One of my experimental analysis is a one way ANOVA. The dependent variable is a count variable (egg counts of an insect). As a prerequisite, I tested the assumptions of residual normality and variance homogeneity on my data. The residual variances are equal, however; my data are non-normal. The scatter plots and histograms suggested that the distribution is not heavily skewed but the results of Shapiro-Wilk and Kolmogorov -Smirnov tests have very low p-values.
The Box-Cox transformation suggested that I do a square root transformation on my data. However, square root transformation does not seem to help. Other transformations also don't help. I can go for Kruskal Wallis test but it being non-parametric, I would lose power. In this case, one of the options is to use a procedure like GENMOD that does not consider data to be normally distributed. I understand that I can analyse my data by defining distribution as Poisson with link=log in Genmod.
My question is, can I call this as analysis of variance with Poisson distributed errors? Also, after the ANOVA, can I report means and standard errors as in regular ANOVA done on normal data? In Poisson distribution, I think the mean and variance are same. However, I am not sure if it is appropriate to report treatment means as the calculation of lsmeans is different.
I would appreciate any insights on this.
You are on the right track with consideration of the use of PROC GENMOD using a log link and the Poisson distribution.
To be sure that the Poisson model is appropriate as opposed to the Negative Binomial, check to make sure that the data are not over-dispersed; the assumption of the Poisson model is that the mean and variance of the outcome are the same. If the variance is variance in the model is greater than you would expect, try scaling the deviance parameter (dscale in model statement) or check using the Negative Binomial distribution and compare the fit (the NB allows for an extra parameter to estimate the variance where the Poisson does not).
SAS allows for a simple test of dispersion using the Deviance and the degrees of freedom; run the poisson model and look in the goodness of fit portion of the outcome. if the deviance value / df is far greater than 1, you may have a problem with dispersion.
I would call this a generalized linear model for count data, rather than ANOVA.
good luck.
Thank you so much for taking time to look at my query and for such a thorough response, Mr. Baker. I highly appreciate your help with this. I tested my Poisson model and looked for fit statistics, and it the value is 1.02. It thus fits a Poisson model to my data. If the value of the fit statistics (deviance/df) is very less such as 0.06 or 0.13 etc. does that indicate underdispersion? Is using negative binomial distribution a good option in this case?
Thank you so much again for your response and insights. It certainly helped a lot.
Underdispersion is a bit of a tricky beast as compared to overdispersion. Overdispersion of count data has been study extensively and the negative binomial distribution works will in most cases of overdispersion. In cases of underdispersion, your variability is less than expected given the Poisson distribution (conditioned on the mean); you may want to check that the model is not over-fit. Do you have a great number of explanatory variables or an interaction in the model that may not be necessary?
If the model does not appear to be over-fit, you may want to look into modeling the data using a distribution other than Poisson. You may consider checking the fit using the Gamma or the Conway-Maxwell (COM) Poisson distribution. It is a generalization of Poisson model that is designed to handle both over and underdispersion quite well. There is not, to my knowledge, a routine in SAS for implementing the COM Poisson method; but surely it could be coded into PROC NLMIXED. There are other methods out there as well.
http://arxiv.org/pdf/1011.2077.pdf
http://onlinelibrary.wiley.com/doi/10.1111/j.1539-6924.2010.01417.x/pdf
Thanks again. I luckily don't have underdispersion in my data that I analyzed with GENMOD but I saw some cases where the fit values were low. I was not sure how to deal with it because there is a lot more of recent literature on overdispersion. Thank you for sharing the articles on Conway-Maxwell Poisson distribution. I was not aware of this approach and it is very helpful to know about it.
Do we look for same fit statistics when usig Proc GLIMMIX? I have some data that are not normal again. But Poisson distribution does not fit well. Also, there is a random term in there so can't use Genmod. If I were to use GLIMMIX with Poisson distributed errors, what should be the value of chi. square/DF? Again is it 1 to consider a good fit? Currently, it is consistently less than 1. It gets lesser if i use NegBin as distribution. If I use Normal distribution with identity link, it is more than 0.50. I am not sure if it is a good fit. do I have underdispersion in this case? Is it better to just use Kruskal-Wallis then instead of fitting another distribution? Many thanks again for your great help!
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