Fungible Correlation Matrices: A Method for Generating Nonsingular, Singular, and Improper Correlation Matrices for Monte Carlo Research

Abstract

For a fixed set of standardized regression coefficients and a fixed coefficient of determination (R-squared), an infinite number of predictor correlation matrices will satisfy the implied quadratic form. I call such matrices fungible correlation matrices. In this article, I describe an algorithm for generating positive definite (PD), positive semidefinite (PSD), or indefinite (ID) fungible correlation matrices that have a random or fixed smallest eigenvalue. The underlying equations of this algorithm are reviewed from both algebraic and geometric perspectives. Two simulation studies illustrate that fungible correlation matrices can be profitably used in Monte Carlo research. The first study uses PD fungible correlation matrices to compare penalized regression algorithms. The second study uses ID fungible correlation matrices to compare matrix-smoothing algorithms. R code for generating fungible correlation matrices is presented in the supplemental materials.

Kanonik Korelasyon Katsayılarının İstatistiksel Önemliliğini Test Etmek için Hangi Test Daha Güvenilirdir?

“…In this study, random numbers generated by the Monte Carlo simulation technique were used (Waller, 2016). Random numbers were generated using the monte1 function of the fungible package in the R (R Core Team, 2019).…”

What are the mathematical bounds for coefficient α?

“…A common feature of these matrices is that their elements sum to k , which we formalize by writing 1 ′ R1 = k . Theory and methods that are described in Waller (2016) can be used to construct correlation matrices in this innumerable set. For instance, as shown in the online supplemental materials, when k = 4, the following matrix will produce α = 0: It merits comment that the eigenvalues of Equation 4—a matrix with unit diagonal values—are positive and sum to k .…”

“…Then, write Rearranging Equation 5, we obtain Because S denotes the sum of the entries in a ( k × k ) correlation matrix, it can be shown that Rearranging Equation 7, we find that Next, 4 after substituting Equation 6 into Equation 8, we have Thus, to generate a data set with k items that yields a known value of α ∈ (0, 1), a researcher can use Equation 9 to quickly determine r¯ij. With this quantity in hand, they can then use theory and methods described in Waller (2016) to generate all correlation matrices that produce the desired α. Note that for fixed c , where c ∈ (0, 1), if R is a k × k correlation matrix ( k ≥ 2) then the set double-struckS=falsefalse{R falsefalse| αfalse(Rfalse)=cfalsefalse} is infinite (where α( R ) denotes the α computed on set member R ).…”

“…Thus, generating an R with all r ij = 0.5866801 will yield an α=.8765. Using the theory described in Waller (2016), we can generate innumerable other matrices that will yield this value of α. For instance, the matrix shown below will also produce α=.8765:…”

“…Each of these matrices belongs to an infinite set of ( k × k ) correlation matrices that will produce a fixed α < 0. Theory and methods described in Waller (2016) can be used to populate the members of each set. We have proved that α has no lower bound.…”