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

peter dalgaard pdalgd at gmail.com
Fri Jun 8 12:52:08 CEST 2012 

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
>> 
>> 
>> 
>> ------------------------------
>> 
>> _______________________________________________
>> R-sig-mixed-models mailing list
>> R-sig-mixed-models at r-project.org
>> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>> 
>> 
>> End of R-sig-mixed-models Digest, Vol 66, Issue 10
>> **************************************************
>> 
> 
> 
> 
> -- 
> 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
> 
> 	[[alternative HTML version deleted]]
> 
> _______________________________________________
> R-sig-mixed-models at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models

-- 
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

More information about the R-sig-mixed-models mailing list