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I used Poisson regression model to model how count of user actions on a website (dependent variable) are explained by website content (independent variables). The dependent variable distribution is shown in this plot.

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As you can see, the distribution is positively skewed and has a long tail. The results of the Poisson regression model from glm in R are shown here:

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As you can see, residual deviance is much greater than the degrees of freedom, so there is overdispersion. Next, I tried Quasipoissson model that gave these results:

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I see that the Quasipoisson model shows that none of the independent variables are significant. I am a little unsure how to interpret these results. Am I handling the overdispersion incorrectly, or am I using independent variables that just do not explain the variation in my dependent variable?

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  • $\begingroup$ Welcome. It could be a convergence issue. But first, I wonder if there is a lot of zeros for subgroups of your covariates. Check the variation in your response for particular covariates such as Active_Floorplans. Try that and see if it fixes the problem. Please follow-up if this doesn’t address your concern. $\endgroup$ Jul 11 '20 at 22:22
  • $\begingroup$ @ThomasBilach After closer inspection of the independent variables, I am noticing both a lot of zeros and some rather large outliers. For example, Active_Floorplans has ~9% zeros and Low_Price has about 4% zeros. This is quite surprising to me due to the nature of this content. However, Active_Photos has about 1% zeros. When I use LPP ~ Active_Photos as the formula in the Poissson and Quasipoisson model, both do show significance for this variable. I would love to understand this at a deeper level if you can explain or point me to some appropriate reading material. $\endgroup$
    – Charles
    Jul 11 '20 at 23:20
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    $\begingroup$ @ThomasBilach Actually, upon closer inspection it appears that Low_Price was the real problem. The Poisson and Quasipoisson models show significance for all variables as long as I drop Low_Price from the model. This variable is discrete/count variable with 4% zeros and a lot of subgroups. Here is a description of that distribution: range = 0-10000, Q1 = 750, Median = 975, Q3 = 1055. $\endgroup$
    – Charles
    Jul 11 '20 at 23:36
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    $\begingroup$ So for certain combinations of covariates, it appears there isn’t enough variation in your response. You might wish to enter a factorized version of a regressor on the right-hand side that has multiple subgroups, only to learn that there is a predominance of zeros within that subgroup. This is the likely diagnosis. After dropping some of the variables, can you estimate both models? $\endgroup$ Jul 12 '20 at 1:07
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    $\begingroup$ @ThomasBilach Yes, I believe you have solved my problem, and I believe I follow your suggestions about regressor factorization. If you want to write your above comments as an answer, I will mark it as solved. I really appreciate your guidance! $\endgroup$
    – Charles
    Jul 12 '20 at 1:40
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Two issues are likely at play here. One is a convergence failure. The other is adjusting for a covariate(s) with low-cardinality. The likely diagnosis is the latter.

For instance, Low_Price and Numeric_Rank have small estimates of uncertainty and consequently large z-values. This should make you suspicious of your model inputs (covariates). For certain combinations of covariates, there appears to be little to no variation in your response variable. I would assess summaries of LPP by covariate. You might observe a lot of 0's for one particular input. To get around this, you might have to consider dropping a particular covariate or aggregating your data to obtain cleaner estimates.

In the comments, I noted that this often occurs when you blindly throw predictors on the right-hand side of your equation. By "factorized" version of a variable, I meant a variable with many categories (levels). See the basic specification below:

poisson <- glm(LPP ~ as.factor(Floor_Plan) + ..., family = "poisson", data = ...)

where Floor_Plan might be a multivalued factor variable with many levels. R will attempt estimation of all levels of this variable. For any particular subgroup, there might be predominance of 0 counts. This is just something to consider as you make your model more complex. I hope this helps with your intuition.

As a final thought, there are other ways to estimate overdispersion. In the AER package you will find the function dispersiontest, which implements a test for overdispersion by Cameron & Trivedi (1990). Here is the documentation.

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