Paired t test: check normality with residuals or paired differences (or both)?

Question

One of the assumptions of the paired t test is that the underlying data is normally distributed. I've read either the paired differences should be normally distributed, or, that the residuals should be normally distributed (which I thought was the case for ANOVAs / regressions).

I've also read that the residuals in this case ARE the paired differences (link), or that the residuals for a paired t test are the paired differences for each pair minus the mean difference overall (link).

So, is there one simple answer to my question: when checking the assumption of normally distributed data for a paired t test, should you be checking the paired differences, or residuals?

Answer 1

The paired differences.

A paired t-test is the same as a one-sample t-test on the differences, so the assumption is that the differences are normally distributed. The residuals are simply the differences minus the mean difference.

n = 10
x = rnorm(n)
y = x + 1 + rnorm(n, 0, 1)
t.test(x, y, paired=T)
# Paired t-test
# 
# data:  x and y
# t = -3.0863, df = 9, p-value = 0.01301
# alternative hypothesis: true difference in means is not equal to 0
# 95 percent confidence interval:
#   -1.9138020 -0.2948852
# sample estimates:
#   mean of the differences 
# -1.104344 

d = y - x
t.test(d)
# One Sample t-test
# 
# data:  d
# t = 3.0863, df = 9, p-value = 0.01301
# alternative hypothesis: true mean is not equal to 0
# 95 percent confidence interval:
#   0.2948852 1.9138020
# sample estimates:
#   mean of x 
# 1.104344