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

--------------------------------------

Leonid Gibiansky, Ph.D.

President, QuantPharm LLC

web: www.quantpharm.com

e-mail: LGibiansky at quantpharm.com

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

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

--------------------------------------

Leonid Gibiansky, Ph.D.

President, QuantPharm LLC

web: www.quantpharm.com

e-mail: LGibiansky at quantpharm.com

tel: (301) 767 5566

Sebastien Bihorel wrote:

Received on Fri Feb 13 2009 - 14:12:39 EST