[R-sig-ME] overdispersion with binomial data?
John Maindonald
john.maindonald at anu.edu.au
Sun Feb 13 00:58:30 CET 2011
"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
>>> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>>
>> _______________________________________________
>> R-sig-mixed-models at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>
> ================================================================
> 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
>
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