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I will conduct a study in which I compare the reaction times (acceptability judgement) to two different types of sentence.

I am assuming that scores will be normally distributed and I will use paired sample t-test but my professor also wants me to state another test for the situation if the scores are not normally distributed? Which test can I use instead of paired sample t-test?

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  • $\begingroup$ It doesn't sound plausible that reaction times are normally distributed. I would expect some long right tail! $\endgroup$
    – kjetil b halvorsen ♦
    Apr 29, 2022 at 12:50
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    $\begingroup$ Here is an interesting-looking paper discussing distributions of reaction time data, with links to the data. $\endgroup$
    – kjetil b halvorsen ♦
    Apr 29, 2022 at 12:56
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    $\begingroup$ (1) If two indep samples of reaction times seem to be of similar shape, perhaps use two-sample (nonparametric) Wilcoxon test. (2) If you think two indep samples might be exponential, then use exact test based on ratio of exponential means. $\endgroup$
    – BruceET
    Apr 29, 2022 at 15:33
  • $\begingroup$ A related post on the psychology SE: psychology.stackexchange.com/questions/16669/… $\endgroup$
    – kjetil b halvorsen ♦
    Apr 30, 2022 at 3:23
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    $\begingroup$ If these data are paired (it sounds like they might be), it's not the population distribution of times you need to be approximately normal but of the pair differences. $\endgroup$
    – Glen_b
    Apr 30, 2022 at 4:26

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Reaction time data will most probably be skewed, with a long right tail. So look for some literature about analyzing reaction time data. One recent tutorial which proclaims

Although RT distributions are not normal (bounded on their inferior side and exhibiting some skewness), psychologists have agreed to consider these distributions normal enough to be processed with these methods. A logarithmic transformation can also be applied to improve Gaussianity.

... which seems strange. But at least it gives one idea: Use paired t-test on the log-transformed data.

One interesting paper, which goes deeper, and have links to all its data, is here. They base some analysis on the exGaussian distribution (distribution of sum of normal and exponential random variables). Some plots of reaction times from that paper

Histograms of reaction times

with the corresponding qq-plots:

QQ-plots of reaction times

and the non-normality is quite clear. Does it look better with a log transformation? :

QQ-plots of log reaction times

and the answer is maybe clear: It does not look much better!

But you have paired data. One way to analyze is letting the pairing variable (person, experimental subject) be a random effect, so you will need mixed models. Mixed models with a non-normal random effects are not very standard ... one paper looking into mixed models for reaction time data is here. (I will try to come back)

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I suggest using the Wilcoxon signed-rank test, which is a non-parametric test for paired observations (so it could be applied to not normally distributed data)

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