overdispersion with binomial data?

This is a useful clarification.  Jared might perhaps have said
"cannot be accounted for in the error term".

What one can do (in the binary glm model) is to fit group as a fixed
effect, then use the deviance that is due to group to estimate
the overdispersion. Or one can use a glmm model with group
as a random effect.

John Maindonald             email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.
http://www.maths.anu.edu.au/~johnm

On 13/02/2011, at 9:27 AM, John Maindonald wrote:

"However, most people seem to ignore overdispersion estimates 

(chi-square/df) if they are less than about 1.5 or so as a practical matter."

If there is large uncertainty in the overdispersion estimate, 

then adjusting for the

overdispersion is to trade bias for that uncertainty.  If the 

argument is that the

bias is preferable, p-values should be adjusted for the long-term 

(over multiple

studies) bias.  For an overdispersion that averages out at around 

1.5, a p-value

that appears as 0.05 becomes, depending on degrees of freedom, around 0.1

Sure, the deviance and Pearson chi-square are commonly quite close.  The
preference for the Pearson chi-square, as against the mean 

deviance, is not

however arbitrary.  The reduced bias is, over multiple analyses, 

worth having.

If the Poisson mean is small, or many of the binomial proportions 

are close to 0

or to 1, it is noticeable.

---------

zp <- rpois(20, 0.25)
summary(glm(zp~1, family=quasipoisson))

. . . .

(Dispersion parameter for quasipoisson family taken to be 0.7895812)

 Null deviance: 13.863  on 19  degrees of freedom
Residual deviance: 13.863  on 19  degrees of freedom
--------

Compare the mean chi-square = 0.73 = 13.86/19 with a Pearson chi-square
estimate (as above) that equals 0.79

The preference for the Pearson chi-square, as against the mean deviance,
is not arbitrary.


John Maindonald             email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.
http://www.maths.anu.edu.au/~johnm

On 13/02/2011, at 5:16 AM, Robert A LaBudde wrote:

I don't believe in tests generally, so I agree with your point 

1) in principle.

However, most people seem to ignore overdispersion estimates 

(chi-square/df) if they are less than about 1.5 or so as a 
practical matter. But if you have a reasonable amount of data and 
get an effect of 10 as in the example given, minor issues as to a 
posteriori testing are irrelevant. Ignoring an apparent 
overdispersion of 10 does not seem sensible.

The question as to whether deviance or Pearson chi-square is 

used is a minor issue, as the two are almost invariably quite close anyway.

In the end, we must all agree that the model must include all 

important effects to be useful.

At 01:04 AM 2/12/2011, John Maindonald wrote:

1) Different types of residuals serve different purposes.

2) I am of the school that thinks it misguided to use the
results of a test for overdispersion to decide whether to
model it.  If there is any reason to suspect over-dispersion
(and in many/most ecological applications there is), this
is anti-conservative.  I judge this a misuse of statistical
testing.  While, some do rely on the result of a test in these
circumstances, I have never seen a credible defence of
this practice.

3) In fitting a quasi model using glm(), McCullagh and Nelder
(which I do not have handy at the moment) argue, if I recall
correctly, for use of the Pearson chi-square estimate.  The
mean deviance is unduly susceptible to bias.

4) Whereas the scale factor (sqrt dispersion estimate) is incorporated
into the GLM residuals, the residuals from glmer() exclude all
random effects except that due to poisson variation.  The residuals
are what remains after accounting for all fixed and random effects,
including observation level random effects.

5) Your mdf divisor is too small.  Your stream, stream:rip and ID
random terms account for further 'degrees of freedom'.  Maybe
degrees of freedom are not well defined in this context?  Anyone
care to comment?  The size of this quantity cannot, in any case, be
used to indicate over-fitting or under-fitting.  The model assumes
a theoretical value of 1.  Apart from bias in the estimate, the
residuals are constrained by the model to have magnitudes that
are consistent with this theoretical value.

6) If you fit a non-quasi error (binomial or poisson) in a glm model,
the summary output has a column labeled "z value".  If you fit a quasi
error, the corresponding column is labeled "t value".  In the glmer
output, the label 'z value' is in my view almost always inappropriate.
To the extent that the description carries across, it is the counterpart
of the "t value" column in the glm output with the quasi error term.
(Actually, in the case where the denominator is entirely composed
from the theoretical variance, Z values that are as near as maybe
identical can almost always [always?] be derived using an
appropriate glm model with a non-quasi error term.)

