RE: Theoretical questions about Beal's M2 method
Dear Sebastian,
The likelihood is not constrained by 1. It is a probability density function -
not a probability.
Br
Lars Erichsen
Modelling and simulation specialist
Experimental Medicine
Ferring Pharmaceuticals
Quoted reply history
-----Original Message-----
From: [email protected] [mailto:[email protected]] On
Behalf Of Sebastien Bihorel
Sent: 13 February 2009 16:00
To: [email protected]
Subject: [NMusers] Theoretical questions about Beal's M2 method
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: [email protected]
<mailto:[email protected]>
Phone: (716) 633-3463 ext. 323
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