# Can one use parametric tests (e.g. ANOVA) if part of the data/ variables meet normality criteria but others not?

I want to compare differences between two groups(n1=26, n2=18) regarding their performance on different tests but also explore how each group performed on different times of these tests. I am therefore inspecting my data for normality to see if I could use t-tests, one-way anova and repeated measures anova. Shapiro-Wilks testing showed that for some variables group1 responses met the normality criteria but group2 not whereas for some other variables it was the opposite. How should I proceed? Use parametric tests for some variables and non-parametric for those that do not meet criteria or non-parametric for all?

• If for some variables at least one of the groups does not follow a normal distribution I suggest that you use non-parametric tests like the Kruskal-Wallis on those variables. Apr 27 '19 at 16:57
• Thank you for the response. A further question would be whether you would be aware of a non-parametric test to detect for the type of interaction between these variables (I am using SPSS).
– elma
Apr 28 '19 at 8:56
• Are you asking about the interaction between two different variables or do you want to compare group 1 with group 2? Apr 28 '19 at 19:32
• Firstly, I want to compare group1-group2. Secondly, I would like to see (if there is an appropriate test) how the pretest performance of either group might predict post-test performance as well what the interactions between the variables are in each group
– elma
Apr 30 '19 at 10:03
• Assuming that non-parametric tests is the way to go could you please indicate any tests that allow for some sort of analysis of the effect of the interaction between different variables? as far as I know Kruskal-Wallis test does not do that.
– elma
May 17 '19 at 11:26

Your sample sizes are quite small, so you probably cannot reasonably rely on statistical tests that appeal to the asymptotic distribution of quantities under large sample sizes. ANOVA tests in parametric models use distributions that occur either from underlying normal data, or by large-sample distributional approximations. These should generally be avoided if you have a small sample size and your underlying data deviates substantially from normal data.

• Overall, inspection of qq plots and skewness/ kurtosis values for most variables (but not for all) seem ok but SW normality tests "fail" for at least half of the variables. What is considered "substantial deviation" from normality? Many thanks in advance
– elma
Apr 30 '19 at 10:00
• That's a bit like asking when a piece of string is considered to be "long". It raises more questions.
– Ben
Apr 30 '19 at 11:13
• Assuming that non-parametric tests is the way to go could you please indicate any tests that allow for some sort of analysis of the effect of the interaction between different variables? as far as I know Kruskal-Wallis test does not do that.
– elma
May 17 '19 at 11:25

If you have two paired measurements (pre-test performance and post-test performance) for each of the two independent groups, you may try to study the evolution of each individual: $$e_j = post_j - pre_j$$

Then, you may apply a Mann-Whitney test to compare the two independent groups of individuals with respect to the evolution. (I'm assuming that the evolution will not follow a normal distribution.)

Regarding the interactions, you may start by computing Spearman's correlations between pairs of variables.

• Thanks, I will definitely look into that.
– elma
May 22 '19 at 8:54
• On a different level would robust Anova be an option for my data?
– elma
May 22 '19 at 8:55