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My experiment is a naming task, in which participants name pictures and their vocal responses are recorded and the onset of vocalization. The data I am dealing with are reaction times in ms and accuracy ( correct answers in percentages. I did the analysis of reaction times and what is left is the accuracy data in which I need your valued help.

Now, just to explain what I have done so far: Each subject has an average score results for four conditions: Condition A, scored 10/15 , B 9/15, C 6/15 , and D 7/15. I transferred the discrete numbers into percentages and intended to use a mixed anova as I have One independent variable with two levels and two dependent variables with two level each.

My questions are:

1- Do I have to run a normality test for these data? the data size is over 30? 2- what if they were not normally distributed? what should I do? 3- Is transferring data into percentage right or wrong?

Many Thanks.

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Sounds interesting! Transferring counts into percentages definitely loses some information: 1/2 is not the same evidence as 100/200, but the difference is lost as a percentage.

ANOVA is pretty popular, but as you note in the question it can have problems. Because you have the classic linguistics/psych use-case of non-independent data (ie grouped by participant), you probably want to jump on the mixed effects model bandwagon if you possibly can. I hear it's all the rage.

Some justification/motivation here:

Jaeger, T. F. (2008). Categorical data analysis: Away from ANOVAs (transformation or not) and towards logit mixed models. Journal of memory and language, 59(4), 434-446.

And R package to do the heavy lifting for you here:

https://cran.r-project.org/web/packages/glmm/vignettes/intro.pdf

Sounds like you have the binomial-input case.

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The comment by @steveLangsford is good. Also, you should always make a habit of checking normality of your statistical tests have it as an assumption. If they are not normal, you can adjust your analyses or use transformations. If they are unmanageably non-normal, the original tests can't be interpreted anyway. Ideally, these decisions and their criteria would be settled before you see the actual data. But it's normal to just transparently disclose what you decided and when in terms of analysis.

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