# Negative Binomial „Ancova” with planned contrasts

My DV is count, obtained from psychological questionnaire (14 questions with options ”Yes”/”No”, each yes is 1 point, thus overall result is count of yeses).

I’ve checked that Poisson distribution isn’t exactly what I’ve got (data is a bit overdispressed) and NB fits better (in comparison to Poisson – log likelihood is closer to 0 and AIC/BIC values are lower. Deviance/df = 1,153; Pearson Chi-square/df = 0,96 – which seem fair enough to me). Although 14 is obviously a bound, no one scored more than 8, so I think it should not be a problem (also all groups had same questionnaire, so the same bound).

I have 4 groups (smallest n=28, biggest n=35), which i need to compare. My hypotesis has a form of planned orthogonal contrast (-3,1,1,1 and 0,1,1,-2). In the best scenario, I would love to also include some covariates in the model, that’s why I mentioned „Ancova”.

I’ve spent last week reading stats coursebooks and articles about count data – I've finished with impression that regression is a fair option. But stats teacher I asked about it said she had no idea what I am speaking about and that I should not contrive, just use Kruks-Wallis.

So I thought that maybe I misunderstood something and that's why I wanted to ask 3 questions:

1. Assuming that distribution of the data is in fact negative binomial, is negative binomial regression an appropriate method to test an alternative hypotesis that groups differ?
2. Assuming that distribution of the data is in fact negative binomial, should negative binomial regression be considered a better fitted test for such comparison than Kruks-Wallis? (more powerfull?)
3. [If answer for 2 former questions was „yes-ish”] Is it right to implement planned orthogonal contrasts (-3,1,1,1 and 0,1,1,-2) by including 2 dummy variables coded for groups just like this (-3,1,1,1 and 0,1,1,-2) as covariates? I’m using SPSS.

1. DV describes one’s Locus of Control
2. That’s my syntax (yet without covariates), if it helps in any way:

GENLIN LOC WITH Contrast1 Contrast2 /MODEL Contrast1 Contrast2 INTERCEPT=YES DISTRIBUTION=NEGBIN(MLE) LINK=LOG /CRITERIA METHOD=FISHER(1) SCALE=1 COVB=MODEL MAXITERATIONS=100 MAXSTEPHALVING=5 PCONVERGE=1E-006(ABSOLUTE) SINGULAR=1E-012 ANALYSISTYPE=3(WALD) CILEVEL=95 CITYPE=WALD LIKELIHOOD=FULL /MISSING CLASSMISSING=EXCLUDE /PRINT CPS DESCRIPTIVES MODELINFO FIT SUMMARY SOLUTION (EXPONENTIATED) /SAVE MEANPRED DEVIANCERESID.

If the data seems like it follows a negative binomial distribution (or an over dispersed Poisson), then it is fine to use a negative binomial regression. You could use an GLM a generalized linear model approach for analyzing the data (here is a good book if you can get hold of it https://www.amazon.com/Negative-Binomial-Regression-Joseph-Hilbe/dp/0521198151). You can specify any contrasts of interest that you want, but you should be aware of the interpretation of the contrasts in this context as differences are usually on the log scale (assuming you use the default log link functions).

You can use the GLM approach and I would suggest reading more on Poisson regressions and the interpretation of contrasts in that context with the log link to see if that suits your purpose. Agresti's Categorical Data Analysis is an excellent book for this.

The negative binomial may be more powerful than then non-parametric Kruskal-Wallis test, but in the end it may depend on the sample sizes that you have in total. If you are interested in the mean differences on the original scale then the Kruskal-Wallis might be more appropriate

• Hi, thank you very much for your response. Speaking of interpretation difficulties. Did you just mean that Beta is on log-scale and therefore usually interpretation of exp(B) is more intuitive (although still different from OLS)? Jul 12, 2017 at 7:26
• Well I mainly want to point out that you need to be careful as to what comparisons you want to derive, whether on the original scale or not and that should determine your model choice. Because a contrast on say two means at the log scale actually translates to a ratio on the original scale. So if you did run a contrast with the GLM based on the betas.. those are actually comparing ratios exp(b1-b2) = exp(b1)/exp(b2) Jul 12, 2017 at 14:17