[R-sig-ME] lme4, cloglog vs. binomial link (peter dalgaard)

Murray Jorgensen maj at waikato.ac.nz
Mon Jun 11 05:26:16 CEST 2012 

Hi Peter and List,

I confess that I have difficulty in seeing the connection with Poisson 
processes. When we are fishing indefinitely we can assume an endless 
supply of fish capture events so a Poisson process seems reasonable. In 
my case of an insect leaving a habitation each 'process' will just be a 
single departure event at some future time which is not exactly observed.

Perhaps I'm thinking about this in the wrong way?

Murray

On 8/06/2012 10:52 p.m., peter dalgaard wrote:
> 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
>>>
>>>
>>>
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>>> **************************************************
>>>
>>
>>
>>
>> --
>> 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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-- 
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      majmurr at gmail.com         Fax 7 838 4155
Phone  +64 7 838 4773 wk    Home +64 7 825 0441   Mobile 021 0200 8350

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