is it correct to say the odds of mortality per unit time is multiplied by exp(0.015) = 1.015113 (~1.5% increase per unit time)
No, because it doesn't model in terms of log-odds. That's what you'd have with a logit link; if you want a model that works in terms of log-odds, use a logit-link.
The complementary-log-log link function says that
$\eta(x) = \log(-\log(1-\pi_x))=\mathbf{x}\beta$
where $\pi_x=P(Y=1|X=\mathbf{x})$.
So $\exp(\eta)$ is not the odds ratio; indeed $\exp(\eta)=-\log(1-\pi_x)$.
Hence $\exp(-\exp(\eta))=(1-\pi_x)$ and $1-\exp(-\exp(\eta))=\pi_x$. As a result, if you need an odds ratio for some specific $\mathbf{x}$, you can compute one, but the parameters don't have a direct simple interpretation in terms of contribution to log-odds.
Instead (unsurprisingly) a parameter shows (for a unit change in $x$) contribution to the complementary-log-log.
As Ben gently hinted in his question in comments:
is it true to say that the probability of mortality per unit time (i.e. the hazard) is increased by 1.5% ?
Parameters in the complementary log-log model do have a neat interpretation in terms of hazard ratio. We have that:
$e^{\eta(x)}=-\log(1-\pi_x) = -\log(S_x)$, where $S$ is the survival function.
(So log-survival will drop by about 1.5% per unit of time in the example.)
Now the hazard, $h(x)=-\frac{d}{dx}\log(S_x)=\frac{d}{dx}e^{\eta(x)}$, so indeed it seems that in the example given in the question, the probability of mortality* per unit of time is increased by about 1.5%
* (or for binomial models with cloglog link more generally, of $P(Y=1)$)
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).
Best Answer
This isn't really a 'discrete' variable, it's a categorical variable. Only the intercept is the odds of a woman (in the reference level party) being elected. The other coefficients are odds ratios. You multiply those odds ratios times the odds in the intercept to get the odds of a woman in the, say, green party being elected (in that case the odds is .3125, or about 24%).
It may help you to read: