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Overdispersion is the phenomenon of having data that is more variable than its model assumes. Overdispersion can occur when the model in question has inseparable mean and variance parameters. If I understand correctly, an overdispersed model yields overly confident regression coefficient estimates. In other words, we might conclude the coefficients are statistically significant when in fact they should not be. My question is, if my fitted coefficients are not statistically significant, is there any sense in dealing with the overdispersion? If we are already overconfident and observe no significant effects, then fixing this overdispersion will only result in even less significant effects, is that true?

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Yes, that is true.

There are only two commonly-used generalized linear model families for which the concept of overdispersion is relevant. These are Poisson regression or binomial regression when the number of trials is greater than one. If the data is genuinely overdispersed then switching from one of these glm regression models to a model that allows for overdispersion will result in larger p-values for the same hypothesis tests.

Note however that it is also possible for data to be underdispersed and, in those circumstances, quasi-Poisson regression or quasi-binomial regression will estimate quasi-dispersions less than one and hence can give smaller p-values than the correponding Poisson or binomial regressions, especially if the number of observations is large.

On the other hand, if you use a mixture model to model the overdispersion then getting smaller p-values is not possible. Commonly used mixture models include negative binomial glms to model overdispersion relative to Poisson or beta-binomial regression to model overdispersion relative to the binomial.

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    $\begingroup$ Thank you for the answer and additional details. Could you elaborate on how the use of a quasi model (including an overdispersion parameter) results in a smaller p-value? I would have thought that the introduction of an overdispersion parameter just has the benefit of an additional variance estimate. $\endgroup$ Apr 12 '20 at 0:12
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    $\begingroup$ @fool126 You seem to be assuming that quasi models have an "overdispersion" parameter, but they do not. They do have a dispersion parameter, but this can be larger or smaller than the dispersion value of 1 assumed by the Poisson or binomial models. A quasi-dispersion >1 represents overdispersion, a quasi-dispersion <1 represents underdispersion. For any glm, the dispersion parameter multiplies the variance function and is analogous to the variance estimate for normal models. $\endgroup$ Apr 13 '20 at 0:15
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Just to add to @GordonSmyth's answer, when you are fitting a quasipoisson or quasibinomial, the variance-covariance matrix is scaled by the dispersion value. This means the standard error of your coefficients are multiplied by sqrt(dispersion). So

For example, we fit a poisson:

library(pscl)
fm_pois <- glm(art ~ ., data = bioChemists, family = poisson)
coefficients(summary(fm_pois))

               Estimate  Std. Error    z value     Pr(>|z|)
(Intercept)  0.30461683 0.102981443  2.9579779 3.096643e-03
femWomen    -0.22459423 0.054613488 -4.1124315 3.915137e-05
marMarried   0.15524338 0.061374395  2.5294487 1.142419e-02
kid5        -0.18488270 0.040126898 -4.6074506 4.076360e-06
phd          0.01282258 0.026397045  0.4857582 6.271386e-01
ment         0.02554275 0.002006073 12.7327095 3.890982e-37

And a quasipoisson:

fm_qpois <- glm(art ~ ., data = bioChemists, family = quasipoisson)
coefficients(summary(fm_qpois))
               Estimate  Std. Error    t value     Pr(>|t|)
(Intercept)  0.30461683 0.139272885  2.1871941 2.898252e-02
femWomen    -0.22459423 0.073859696 -3.0408225 2.426991e-03
marMarried   0.15524338 0.083003199  1.8703301 6.175917e-02
kid5        -0.18488270 0.054267922 -3.4068506 6.859925e-04
phd          0.01282258 0.035699564  0.3591803 7.195436e-01
ment         0.02554275 0.002713028  9.4148462 3.777939e-20

sqrt(summary(fm_qpois)$dispersion)
[1] 1.352408

You can work out 1.352408 * the standard error of coefficients from poisson model is equal to the standard error of coefficients from the quasipoisson.

The one exception I can think of is when your overdispersion is caused by zero counts, in that case, if you do a zero-inflated model, some of the estimates might change.

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    $\begingroup$ Thank you, this answers my follow-up question. (I think I can only check one correct answer unfortunately.) $\endgroup$ Apr 12 '20 at 4:51
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    $\begingroup$ No worries, glad it was useful for you :) $\endgroup$
    – StupidWolf
    Apr 12 '20 at 14:06

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