lmer and a response that is a proportion

-----Original Message-----
From: Cameron Gillies [mailto:cgillies at ualberta.ca] 
Sent: Sunday, December 03, 2006 6:31 PM
To: Prof Brian Ripley; John Fox
Cc: r-help at stat.math.ethz.ch
Subject: Re: [R] lmer and a response that is a proportion

Dear Brian and John,

Thanks for your insight.  I'll clarify a couple of things 
incase it changes your advice.

My response is a ratio of two measures taken during a bird's 
path, which varies from 0  to 1, so I cannot convert it 
columns of the number of successes.  It has to be reported as 
the proportion.  I could logit transform it to make it 
normal, but I am trying to avoid that so I can analyze it directly.

The subjects are individual birds and I have a range of 
sample sizes from each bird (from 8 to >200, average of about 
75 measurements/bird).

Thanks!
Cam


On 12/3/06 3:47 PM, "Prof Brian Ripley" <ripley at stats.ox.ac.uk> wrote:

On Sun, 3 Dec 2006, John Fox wrote:

Dear Cameron,

-----Original Message-----
From: r-help-bounces at stat.math.ethz.ch 
[mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Cameron 
Gillies
Sent: Sunday, December 03, 2006 1:58 PM
To: r-help at stat.math.ethz.ch
Subject: [R] lmer and a response that is a proportion

Greetings all,

I am using lmer (lme4 package) to analyze data where the 

response is 

a proportion (0 to 1).  It appears to work, but I am wondering if 
the analysis is treating the response appropriately - 

i.e. can lmer 

do this?

As far as I know, you can specify the response as a proportion, in 
which case the binomial counts would be given via the weights 
argument -- at least that's how it's done in glm(). An alternative 
that should be equivalent is to specify a two-column matrix with 
counts of "successes" and "failures" as the response. 

Simply giving 

the proportion of successes without the counts wouldn't be 

appropriate.

I have used both family=binomial and quasibinomial - is one more 
appropriate when the response is a proportion?  The coefficient 
estimates are identical, but the standard errors are larger with 
family=binomial.

The difference is that in the binomial family the 

dispersion is fixed 

to 1, while in the quasibinomial family it is estimated as a free 
parameter. If the standard errors are larger with family=binomial, 
then that suggests that the data are underdispersed 

(relative to the 

binomial); if the difference is substantial -- the factor 

is just the 

square root of the estimated dispersion -- then the 

binomial model is 

probably not appropriate for the data.

John's last deduction is appropriate to a GLM, but not 

necessarily to 

a GLMM. I don't have detailed experience with lmer for 

binomial, but I 

do for various other fitting routines for GLMM.  Remember 

there are at 

least two sources of randomness in a GLMM, and let us keep 

it simple 

and have just a subject effect and a measurement error.  Then if 
over-dispersion is happening within subjects, forcing the binomial 
dispersion (at the measurement level) to 1 tends to increase the 
estimate of the subject-level variance component to 

compensate, and in 

turn increase some of the standard errors.

(Please note the 'tends' in that para, as the details of 

the design do 

matter.  For cognescenti, think about plot and sub-plot 

treatments in 

a split-plot design.)