John Maindonald             email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.
http://www.maths.anu.edu.au/~johnm

On 12/02/2011, at 12:58 PM, Colin Wahl wrote:

In anticipation of the weekend:
In my various readings(crawley, zuur, bolker's ecological 

models book, and

the GLMM_TREE article, reworked supplementary material and R 

help posts) the

discussion of overdispersion for glmm is quite convoluted by different
interpretations, different ways to test for it, and different 

solutions to

deal with it. In many cases differences seem to stem from the 

type of data

being analyzed (e.g. binomial vs. poisson) and somewhat 

subjective options

for which type of residuals to use for which models.

The most consistent definition I have found is overdispersion 

is defined by

a ratio of residual scaled deviance to the residual degrees of 

freedom > 1.

Which seems simple enough.

modelB<-glmer(E ~ wsh*rip + (1|stream) + (1|stream:rip), data=ept,

family=binomial(link="logit"))

rdev <- sum(residuals(modelBQ)^2)
mdf <- length(fixef(modelBQ))
rdf <- nrow(ept)-mdf
rdev/rdf #9.7 >>1

So I conclude my model is overdispersed. The recent consensus 

solution seems

to be to create and add a individual level random variable to the model.

ept$obs <- 1:nrow(ept) #create individual level random variable 1:72
modelBQ<-glmer(E ~ wsh*rip + (1|stream) + (1|stream:rip) + (1|obs),
data=ept, family=binomial(link="logit"))

I take a look at the residuals which are now much smaller but 

are... just...

too... good... for my ecological (glmm free) experience to be 

comfortable

with. Additionally, they fit better for intermediate data, which, with
binomial errors is the opposite of what I would expect. Feel 

free to inspect

them in the attached image (if attachments work via mail 

list... if not, I

can send it directly to whomever is interested).

Because it looks too good... I test overdispersion again for 

the new model:

rdev/rdf #0.37

Which is terrifically underdispersed, for which the consensus 

is to ignore

it (Zuur et al. 2009).

So, for my questions:
1. Is there anything relevant to add to/adjust in my approach thus far?
2. Is overdispersion an issue I should be concerned with for binomial
errors? Most sources think so, but I did find a post from 

Jerrod Hadfield

back in august where he states that overdispersion does not exist with a
binary response variable:
http://web.archiveorange.com/archive/v/rOz2zS8BHYFloUr9F0Ut (though in
subsequent posts he recommends the approach I have taken by using an
individual level random variable).
3. Another approach (from Bolker's TREE_GLMM article) is to 

use Wald t or F

tests instead of Z or X^2 tests to get p values because they 

"account for

the uncertainty in the estimates of overdispersion." That 

seems like a nice

simple option, I have not seen this come up in any other 

readings. Thoughts?

Here are the glmer model outputs:

ModelB
Generalized linear mixed model fit by the Laplace approximation
Formula: E ~ wsh * rip + (1 | stream) + (1 | stream:rip)
Data: ept
AIC BIC logLik deviance
754.3 777 -367.2    734.3
Random effects:
Groups     Name        Variance Std.Dev.
stream:rip (Intercept) 0.48908  0.69934
stream     (Intercept) 0.18187  0.42647
Number of obs: 72, groups: stream:rip, 24; stream, 12

Fixed effects:
       Estimate Std. Error z value Pr(>|z|)
(Intercept) -4.28529    0.50575  -8.473  < 2e-16 ***
wshd        -2.06605    0.77357  -2.671  0.00757 **
wshf         3.36248    0.65118   5.164 2.42e-07 ***
wshg         3.30175    0.76962   4.290 1.79e-05 ***
ripN         0.07063    0.61930   0.114  0.90920
wshd:ripN    0.60510    0.94778   0.638  0.52319
wshf:ripN   -0.80043    0.79416  -1.008  0.31350
wshg:ripN   -2.78964    0.94336  -2.957  0.00311 **

