2 factorial experiment (2 by 3). DV is NOT normally distributed

Question

I just conducted an 2 factorial experiment that has 6 conditions (2 by 3). Specifically, my design is: IV1 = prior positive information (positive in A domain vs. control vs. positive in B domain) IV2 = negative information (negative in A domain vs. negative in B domain) DV = Blame judgment (7 point Likert type scale)

I assumed that if prior positive domain and negative domain do not match, prior positive domain can play a shield role to block negative information (i.e., if the target person has prior positive quality in A domain and later engage in negative behavior in B domain, people do not really blame the target).

The results of two way ANOVA supported some of my assumptions. However, I just found out that my DV is not normally distributed (Negatively skewed so much). The reason can be related to my research context. People may immediately place 7 in the scale (i.e., blame) when they receive negative information. I tied to transform the data (log, centered, standardized), but it was not corrected enough to reach normal distribution.

Then my questions are: 1. Is the two way ANOVA a wrong statistic technique to use in this situation? 2. If it is wrong, which statistical technique is appropriate?

NOTE: I am using SPSS and AMOS.

Thank you very much in advance for your help and support!

Answer 1

Clearly the assumptions required for the usual normal-theory tests don't hold.

However, the question is not so much "are the assumptions actually true" -- I find it doubtful that they ever really do hold -- as "how much does it impact the inferential procedures of interest?"

This question is equally applicable to nonparametric procedures; they're not assumption-free either and a similar question of amount of impact will arise. In respect of some assumptions common nonparametric procedures may even be more sensitive.

When we speak of impact on inference, in relation to hypothesis tests this will generally be consideration of the effect on actual significance level and on power.

The two are intimately related; moving the true type I error rate will move the power curve.

The impact of the normality assumption in ANOVA tends to reduce with sample size and under some mild conditions might be effectively ignored at very large sample sizes; the impact of unequal variance or dependence does not tend to go away in the same fashion.

One option you might look at is to model the DV as ordered categorical and tailor the model (and subsequently, the formal hypotheses) to that.

You might consider ordered logistic regression (ordered logit, proportional odds)

Another potentially suitable model is the stereotype model which is related to multinomial logistic regression (which itself is suitable for nominal data but is also at least sometimes used with ordered data).

(See also this post)