overdispersion with binomial data?
"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
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