RE: Theoretical questions about Beal's M2 method

________________________________ From: [email protected] [mailto:[email protected]] On Behalf Of Jun Shen Sent: Sunday, February 22, 2009 8:09 PM To: Sebastien Bihorel Cc: Leonid Gibiansky; [email protected] Subject: Re: [NMusers] Theoretical questions about Beal's M2 method Sebastien, I have been reading the BQL papers recently. Here is what I think. 1. M2 can be regarded as a truncated distribution method. You can look at it this way. Any probability density function (pdf) MUST be integrated into a cumulative distribution function (cdf) of 1 on its entire sample space, which is negative infinity to positive infinity for a normal distribution. If the BQL observations are discarded, the sample space is truncated from negative infinity to QL. This results in a cdf less than 1. In order to bring the cdf back to 1 on the new sample space (from QL to positive infinity), the pdf function has to be corrected by ( 1- phi((QL-f(t)/g(t))). Wikipedia has an exellent entry to explain the math ( http://en.wikipedia.org/wiki/Truncated_distribution) 2. The question I have is from M3 and M4 where the likelihood of an uncencored observation is expressed in pdf and the likelihood of a cencored observation is in cdf. Well, you can still combine everything together to get an objective function. But what is the definition of likelihood here exactly, pdf or cdf? 3. If we use cdf to represent the contribution of censored observations, then the smaller concentrations will have larger contribution to the objective function. For example, in M3 the likelihood of a censored observation is l(t)=phi((QL-f(t))/sqrt(g(t))). The smaller the f(t), the higher the phi(). Does that make sense? I believe the clear undersdanding on these theoretical concepts is just as important as to know those "know-how". Appreciate more comments. -- Jun Shen PhD PK/PD Scientist BioPharma Services Millipore Corporation 15 Research Park Dr. St Charles, MO 63304 Direct: 636-720-1589 On Mon, Feb 16, 2009 at 4:22 PM, Sebastien Bihorel <[email protected]> wrote: 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.