I determined some parameter in a group of subjects under two conditions, such that each subject was tested in both conditions. I want to know if the condition has an effect on the parameter. My design is obviously paired. However, previous experience with this experiment makes me believe that the subject's response is independent during each measurement. Thus, I assume that subjects are independent and not paired. Accordingly, I decide to analyze the data with an unpaired t-test. But out of curiosity, I also run a paired t-test, with the following results:
P value for the paired T-test is 0.08 (no difference). P value for the unpaired T-test is 0.04 (significant difference).
I am confused because I thought that a paired test should always give lower p-value (or equal in the worst case) than an unpaired test, by eliminating between-subject variability and increasing power.
1) When can a paired test produce higher p-value that an unpaired test? What does my result mean in terms of sources of variation? It seems that pairing the subjects not only does not eliminate between-subject variability, but adds some kind of a new variability.
2) Can this result tell me something about my assumption that subjects are independent? Can it invalidate this assumption?
3) In summary, is it always a matter of choice to pair or not to pair subjects during analysis? Or is it formally incorrect to use an unpaired test when subjects are logically paired?