ModelBQ

Generalized linear mixed model fit by the Laplace approximation
Formula: E ~ wsh * rip + (1 | stream) + (1 | stream:rip) + (1 | obs)
Data: ept
AIC   BIC logLik deviance
284.4 309.5 -131.2    262.4
Random effects:
Groups     Name        Variance Std.Dev.
obs        (Intercept) 0.30186  0.54942
stream:rip (Intercept) 0.40229  0.63427
stream     (Intercept) 0.12788  0.35760
Number of obs: 72, groups: obs, 72; stream:rip, 24; stream, 12

Fixed effects:
       Estimate Std. Error z value Pr(>|z|)
(Intercept)  -4.2906     0.4935  -8.694  < 2e-16 ***
wshd         -2.0557     0.7601  -2.705  0.00684 **
wshf          3.3575     0.6339   5.297 1.18e-07 ***
wshg          3.3923     0.7486   4.531 5.86e-06 ***
ripN          0.1425     0.6323   0.225  0.82165
wshd:ripN     0.3708     0.9682   0.383  0.70170
wshf:ripN    -0.8665     0.8087  -1.071  0.28400
wshg:ripN    -3.1530     0.9601  -3.284  0.00102 **


Cheers,
--
Colin Wahl
Department of Biology
Western Washington University
Bellingham WA, 98225
ph: 360-391-9881
<ModelComp2.png>_______________________________________________
R-sig-mixed-models at r-project.org mailing list

================================================================
Robert A. LaBudde, PhD, PAS, Dpl. ACAFS  e-mail: ral at lcfltd.com
Least Cost Formulations, Ltd.            URL: http://lcfltd.com/
824 Timberlake Drive                     Tel: 757-467-0954
Virginia Beach, VA 23464-3239            Fax: 757-467-2947

"Vere scire est per causas scire"
================================================================

Just to let you know: I love and respect your body of work, Prof. Maindonald, and your last comments are particularly pithy for the cognescenti.

However, if you follow the whole thread, how we got to this point is somewhat of a "chicken or egg?" problem.

If we always include "Group" in the model to check for overdispersion, and accept it if it tests positive but not if it doesn't, this biases our result in a particular way. If we exclude "Group", but test based on Pearson chi-square, this biases our result in the another way.

Also, in neither case must the overdispersion fit the assumption involved (e.g., linear additive effect).

I believe any type of testing to choose among models to use biases the choice of models. I much prefer choosing models based upon subject matter expertise instead.

I also prefer to base decisions on the size of an effect observed rather than upon its statistical detectability (significance) in a particular study.

In the present case, I'd feel uneasy about any model that didn't include a "Group" effect, whether or not it was detectable (significant). That's because it must be present logically, even if small. So the logical plan is to include and estimate it.

At 07:04 PM 2/12/2011, John Maindonald wrote:

This is a useful clarification.  Jared might perhaps have said
"cannot be accounted for in the error term".

What one can do (in the binary glm model) is to fit group as a fixed
effect, then use the deviance that is due to group to estimate
the overdispersion. Or one can use a glmm model with group
as a random effect.

John Maindonald             email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.
http://www.maths.anu.edu.au/~johnm

On 13/02/2011, at 9:27 AM, John Maindonald wrote:

"However, most people seem to ignore overdispersion estimates (chi-square/df) if they are less than about 1.5 or so as a practical matter."

If there is large uncertainty in the overdispersion estimate, then adjusting for the
overdispersion is to trade bias for that uncertainty.  If the argument is that the
bias is preferable, p-values should be adjusted for the long-term (over multiple
studies) bias.  For an overdispersion that averages out at around 1.5, a p-value
that appears as 0.05 becomes, depending on degrees of freedom, around 0.1

Sure, the deviance and Pearson chi-square are commonly quite close.  The
preference for the Pearson chi-square, as against the mean deviance, is not
however arbitrary.  The reduced bias is, over multiple analyses, worth having.
If the Poisson mean is small, or many of the binomial proportions are close to 0
or to 1, it is noticeable.

---------

zp <- rpois(20, 0.25)
summary(glm(zp~1, family=quasipoisson))

. . . .

(Dispersion parameter for quasipoisson family taken to be 0.7895812)

 Null deviance: 13.863  on 19  degrees of freedom
Residual deviance: 13.863  on 19  degrees of freedom
--------

Compare the mean chi-square = 0.73 = 13.86/19 with a Pearson chi-square
estimate (as above) that equals 0.79

The preference for the Pearson chi-square, as against the mean deviance,
is not arbitrary.


