Back to the Top
I beg some guidance on two issues:
What is the standard US and European PK terminology for these two
coefficients of variation?
a) sqrt[(Var(x))/(arithmetic mean)]
b) sqrt([exp(Var(log(x)]-1)
Thanks
Carl Dmuchowski
Back to the Top
I never used the word geometric coefficient of variation vs
arythmetic coefficient of variation but it seems that is what we have
here.
Back to the Top
"Dmuchowski, Carl (by way of David Bourne)" wrote:
>
> I beg some guidance on two issues:
>
> What is the standard US and European PK terminology for these two
> coefficients of variation?
>
> a) sqrt[(Var(x))/(arithmetic mean)]
In New Zealand I would call this "nonsense" (the dimensions of the
numerator and denominator are different).
Perhaps you meant sqrt[Var(x)]/(arithmetic mean)? I would call that a
coefficient of variation.
> b) sqrt([exp(Var(log(x)]-1)
I don't recognize this as anything I know about.
Perhaps you could explain why you are asking for guidance on these
strange expressions?
Nick
--
Nick Holford, Divn Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand
email:n.holford.at.auckland.ac.nz
http://www.health.auckland.ac.nz/pharmacology/staff/nholford/
Back to the Top
With respect to term b):
If X has a two-parameter lognormal distribution with parameters m and s^2
(i.e. log(X) has a normal distribution with mean m and standard deviation
s), then the mean and variance of X are
E(X) = exp(m + s^2 / 2)
Var(X) = exp(2m +s^2)[exp(s^2)-1]
where s^2 is s squared.
Therefore, the coefficient of variation of X is
CV(X) = sqrt[Var(X)] / E(X) = sqrt[ exp(s^2) - 1 ].
Perhaps this is what you had in mind.
Jorn
Jorn Attermann, MSc, PhD jorn.aaa.biostat.au.dk
Department of Biostatistics, University of Aarhus
Vennelyst Boulevard 6, DK-8000 Aarhus C, DENMARK
Back to the Top
This is to add a word to the discussion about coefficients of
variation. The question of Carl refers e.g. to the FDA Guidance for
Industry "Statistical Approaches to Establishing Bioequivalence"
(http://www.fda.gov/cder/guidance/3616fnl.pdf), where the
recommendation is :
"that bioequivalence measures (e.g., AUC and Cmax) be
log-transformed"... and "geometric means (antilog of the means of the
logs) should be calculated for selected bioequivalence measures".
The guidance recommends that PK parameters be presented as mean, SD,
geometric means and coefficient of variation, but does not make any
formal statement about the calculation of this coefficient of
variation. There are two options for this :
- either the arithmetic one : CV = SD/mean = Sqrt(Var(x))/mean (as
corrected by Nick)
- or the geometric one : for this I personally use CV =
Exp(SD(Log(x)))-1 rather than the formula quoted by Carl. Both are
however asymptotically equivalent for not too large SDs. The antilog
of the SD of the logarithms has been termed "percentage standard
deviation" by Snedecor & Cochran, who mention this calculation in
their classical textbook (Statistical methods, Iowa State Univ Press,
8th ed, 1989, p. 290-1). For example, if you consider the 3 data 80,
100 and 125, their geometric mean is 100 and the value of
Exp(SD(Log(x))) is 1.25, meaning that the data can be summarized as
100 times-or-divided-by 1.25, meaning a CV of 25%. (Notice that 75,
100 and 125 would have an arithmetic CV of 25%, while 78.2, 100 and
127.9 would have 25% of the CV calculated according to the formula
quoted by Carl, which is the arithmetic CV of a lognormally
distributed variable, as recalled by Jorn).
Thierry Buclin
Division of Clinical Pharmacology,
University Hospital of Lausanne (Switzerland)
Back to the Top
The problem is some of my European colleagues call
'sqrt([exp(Var(log(x)]-1)' the geometric coefficient of variation and I have
been unable to unearth a legitimate academic reference for describing the
coefficient of variation of a lognormal distribution this way. I wanted to
gauge how the discussion group referred to these two expressions for the
coefficient of variation, which are used as summary statistics for AUC and
Cmax when reporting to regulatory authorities. I have only seen the
'sqrt([exp(Var(log(x)]-1)' referred to as the coefficient of variation of
the multiplicative model in the International Journal of Clinical
Pharmacology, Therapy and Toxicology (Diletti et al). My European colleagues
claim Dr. Cawello refers to the 'sqrt([exp(Var(log(x)]-1)' as the geometric
coefficient of variation in his book, Parameters for Compartment Free
Pharmacokinetics, but I have not been able to corroborate this because I
cannot locate a copy of his book and they appear to be reluctant to share
their copy with me.
Back to the Top
A "legitimate" reference can be found in Statistical Distributions by
NAJ Hastings and JB Peacock, Halsted Press, New York, 1975. They
refer to is simply as the coefficient of variation.
Pete Bonate
Back to the Top
Dear colleagues,
With respect to the topic of means and standard deviation of a
log-normal
distribution, I would like to add the following to continue the
discussion.
