# RE: Error models for log-transformed data

From: Martin Bergstrand <martin.bergstrand>
Date: Tue, 28 Apr 2009 13:48:03 +0200

Dear Kelong and NMusers,

We will try to answer the questions by Kelong Han and also make some

>> 1. What is the rationale of fixing \$SIGMA 1?

The error model suggested by Mats Karlsson could as well be written with the
estimation of an ERR(1) and ERR(2) (estimation of the variance for two
SIGMA). However it is often convenient to fix the variance for SIGMA to 1
and instead estimate a scale-factor for that variance. This scale-factor can
in the simple case of an additive error be a single THETA(x). In the case of
an additive + proportional error model on the normal scale, the scale-factor
is a function of two estimated parameters THETA(x), THETA(y) and the model
prediction F.

W = SQRT(THETA(x)**2 + THETA(y)**2/F**2) Y = LOG(F) + W*ERR(1) [\$SIGMA 1
FIX]

There are a number of practical reasons for estimating a scalar for SIGMA
(W) rather than SIGMA itself. The two reasons that comes to mind immediately
is to be able to calculate individual weighted residuals (see IWRES below)
and to use the M3 or M4 methods for handling censored data (Beal SL, 2001).

RES = DV - IPRED
IWRES = RES/W

>> 2. What an error structure on the normal scale is this "double
exponential error model" equivalent to?

The simple answer is that it is not equivalent to any model on the normal
scale. It has some similarities to the combined additive and proportional
error model but it does not predict negative concentrations and assumes a
slight bias in the model predictions due to the addition of M (Y =
LOG(F+M)...). ERR(1) can be fairly well translated to a proportional error
on the normal scale but ERR(2) can't be directly interpreted as an additive
component.

>> 3. Compared to the simplest error model Y=LOG(F)+ERR(1), the two error
THETA's accounted for in the calculation of the objective function value?

Addition of parameters to the error model seems to follow the chi-squared
distribution (Silber HE et al, 2009) i.e. with the addition of one parameter
a drop in OFV of 3,84 corresponds to a 5% significance. The Karlsson is a
nested model to the simple additive error model and the "double exponential
error model" could probably be said to be "almost nested". However my
personal opinion (Martin's) is that the development of a residual error
model is best guided by the gof-plots.

As a final remark to this answer I would like to point out that all models
suggested this far for approximating a combined additive and proportional
error model (on normal scale) for log-transformed data has drawbacks. I have
already pointed out that the "double exponential error model" does introduce
a bias due to the addition of the parameter M. The model suggested by Mats
Karlsson on the other hand is only a good approximation for the cases when F
> THETA(y). If F << THETA(y) the approximation will give rise to increasing
mean absolute error and unrealistic predictions (Y). From personal
experience (Martin's) this is primarily a problem in case of simulations.
One might argue that also the combined additive and proportional error model
on the normal scale has its drawbacks. Especially so since it is often
applied to bioanalytical data for which non-random censoring of estimated
negative concentrations is performed.

Kind regards,

Martin Bergstrand, Andrew Hooker and Joakim Nyberg
-----------------------------------------------
Department of Pharmaceutical Biosciences,
Uppsala University
-----------------------------------------------
P.O. Box 591
SE-751 24 Uppsala
Sweden
-----------------------------------------------

-----Original Message-----
From: owner-nmusers
Behalf Of Han, Kelong
Sent: den 28 april 2009 00:16
To: nmusers
Subject: [NMusers] Error models for log-transformed data

Dear NMusers,

I am working on a PK model using log-transformed data. I have read previous
discussions on NMusers regarding this, and they are really helpful, but I am
still a little bit confused about the following questions. I would greatly
appreciate it if someone could make it clear:

1. Dr. Mats Karlsson suggested
Y=LOG(F)+SQRT(THETA(x)**2+THETA(y)**2/F**2)*ERR(1) with \$SIGMA 1 FIX as an
equivalent error structure to the
additive+proportional error model on the normal scale. What is the rationale
of fixing \$SIGMA 1?

2. Dr. Stu Beal and Dr. William Bachman suggested the "double exponential
error model": Y = LOG(F+M) + (F/(F+M))*ERR(1) + (M/(F+M))*ERR(2) without
fixing \$SIGMA. The Goodness-of-Fit plot looks slightly better using this
error model in my study. What an error structure on the normal scale is this
"double exponential error model" equivalent to?

3. Compared to the simplest error model Y=LOG(F)+ERR(1), the two error
THETA's accounted for in the calculation of the objective function value?
This especially bothers me because the "double exponential error model"
leads to a lower OFV compared to Y=LOG(F)+ERR(1) (also slightly better
Goodness-of-Fit plot) in my study.

Sorry for the length. Would anyone please give me some explanations or
references?

Thanks a lot!

Kelong Han

PhD Candidate=
Received on Tue Apr 28 2009 - 07:48:03 EDT

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