From: Lars.Erichsen

Date: Fri, 13 Feb 2009 19:45:23 +0100

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

-----Original Message-----

From: owner-nmusers

Sent: 13 February 2009 16:00

To: nmusers

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"A(BB) / 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"A(BB) 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

<mailto:sebastien.bihorel

Phone: (716) 633-3463 ext. 323

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Received on Fri Feb 13 2009 - 13:45:23 EST

Date: Fri, 13 Feb 2009 19:45:23 +0100

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

-----Original Message-----

From: owner-nmusers

Sent: 13 February 2009 16:00

To: nmusers

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"A(BB) / 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"A(BB) 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

<mailto:sebastien.bihorel

Phone: (716) 633-3463 ext. 323

**********************************************************************

Proprietary or confidential information belonging to Ferring Holding SA or to one of its affiliated companies may be contained in the message.

If you are not the addressee indicated in this message (or responsible for the delivery of the message to such person), please do not copy or deliver this message to anyone.

In such case, please destroy this message and notify the sender by reply e-mail. Please advise the sender immediately if you or your employer do not consent to e-mail for messages of this kind.

Opinions, conclusions and other information in this message represent the opinion of the sender and do not necessarily represent or reflect the views and opinions of Ferring.

**********************************************************************

Received on Fri Feb 13 2009 - 13:45:23 EST