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Can
we eliminate carry-over?
There
is a simple solution for the AB/BA design and that is simply to
ignore the second period data altogether and use the first period
data only, comparing them between groups as in any parallel group
trial. Of course, to do so defeats the whole purpose of the cross-over
trial and the resulting estimator, which shall be referred to
as PAR, lacks precision and any test using it lacks power.
If we
compare CROS and PAR, then the former is efficient but potentially
biased whereas the latter is unbiased but potentially inefficient.
We thus have a clear example in choosing between them of what
is referred to in statistics as a bias variance trade-off.
More
complex designs
Consider
the design in four periods using the two sequences PPVV/ VVPP.
Suppose that in a preliminary step to analysis we calculate the
so-called cell means corresponding to the cross-classification
of periods and sequences. Since we have four periods and two sequences
we have eight possible cross-classifications: from first sequence
first period through to second sequence fourth period. To calculate
the cell means, we collect our results together in these eight
sets of values. Having obtained these eight sets of measurements
we then calculate a mean for each set. These eight means are the
cell means.
An intuitively
obvious estimate of the treatment effect is given by weighting
these cell means as follows
-1/4,
-1/4, 1/4, 1/4
in the
first sequence, and
1/4,
1/4, -1/4, -1/4
in the
second, and then adding the results together. If we have exactly
the same number of patients on the first sequence as on the second,
we could achieve the same result by simply taking the mean of
all results measured under verum and subtracting the mean of all
results measured under placebo. Thus under those circumstances
using these cell means would be equivalent to calculating a mean
treatment difference. The advantage of using this more complex
procedure is that if we have unequal numbers of patients on the
two sequences this procedure will give the two sequences equal
weight, rather than every patient, and this will remove any bias
due to the period effect.
However,
the point of this scheme is really to introduce a more complex
possibility for using the cell means. Such a scheme of weights
is given in the table below.
| Table
3.1: Complex weight scheme
|
|---|
|
| Period
|
|---|
| Sequence
| 1
| 2
| 3
| 4
|
|---|
| PPVV | P
-6/20 | P
-4/20 | V
7/20 | V*
3/20 |
| VVPP | V
6/20 | V*
4/20 | P*
-7/20 | P
-3/20 |
|
These
weights appear rather curious and arbitrary but they share the
following properties with the simpler scheme. 1) They add to zero
in any sequence. 2) They add to zero in any period. 3) They add
to one over the cells labelled V and to minus one over the cells
labelled P. Thus period and patient effects are eliminated and
the treatment effect is recovered. The scheme is, however, less
efficient than the simpler scheme, as the sum of the squares of
the weights, to which (under the simplest reasonable set of assumptions)
the variance will be proportional, is 0.55 rather than 0.5. However,
the scheme has another feature of interest. When added over the
periods immediately following treatment with the verum they come
to zero. (The relevant cells have been marked with a * and the
sum of the corresponding weights is 3/20 + 4/20 -7/20 = 0.) Thus,
if carry-over lasts for one period only and depends only on the
preceding treatment, the carry-over effect will be eliminated.
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