Simulation Methods for Categorical Variables

Abstract

Abstract The simulation of real‐world processes is a strong and legitimate tool in the hand of the research worker. The parameters and variables used in simulation models can be discretely or continuously distributed. A random variable that can assume the discrete values 0, 1, 2,…, n has a discrete distribution. Under many conditions, discretely distributed variables are called categorical variables . Most simulations involve using computers. Researchers write programs using general purpose software (e.g., FORTRAN, C++, and Excel). In applied research, the discrete distribution, that is, the set of probabilities of patterns of variable categories, is almost always found by analyzing the data of a sample. One typically starts from an existing uniform random number generator that a programming environment such as R, MS Excel, Turbo Pascal, C, FORTRAN, or C++ makes available. The random numbers from these generators are then transformed such that the desired distribution results. The probabilities of the categories of the targeted distribution must be determined a priori . The transformation of the uniformly distributed random number from the [0, 1] interval to any discretely distributed random variable with n classes (=categories) is a mapping of the [0, 1] interval onto the probability axis of the cumulative sum distribution. The programming language C (or its newer version C++) uses the full speed of the processor of a computer. The method of generation of the categorical numbers involves using a mapping function. This function maps random numbers with uniform distribution onto a set of categories. One can reach a further acceleration of the generation of random categorical numbers by the use of normally { N 0;1} distributed random numbers. Instead of n drawings of single events with a given probability to get a random frequency, one computes only one normally distributed random frequency. The statistical summary describes the result of the simulation. In an example with the two‐dimensional cross‐classifications, the summary presents the number N A of times the null hypothesis was rejected in relation to the total number of all simulation trials, N R .