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Theoretical questions about Beal's M2 method

From: Sebastien Bihorel <Sebastien.Bihorel>
Date: Fri, 13 Feb 2009 09:59:40 -0500

Dear colleagues,

In a paper dated from 2001, Dr. Beal presented several methods to handle
data below the quantification limit (Journal of Pharmacokinetics and
Pharmacodynamics, Vol. 28, No. 5, October 2001), including the M2 method
that can be implemented in NONMEM 6 via the YLO functionnality. I would
like to submit some questions to the list about the theory associated to
the M2 method.

I quote:
"...the BQL observations can be discarded, and under the assumption that
all the D(t) [the distribution of residual errors] are normal, the
method of maximum conditional likelihood estimation can be applied to
the remaining observations (method M2). With this method, the likelihood
for the data, conditional on the fact that by design, all (remaining)
observations are above the QL, is maximized with respect to the model
parameters. The density function of the distribution on possible
observations at time t, evaluated at y(t), is 1/sqrt( 2*pi*g(t) ))*exp(
-0.5*( y(t)-f(t) )^2/g(t) ) and the probability that an observation at
time t is above the QL is 1- phi((QL-f(t)/g(t))), where phi is the
cumulative normal distribution function. Therefore, conditional on the
observation at time t being above QL, the likelihood for y(t) is the ratio:
l(t)=(1/sqrt( 2*pi*g(t) ))*exp( -0.5*( y(t)-f(t) )^2/g(t) ) /( 1-
phi((QL-f(t)/g(t))) [equation 1]"

Now, lets A and B be two events. The probability of A, given B is:
p(A|B) = p(A∩B) / p(B)

In the context of Dr. Beal's paper, I interpret A as simply the
observation y(t) and B as the fact that y(t) is above QL, and thus have
the following questions about equation 1:
- it looks like p(A∩B) in equation 1 simplifies to the probability of
y(t) given the model parameters, i.e. p(A). Which part of the problem
allows this simplification?
- how can l(t) be constrained between 0 and 1 if both numerator and
denominator can vary between 0 and 1?

Any comment from nmusers will be greatly appreciated.

*Sebastien Bihorel, PharmD, PhD*
PKPD Scientist
Cognigen Corp
Email: sebastien.bihorel
Phone: (716) 633-3463 ext. 323
Received on Fri Feb 13 2009 - 09:59:40 EST

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