Thank you so much, that was really helpful.

Then I'd look for whether robust standard errors can be used also in a mixed ANOVA.

Actually this is a great suggestion, and exactly what is recommended here either to do a transformation

or use the WRS2 package for a robust ANOVA (

https://www.datanovia.com/en/lessons/mixed-anova-in-r/#check-assumptions)

Don't you think that this pursuit of transformation on purely mathematical grounds is completely

detached from the substantal questions for which the study is done?

As I totally agree with your line of argumentation, I potentially will look into the robust mixed ANOVA with WRS2 Package.

Our dependent variable is somewhat strange. It is a sum score build on a questionnaire measuring an attitude.

The authors of the original questionnaire suggested to first standardize (z-transform) all individual items and then add them up.

This is why we have naturally negative value in the dependent variable, and a mean of somewhat around zero. (Zero for all the individual z-transformed items, but of course not for the sum of all items). So in response to your question, I guess there is no such thing as a natural null point (or absolute scale).

As I am intrigued and highly motivated in understanding the statistics behind and reasoning behind the analyses decisions, I am still wondering

why it is, that:

have the same size, and/or the difference of variances is not very large,

then you can proceed

I understand your point that inferential hypothesis tests like the Levene's test always

depend on sample size (thus power). Is that the main point? Is there any reference one could cite on that?

**Is this exemplary sentence for the results section correct / acceptable?**
Homogeneity of the residual variances at the post measurement time point was rejected, Levene's test

*F*(2,142) = 2.47,

*p* = 0.047.

However, due to roughly equal group sizes (Stevens, 1999) and only moderate differences in the residual variances, we

continued with the analyses.

(Of course with the addition of a sentence or paragraph in the discussion).

Additionally I am looking for a way to classify or judge the differences in the variances in the Levene's test.

E.g., an effect size for the differences in the variances.

I found this

https://stats.stackexchange.com/que...for-difference-between-variances-levenes-test
hinting to the variability ratio (Nakagawa et al., 2015), could it be a good addition to the sentence above, to provide an effect size

for the Levene's test, in order to quantify / justify the decision of a "not to large" difference in the residual variances?

Nakagawa, S., Poulin, R., Mengersen, K., Reinhold, K., Engqvist, L., Lagisz, M., & Senior, A. M. (2015). Meta‐analysis of variation: ecological and evolutionary applications and beyond. Methods in Ecology and Evolution, 6(2), 143-152.

PS: I also ordered the 3. ed. (2013) of the Stevens book from our university library, and am still waiting but from the preview one can obtain over at google books, it seems the referenced rule of thumb (1:1.5) have been moved to another chapter (or page), or removed

https://books.google.de/books?id=cv...aXv3Wn8Z&lr&hl=de&pg=PA74#v=onepage&q&f=false
**As an Addition to future readers:**
I will be inspecting Harltey's Fmax (variance ratio), as explained in Andy Field's (2009) book "

*As with the K–S test (and other tests of normality), when the sample size is large, small differences in group variances can produce a Levene’s test that is significant (because, as we saw in Chapter 1, the power of the test is improved). A useful double check, therefore, is to look at Hartley’s FMax, also known as the variance ratio (Pearson & Hartley, 1954). (...) 10 is more or less always going to be non-significant, with 15–20 per group the ratio needs to be less than about 5, and with samples of 30–60 the ratio should be below about 2 or 3*" (Field, 2009; Chapter 5.6 p. 150)

**Interpretation of Hartley's Fmax in our case:**
In our case would be Fmax = 1.86, so smaller than the critical values (obtained from a table of 2.78), thus, we can proceed with the ANOVA?

Field, A. (2009).

*Discovering statistics using SPSS*. Sage PublicationsSage CA: Thousand Oaks, CA.

All the best

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