# Is a heavy skew a problem if the same is expected for the population? And: Can bootstrapping help to confirm findings?

In the context of an experiment, I have eight groups (2x2x2 between subjects) and the n per group ranges between 23 and 29. My variable of interest is how many critical details (range from 0 to 5) are provided in a free report. Half of the groups have skew and kurtosis values that are within the range of what I have read is considered to be acceptable (skew: +/- 0.80; kurtosis +/- 2.00). For the other groups, the distribution is skewed negatively with the most common value being the highest value possible. However, this is not a surprise; these groups are actually expected to provide most or all of the details and thus achieve very high values because of one of the experimental factors.

I planned to run an ANOVA and have two questions because of the situation described above:

(1) If the skewed distribution is what would be expected (i.e. considered to be normal), is there still a problem in terms of running an ANOVA? (And, if so, I'd also be interested in understanding why.) Also, it may be relevant to know that the homogeneity of variance assumption is violated too.

(2) If it is a problem, might bootstrapping be an option? I am not aware of a non-parametric alternative to a 2x2x2 between subjects ANOVA.

• The normality assumption doesn't go away simply because your skewed sample is representative. The assumption relates to finding the distribution of the test statistic under the null hypothesis so you can calculate p-values; "understanding why" it matters, and the manner and extent to which it may matter is answered (at least briefly) in a number of posts already; e.g. see the discussion here (with a skew example). You should be able to use search to find more. – Glen_b May 14 '17 at 18:07
• @Glen_b tanks for your response and the link. I see. As far as I understand, this also means that bootstrap won't resolve the problem. I tried data transformation but to no avail. I have now thought of two approaches for non-parametric analyses to try and work around the problem: Kruskal-Wallis over the eight groups and then, if significant, individual comparisons, or ordinal regression. Alternatively, I thought about using CIs for interpretation, but again the lack of normality is a problem so that won't work either. Any thoughts would be appreciated. – grey May 15 '17 at 0:52
• Since your response is a count variable (number of critical details out of 5), it might be feasible to look at a binomial or quasi-binomial generalized linear model (GLM). You may need a mixed model (GLMM), depending on how you view the sample you have in relation to the population of interest and the particular inferences you need to deal with. – Glen_b May 15 '17 at 1:38
• would you present a sample of data ? and state your key objective ? Are you interested in technique of data analysis for your data or you have express interest in assumptions ? – Subhash C. Davar Jul 29 '18 at 14:31

## 1 Answer

The normality assumption doesn't go away simply because your skewed sample is representative. The assumption relates to finding the distribution of the test statistic under the null hypothesis so you can calculate p-values; "understanding why" it matters, and the manner and extent to which it may matter is answered (at least briefly) in a number of posts already; e.g. see the discussion here (with a skew example). You should be able to use search to find more.

Since your response is a count variable (number of critical details out of 5), it might be feasible to look at a binomial or quasi-binomial generalized linear model (GLM). You may need a mixed model (GLMM), depending on how you view the sample you have in relation to the population of interest and the particular inferences you need to deal with.

• I've copied these comments by @Glen_b as a community wiki answer because they are, more or less, an answer to this question. We have a dramatic gap between answers and questions. At least part of the problem is that some questions are answered in comments: if comments which answered the question were answers instead, we would have fewer unanswered questions. – mkt - Reinstate Monica Jan 17 '19 at 8:25