John Maindonald             email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.
http://www.maths.anu.edu.au/~johnm

On 13/02/2011, at 5:16 AM, Robert A LaBudde wrote:

I don't believe in tests generally, so I agree with your point 1) in principle.

However, most people seem to ignore overdispersion estimates (chi-square/df) if they are less than about 1.5 or so as a practical matter. But if you have a reasonable amount of data and get an effect of 10 as in the example given, minor issues as to a posteriori testing are irrelevant. Ignoring an apparent overdispersion of 10 does not seem sensible.

The question as to whether deviance or Pearson chi-square is used is a minor issue, as the two are almost invariably quite close anyway.

In the end, we must all agree that the model must include all important effects to be useful.

At 01:04 AM 2/12/2011, John Maindonald wrote:

1) Different types of residuals serve different purposes.

2) I am of the school that thinks it misguided to use the
results of a test for overdispersion to decide whether to
model it.  If there is any reason to suspect over-dispersion
(and in many/most ecological applications there is), this
is anti-conservative.  I judge this a misuse of statistical
testing.  While, some do rely on the result of a test in these
circumstances, I have never seen a credible defence of
this practice.

3) In fitting a quasi model using glm(), McCullagh and Nelder
(which I do not have handy at the moment) argue, if I recall
correctly, for use of the Pearson chi-square estimate.  The
mean deviance is unduly susceptible to bias.

4) Whereas the scale factor (sqrt dispersion estimate) is incorporated
into the GLM residuals, the residuals from glmer() exclude all
random effects except that due to poisson variation.  The residuals
are what remains after accounting for all fixed and random effects,
including observation level random effects.

5) Your mdf divisor is too small.  Your stream, stream:rip and ID
random terms account for further 'degrees of freedom'.  Maybe
degrees of freedom are not well defined in this context?  Anyone
care to comment?  The size of this quantity cannot, in any case, be
used to indicate over-fitting or under-fitting.  The model assumes
a theoretical value of 1.  Apart from bias in the estimate, the
residuals are constrained by the model to have magnitudes that
are consistent with this theoretical value.

6) If you fit a non-quasi error (binomial or poisson) in a glm model,
the summary output has a column labeled "z value".  If you fit a quasi
error, the corresponding column is labeled "t value".  In the glmer
output, the label 'z value' is in my view almost always inappropriate.
To the extent that the description carries across, it is the counterpart
of the "t value" column in the glm output with the quasi error term.
(Actually, in the case where the denominator is entirely composed
from the theoretical variance, Z values that are as near as maybe
identical can almost always [always?] be derived using an
appropriate glm model with a non-quasi error term.)

John Maindonald             email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.
http://www.maths.anu.edu.au/~johnm

On 12/02/2011, at 12:58 PM, Colin Wahl wrote:

In anticipation of the weekend:
In my various readings(crawley, zuur, bolker's ecological models book, and
the GLMM_TREE article, reworked supplementary material and R help posts) the
discussion of overdispersion for glmm is quite convoluted by different
interpretations, different ways to test for it, and different solutions to
deal with it. In many cases differences seem to stem from the type of data
being analyzed (e.g. binomial vs. poisson) and somewhat subjective options
for which type of residuals to use for which models.

The most consistent definition I have found is overdispersion is defined by
a ratio of residual scaled deviance to the residual degrees of freedom > 1.

Which seems simple enough.

modelB<-glmer(E ~ wsh*rip + (1|stream) + (1|stream:rip), data=ept,

family=binomial(link="logit"))

rdev <- sum(residuals(modelBQ)^2)
mdf <- length(fixef(modelBQ))
rdf <- nrow(ept)-mdf
rdev/rdf #9.7 >>1

So I conclude my model is overdispersed. The recent consensus solution seems
to be to create and add a individual level random variable to the model.

ept$obs <- 1:nrow(ept) #create individual level random variable 1:72
modelBQ<-glmer(E ~ wsh*rip + (1|stream) + (1|stream:rip) + (1|obs),
data=ept, family=binomial(link="logit"))

I take a look at the residuals which are now much smaller but are... just...
too... good... for my ecological (glmm free) experience to be comfortable
with. Additionally, they fit better for intermediate data, which, with
binomial errors is the opposite of what I would expect. Feel free to inspect
them in the attached image (if attachments work via mail list... if not, I
can send it directly to whomever is interested).

