Generating random correlation matrices with fixed values: An application to the evaluation of multivariate surrogate endpoints

Abstract

When assessing surrogate endpoints in clinical studies under a causal-inference framework, a simulation-based sensitivity analysis is required, so as to sample the unidentifiable parameters across plausible values. To be precise, correlation matrices need to be sampled with only some of their entries identified from the data, known as the matrix completion problem. The positivedefiniteness constraints are cumbersome functions involving all matrix entries, making this a * To whom correspondence should be addressed.

Univariate Versus Multivariate Surrogates in the Single-Trial Setting

“…Even though in most practical settings the use of 3 to 4 surrogates may be sufficient, it would be worthwhile to further explore alternative algorithms (i) that allow for an increase in the dimension of Σ for which valid solutions can be found, and (ii) that increase the speed by which valid solutions for Σ are found. This is a topic of ongoing research (Flórez et al, 2018).…”

An efficient algorithm to assess multivariate surrogate endpoints in a causal inference framework

“…This method is computationally fast, even in the joint evaluation of a large number of surrogates, and easy to implement in standard statistical software packages (see the ICA.ContCont.MultS.PC function in Surrogate R package). In the present work, we show, based on theoretical arguments and simulations, that in the presence of non-informative surrogates, Flórez et al (2020)'s proposal struggles to completely cover the space of the ICA values compatible with the data. Indeed, even after one million runs, the algorithm produces an artificial shrinkage of the range of the ICA values when non-informative surrogates are jointly evaluated, leading to the misguided belief that adding these outcomes improves surrogacy.…”

“…To address this problem, Flórez et al (2020) built upon the work of Joe (2006) and Lewandowski et al (2009), and proposed a new algorithm for carrying out the simulation-based analysis. This method is computationally fast, even in the joint evaluation of a large number of surrogates, and easy to implement in standard statistical software packages (see the ICA.ContCont.MultS.PC function in Surrogate R package).…”

“…For example, it increments the cost of the trial, puts patients through unnecessary clinical tests, and/or increases the chances of missing data. Therefore, by adding extra information to the matrix completion problem, we introduce an improved version of Flórez et al (2020)'s algorithm that substantially mitigates the negative effect of adding non-informative surrogates on the range of the ICA values, without losing computational efficiency. The new proposal has the potential of allowing the evaluation of high-dimensional surrogates at an affordable computational cost.…”

“…In Section 5, the effect of adding non-informative surrogates to the surrogacy assessment is illustrated. The modification of Flórez et al (2020)'s algorithm is presented in Section 6. The algorithms are compared using a simulated scenario and the case study, in Sections 7 and 8, respectively.…”