K. Ogungbenro, G. Graham and L. Aarons
School of Pharmacy and Pharmaceutical Sciences, University of Manchester, Manchester
There is a need for pharmacokinetic(PK) studies just like any other study to include an adequate number of subjects in order to have the required level of power. However there are no general tools for determining sample sizes for population pharmacokinetic studies. The principles of Generalized Estimating Equations (GEE) for Generalized Linear Models (GLM) have been widely used in biostatistics for longitudinal data analysis and recently used to calculate the minimum sample size required based on the exponential family of distributions. Our intention was to extend the principles of GEE to nonlinear models in PK studies since these also involve longitudinal data.
GEE uses the first two moments of the likelihood to specify the model. Suppose measurements are scheduled for a set of time points for all individuals enrolled in the study and tj is the jth time ( j=1,….,T). The vector of expected values in the sth subpopulation is given by . Let h( ) be the link function (h( )= ) and g( ) is the variance function where Xs is the design matrix and is the parameter vector. Under GEE the covariance matrix among repeated measures is given by Vs= where is the dispersion parameter, As=diag[g( )…….g( )] and R is the working correlation matrix among repeated measurements. Equations to solve for the number of subjects using the functions above can therefore be derived so that an hypothesis of the form H0 : versus H1 : can be tested at a particular level of (type I error) and 1- (power, =type II error) where H is an h´r matrix of full row rank and h0 is the null hypothesis value for the (contrast of ) parameter values (h is the number of for hypothesis testing).
A number of approximations have been proposed for nonlinear models, and nonlinear models are often linearised for the purpose of analysis. These include using the Jacobian matrix (Js) in place of the design matrix and the covariance matrix (Vs) as described by mixed effects modelling for the different error models. The method was applied to a one compartment IV bolus model with different values for clearance in two populations with the aim of calculating the minimum number of subjects required to detect the difference. The results of GEE were compared to the results obtained by applying the likelihood ratio test (LRT) and Wald test to the same problem and GEE produced a good estimate of power in all cases.
GEE was applied to the example described by White et al (1992), which involved an investigation using the LRT and confidence intervals to compare the clearance between two populations. The results of GEE were comparable to the results of White et al (1992).
References:
[1] J.Rochon. Applications of GEE Procedures for Sample size Calculation in Repeated Measures Experiments. Statistics in Medicine, 17: 1643-1658 (1998).
[2] P.I.D. Lee. Design and Power of a population Pharmacokinetics Study, Pharmaceutical Research, 18(1): 75-82 (2001).
[3] M. Davidian, and D.M. Giltinan. Nonlinear Models for Repeated Measurement Data, 1st Edition, Chapman and Hall, London, (1995).
[4] S.L. Zeger, and K.Liang. Longitudinal Data Analysis for Discrete and Continuous Outcomes, Biometrics, 42: 121-130 (1986).
[5] D.B. White, C.A. Walawander, D.Y Liu and T.H Grasela, Evaluation of Hypothesis Testing for Comparing Two Populations Using NONMEM Analysis, J. Pharmacokinet. Biopharm, Feb. 20(3) : 295 – 313 (1992)
Reference: PAGE 13 (2004) Abstr 467 [www.page-meeting.org/?abstract=467]
Poster: poster