This blog claims that the fact that the mean and variance of Poisson distribution are equal can cause problem. Could you please elaborate why this is the limitation and how it can affect models?
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2 Answers
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Just read the next two sentences:
There is no way to increase the variance without increasing the mean. Unfortunately, in many data sets the variance is larger than the mean.
If you model some phenomenon with a Poisson distribution, you are tacitly imposing this constraint that the mean and variance must be the same. If your real-life phenomenon does not exhibit this property, then it may not be a good idea to model it with the Poisson distribution.
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When mean and variance are equal, variance increases as mean increases.
Problem in fitting poisson GLM : Overdispersion
Many a time data admit more variability than expected under the assumed distribution. The greater variability than predicted by the generalized linear model random component reflects overdispersion.
Source : https://web.archive.org/web/20130621133920/https://onlinecourses.science.psu.edu/stat504/node/162
Why use poisson GLM : In the case of linear models, sometimes you observe a small difference between fitted and actual values (desired) when fitted value is low and a large difference between fitted and actual values (not desired) when fitted value is high. This is called heteroscedasticity giving a funnel shaped plot between residuals and fitted values - try plot(lm) function in R.
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1$\begingroup$ This answer seems to miss the mark. The Poisson distribution usually concerns count data. Ordinary least squares, as implemented by
lm, is typically inappropriate: a generalized linear model, such as implemented byglm, is more suitable. The residuals will differ from what you show here, and so does their analysis. Furthermore, Poisson regression is perfectly capable of handling heteroscedasticity--any nonzero trend will (almost by definition) be heteroscedastic. The issue is that it might not be a sufficiently flexible method for modeling the heteroscedasticity in some cases. $\endgroup$– whuber ♦Commented Sep 29, 2017 at 19:38 -
$\begingroup$ @whuber I seem to have explained reasons for using Poisson, instead of problems associated with it. Can you please explain how residuals in a GLM are interpreted ? $\endgroup$– sree22Commented Sep 30, 2017 at 3:42
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$\begingroup$ Search for "deviance residuals." $\endgroup$– whuber ♦Commented Sep 30, 2017 at 12:59