To the Editor.-I read with interest the article by Bailey and Gregg on population pharmacodynamic analysis. They have presented a technique to analyze binary response data obtained from multiple patients with one data point per patient to distinguish between intra- and interpatient variability. However, an analysis of the mathematical basis on which the authors based their technique has revealed a fundamental flaw, which in my opinion invalidates the conclusions of their paper.
Basically, the parameters defining the intra- and interpatient variability are not both identifiable. To clarify this, let the concentration threshold above which there is a response to a certain stimulus be lognormally distributed so that Equation 1where C50 is the median value for one patient and epsilon has a normal distribution with mean zero and variance alpha2. To account for interpatient variability, assume that the C50 is also lognormally distributed so that Equation 2where (C50) is the median C50 of the population and delta has a normal distribution with median zero and variance omega2. Now the probability of a response can be shown to be equal to Equation 3where Phi denotes the cumulative standardized normal distribution. Notice that this equation is equivalent with eq.(A-5) in when alpha = 1/gamma is substituted. The parameters alpha and omega are not both identifiable because there is an infinite number of combinations of values of alpha and omega for which [radical](alpha2+ omega2) has the same value. Only the total variance (alpha2+ omega2) can be estimated.
The problem of the unidentifiable parameters originates from the fact that both epsilon and delta are assumed to be normally distributed, but this is a natural assumption. For other distributions, the parameters may be theoretically identifiable, but they probably will be poorly estimable. Moreover, the data analysis should not be critically dependent as to the assumptions on the distributions because their forms are unknown. Therefore, it appears to be imperative to acquire multiple measurements per patient to obtain information about the nature of the intrapatient variability.
Erik Olofsen, M.Sc.
Department of Anesthesiology; University Hospital Leiden; P.O. Box 9600; 2300 RC, Leiden; The Netherlands
(Accepted for Publication September 10, 1997.)