Because it looks too good... I test overdispersion again for the new model:

rdev/rdf #0.37

Which is terrifically underdispersed, for which the consensus is to ignore
it (Zuur et al. 2009).

So, for my questions:
1. Is there anything relevant to add to/adjust in my approach thus far?
2. Is overdispersion an issue I should be concerned with for binomial
errors? Most sources think so, but I did find a post from Jerrod Hadfield
back in august where he states that overdispersion does not exist with a
binary response variable:
http://web.archiveorange.com/archive/v/rOz2zS8BHYFloUr9F0Ut (though in
subsequent posts he recommends the approach I have taken by using an
individual level random variable).
3. Another approach (from Bolker's TREE_GLMM article) is to use Wald t or F
tests instead of Z or X^2 tests to get p values because they "account for
the uncertainty in the estimates of overdispersion." That seems like a nice
simple option, I have not seen this come up in any other readings. Thoughts?




Here are the glmer model outputs:

ModelB
Generalized linear mixed model fit by the Laplace approximation
Formula: E ~ wsh * rip + (1 | stream) + (1 | stream:rip)
Data: ept
AIC BIC logLik deviance
754.3 777 -367.2    734.3
Random effects:
Groups     Name        Variance Std.Dev.
stream:rip (Intercept) 0.48908  0.69934
stream     (Intercept) 0.18187  0.42647
Number of obs: 72, groups: stream:rip, 24; stream, 12

Fixed effects:
       Estimate Std. Error z value Pr(>|z|)
(Intercept) -4.28529    0.50575  -8.473  < 2e-16 ***
wshd        -2.06605    0.77357  -2.671  0.00757 **
wshf         3.36248    0.65118   5.164 2.42e-07 ***
wshg         3.30175    0.76962   4.290 1.79e-05 ***
ripN         0.07063    0.61930   0.114  0.90920
wshd:ripN    0.60510    0.94778   0.638  0.52319
wshf:ripN   -0.80043    0.79416  -1.008  0.31350
wshg:ripN   -2.78964    0.94336  -2.957  0.00311 **

ModelBQ

Generalized linear mixed model fit by the Laplace approximation
Formula: E ~ wsh * rip + (1 | stream) + (1 | stream:rip) + (1 | obs)
Data: ept
AIC   BIC logLik deviance
284.4 309.5 -131.2    262.4
Random effects:
Groups     Name        Variance Std.Dev.
obs        (Intercept) 0.30186  0.54942
stream:rip (Intercept) 0.40229  0.63427
stream     (Intercept) 0.12788  0.35760
Number of obs: 72, groups: obs, 72; stream:rip, 24; stream, 12

Fixed effects:
       Estimate Std. Error z value Pr(>|z|)
(Intercept)  -4.2906     0.4935  -8.694  < 2e-16 ***
wshd         -2.0557     0.7601  -2.705  0.00684 **
wshf          3.3575     0.6339   5.297 1.18e-07 ***
wshg          3.3923     0.7486   4.531 5.86e-06 ***
ripN          0.1425     0.6323   0.225  0.82165
wshd:ripN     0.3708     0.9682   0.383  0.70170
wshf:ripN    -0.8665     0.8087  -1.071  0.28400
wshg:ripN    -3.1530     0.9601  -3.284  0.00102 **


Cheers,
--
Colin Wahl
Department of Biology
Western Washington University
Bellingham WA, 98225
ph: 360-391-9881
<ModelComp2.png>_______________________________________________
R-sig-mixed-models at r-project.org mailing list

================================================================
Robert A. LaBudde, PhD, PAS, Dpl. ACAFS  e-mail: ral at lcfltd.com
Least Cost Formulations, Ltd.            URL: http://lcfltd.com/
824 Timberlake Drive                     Tel: 757-467-0954
Virginia Beach, VA 23464-3239            Fax: 757-467-2947

"Vere scire est per causas scire"
================================================================

================================================================
Robert A. LaBudde, PhD, PAS, Dpl. ACAFS  e-mail: ral at lcfltd.com
Least Cost Formulations, Ltd.            URL: http://lcfltd.com/
824 Timberlake Drive                     Tel: 757-467-0954
Virginia Beach, VA 23464-3239            Fax: 757-467-2947

"Vere scire est per causas scire"
================================================================