A Note on the General Solution to Completing Partially Specified Correlation Matrices

ABSTRACT

Partially specified correlation matrices (not to be confused with matrices with missing data or EM-correlation matrices) can appear in research settings such as integrative data analyses, quantitative systematic reviews or whenever the study design only allows for the collection of certain variables. Although approaches to fill in these missing entries have been considered for special cases of low-dimensional matrices, a general approach that can handle correlation matrices of arbitrary size and number of missing entries is needed. The present article relies on the theory of convex optimization and semidefinite programming to derive a semidefinite program that can offer researchers a mathematically principled approach to fill in the missing entries. An easy-to-use function in the R programming language is also presented that implements the theory derived herein.

KEYWORDS:

• Semidefinite programming
• missing correlation
• positive definiteness
• covariance matrix

Notes

1. It is important to remind the reader that this is not a “missing data” problem where an imputation procedure is needed to estimate the EM-covariance matrix. In the present case, the correlations are missing because data on the variables was not collected or is not available as part of the design

2. One can set $s$ arbitrarily large and the corresponding matrix $A(x)+sI$ will be positive semidefinite