An extension of Satterthwaite's approximation applied to pharmacokinetics

Satterthwaite's approximation for the degrees of freedom of a linear combination of independent mean squares is extended to the case that the mean squares are correlated. The mean squares are sample variances where some of the experimental units have been used in more than one sample. The motivation for such an extension comes from pharmacokinetics. The observations, taken at different time points from a set of animals, are blood drug concentrations. Some animals were sampled at more than one time point. A linear combination of sample means provides an estimate of the population mean area under the concentration-versus-time curve, which is an indicator of drug exposure. An associated linear combination of sample variances provides an estimate of the variance of the area estimator. The behavior of confidence intervals based on the approximation was studied by simulation. The confidence interval for the population mean, constructed by assuming that the variance estimator has a chi-square distribution with the computed degrees of freedom, achieved close to its nominal 95% coverage, justifying the extension of Satterthwaite's approximation.