I am a teacher in a language program at a university, and I was interested in investigating whether or not there is a correlation between the number of sessions a student spends in our program and their performance on the TOEFL (an English proficiency test). I have test scores from students who took this test before entering our program, and test scores from the same students who took the test again after spending 1 - 4 sessions in our program.
To be clear, these students did not take the test again after each session, so for any given student, I only have their pre-program test score and an additional score from when they took the test again (either after 1, 2, 3, or 4 sessions in our program).
Is there a way to analyze this data to learn whether or not the time spent in our program correlates to an improve in performance on the TOEFL test? My experience with statistics of this kind is limited, but I thought I could use the Kruskal-Wallis test to analyze the gain scores between pre and post tests between the four groups of students (students who spent either 1, 2, 3, or 4 sessions in our program) to see if there is a meaningful connection. A co-worker thought I should use Kendall's tau coefficient instead, which I have not used before.
Would either of these statistics be appropriate for this situation? This is not data from an experiment, and the students are all from different populations and took this test in different locations after studying with different teachers in our program, so I realize that there are many confounding variables. Any guidance would be greatly appreciated!