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We have a 3-period, 3-sequence cross over design in which we would like
to analyze for food effects. The study design:
Sequence 1: A B C
Sequence 2: B C A
Sequence 3: C A B
where: A is fasted
B is low-fat meal
C is high-fat meal
7 people are assigned to each sequence. For period 1, 12 people (4 per
sequence) will be dosed on day W and 9 people (3 per sequence) will be
dosed on day X. All 21 people will be dosed for period 2 on day Y and
all 21 people will be dosed for period 3 on day Z.
Normally we would use the following model in which period has 3
levels:
y = sequence + subject(sequence) + period + treatment
For the above study, we are considering using the same model in which
period has 4 levels. Do you think this will work or do you have any
other suggestions?
Thank you for your time.
Jack Jenkins
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Dr.Jenkins,
One possible statistical model is repeated-measures analysis
of variance. In this case the sequences would be the different
treatments, and the differences in bioavailability would be the
treatment effects. If the measured treatment effect (F) does not
have a normal distribution, then a nonparametric approach in the
form of a Friedman Statistic could be applied.
Recent statistic books such as Daniel's Biostatistics or Glantz's
Primer of Biostatistics have chapters on repeated-measures analysis
of variance and the use of the Friedman statistic in the nonpara-
metric case.
If there is covariate affecting bioavailability in addition to
the treatment (diet sequence), then an analysis of covariance study
design could be used. This is found in more advanced statistic books
such as Dowdy's Statistics for Research.
Mike Leibold, PharmD, RPh
ML11439.aaa.goodnet.com
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I think as soon as we have enough wash-out period between day X and day Y
or day W and day Y, we don't need treat that model as 4-level. Otherwise
the whole cross over design and model will be ruined.
Yamei Wang
Statistical Programmer/Biostatistician
Kendle International Inc.
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A good site for stats info and online calculator scripts is:
http://member.aol.com/johnp71/javastat.html
Carol A. Roby, PharmD, MS
Clinical Pharmacist
St. Agnes Hospital
900 Caton Avenue
Baltimore, MD 21229
410-368-3109
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Hi Jack/All,
I thought I would hopefully be able give you some guidance regarding
your query. My thoughts are split into two parts. Firstly, some
personal thoughts on why you should look for a period effect in
general, and the second part I hope is the analysis I think you
should undertake.
Why look for period effect
--------------------------
There is often no good reason to suspect a period (or centre) effect
in a PK or biomarker study. These effects shouldn't, in theory,
exist. There is often no true physiological reason for these
differences. Unfortunately, in the real world these differences
frequently exist in the data derived from multi-centre or multiple
period studies, and the magnitude of these effects can be very large.
The obvious suspects are differences in the aquisition, storage,
transfer and analysis of the samples. This potential additional
variability/bias has to be considered. If my biomarker has a CV of
20% based on 50 subjects from a one centre study, will it be 20% if I
had recruited just one subject from each of 50 centres ? In theory,
perhaps yes, in practice, perhaps no.
Analysis
--------
This I hope is a fair re-ordering of your data, which may clarify
some of my comments. I referred to the two occasions for the first
period as w and x, as per your email. The first 12 subjects as
cohort 1, and subjects 13-21 as cohort 2. For simplicity, I have
called the first 4 subjects 1-4, although I know you would have
randomised the subjects within cohorts.
Data per1 per2 per3
Occ w x y z
sub 1-4 a . b c
sub 5-8 b . c a
sub 9-12 c . a b
sub 13-15 . a b c
sub 16-18 . b c a
sub 19-21 . c a b
My opinion is that w and x should be linked under the per 1 banner,
as two levels under per 1. These two levels are intrinsically more
similar to each other than per 2 and per 3 (the first time the
subjects enter the study). This "w versus x" contrast is of interest
in the preliminary analysis.
Some things become more obvious with this layout
i) differences between 'w' and 'x' is a between subject comparison.
ii) differences between sequence is a between subject comparison.
iii) differences between period is a within-subject comparison.
iv) differences between tmt's is a within-subject comparison.
Thus the 'technical' ANOVA (and degrees of freedom) can be written as:
d.f.
w versus x 1 (compare to subject error)
sequence 2 (compare to subject error)
Subject 17
tmt 2 (compare to residual error)
per 2 (compare to residual error)
total 24
Residual (+ everything else(e.g. interactions)) = 2 d.f. for each
within comparison for each subject less 2 for period, less 2 for tmt.
Hence 2 * 21 - 2 - 2 = 38. Hence the good news is the d.f. add up,
and we have N-1 (63-1) total d.f. = 62 (24 from above, and 38
residual).
Our 'weak' between subject contrast of the 'w versus x' to the
subject to subject error, will guide us to the appropriateness of
investigating period effects at the within-subject level. The
sequence effect should (I haven't checked) be orthogonal to the 'w
versus x' comparison.
The reason I wrote 'technical' above the ANOVA table is that I'm not
completely sold on this analysis. The confounding with cohort,
combined with the low power of this contrast, doesn't seem great.
Similarly, I do not like a 'sequence' effect, because I think the
presence or absence of this effect has more to do with whether 7
subjects happen to be giving a higher response than 7 others, rather
than anything to do with the sequence of treatments they actually
received. However, I think your consideration and definition of this
problem was wholly merited.
I hope this helps,
good luck
RA Fisher
p.s.
I would use the reverse williams square for subjects 13-21. That is,
sequences "a c b", "c b a", and "b a c". This ensures that any
potential differencial carry-over is minimised.
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