Analyze non-normally distr. data from within-subjects-design

#1
Dear all,
I conducted a study where I asked participants to rate music, text and combined stimuli of these music und text stimuli on a likert-type scale. The music and text could each have one out of 4 different emotions that they should try to convey and the combined stimuli could have a combination of these, so that in the end there were 4 songs without lyrics (control variable 1), 4 spoken lyrics (control variable 2) and 16 combinations (dependent variables). Every participant had to rate every single one of these stimuli in a randomized order. The rating was done by asking for how strongly all the four possible emotions were conveyed on a five point likert scale ranging from "not at all" to "very much". It was possible to say that there was happiness as well as sadness conveyed by the song at the same time using these unipolar scales.

The data that I got from this was not at all normally distributed. For example, a happy song with happy lyrics had a very left skewed rating to the "very much" side for the happiness-rating (in other words 80% or more chose the highest or second to highest rating, meaning its a happy song) and a very right skewed sadness-rating to the "not at all" side (80% or more chose the lowest rating, meaning it's not a sad song).

Is it right to assume that I cannot use an ANOVA to test for difference between the song-only, lyrics-only and combined-ratings? And if so, would a Friedman-Test be appropriate or is there a more fitting test? It sounded to me like a Friedman-Test was usually used for a repeated measure design. Could the questions for the control variables, which in my case are the song-only and lyrics-only stimuli, be seen as a control test and i can therefore use the Friedman-Test?

Thanks in advance,
Martin

PS: Here you can see the distribution of the example I mentioned, happy song and happy lyrics and a combination of the two, ratings for how "happy" the stimuli were:
ratings_happy.jpeg
 
#3
Thank you, that's helpful! Sorry that these are probably questions that have been asked countless times, but I'm fairly new to the statistics game and have to learn most of it by myself. So thank you that there was no rant, I'm always eager to learn and know that I do a lot of mistakes still :) I'm aware of the problem of ordinal data with metric models, but as my professor said I shouldn't worry too much in this case and that there are lots of debates about the topic, I thought it would be okay at least for this study.
Sadly, the residuals of the ANOVA of my example data I was talking about has no normality either. It looks something like this: qq_anova.jpeg residual_density.jpeg
And a quick Shapiro Test says the same (p< 0.05). So would it then be more appropriate to use a Friedman-Test?

(The R Code looks something like this: happy.anova <- aov(Combo ~ Song + Lyrics, data = df), which hopefully is right in this case.)
 

spunky

Doesn't actually exist
#4
Sadly, the residuals of the ANOVA of my example data I was talking about has no normality either
Why would you expect the residuals to be normally distributed? Your dependent variable is bounded from 1 to 5 in discrete intervals. Just look at your quantile plot and see how they move in steps as opposed to a smooth, continuous curve.

In any case... you're in one of those cases where I'm honestly not sure how I'd proceed without a more through analysis like a Monte Carlo simulation to see how off you'd be if you were to do one instead of the other. I guess in the meantime, just to be safe,the Friedman test may not be a bad idea.

Now, after re-reading your original post I'm not entirely sure of your design... is this a One-Way ANOVA with 2 levels of the factor? Like your factor is "song" and the levels are "yes lyrics" and "no lyrics"? What are these "16 combinations (dependent variables)"? When I read 16 dependent variables it makes me think that 16 ANOVAs need to be ran? But then you have that as a whole separate panel in your ggplot2 graph so i'm confused.
 
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