It would be unconventional, but not improper, to use compute the
Repeated Measures analysis as I described, and essentially achieve
a one-tailed, paired t-test by dividing the observed p-value by two.
You have not described the Treatments or the measurements or the
logic of the experiment, so, maybe you do have reason to ignore Order
and reason to use a one-tailed (paired t) test.
I hope that you recognize that using a one-tailed test in an experiment
is also unconventional. Some journal editors and reviewers are more
opposed than others. One criterion that I have suggested to consultees:
Would it be totally uninteresting and not worth reporting if the result were
significant in the wrong direction? - For some experimental Treatments,
that might be the case, but that still leaves you with reader prejudice.
You will avoid criticism and complaint if you increase your sample by a
bit to achieve the same power for a two-tailed test. My memory of
applying a one-tailed test in clinical research is, I did it ONCE: It was for
testing an ancillary, a-priori hypothesis, which we wrote up in the grant
proposal. It was to be performed on a subsample in a larger (expensive)
study, so there was no realistic chance of increasing the N; and "one-tailed"
was specified in order to achieve a half-decent power (60% or 70%). The
journal accepted it because the grant proposal proved we were not making
up the argument after the fact.
Rich Ulrich