My DV is a score between 0 and 6 (based on the number of right/wrong answers given to 6 questions, so only integers are possible). I first thought I could consider it as a continuous variable and I conducted a linear regression. The problem is that my residuals are not normally distributed. They kind of follow a normal curve, but with a lot of peaks and valleys around the curve (see picture). So I thought I might want to try a (quasi)Poisson regression, but my data do not follow a Poisson distribution (as attested by a Kolmogorov-Smirnov test), and the mean is larger than the variance, so my data are underdispersed.
What should I do to be able to conduct a valid regression?
Thank you!
Both quasi-Poisson and negative binomial regressions allow some flexibility in the relationship between the mean and variance. Hermite regression is more flexible with the distribution of data.
If you are concerned with over- or under-dispersion of the data, you should find a suitable check for this. Don't just look at the mean and the variance of the raw data.
BTW, I think if you changed the number of bins in your figure, that the histogram would follow the curve pretty well.
But in any case, I wouldn't consider counts from 1 to 6 to be continuous data.
The distribution of your response is binomial,
-> binomial regression, aka. logistic regression
https://www.stat.cmu.edu/~cshalizi/uADA/12/lectures/ch12.pdf
Jochen,
6 questions may by different? some of them are easy to answer, some - hard. So, we have different probability p for each right answer.
may be logisitic regression with random effect?
I also agree with Jochen and Olga. Treating the dependent variable as a binomial variable makes more sense.
Based on the information you provided, I will try the generalized Poisson (GP) regression model. This is one of the few models that allow under-dispersion. One thing about the GP model is that the value of DV is truncated above since the dispersion parameter is negative. A reference for the GP model is Famoye (1993) "Restricted GP regression model" Communications in Statistics-Theory and Methods, 22(5), 1335-1354. Good luck.
Dear all,
Sorry for coming so late back to you, and thank you for your help. I will try to get more acquainted with the different tests you've suggested.
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