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Say you have a group of 30 students and you measure each individual's performance on a test at 4 intervals throughout the year. (For the purpose of this investigation, assume the tests taken are identical.)

What tools can I use to investigate the change in their performance over time?

In particular, I want to see if there is a positive correlation between the marks and the time intervals (if students have generally improved over time). It would also be interesting to see if the amount of improvement observed varies depending on initial marks (do students who begin with a higher mark tend to improve more than students with lower marks?)

My first thought was to fit some sort of simple regression model, but with each observation being "linked" to a particular student, it wouldn't be accurate to treat them all as random measurements.

Peta Hillier
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  • You seem to be meaning rather some kind of causal inference then correlation, aren't you? – Tim Oct 19 '19 at 10:07
  • @Tim Not quite, at the moment I just want to see if there has been any change over the course of that time and what general difference that might be, not necessarily what caused that difference, hence analysing only the observations themselves (initially) rather than factoring in any other predictors. – Peta Hillier Oct 19 '19 at 11:35
  • You are vague about the measure of 'improvement'. Would it be enough if scores $D_i = X_{4i} - X-{1i}$ (last minus first exam scores) are significantly greater than 0? Then maybe a paired t test on first vs last test. Or maybe you want something like $C_i = 3X_{4i} + X_{3i} - X_{2i} - 3X_{1i}$ signif above 0 (one sample t test on $C_i$'s). A really bad score on 3rd exam could ruin a student's C-score. – BruceET Oct 20 '19 at 02:41

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