# How to analyze this positively skewed data?

I am having trouble analyzing my dataset consisting of the sumscores of a questionnaire. For each item, subjects had to indicate whether they performed this behavior 'never', 'sometimes', or 'often', which was recoded into '0', '1', and '2' resp., and then summed. I have three categorical IV's, and I would like to correct for age. A GLM yielded highly non-normal residuals, so now I am looking for alternatives. I am considering a negative binomial GLzM (Poisson is out of the question, since the data are overdispersed), but I am not sure if this is suitable, since the data are not count data in the strict sense?

I am a little hesitant about nonparametric tests, as I fear these will have less power.

A histogram of my data

Edit: any transformation I tried did not result in (near-)normality, and based on the following article I would like to try another approach (e.g. negative binomial): http://www.r-bloggers.com/do-not-log-transform-count-data-bitches/

• I'd focus more on the fact that your response is bounded as the limits of the sum of scores that are each $(0, 1, 2)$ are $0$ and $2 \times$ the number of items being summed. A first approximation is thus a binomial distribution. It shouldn't too much whether that is a really good fit to the marginal distribution. A binomial distribution can certainly be skewed. (Detail: you say you have 3 categorical variables, but your scores vary from 0 to 12; presumably you are referring to categorical predictors and are summing 6 items.) – Nick Cox Jan 28 '16 at 11:23
• Consider that a mean of 0 implies a variance of 0 as does also a mean of 12. The variance will be highest for intermediate values. This variance-mean relationship doesn't match Poisson or negative binomial at all. Nor is the normal really a target here. – Nick Cox Jan 28 '16 at 11:26
• These aren't counts strict sense, for which there is no defined upper limit, but counted fractions (0/12, ...., 12/12) or approximations to measured grades. – Nick Cox Jan 28 '16 at 11:29
• Dear Nick, thanks for your reply. Good point! The categorical variables are my independent variables, my dependent variable is the sum of 12 items (0, 1, 2), hence the range is theorethically from 0 until 24, but your point still stands. – NielsL Jan 28 '16 at 11:32
• Could you suggest a proper analysis technique for these data? I am working in SPSS, and unfortunately not too comfortable with statistics. – NielsL Jan 28 '16 at 11:33

It could make a difference what you are looking to obtain from your modeling. Precise results for coefficients, standard errors, predicted values for individual cases, or p-values? For those, the non-normal residuals from your original glm will be problematic. But maybe you are looking for a more impressionistic sense of the relative role played by different predictors, or of the degree to which this outcome can even be predicted. In such as case that glm could be informative and helpful.

You could also consider multinonimal logistic regression, after converting your dependent variable into about 4 categories. It sacrifices some information, but even so, it may give you useful results that are more defensible given the more relaxed assumptions of this approach.

(I think you'd agree that null hypothesis significance testing was not designed to handle variables like this DV that have been manipulated ad hoc. A p-value would tell us how often chance alone would produce such-and-such a result. That becomes less convincing to many audiences the farther we get from concrete results (in this case, original survey responses) or from scale scores that have been validated as indicative of established constructs.)

Possible duplicate

It can often be useful to transform the data before processing to make it more normal. Some common transformations would be to take the natural log of the data or apply a power (e.g. data = data.^0.5) until the data fits a more appropriate distribution for parametric statistical tests.

• Thanks for your reply! I should add that I did consider transformations, however I did not manage to get the data normal (after trying several power transformations and Blom transformations). Also, I figured negative binomial may be more suitable based on this article (r-bloggers.com/do-not-log-transform-count-data-bitches) – NielsL Jan 28 '16 at 11:24
• This is good advice in general, but not I think a good way to go here. It seems that the possible values are just 0, .., 12 and probably best kept as such. – Nick Cox Jan 28 '16 at 11:27
• Note that if you're going to use someone else's work (in this case, that's my plot you used in your answer) you are required to give credit; a link alone is not sufficient. If you think a post is a duplicate the appropriate action is to flag or vote to close (if you have sufficient privilege). – Glen_b -Reinstate Monica Mar 24 '19 at 2:32