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I'm having difficulty finding out what I can do with my data set. It comes from a french reading test that have 30 multiple choice questions.

For each of these questions:

  1. I have the facility value for the items on the test, wich is the percentage of right answers to the questions.
  2. I have the perception of the difficulty by the test takers, on a likert scale (Really easy, Easy, Medium, Hard, Really Hard).

Here is an example of what my data looks like:

Example of my data

I want to know if there is a correlation between the facility value and the perceived facility of the items... In other words, if the easy questions are perceived as easy ones. I don't really know what to do with the data... Should I convert the scores of the likert scale? Should I do a "mean" for each item?

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I recommend calculating Kendall's $\tau$. This effect size estimate is appropriate for ordinal data and permits nonparametric hypothesis testing if you're so inclined. No conversion of your data should be necessary if your Likert ratings are already stored in an ordered fashion. You can convert them to 1–5 if not and correlate these values with your percentage variable.

In , the command is cor(x,y,method='kendall') where x and y are your two variables. To perform a hypothesis test, just change cor to cor.test. I prefer to interpret $\tau$ on the scale of Pearson's r, as I am more familiar with that scale and expect most audiences to be as well. To convert $\tau$ (which is always smaller) to r, use $r = \sin\big(\tau\cdot\frac \pi 2 \big)$ as I mentioned in another answer.

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  • $\begingroup$ Thank you for your answer! Unfortunately, I'm not able to code in R (but I'm learning, though!). I also have a question. With the Kendall's tau, do I have to calculate this value for each Item? $\endgroup$ – user54026 Aug 14 '14 at 15:00
  • $\begingroup$ You can definitely do this in R without any previous experience. Only basic commands will be necessary. You could probably calculate Kendall's tau by hand if you have to. It sounds like you have 30 observations of two paired variables: one observation per item for each variable. That would mean calculating Kendall's tau one time across all of your items. Please edit if I'm misunderstanding the structure of your data (e.g., if each participant has different facility values for different items, you might have a hierarchical structure). $\endgroup$ – Nick Stauner Aug 14 '14 at 23:50
  • $\begingroup$ I edited my question to illustrate with an example of what my data looks like. When the facility value is high, the item is easy. I would like to know if people in general perceived the right degree of difficulty for the items... If I had only one value each time on the likert scale, I would have paired values. But it the answers on the likert scale are not that easily interpreted... I hope I am clear enough, as english is not my first language (I speak french). Again, thanks a lot for the help, I appreciate it a lot. $\endgroup$ – user54026 Aug 15 '14 at 14:39

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