Simulating correlated count data

In this study we compare two techniques for simulating count-valued random n-vectors Y with specified mean and correlation structure. The first technique is to use a lognormal-Poisson hierarchy (L-P method). A vector of correlated normals Z is generated and transformed to a vector of lognormals X. Then, Y is generated as conditionally independent Poissons with means X-i. The L-P method is simple, fast, and familiar to many researchers. However, the method requires each Y-i to be overdispersed (i.e., sigma(2) > mu), and only low correlations are possible with this method when the variables have small means. We develop a second technique to generate the elements of Y as overlapping sums (OS) of independent X-j's (OS method). For example, suppose X, X-1, and X-2 are independent. If Y-1 = X + X-1 and Y-2 = X + X-2, then Y-1 and Y-2 are correlated because they share the common component X. A generalized version of the OS method for simulating n-vectors of two-Parameter count-valued distributions is presented. The OS method is shown to address some of the shortcomings of the L-P method. In particular, underdispersed random variables can be simulated, and high correlations are feasible even when the means are small. However, negative correlations cannot be simulated with the OS method, and when n > 3, the OS method is more complicated to implement than the L-P method.