How do I compare between-test changes in mean across groups with different starting means

I have 10 groups that took a test at two different points in time. Each group had a different starting mean. i think my null hypothesis is that all groups improved the same amount relative to their first test mean. I am confused about how to actually compare the differences in means across the groups.
Test1 means:
Group A: 85
Grp B: 78
Grp C: 79

Test2 means:
Grp A: 87
Grp B: 81
Grp C: 82

Because the full test score data is normally distributed, intuitively I understand this to mean that the difference between an 85 and 87 is bigger than the difference between a 79 and an 82 (i.e. each additional point of improvement above the mean is "harder" than the last point of improvement). How do I quantify this idea so that I can actually compare the differences in improvements across groups?


TS Contributor
Do you have the individual data from all group members at tests 1 and 2, respectively?
And how large are the 10 groups?

With kind regards



Less is more. Stay pure. Stay poor.
What is the difference between groups?

The best approach may be linear regression controlling for group variable and starting values. you will need to set a reference group, usually the group with the lowest value or you could conduct pairwise comparisons, but in either approach you should control for pairwise false discovery given you are making multiple comparisons.

I am guessing the differences in starting values js related to non-randomization of group assignment. Also is the time increment between all measurements equal? Lastly, what does your outcome represent, its not a percentage score is it?
The difference between groups is that each group is a different school. So the differences in starting values is definitely related to non randomization of group assignment. Time increment in between all measurements is equal. And the outcome represents a raw score that just happens to be a score out of 100 points.


Less is more. Stay pure. Stay poor.
One of the least frown upon approaches is to model time two scores using group and initial score. However, using linear regression for a bounded outcome (contained within 0-100) can be problematic. In particular, if you slap confidence intervals on the outputted results the value may be beyond 100%, which would be impossible. The alternative is to use beta regression which is not as interpretable to most people as linear regression.