Simulating longer vectors of correlated binary random variables via multinomial sampling

Abstract

The ability to simulate correlated binary data is important for sample size calculation and comparison of methods for analysis of clustered and longitudinal data with dichotomous outcomes. One available approach for simulating length n vectors of dichotomous random variables is to sample from the multinomial distribution of all possible length n permutations of zeros and ones. However, the multinomial sampling method has only been implemented in general form (without ?rst making restrictive assumptions) for vectors of length 2 and 3, because specifying the multinomial distribution is very challenging for longer vectors. I overcome this di?culty by presenting an algorithm for simulating correlated binary data via multinomial sampling that can be easily applied to directly compute the multinomial distribution for any n. I demonstrate the approach to simulate vectors of length 4 and 8 in an assessment of power during the planning phases of a study and to assess the choice of working correlation structure in an analysis with generalized estimating equations. The ability to simulate correlated binary data is important for sample size calculation and comparison of methods for analysis of clustered and longitudinal data with dichotomous outcomes. One available approach for simulating length n vectors of dichotomous random variables is to sample from the multinomial distribution of all possible length n permutations of zeros and ones. However, the multinomial sampling method has only been implemented in general form (without first making restrictive assumptions) for vectors of length 2 and 3, because specifying the multinomial distribution is very challenging for longer vectors. I overcome this difficulty by presenting an algorithm for simulating correlated binary data via multinomial sampling that can be easily applied to directly compute the multinomial distribution for any n. I demonstrate the approach to simulate vectors of length 4 and 8 in an assessment of power during the planning phases of a study and to assess the choice of working correlation structure in an analysis with generalized estimating equations.

Simultaneous Inference for Multiple Proportions: A Multivariate Beta-Binomial Model

“…In particular, the question which correlation structures are admissible for an m-dimensional Beta distribution can directly be transferred to the same question regarding an m-dimensional Bernoulli distribution (Hailperin, 1965;Chaganty and Joe, 2006). Similar considerations have been made regarding the question how to generate correlated binary data (Leisch, Weingessel and Hornik, 1998;Xue et al, 2010;Preisser and Qaqish, 2014;Shults, 2017). The distinctive feature of this work is thus the different (Bayesian) setting and the focus on statistical inference.…”

“…Hereby R A is the m × m matrix with all rows identical and equal to α = diag(A) and C A = R A . The derived conditions (MB) and (FB) or variations thereof have appeared several times in the relevant literature (Leisch, Weingessel and Hornik, 1998;Shults, 2017).…”

“…The bounds on the derived moment matrix A induced by (MC) imply bounds on the correlation matrix R because the elements of A are monotone in the according elements of R. The construction and the according conditions translate to a multivariate Bernoulli distribution with minor modifications. In this context, several works have illustrated the bounds on the correlation coefficients for low dimensions (Prentice, 1988;Preisser and Qaqish, 2014;Shults, 2017). For instance, for m = 2, (FB) implies…”

“…The distinctive feature of this work is thus the different (Bayesian) setting and the focus on statistical inference. Moreover, to the best of the author's knowledge, the necessary and sufficient moment conditions that are provided in section 2.1 have not been mentioned in this form in the literature despite the fact that many authors have recognized the connection to linear programming (Madsen, 1993;Fontana and Semeraro, 2018;Shults, 2017).…”

Characterization of multivariate Bernoulli distributions with given margins

“…The problem of the simulation of correlated binary data is extensively addressed in the statistical literature, e.g. [3], [6], [15] and [9]. Simulation studies are a useful tool for analysing extensions or alternatives to current estimating methodologies, such as generalised linear mixed models, or for the evaluation of statistical procedures for marginal regression models ( [13]).…”