What is the p-value for paired t-test if the two set of data are identical? As in the title, I have two sets of data that are identical. I performed a paired t-test using R. 
The p-value for the difference estimated (which is essentially 0) is NaN. 
I wonder if there can be a theoretical p-value for this case.
 A: *

*If the two sets of data are identical because the variables are defined on a discrete set of values, then the assumptions of the t-test are false, since the random variables aren't even continuous.
As such, any normal-theory calculation would not yield the correct p-values. 
I think the "correct" t-test p-value would be NaN.

*Alternatively, consider the degenerate case where continuous variables have finite variance but the differences have variance 0. Then the t-statistic would be 0/0; again I'd say that's "correctly" NaN. (edit: more details of that argument are in a comment)

*However, if you (for example) assumed some discrete distribution for the differences and derived say a likelihood ratio test, or did a permutation test, you'd get a (legitimate) p-value of 1, since anything but zero-differences would be "more extreme" than all-0-differences.
So I think justifiable p-values for a t-test will actually be what R gave you, while justifiable p-values if you have discrete variables and choose a more appropriate test would generally be 1.
A: The null hypothesis for a paired t-test is that the mean difference of your two paired samples is equal to zero.
Considering you have identical values for each pair, the difference should always be equal to zero, thus failing to reject the null hypothesis.
This would mean p-value = 1 or NaN (0/0), but this doesn't really imply you should conclude anything.
A p-value of less that 0.05 will suggest the existence of a significantly different mean for each sample.
