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Dear Colleagues,
Generally, most PK parameters are log-normally
distributed including Cmax that is why there is a
tendency to report the Cmax of several subjects as
geometric mean. However, we believe plasma
concentrations are normally distributed continuous
variable and adequately characterized by arithmetic
mean. I see a little bit of controversy here. Would
some one please provide some explanation?
Rostam
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I often found plasma concentrations are log normally distributed.
Xia
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Thanks Xia, what statistical parameter do you use as a
measure for the center of a distribution for each time
point? Arithmetic mean, median, geometric mean?
Rostam
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Before the exploratory analysis, I would like to see how plasma
concentration distribution looks like. Usually, after log transform, the
distribution is very close to normal. This can be seen if using Q-Q
plot.
If log normally distributed, then geometric mean is a good statistics
for
center location. Please correct me, if I am wrong.
Thanks
Xia
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Dear Rostam,
Under the (usually fair) assumption of log normality, the arithmetic
mean is a biased estimate for the center. Both median and geometric
mean are unbiased estimates for the center but the geometric mean is
more 'precise' (smaller variance) though less robust for outliers. The
geometric mean may seem to be the best choice but I do not favour using
the geometric mean in case some points may be 'zero-like' (e.g. below
the detection limit) since this complicates things. Hence I propose to
use the geometric mean and median for Cmax but take the median as
primary center statistic for the individual time points.
I did a small simulation exercise by drawing 10,000 samples from the
standard normal. Anti-logging this sample yields a lognormal
distribution with center 1.
I obtained :
Arithmetic mean = 1.642 (biased !)
Median=0.999 (unbiased)
Geometric mean=0.988 (unbiased)
There are a number of diagnostic tools to see if a sample is drawn from
a (log) normal distribution but I should not 'overdiagnose',
particularly not when the sample size is small.
Regards,
Paul Meyvisch
Statistician
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Dear Paul,
Thanks for your e-mail, very helpful indeed. But I
have never seen the average time-concentration profile
for group of patients express as median concentration
vs time. These data often presented as MeanąSD or SE
and I mean arithmetic mean. So, may be other s can
comment on this? How do regulators likes to see the
data presented?
On the same subject, Xia, suggested that usually,
after log transform, the distribution is very close to
normal. This can be seen if using Q-Q plot. I was
wondering why Q-Q plot and not normal probability
plot? Any comments?
Rostam
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Dear Rostam,
In my work for several CROs, we were in most cases used to present only
a
geometric mean curve. Statistics like 95% confidence interval, min, max
and
median for individual time points were calculated and only presented as
numbers. In some cases, sponsors required median curves and back
calculated
SDs.
Aritmethic means are not OK. Try it yourself: Put the arithmetic curve,
the
geometric curve and the median curve in one graph and you will see that
the
median curve and geometric curve are more or less equal while the
arithmetic
curve is clearly different. This is an indiciation that the data is
log-normally distributed. Also normal probability plots, will confirm
this.
As far as I know, regulators are used to see geometric mean curves. So,
I
would advice to create them.
Kind Regards
Berber Snoeijer
Netherlands
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Dear Willi,
In your textbook 'Parameters for Compartment-free
Pharmacokinetics' (Shaker Publishing House,
Aachen,Germany 1999) in Chapter 8, Table 8.1 you have
indicated the following values/statistics should be
reported for individual plasma conc at each time
point: arithmetic mean, SD, median, min and max.
Do you agree with recent discussion between Berber and
I that geometric mean should also be reported and
perhaps it is more relevant? I noted all the CROs that
I have worked with only report arithmetic mean ą SD
(not SE). Is this because most people are used to
arithmetic mean? Please comment. Thanks
Rostam
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