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Re: Theoretical questions about Beal's M2 method

From: Leonid Gibiansky <LGibiansky>
Date: Fri, 13 Feb 2009 14:12:39 -0500

You can view it as:

p(y ∩ y > LLQ) = 0 when y < LLQ
p(y ∩ y > LLQ) = p(y) when y > LLQ

Another way to look on this is to say that
p(y | y > LLQ) is proportional to p(y) and should integrate to 1

integral(p(y)) over y > LLQ is ( 1- phi((LLQ-f(t)/g(t))) that
immediately leads to l(t) below.

As to the 0 to 1 restriction, l(t) is the density, not probability. It
should integrate to one but can be smaller or greater than 1 (any
positive number).


Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
e-mail: LGibiansky at
tel: (301) 767 5566

Sebastien Bihorel wrote:
> 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.
Received on Fri Feb 13 2009 - 14:12:39 EST

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