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Recently there was considerable discussion of confirming linearity. I
suggested the Lack of Fit test that requires replicate measurements
at each level of the independent variable which I learned from the
discussion is not done in Pk work.
Since then I thought perhaps the following might be useful.
For a linear relationship, y = x^1.
For a near linear relationship, the empirical model, y = x^(1+-a),
can be fitted to the data.
The value of 'a' then represents the deviation from linearity.
Acceptable limits of 'a' might be defined based on the intended
application of the data.
Regards,
Stan Alekman
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Dear Stan,
How do you propose to use this empirical model? I assume you would
test your data first to see whether a+/-95%CI (for p<0.05) is within
a pre-defined range (I have seen recommended somewhere a +/- 0.03,
but not clear to me what the convention would be/is). Then, if a +/-
95%CI is indeed within the said range, the data are assumed to be
linear. Would you then pre-set a=0 and re-fit according to y=ax+b? I
think you have omitted the intercept for clarity.
TIA
Frederik Pruijn
Frederik B. Pruijn PhD MSc (Senior Research Fellow)
Experimental Oncology Group
Auckland Cancer Society Research Centre
Faculty of Medical and Health Sciences
The University of Auckland
Private Bag 92019
Auckland
New Zealand
Phone: +64-9-3737 599 x86939 or x86090
Fax: +64-9-3737 571
E-mail: f.pruijn.aaa.auckland.ac.nz
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Dear Stan,
several approaches to the check of linearity are presented in the
literature. The approach you proposed is known as the "power model".
Other
approaches are for example
y = ax^1 + bx^2 + cx^3 + .....
Deviations from linearity may be considered to be statistically
significant, if the coefficients b, c etc. are different from zero.
A good paper is
Gough K, Hutchinson M, Keene O, Byrom B, Ellis S, Lacey L, McKellar J.
Assessment of dose proportionality: Report from the Statisticians in the
Pharmaceutical Industry / Pharmacokinetics UK Joint Working Party.
Drug Inf J 1995; 29:1039-1048
Regards,
Peter Wolna
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Peter,
Thanks for the reference.
When I fit raw data, I always check whether higher orders improve the
model. If not, I stick with a linear model. Of course in my work I
make replicate measurements at each calibrator level so I rely on the
Lack of Fit test for any model I fit. That is the utility of this test.
I fit dissolution processes and others where many controlling factors
come into play. Some fitted models are rather involved expressions
but I never accept one that is not known in the literature for a
physical process I can review and understand. (I am not so arrogant
as to believe that I might observe an original, heretofore unobserved
model.)
The suggestion I made is somewhat different from the power model in
that I propose finding the exponent of x that fits the data best. It
is not likely to be exactly one for true linear relationships but
very close to one. I suggest that the difference between the observed
exponent and one may be considered a deviation from linearity. For
the intended use of the model, a tolerance, the difference from
observed and one must be earlier established. In this way, one may be
comfortable the linear model may be adequate for the specific
application. The tolerance may be established by simulation or simple
trial and error estimations.
This suggestion was addressed to those who posed the question, how
far from linearity may we be with data and safely use a linear
relationship. There were replies based on various statistical
parameters. This suggestion, based on a pre-established tolerance for
the intended application, may meet that need.
Regards,
Stan Alekman
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