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) / 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. -- *Sebastien Bihorel, PharmD, PhD* PKPD Scientist Cognigen Corp Email: [email protected] < mailto: [email protected] > Phone: (716) 633-3463 ext. 323