Reliability of pharmacodynamic analysis by logistic regression: mixed-effects modeling

Many pharmacologic studies record data as binary, yes-or-no, variables with analysis using logistic regression. In a previous study, it was shown that estimates of C50, the drug concentration associated with a 50% probability of drug effect, were unbiased, whereas estimates of gamma, the term describing the steepness of the concentration-effect relationship, were biased when sparse data were naively pooled for analysis. In this study, it was determined whether mixed-effects analysis improved the accuracy of parameter estimation.

Pharmacodynamic studies with binary, yes-or-no, responses were simulated and analyzed with NONMEM. The bias and coefficient of variation of C50 and gamma estimates were determined as a function of numbers of patients in the simulated study, the number of simulated data points per patient, and the "true" value of gamma. In addition, 100 sparse binary human data sets were generated from an evaluation of midazolam for postoperative sedation of adult patients undergoing cardiac surgery by random selection of a single data point (sedation score vs. midazolam plasma concentration) from each of the 30 patients in the study. C50 and gamma were estimated for each of these data sets by using NONMEM and were compared with the estimates from the complete data set of 656 observations.

Estimates of C50 were unbiased, even for sparse data (one data point per patient) with coefficients of variation of 30-50%. Estimates of gamma were highly biased for sparse data for all values of gamma greater than 1, and the value of gamma was overestimated. Unbiased estimation of gamma required 10 data points per patient. The coefficient of variation of gamma estimates was greater than that of the C50 estimates. Clinical data for sedation with midazolam confirmed the simulation results, showing an overestimate of gamma with sparse data.

Although accurate estimations of C50 from sparse binary data are possible, estimates of gamma are biased. Data with 10 or more observations per patient is necessary for accurate estimations of gamma.