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Hi,
What would the scientific reason be for a weighted curve (1/y*y using
weighted least squares) being the best fit for certain assay
calibration curves and not others? Both assays are HPLC-based, the one
using UV detection and the other MS.
Thanks.
Jennifer Norman
Division of Pharmacology
University of Cape Town
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The following message was posted to: PharmPK
Jennifer,
Weighted least squares regression should be used when the error term
variance is not constant for all observations.
In HPLC methods, low concentration has high variance and high
concentration
has low variance.
In Chromatographic methods the calibrations curves are generally
evaluated
by ordinary linear (unweight) regression. This technique might be less
accurate in the LOWER part of calibration curve, because the regression
equation to a great extent curve is determined by values in the high
concentration range.
A consequence of this is that when the concentration of the standards
are
back calculated using the standard line parameters (y=ax+b) there may
be a
large discrepancies between the actual concentrations and the back
calculated concentrations, particularly at the LOWEST concentrations.
So, the problem may be overcome in a number of ways. One method is to
use
weighted regression (1/x;1/x2;1/Y or 1/y2)
I hope this help,
Daniel Rossi de Campos
Technical Consultant of BE - Brazil
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Dear Daniel,
Your explanation on use of weighted calibration curves is very nice. In
the
end you said one can use weighted regression (1/x;1/x2;1/Y or 1/y2) for
not
so linear curves to have a more appropriate values. But, my doubt is
when to
use what, i.e., are there any criteria to chose the exact term.
To be more detailed, commonly the non linearity in the calibration curve
could be due to poor extraction (sometimes combined with interference)
and/or signal saturation. Generaly, the former affects the lower range
of
curve and the later affects the higher range. My question is which
weighting
term to use when.
Thanks in advance,
Kasiram Katneni.
PhD student,
Victorian College of Pharmacy
Monash University, Australia.
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The following message was posted to: PharmPK
Dear Daniel and Karisam,
Two short comments on your notes on weighted calibration curves:
Daniel wrote:
> Weighted least squares regression should be used when the error
> term variance is not constant for all observations.
This is correct.
> In HPLC methods, low concentration has high variance and
> high concentration has low variance.
No, low concentrations will show low variance and high variance will
show
high variance. Perhaps you mean that the coefficient of variation at low
concentrations is generally higher than for high concentrations.
However, if
this difference is really relevant (say, more than a factor 2 or so),
the
LOQ may be chosen too low.
A weighting scheme should be chosen to correct for the differences in
variance of low and high concentrations. The conclusion that weighted
regresssion analysis should be applied is correct!
Karisam wrote:
> To be more detailed, commonly the non linearity in the calibration
> curve
> could be due to poor extraction (sometimes combined with interference)
> and/or signal saturation.
IMHO, this is a completely different topic. Nonlinearity of calibration
curves has nothing to do with differences in variance at low and high
concentration, and weighting is definitely not the solution to this
problem.
In this case the mathematical relationship between signal and
concentration
should be adapted from a linear to a more complex relationship.
In some cases, however, both problems (and even a third, see below) can
be
solved satisfactorily by a single technique. Sometimes the nonlinearity
of
the calibration curve can be treated by a logarithmic transformation.
Instead of a calibration curve
RR = a + b . X
where RR is the response ratio (signal) and X is the amount of analyte
in
the calibration sample, use the following equation:
RR = a . X^b
or after logarithmic transformation at both sides:
log RR = log a + b . log X
representing a straight line.
This transformation can be used as an alternative if the curve of log RR
versus log X is reasonably close to a straight line.
The logarithmic transformation has been described and applied in our
paper:
Kleef UW, Proost JH, Roggeveld J, Wierda JMKH. Determination of
rocuronium
and its putative metabolites in body fluids and tissue homogenates. J
Chromatogr 1993;621:65-76.
If the coefficient of variation at low and high concentrations is
broadly
the same, the logarithmic transformation results in equal variances
over the
entire concentration range, thus complying to the requirement for linear
regression and circumventing the problem of weighting.
The logarithmic transformation may also solve another problem. Linear
regression analysis is valid only if the X-values are (broadly) equally
spaced on the X-axis. A calibration curve of equally spaced
concentrations
is not the most efficient way in HPLC analysis (unless the concentration
range is rather narrow). If the distance between concentrations is
gradually
increasing with concentration (much more efficient), the logarithmic
transformation results in broadly equally spaced log X values, thus
complying to the requirement of linear regression.
Any comment to my view is welcomed.
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics and Drug Delivery
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
tel. 31-50 363 3292
fax 31-50 363 3247
Email: j.h.proost.-at-.farm.rug.nl
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The following message was posted to: PharmPK
Hi Jennifer,
Look for the article "Linear regression for calibration lines revisited:
weighting schemes for bioanalytical methods" of A.M. Almeida, M.M.
Castel-Branco, A.C. Falcao in J. Chromatogr. B, 774 (2002) 215.
They made it easy to justify the weighting and the decision between
weigting
factors 1, x^0.5, y^0.5, x, y, x^2 and y^2.
Joachim
Joachim Ossig, PhD
Grünenthal GmbH
Department of Pharmacokinetics (FO-PK)
52099 Aachen, Germany
Office: Zieglerstr. 6, 52078 Aachen, Germany
Tel.: +49-(0)241-569-2409
Fax.: +49-(0)241-569-2501
Mailto:Joachim.Ossig.aaa.grunenthal.de
http://www.grunenthal.com
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The following message was posted to: PharmPK
Kasiram,
Generally, we use the weighted LINEAR regression (1/x2). This is the
"true"
weighted regression because x2 is the variance. 1/x is an alternative
but it
do not improve your results more than ordinary regression.
Regards
Daniel
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Generally if the curve range is large, if there is large error at the
low end, try 1/y then 1/y2, these are the most appropriate, since they
weight the response. You will alos have to exxamine the coefficeint of
determination since that will get weaker as you weight the data. If
the proble is at the high end you may need to weight using the log.
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The following message was posted to: PharmPK
Dear Karisam
The appropriate election of the weighting term depends on the variance
model of your analytical technique. A simplified linear variance model
as var = a.C determines a weighting term of 1/C. However a potential
variance model var = a.Cb determines a weighting term of 1/Cb. When
the analytical technique as a constant variation coefficient this
implies that b=2 and then a weighting term of 1/C2 may be used.
Dr. J.M. Lanao
Dpt. Pharmacy and Pharmaceutical Technology
University of Salamanca
Spain
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May I suggest that the following classical paper be consulted as an
alternative to the range of ad hoc weighting suggestions that have been
made. It proposes a method for a more general model for the residual
error variance and allows model selection based on the use of the
extended least squares objective function.
Peck CC, Beal SL, Sheiner LB, Nichols AI. Extended least squares
nonlinear regression: A possible solution to the "choice of weights"
problem in analysis of individual pharmacokinetic parameters. Journal
of Pharmacokinetics and Biopharmaceutics 1984;12(5):545-57.
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New
Zealand
email:n.holford.-a-.auckland.ac.nz tel:+64(9)373-7599x86730 fax:373-7556
http://www.health.auckland.ac.nz/pharmacology/staff/nholford/
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The following message was posted to: PharmPK
Dear All,
I need a step by step procedure for performing weighted least squares
regression using WinNonlin ver 3.3, using the preselected weighting
scheme 1/Y*Y
Thanks very much.
Jennifer Norman
Quality Assurance
Division of Pharmacology
University of Cape Town
Tel. +27 (0)21 406 6498
Fax. +27 (0)21 448 1989
[Another 10-12 messages in the queue to send out but I might let this
group go out first - db]
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