Could someone advise me on how to interpret the estimates from a logistic regression using a cloglog link?
I have fitted the following model in lme4:
glm(cbind(dead, live) ~ time + factor(temp) * biomass,
data=mussel, family=binomial(link=cloglog))
For example, 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).
In other words are the estimates obtained in a cloglog expressed in log odds as is the case for a logit logistic regression?
Best Answer
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.
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:
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 $P(Y=1)$)