Analysing discrete data from a factorial experimental design The dependent variable in my data can take on the values 0-7 and is, therefore, a discrete variable. I have data from a number of experiments with one, two, and three-way factorial designs. 
The previous literature on my topic has just used ANOVA of analyse this type of data. This seems clearly wrong to me and my data rarely meet the assumptions of ANOVA which suggests all the published research on this topic has used the wrong statistical models. 
My issue is that I don't know what the best approach to analysing this data is. What is the most appropriate model for analysing discrete data using a factorial design? I'm using R to analyse the data.
 A: Given that prior research has analyzed comparable data using ANOVA, I am assuming that the 8-point scale is at least ordinal. That it is discrete is not problematic. Does this scale measure a latent characteristic that is continuous or is it measuring something that only takes on these eight values? Can you assume, even if only roughly, that there are equal intervals between the levels of the scale? What is your sample size and is the sample size balanced across conditions? Answers to all of these questions effects whether an ANOVA approach is "clearly wrong" as you state or a reasonable approach, even if not strictly ideal.
Recall that given the Central Limit Theorem, as sample size increases, the sampling distribution becomes normal even with data from a population that is non-normally distributed. This also extends to discretized data (hence the ability to use a z-test on proportions given a large sample size -- although with modern computers there is little to no need to do so). A consequence of this is that ANOVA (or OLS regression) will often produce results that are in agreement with more complex models with assumptions that more closely match the data, such as an ordered logistic regression model.
Thus, an ANOVA approach is not "clearly wrong" even if it is not precisely correct. I recommend that you compare the results from an ANOVA model to an ordered logistic regression model. For the ordered logistic regression model, take care in how you parameterize your factors and interactions. I recommend effect coding to get a model that more closely aligns with a simple ANOVA model. If the design isn't balanced, you also need to build the model in stages to simulate a Type II sums-of-squares analysis. If the ordered logistic regression model and ANOVA lead to comparable substantive interpretations, go with the latter as it is easier to interpret and easier for most readers to understand.
