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I'm running a regression where the dependent (response) variable is a highly dispersed (slightly zero-inflated) count and the explanatory (independent or predictor) variables are continuous, counts as well as binary.

I read a couple of interesting contributions on this topic on SO (here, here, and here as well as the articles / blogs they refer to), yet they generally deal either explicitly or implicitly with response variables that are continuous. So I have a couple of questions. Any help and (partial) answers would be much appreciated.

  1. Given overdispersion of the response variable (variance >> mean) and a mean that varies over time, what is the best way to 'change' the response variable?
  2. @Gung argued in one of his/her contributions that centering or scaling predictors is especially useful when you plan to interact them with other variables or in case of quadratic terms. This would reduce collinearity. Now in my data, I find that the correlation between the mean-centred variables and their quadratic terms is indeed smaller (from .88 to .80), however, the correlation between the different mean-centred variables x1_centered and x2_centered is actually bigger. Hence I'm not sure about which effect is worse...
  3. It is often suggested (e.g. Angrist & Pischke, 2009 - Mostly harmless econometrics) to mean-center the response variables in panel data as this would account for fixed effects. But this raises 2 questions:
    1. Which mean do you use in panel data? Using the mean of the entire sample would imply knowing the future before it unfolds. Using a rolling mean that includes the lagged variables for each year of regression might work but I have never seen it done in a paper.
    2. What is the interpretation of a mean-centered count variable. In my case I look at patent citations and mean-centered would mean I have negative values suddenly
  4. Finally, by mean-centering the response count variable (i.e. if there is a good reason to do so, I have to move away from a negative binomial regression because the nature of the response variable changes from positive integers to positive and negative non-integers. As such, I think the points made by @MansT regarding least squares estimators (see here) are not valid (I don't imply @MansT argues they are!!). Clearly, IF I have to change the regression (from glm, familiy = quasibinomial()) to something else, this will affect all the betas.

Ok, I know it's a long question so I can only hope some of you can contribute to parts of it.

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