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. Team Leader, Pharmacometrics Office of Clinical Pharmacology Office of Translational Science Center for Drug Evaluation and Research U.S. Food and Drug Administration Phone: 301-796-1624 Email: [email protected] "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: [email protected] [mailto:[email protected]] On Behalf Of Sebastien Bihorel Sent: Monday, February 16, 2009 5:22 PM To: Leonid Gibiansky Cc: [email protected] 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: [email protected] <mailto:[email protected]> Phone: (716) 633-3463 ext. 323 Leonid Gibiansky wrote: > 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. >>
