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I have a very straight-forward question:
If the difference of scores is normally distributed, the sample distributions don't matter in a paired t-test, right?
Therefore, If the samples are not normally-distributed, but their differences are (Scores 2 - Scores 1), then it's ok to run the parametric test?
I've seen some related questions, but not this specifically one (here here , for example).
You are correct. A paired t-test is conducted on the differences of the paired scores. It doesn't look at the individual scores in any way.
A paired t-test is precisely the same as a one-sample t-test on the differences of the pairs.
Consider the following observations, A and B. While the distributions of each are skewed, the differences are relatively symmetric and bell-shaped in distribution.
The results of the paired t-test and one-sample t-test are the same.
A = (1, 2, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7)
B = (0.98, 2.10, 3.84, 4.11, 3.02, 3.01, 5.67, 5.07, 6.20, 5.67, 6.77, 7.61, 6.15, 9.32, 8.41, 7.71, 8.64, 8.49, 8.15, 9.00, 8.80, 7.85, 8.90)
In R:
A = c(1, 2, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7)
B = c(0.98, 2.10, 3.84, 4.11, 3.02, 3.01, 5.67, 5.07, 6.20, 5.67, 6.77, 7.61,
6.15, 9.32, 8.41, 7.71, 8.64, 8.49, 8.15, 9.00, 8.80, 7.85, 8.90)
hist(A)
hist(B)
Difference = A - B
hist(Difference)
t.test(A, B, paired=TRUE)
t.test(Difference)
### Paired t-test
### t = -3.3339, df = 22, p-value = 0.00301
###
### One Sample t-test
### t = -3.3339, df = 22, p-value = 0.00301
