Simulating Cross-Omics Dependence

Why Specify Cross-Omics Dependence?

A simulation can have realistic values within each omics modality and still be a poor benchmark if it gets the relationships between modalities wrong. For example, a method that combines molecular measurements may behave differently when matched features share substantial correlation than when the modalities are nearly independent. A simulation should therefore specify not just the separate distributions of each modality, but also the dependence between measurements made on the same individual.

This tutorial focuses on generating and checking that dependence. It does not specify outcome-related or modality-specific signals: those are separate design choices. Assume that the rows of the modality matrices refer to the same \(n\) individuals, as required for a cross-modal association to be meaningful. This relies on the matched-cohort setup from the earlier material on generating matched multi-omics cohorts.

A dependence specification should be stated in terms of a target: for example, a particular cross-modal correlation for selected feature pairs, or a pattern of cross-modal covariance across blocks of features. The target is a population property of the generating mechanism. A finite simulated data set will generally show a different empirical covariance or correlation because of sampling variation.

Generating Dependence With a Shared Latent Variable

A direct construction uses a shared random variable to induce dependence. For each individual \(i\), generate a latent variable \(Z_i\) with mean zero and variance one. For standardized features, use

Here \(\lambda_j\) and \(\gamma_k\) are loadings between \(-1\) and \(1\). Assume that the \(Z_i\) are independent across individuals; all residuals \(\varepsilon_{ij}\) and \(\eta_{ik}\) are mutually independent, independent of all \(Z_i\), and have mean zero and variance one. A convenient distributional choice is standard normal for every variable in the construction. The square-root residual coefficients ensure that every feature has variance one.

The shared \(Z_i\) induces cross-modal dependence. Since the residuals are independent,

Because both features have variance one, this is also their correlation. Within a modality, the same construction gives \(\operatorname{Cov}(X_{ij},X_{i\ell})=\lambda_j\lambda_\ell\) for distinct features, and similarly for \(Y\). Thus a single shared factor creates a structured, rank-one cross-modal covariance block, not arbitrary pairwise correlations. Loading signs matter: features with loadings of opposite signs have negative cross-modal correlation.

For more than one shared factor, write \(Z_i\) as a vector with covariance matrix \(I\), and let loading vectors be \(a_j\) and \(b_k\). Provided the residual variances are nonnegative, a model of the form \(X_{ij}=a_j^\mathsf{T}Z_i+\varepsilon_{ij}\) and \(Y_{ik}=b_k^\mathsf{T}Z_i+\eta_{ik}\) gives cross-modal covariance \(a_j^\mathsf{T}b_k\). The number and arrangement of factors therefore determine the structure that can be represented. This remains a structured model: choosing loadings does not grant independent control over every entry of the cross-modal matrix.

Generating a Specified Joint Covariance

When the target is a particular covariance pattern, a direct multivariate-normal construction can be more flexible. Write the combined feature vector for individual \(i\) as \(W_i=(X_i^\mathsf{T},Y_i^\mathsf{T})^\mathsf{T}\), and specify

The within-modality covariance matrices are \(\Sigma_X\) and \(\Sigma_Y\); the cross-modal covariance matrix is \(C\). A necessary and sufficient condition for this to be a valid covariance matrix is that \(\Sigma\) be symmetric and positive semidefinite, meaning \(v^\mathsf{T}\Sigma v\geq 0\) for every vector \(v\). If \(\Sigma\) is positive definite, a Cholesky factorization \(\Sigma=LL^\mathsf{T}\) permits generation by drawing \(G_i\sim N(0,I)\) and setting \(W_i=LG_i\). For a positive semidefinite matrix that is not positive definite, a suitable eigenvalue-based square root can be used instead.

Verifying the Dependence in Simulated Data

Verification has two parts. First, check the specification itself: confirm that the intended covariance matrix is valid, that loadings imply the intended population covariances, and that all modalities use the same individual ordering. Second, check simulated output: estimate cross-modal covariance or correlation and compare it with the target. These checks answer different questions. A valid model can produce a finite sample whose empirical dependence differs noticeably from the target, while a close empirical match in one sample does not prove that the simulation code implements the intended model in general.

For feature \(j\) in \(X\) and feature \(k\) in \(Y\), the empirical covariance and correlation are

Compare the statistic that matches the target: use covariance for a covariance specification and correlation for a correlation specification. Also inspect the entire cross-modal matrix where feasible. Checking only its average can conceal incorrect signs, misplaced associations, or a block of unintended dependence. For a simulation study, repeat the check across independent generated data sets and summarize the distribution of empirical dependence; this distinguishes systematic implementation errors from ordinary finite-sample fluctuation.

Diagnostics, Comparisons, and Limitations

When a check fails, first separate a specification problem from a sampling problem. A non-positive eigenvalue in a proposed joint covariance is a structural failure: that matrix cannot be used as the covariance of the intended nondegenerate multivariate normal. An empirical discrepancy in a valid simulation may instead be finite-sample variation. Further diagnostic clues include:

• Incorrect or permuted cross-modal associations: inspect the individual row alignment and the indexing used to construct the cross-modal block.
• Unexpectedly large associations: check for duplicated variables, residuals that are not independent as intended, or an incorrectly scaled latent factor.
• Unexpected within-modality correlations: shared factors generally induce within-modality dependence as well as cross-modal dependence. Compare both blocks with the model’s implied covariance.
• Numerical failures in matrix factorization: inspect symmetry and eigenvalues, and distinguish small numerical tolerances from genuinely negative eigenvalues. Do not silently change the target matrix without documenting the change.
• Extreme empirical correlations in small samples: remember that sample correlations fluctuate; check repeated simulations rather than treating a single draw as a precise estimate of the population target.

The latent-variable construction is compact and interpretable: loadings directly describe how strongly features relate to shared variation, and valid covariance follows from the construction. Its limitation is structural: a small number of factors cannot express every arbitrary cross-modal pattern, and shared factors also induce within-modality structure. Direct covariance specification offers finer control over particular entries, but requires a valid full covariance matrix and can become difficult to design as feature dimensions grow. Neither construction automatically guarantees realistic marginal distributions, dependence beyond the chosen model, or an appropriate biological interpretation. These are assumptions to assess for the intended benchmark, not properties supplied by the formulas alone.

A common mistake is to set each desired cross-modal correlation separately and assume the resulting matrix must be valid. Pairwise values can each lie between \(-1\) and \(1\), yet the full joint covariance can still fail positive semidefiniteness. The correction is to validate the complete covariance matrix, or use a generative construction whose assumptions guarantee a valid joint distribution. Another mistake is to treat a simulated sample correlation as the population target: the target belongs to the generating mechanism, while the observed statistic is a random estimate that varies across repetitions.