Of course, yes.Kendal's tau can be used to measure the correlation between a binary variable and an ordinal variable.

Dear José Ribeiro,

What do you think about comparing the groups of students according to their grades just with Mann Whitney U test? In my opinion it would be more justified.

That is true that I saw several times a correlation between an ordinal variable and even a sex - where there is no order at all! It all depends on the interpretation and description of the results.

If you decide to do a correlation, please differentiate the three nonparametric correlation coefficients: Spearman's rho, Kendall's tau and Gamma.

Spearman's rho is similar to Kendall's tau as far as you consider statistical power and assumptions. The difference is only in interpretation: Spearman's rho can be thought of as the regular Pearson product-moment correlation coefficient; that is, in terms of proportion of variability accounted for. On the other hand, Kendall's tau represents a probability; that is, it is the difference between the probability that the two variables are in the same order in the observed data versus the probability that the two variables are in different orders.

Gamma statistic is preferable to Spearman's rho or Kendall's tau when the data contain many tied observations (ranks).

Good luck in your research!

Michal

You could go straight to regression. If you ran a binary logistic model you could asses if performance (ordinal) can 'predict' the likelihood of the students being in the same class , or not (binary). Be careful of sample size, though. 

First of all, Ette, thank you very much for reasserting my own conviction. I too think that the correlation between categorical variables (one ordinal and the other dichotomic, but still ordinal) can be expressed with the help of Kendall's tau correlation coefficient. The problem - to me - is the choice of the better coefficient.

Michal, your answer offered me a great number of paths. I totally agree with you that Mann-Whitney (or Wilcoxon-Mann-Whitney) it is a great choice, as it enables to say which one of the two dichotomic options (YES/NO in my example) has the best scores (given in the ordinal 1-5 scale). Why have I considered computing a correlation coefficient, if Mann-Whitney test gives a straight answer?

The main reason is that (I choose a very poor example, I regret) I am also thinking of trying the opposite, that is, compare the proportion of YES/NO given a certain value of the ordered 1-5 scale and, in that case, Mann-Whitney can’t be used

So, in alternative to the Kruskal-Wallis test (which is less informative), I was considering the use of non-parametric correlation coefficients, like the Kendall's tau-b, to account for tied values. I know Kendall-c is doesn’t take ties into account, but I supposed Kendall-b would perform better, if many observations are ranked together. Do you think Gamma deals better with that problem?

And, finally, Jean, you gave me a good option too. I am not used with logistic regression, but in this case perhaps I would try it and see the results.

Many thanks to all of you,

José

Thank you, Jose, for your feedback.

I found this discussion valuable as we all see there may be a great variety of tests to estimate if there is "something" in your research. My conclusion is: "the research question should be the prerequisite for the correct choice of the test". I mean it depends which question would you like to answer. And it is vitally important to chose the test (or a battery of tests) a priori i.e. during the time of planning the experiment - in order not to account for "type III error".

I used Gamma only once in my research. It was the situation where there were plenty of ties (up to 70%), and in this situation it was a little bit more powerful than Spearman rho. In my opinion the chose of Gamma should depend on your data. The other question is what is "many" ties. I have never heard a good answer. Maybe you have some proposal?

Regards,

Michal

Michal, I deeply aknowledge your help and expertise in this interesting ongoing discussion.

I never used Gamma and my option for Kendall’s tau-b is mainly based in my reading of “Agresti_1976_J. Am. Stat. Assoc._The effect of category choice on some ordinal measures of association”, a justification that of course if far from sufficient to decide between the two tests.

Just another thing... I don't how that works in other countries, but in Portugal the 1-5 scale employed in grading students' achievement results from the agglutination in five unequal categories of the 0-100 (percentage) scale. That is, if a student in a ordinary test or a national exam has between 0 and 19%, we gave him a 1; between 20 and 49%, a 2; between 50 and 69%, a 3; between 70 and 89%, a 4; finally, between 90 and 100%, a 5.

Correspondently, if a student has a 3 or more, he passes the exam. Strangely, the aritmetic mean (???) of these 1-5 values (and not of the 0-100% scale) is often used by Ministry of Education to rank schools by their performance in national examinations in Mathematics and Portuguese, which is obviously wrong.

Best regards,

José

Dear Jose. If you may include the percentages of points the students were given -  instead of the grades, it could increase the statistical power of your analysis and make the variable more continuous (however not fully).

Michal, I totally agree with you, that's why I don't understand why the so-called school rankings (that lately have been in the fashion in the Portuguese media) are based in school's aritmetic means of 1-5 grades from national examinations and not in the corresponding values (although not entirely continuous, as you correctly pointed out) of the percentages of points.

I admit that a value of 3.03 is better to remind than a 59.37%, but still...

This can be applied. Changing dichotomous data to ordinal data has no effect on the analysis. The analysis becomes ordinal by ordinal if the other variable is ordinal!