Re: 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.