Jorn Attermann wrote a clear message:
> If X has a two-parameter lognormal distribution with parameters m and
> s^2
> (i.e. log(X) has a normal distribution with mean m and standard
> deviation
> s), then the mean and variance of X are
>
> E(X) = exp(m + s^2 / 2)
>
> Var(X) = exp(2m +s^2)[exp(s^2)-1]
>
> where s^2 is s squared.
>
> Therefore, the coefficient of variation of X is
>
> CV(X) = sqrt[Var(X)] / E(X) = sqrt[ exp(s^2) - 1 ].
This is correct indeed. E(X) is here the arithmetic mean. Assuming that
a
log-normal distribution applies to the data, however, the geometric
mean is
a more appropriate measure of the central tendency of the data, since it
coincides with both the expected value of modus and median (just as the
arithmetic mean coincides with modus and median in a normal
distribution):
'Mean' = Geometric mean = Exp(m)
This is both simple and comprehensible.
Please note that the calculated mean E(X), and Var(X) and CV(X), refer
to
the normal (i.e. not log-transformed) scale. They refer to a skewed
distribution. This implies that these values cannot be used for further
statistical calculations, e.g. the calculation of confidence intervals,
or
as priors in a Bayesian analysis. One may say that E(X) and Var(X) are
the
moments of the normal distribution that approaches the log-normal
distribution with moments m and s^2 as good as possible. However, it is
not
an accurate representation of the true distribution, in particular in
the
tails.
Assuming that a log-normal distribution applies to the data, it would be
more appropriate 'to stay as much as possible in the log-transformed
world'
.. One should use m and s for any further statistical procedure, and
perform
the back-transformation at the end. This is also a good recipe to avoid
mistakes and misunderstanding of the log-normal distribution (it is
'completely normal' in the log-transformed world).
This applies also to the use of priors in a Bayesian analysis. Suppose
that
we have done a population analysis, assuming a log-normal distribution
of
the parameters within the population. For one particular parameter we
get: m
= 4 and s = 0.2.
According to the aforementioned equations, E(X) = 55.70, Var(X) =
126.62,
S(X) = 11.25 and CV(X) = 0.2020.
If we want to use these results as a prior in a subsequent Bayesian
analysis, e.g. for Therapeutic Drug Monitoring in a new patient, we
should
not use the latter values, but we should use m = 4 and s = 0.2, and
perform
the analysis assuming a log-normal distribution of the parameter within
the
population.
Since the actual value of m does not have a 'proper look' because of the
logarithmic transformation, m and s can be back-transformed for the
'normal'
world by the following:
'Mean' = Geometic mean = Exp(m) = 54.60
'SD' = s . Mean = 10.92
'CV' = s = 0.2
These values can be easily converted to the log-transformed world. Using
E(X) and Var(X) would require to reverse the equations from which they
were
derived. In my view, this is needlessly complicated.
Alternatively, one might use E(X) and Var(X) assuming a normal
distribution
of the parameter within the population. Although illogical, and not
accurate, it may be expected that it will introduce only minimal bias.
On
the other hand, it would be incorrect to use the geometric mean and
'SD' and
assume a normal distribution, or to use Log(E(X)) and CV(X) and assume a
log-normal distribution. In both cases, a bias of about 2% (55.70 vs
54.60)
would be introduced. The bias rapidly increases with increasing s (or
CV).
Actually, both approaches are not really different, as long as we use
the
data correctly, according to their meaning. The geometric mean should
not be
interpreted as 'the mean' (since in normal language 'the mean' refers
to the
arithmetic mean), and E(X) should not be used in further statistical
calculations.
I would appreciate comments on this view.
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics and Drug Delivery
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
Email: j.h.proost.at.farm.rug.nl
Back to the Top
Dear Dr. Buclin,
In your message on coefficient of variation you wrote:
> - or the geometric one : for this I personally use CV =
> Exp(SD(Log(x)))-1 rather than the formula quoted by Carl. Both are
> however asymptotically equivalent for not too large SDs.
Please note that this formula is not correct.
The correct form was given by Jorn Attermann:
CV(X) = sqrt[Var(X)] / E(X) = sqrt[ exp(s^2) - 1 ].
where s = SD(Log(x)) in your equation.
The alternative approach, as explained in my previous message, is
CV(X) = s
In your example of three values 80, 100, 125, it follows that s =
0.223.
Your equation would yield indeed 0.25. Jorn Attermann's equation yields
0.226. A simple view at the data learns that the CV must be somewhere
between 20% (the deviation of 80) and 25% (the deviation of 125).
In the example in my previous mail (SD(Log(x)) = s = 0.2), this would
yield
a value of 0.2214, which overestimates the CV by 10%.
The deviation in your equations are even much larger than the difference
between the various estimates mentioned in my previous mail, when using
erroneous combinations of normal and log-normal values!
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics and Drug Delivery
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
Email: j.h.proost.aaa.farm.rug.nl
PharmPK Discussion List Archive Index page
Copyright 1995-2010 David W. A. Bourne (david@boomer.org)