Simulation and evaluation of bivariate Beta distribution for interval response variables
Francois Mercier, Amy Racine
Novartis Pharma AG
Objectives: In certain clinical trials, the response variable is restricted into an interval (e.g. visual analogue scale VAS). This response may be measured repeatedly during the trial. A natural distribution for such a variable is a multivariate Beta distribution. For clinical trial simulations, it is therefore necessary to generate data from such a distribution in order to perform inference/predictions.
Methods: We assume a clinical trial in which a response variable Y, characterized by an interval distribution, is evaluated at both early (Y1) and late (Y2) stages. The clinical trial may have been designed in such a way that at the occasion of an interim analysis, one wants to predict Y2 based on the available Y1 data for decision making (e.g. futility or success of the trial).
With such a possibly skewed distribution of interval data, it is not recommended to use e.g. a truncated bivariate normal distribution for simulation, but rather a bivBe (bivariate Beta) distribution.
Generation of univariate Beta-distributed random variable is straightforward, but generating pairs of correlated Beta-distributed random variables is more complex since there is no natural multivariate extension of univariate Beta distribution (Johnson and Kotz ).
It is proposed to use a Dirichlet distribution to simulate outcomes from a bivBe distribution. The Dirichlet distribution can be generated via a set of four independent Gamma distributions.
Various methods are also presented for the evaluation of the response (to treatment) at the interim analysis.
Results: The data generation is implemented in SAS or R, whereas the evaluation and prediction of data generated from a bivBe distribution has been implementation in WinBUGS.
 Johnson N.L., Kotz S. 1976. Distributions in statistics: Continuous multivariate distributions. Wiley, New York.