lme4, cloglog vs. binomial link (peter dalgaard)

Hi Murray,

I think this is pretty strongly related to proportional hazards modelling. If you are looking at it from a Poisson (*) process point of view, the rate of event per unit of time when observing a number of independent processes should be proportional to time and the number  of processes, and the probability of at least one event in a fixed length of time T is then 1 - exp(- n T lambda) (or 1 - exp (-n Lambda(T)) if you have a time-varying intensity, Lambda being the integrated intensity).

-pd

(*) Could be fun if this was actually about fish...

On Jun 6, 2012, at 23:21 , Murray Jorgensen wrote:

*Hi Peter, Tibor et al.

I came across an ecological situation recently where a cloglog link seemed
to be called for. I won't remove the context to the following explanatory
note that I wrote but I'm sure the same kind of situation could be
reasonably common:


We wish to explore the probability of one or more females departing a
cavity between two site visits as a function of the habitation state of the
cavity at the first visit. More strictly we study the probability of a
decrease in the number of females inhabiting the cavity between the two
visits. Clearly this probability will be zero if no females inhabit the
cavity at the first visit. More generally the probability will be larger as
the number of female inhabitants increase as each has the opportunity to
depart.

Although this dependance on the initial number of females is part of what
we want to study we are more interested in questions such as the influence
of the initial number of males on the probability of female decrease. We
are indeed also interested in the effect of the initial numbers of females
on the probability of female decrease, but more in the sense that we would
like to know whether this is greater than, less than or equal to what would
be predicted by a simple model.

One naive model that could be considered is that the decrease probability
would be proportional to the number of females. This might work if the
decrease probability was very low but for larger decrease probabilities
would predict decrease probabilities greater than one. A less naive model
would assume that each female departs with the same probability,
independently of the other females.
Then if the probability of a single female departing is $p$ and there are
$x$ females in the cavity the probability of 1 or more departing is $p_x =
1 - (1-p)^x$.

The link function for the complementary log-log link is $\eta =
\log(-\log(1-p))$. To examine the effect of multiple initial females we
evaluate this at $p_x$.

                 1-p_x  =  (1-p)^x
           \log(1-p_x)  =  x\log(1-p)
    \log(-\log(1-p_x))  =  \log(x)+\log(-\log(1-p))

Thus the effect of an initial habitation of $x$ females is a shift of
$\log(x)$ on the linear predictor scale if a complementary log-log link is
used in a GLM or GLMM for the probability of female decrease. This means
that the naive model can be accommodated by including $\log(x)$ as an
offset.  If $x$ were also included as a covariate, a significant
coefficient would indicate a departure from the naive model.

Regards, Murray

*

Message: 5
Date: Wed, 6 Jun 2012 22:54:16 +0200
From: peter dalgaard<pdalgd at gmail.com>
To: Tibor Kiss<tibor at linguistics.rub.de>
Cc: r-sig-mixed-models at r-project.org
Subject: Re: [R-sig-ME] lme4, cloglog vs. binomial link
Message-ID:<7658F572-AEE2-4127-AB77-321B5B6C3D69 at gmail.com>
Content-Type: text/plain; charset=us-ascii


On Jun 4, 2012, at 13:07 , Tibor Kiss wrote:

[...snippage...]
My questions are as follows:

1. Is it correct to assume that given a cloglog link, the less frequent

response should be considered the success?

No, cloglog is asymmetric, so it will make a difference which outcome is
considered success, but there is no mathematical reason to choose between
them. In survival data, the cloglog comes out of the proportional hazards
model when you have death within a fixed time period as the response (exact
date of death not recorded). In that case, death is "success" (!);
hopefully, it is the least likely outcome, but it might not be. If cloglog
is just used as a generic link function, then no such logic applies.

2. Is it correct to conclude that the changes in the model have led to

less influence of the random factor?

No. The scales are different. At the very least, you need to somehow
compare it to the fixed effects on the same scale.

3. How shall I react to the increase in AIC?

(Or, equivalently, the deviance). The cloglog link model seems to give the
worse fit to data.

A final question, which may not have an answer at all: I am most curious

to learn about possible modifications of the model so that an observed
random effect can be minimized (while its presence cannot be denied).

First, is that desirable, and why? The only logic, that I can think of, is
that you want to get the fixed-effect part of the model right, so that the
error is not mistakenly taken as part of the random variation.

--
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com



------------------------------

--
Dr Murray Jorgensen      http://www.stats.waikato.ac.nz/Staff/maj.html
Department of Statistics, University of Waikato, Hamilton, New Zealand
Email: maj at waikato.ac.nz    majorgensen at ihug.co.nz      Fax 7 838 4155
Phone  +64 7 838 4773 wk    Home +64 7 825 0441   Mobile 021 0200 8350

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