Reporting A Multi-Omics Factor Analysis

Why Reporting Is Part of the Analysis

A multi-omics factor analysis can yield scores and loading profiles that are difficult to interpret without knowing how they were obtained. A statement such as “three factors were identified” leaves important questions unanswered: Were all factors shared across omics blocks? Which likelihood was fitted? Were measurements transformed or standardized? How were rotations and signs fixed? What evidence suggests that the factors are adequately fitted and stable?

A useful report makes these choices explicit. It lets a reader understand what the factors represent under the fitted model, judge whether the fit is adequate for the intended interpretation, and reproduce the analysis. Reporting does not make a factor biologically causal: a factor summarizes patterns of association under a statistical model.

State the Model Before Reporting Its Results

Consider \(M\) omics blocks measured on the same \(n\) subjects. Let \(X_i^{(m)}\) be the \(p_m\)-dimensional vector for subject \(i\) in block \(m\). One possible model has shared latent scores \(z_i\), block-specific scores \(u_i^{(m)}\), and residuals:

Here, \(\mu^{(m)}\) is a block-specific mean vector; the columns of \(\Lambda_s^{(m)}\) are shared-factor loadings for block \(m\); and the columns of \(\Lambda_p^{(m)}\) are private-factor loadings for that block. The score vectors describe subject-level positions on the factors. This is one representative structure, not a universal specification: a fitted model might omit private factors, impose sparsity, use another residual covariance structure, or model transformed observations.

This model requires the stated distributional assumptions and the specified covariance structure to be plausible enough for the analysis. It also assumes that the observations being treated as independent subjects are appropriately defined. If the fitted model uses non-Gaussian data, correlated residuals, or a different score distribution, the report should state that model and its corresponding likelihood instead; it should not describe the Gaussian likelihood above as though it applied.

For the Gaussian model, the log likelihood is

This is the marginal or observed-data log likelihood: the latent scores have been integrated out. A fitting procedure may instead report a conditional likelihood given estimated scores, a variational bound, or another objective. These are not interchangeable quantities. State what was optimized and what the reported number represents.

What a Reproducible Report Should Specify

A report should identify the data entering the model before describing the fitted factors. Name each omics block, its number of subjects and features, the subject-matching rule, and any exclusions or missing-data handling. State the transformations, normalization, centering, scaling, and feature filtering applied to each block. In particular, tell readers whether scaling occurred within each block and whether the factor model used transformed values or the original measurement scale.

Report the number of shared factors and the number of private factors in each block separately. “Four factors” is ambiguous when some are shared and others private. Also report the estimation method, software and version, relevant convergence tolerances, initialization or restart strategy, and any priors or tuning parameters. If a method returns a point estimate, say whether it is a maximum-likelihood estimate, a penalized estimate, or another estimate.

Factor loadings and scores can be non-identifiable without constraints. For example, replacing a loading matrix \(\Lambda\) by \(\Lambda Q\) and scores \(Z\) by \(ZQ\), for an orthogonal matrix \(Q\), leaves the fitted product \(Z\Lambda^\mathsf{T}\) unchanged. Factor signs can also be reversed together without changing the fit. Consequently, the labels “Factor 1” and “Factor 2” do not by themselves identify unique scientific quantities.

State the constraints or conventions used to address this ambiguity: for example, fixed score variances, a specified rotation, triangular loading constraints, or a post-fit alignment and sign rule. A post-fit rotation may aid interpretation, but readers should be told whether it changes the fitted model, is only a display convention, or is part of the estimation procedure. The constraints should be consistent with the model and adequate to identify the reported parameterization; do not imply that any particular constraint resolves every possible ambiguity.

Likelihoods, Factor Counts, and Comparisons

When comparing candidate factor counts, give the candidate models, their likelihood values, and the rule used to choose among them. For models fitted by maximum likelihood to the same observations using comparable likelihoods, common criteria include

where \(\ell\) is the maximized log likelihood and \(q\) is the number of free parameters under the fitted, identified model. Lower values are preferred by these criteria, but neither criterion proves that a factor is real or uniquely interpretable. Parameter counting must respect the model’s constraints. For a marginal likelihood that integrates latent scores, do not count each subject’s latent score as a free parameter merely because scores are later estimated or predicted. If models use different observations or incomparable objective functions, a direct AIC or BIC comparison may not be justified.

Diagnostics, Stability, and Uncertainty

A fitted factor model should be accompanied by evidence about whether its assumptions and interpretation are adequate for the intended use. Report the diagnostics actually examined and summarize important findings. These may include residual patterns within and across blocks, large residual associations, convergence behavior, and checks of whether replicated or simulated data reproduce relevant features of the observations. A single favorable summary statistic cannot rule out misspecification. If residual dependence remains where the model assumes independence, or if results change substantially under reasonable model adjustments, say so and qualify the interpretation.

Separate three questions that are sometimes collapsed into “the factors are robust.” First, fit: does the model represent important features of the observed data? Second, stability: are the estimated loading profiles or subject scores similar under resampling, perturbing the feature set, or changing reasonable settings? Third, uncertainty: how imprecisely are particular estimated quantities determined? The earlier material on residual diagnostics, posterior predictive checks, resampling stability, and uncertainty in factor interpretation supplies tools for these distinct questions; a report should state which tools were used and what each showed.

For stability summaries, identify the perturbation scheme, number of replicates, and how factors were matched across fits. Matching matters because factors can switch order, change sign, or rotate. Give a quantitative summary, such as the proportion of replicates in which a factor could be matched and the distribution of matched loading-profile correlations. For uncertainty, report the method and the quantity being summarized: for example, a bootstrap interval for a loading, or a posterior credible interval under a specified model. A confidence interval and a credible interval have different interpretations and should be named correctly.

Comparison, Limitations, and Common Reporting Errors

Different reporting choices answer different questions. AIC and BIC compare candidate likelihood-based models with complexity penalties; resampling examines sensitivity to data perturbations; intervals summarize uncertainty under a specified inferential procedure; residual and predictive checks assess aspects of fit. None substitutes for the others. A model with the preferred information criterion may still have poor residual behavior, and a stable factor may still be an artifact of shared preprocessing or an unsuitable model.

A common mistake is to report only the number of factors and a plot of the largest loadings, then call those factors “validated.” That reasoning confuses a readable display with evidence about fit, stability, and uncertainty. The correction is to report the model and estimation choices, the identification convention, the diagnostics performed, and quantitative stability and uncertainty summaries. A second mistake is to compare conditional likelihoods from one fit with marginal likelihoods from another as if they were the same quantity. Name the likelihood or objective precisely and compare only compatible quantities.

Factor interpretation also has limits. Loadings depend on the fitted model, preprocessing, feature set, and identification convention. Rotated factors can be useful summaries without being unique biological entities. If a factor is weakly matched across resamples, has broad uncertainty, or changes markedly when reasonable settings change, state that limitation plainly rather than assigning it a confident biological label.