Simulating dependent discrete data

Abstract

This article describes a method for simulating n-dimensional multivariate non-normal data, with emphasis on count-valued data. Dependence is characterized by either Pearson correlations or Spearman correlations. The simulation is accomplished by simulating a vector of correlated standard normal variates. The elements of this vector are then transformed to achieve the target marginal distributions. We prove that the method corresponds to simulating data from a multivariate Gaussian copula. The simulation method does not restrict pairwise dependence beyond the limits imposed by the marginal distributions and can achieve any Pearson or Spearman correlation within those limits. Two examples are included. In the first example, marginal means, variances, Pearson correlations, and Spearman correlations are estimated from the epileptic seizure data set of Diggle et al. [P. Diggle, P. Heagerty, K.Y. Liang, and S. Zeger, Analysis of Longitudinal Data, Oxford University Press, Oxford, 2002]. Data with these means and variances are simulated to first achieve the estimated Pearson correlations and then achieve the estimated Spearman correlations. The second example is of a hypothetical time series of Poisson counts with seasonal mean ranging between 1 and 9 and an autoregressive(1) dependence structure.