# RE: Theoretical questions about Beal's M2 method

From: Wang, Yaning <yaning.wang>
Date: Wed, 18 Feb 2009 13:05:44 -0500

Bihorel:
If you are familiar with the concept of hazard in survival analysis, this conditional likelihood should be straightforward. A slight modification of h(y)=f(y)/S(y) to h(y)=f(y)/S(LOQ) where (y>=LOQ) will give you the expression used by M2: l(y)=f(y)/S(LOQ) at a specific time t. You should ignore t in the M2 expression and think of it as l(y)=P(Y=y|Y>=LOQ) where y>=LOQ at time t. P(Y=y|Y>=LOQ) is not a discrete probability but a probability density because y is a continuous variable.
I think P(Y=y, Y>=LOQ) can be simplified to P(Y=y) because there is an inherent restriction: y>=LOQ.

Yaning

Yaning Wang, Ph.D.
Office of Clinical Pharmacology
Office of Translational Science
Center for Drug Evaluation and Research
Phone: 301-796-1624
Email: yaning.wang

"The contents of this message are mine personally and do not necessarily reflect any position of the Government or the Food and Drug Administration."

-----Original Message-----
From: owner-nmusers
Sent: Monday, February 16, 2009 5:22 PM
To: Leonid Gibiansky
Cc: nmusers
Subject: Re: [NMusers] Theoretical questions about Beal's M2 method

Thanks Leonid,

However, there is still one point that is unclear to me. You have
demonstrated that p(y | y > LLQ) = p(y) / p(y>LOQ), given the
assumptions of the text. Now, this is a discrete probability, while l(t)
is a likelihood... How can one mathematically demonstrate the expression
of l(t) used by Dr. Beal starting from the previous expression of p(y |
y > LLQ)?

*Sebastien Bihorel, PharmD, PhD*
PKPD Scientist
Cognigen Corp
Email: sebastien.bihorel
<mailto:sebastien.bihorel
Phone: (716) 633-3463 ext. 323

Leonid Gibiansky wrote:
> You can view it as:
>
> p(y \$B"A(J y > LLQ) = 0 when y < LLQ
> p(y \$B"A(J 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"A(JB) / 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(JB) 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 Wed Feb 18 2009 - 13:05:44 EST

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