Statistical Methods in Medical Research 4ed

632 Clinical trials
Treatment period interaction, and carry-over
The analysis illustrated in Example 18.4 assumes that the response is affected
additively by treatment and period effects, i.e. that there is no treatment period
(TP) interaction. The terms g AB and g BA in (18.4) represent such an interaction.
Note that it is only their difference that matters: if these two terms were equal
they could be absorbed into the period parameter p2 and we should be back with
the no-interaction model. There are various possible reasons for an interaction:
1 If the wash-out period is too short, the response in period 2 may be affected
by the treatment received in period 1 as well as that in period 2.
2 Even if the wash-out period is sufficiently long for a period 1 drug to be
eliminated, its psychological or physiological effect might persist into period
2.
3 If there is a strong period effect, changing the general level of response from
period 1 to period 2, the treatment effect might be changed merely because it
varies with different portions of the scale of measurement.
A test for the existence of a TP interaction may be obtained as follows.
Denote the sum of the two responses for subject j in group i by
sij ˆ yij1 ‡ yij2:
If there is no TP interaction, the expectation of sij for this subject is, from (18.3)
and (18.4), 2m ‡ jij ‡…tA ‡ tB†‡…p1 ‡ p2†. Since the subjects were assigned
randomly, the jij can be taken to be randomly distributed with the same mean
(which can be taken as zero, since the overall mean is accounted for by the term
m in (18.4)), and a variance s2 0 , say. The expectations of s1j and s2j are thus equal.
If, on the other hand, there is a TP interaction, the expectations will differ: if
gAB > gBA, for instance, as might happen if the two treatments had different
carry-over effects, the expectation of s1j will exceed that of s2j. We can therefore
test the difference between the two mean values of sij, s1 and s2, again using a
two-sample t test. Here, however, the within-group variance estimate s2 s is that of
the sij, not (as before) the dij. Since the sij vary in part because of the variability
between subjects, we should expect s2 s > s2 d .
Example 18.4, continued
The values of sij are given in Table 18.5. The two means are s1 ˆ 13 412, s2 ˆ 16 583. The
pooled variance estimate is s2 s ˆ 41 890 (substantially larger than s2 d , as expected) and
SE…s1 s2† ˆ …41 890† 1
r
1
‡
17 12
ˆ 2 440:
Thus, t ˆ…13 412 16 583†=2 440 ˆ 1 30 on 27DF (P ˆ 0 20). There is no clear
evidence of a TP interaction, and the earlier analysis therefore seems reasonable.