I am trying to perform a GLM analaysis using R for an outcome that is:
(Numerical Rating Scale for Pain: 0 - 10)
I have a set of demographic factors, age, sex etc, that I want to input as factors for the GLM.
I understand that Gaussian might not be the best option (since bounded by 0) but am not sure if I should choose Gamma (since this is not continuous) or Poisson (since the outcome is not counts)
the data is very much skewed:
thanks
s
This answer elaborates on some discussion in comments on the answer from Nick Cox.
Your situation might be handled by a multi-category extension of binomial regression: ordinal regression. You model the probability of moving from one category to the next in a way that takes advantage of the ordering among the outcome categories.
This UCLA web page illustrates ordinal logistic regression, based on a "proportional odds" (PO) assumption for moving up the scale. I don't know whether that assumption will hold for your data, but the page does show how to evaluate it.
Also, as Frank Harrell points out in Section 13.3.3 of his Regression Modeling Strategies book, a PO model can sometimes work well even if the assumption isn't met. In this answer to a question on highly skewed data that take only a few values with clumping at one end, he says:
When the dependent variable Y has a beautiful distribution I still recommend it be modeled using a Y-transformation-invariant semiparametric ordinal regression model such as the proportional odds model. With your Y, the need for a semiparametric model is even greater. Semiparametric models handle arbitrary clumping of Y values, bimodality, floor effects, ceiling effects, and outliers. Such models are also very efficient.
The orm() function in Harrell's rms package allows for ordinal regression with link functions other than the logit, and Section 13.4 of his book shows how to implement a "continuation ratio" method that sometimes works better than a PO model. That provides you some flexibility in how to proceed.
With a PO model you can often model, without overfitting, almost as many parameters as you can with linear regression. Section 4.4 of Harrell's book and course notes provides an estimate of the effective sample size that takes the distribution of cases among categories into account. Your sample size of about 200 would be reduced to an effective sample size of about 180 on that basis, so you should be able to estimate about 12 regression coefficients.
I would use logit link and binomial family, without too much queasiness.
There will be, or should be, switches in your software to indicate bounds of 0 and 10, or more generally to work with the kind of data You are using R, but I won't attempt even broad coding advice.
I support what I take to be a widespread view that getting the link right (here to respect the bounded nature of the outcome) is more crucial than whichever family you specify. There is always small print about which choices give best guesses at standard errors and P-values.
In practice, you may get very similar predicted values even with gamma and Poisson families, but the inferential details could vary quite a lot. But watch out: with some choices you might get absurd negative predictions for your outcome, given a marginal distribution like that.
In sum: you're focusing on which error family to specify, but the link function is the first choice to make.
On a different level: this could be very awkward data for other reasons, depending on how far people reporting no pain could be qualitatively as well quantitatively different.
