Methods to Simulate Correlated Binomial Random Variables - Lai, Winfield (My University, 2021)
Single nucleotide polymorphisms (SNPs) have been involved in describing the risk a person is at for developing diseases. Simulating a collection of d correlated autosomal biallelic SNPs is useful to acquire empirical results for statistical tests in settings such as having a low sample size. A collection of d correlated autosomal biallelic SNPs can be modeled as a random vector X = (X1,...,Xd) where Xi ∼ binomial(2, pi) and pi is the minor allele frequency for the ith SNP. The pairwise correlations between components of X can be specified by a d ×d symmetric positive definite correlation matrix having all diagonal entries equal to one. Two versions of a novel method to simulate X are developed in this thesis; one version is based on generating correlated binomials directly and the other is based on generating correlated Bernoulli random vectors and summing them component wise. Two existing methods to simulate X are also discussed and implemented. In particular, a method involving the multivariate normal by Madsen and Birkes (2013) is compared to our novel methods for d ≥ 3. Our novel binomial method has a different variance for the Fisher transformed sample correlation than the other two methods. Overall, if the target pairwise correlations are smaller than the lowest upper bound possible and the number of SNPs is low, then our novel Bernoulli method works the best since it is faster than the Madsen and Birkes method and has comparable variability and bias for sample correlation.
Keywords: Statistics; Binomial; Correlation; Simulation; Multivariate