Matching Omics Marginal Distributions

Why Marginal Distributions Matter

A simulation can contain the intended shared signals and cross-omics associations yet still be unrealistic if its individual measurements have the wrong scale or shape. For example, a continuous assay may be positive and strongly right-skewed, a count assay may have many small values and a long upper tail, and a composition must sum to one. Replacing all of these with normally distributed values can produce impossible measurements and misrepresent the extreme observations that affect many analysis methods.

Here, the goal is to match each feature’s marginal distribution: the distribution of that feature considered by itself, without specifying its joint relationship with other features. For subject \(i\), feature \(j\), and modality \(m\), write the measurement as \(X_{ij}^{(m)}\). A target marginal \(F_j^{(m)}\) describes the distribution intended for that feature across subjects. Matching these targets is one component of simulation design, not a substitute for the cross-omics dependence considered in the earlier material on Simulating Cross-Omics Dependence.

A useful target is more specific than “realistic.” State the measurement scale, allowable values, and quantities to be matched. These may include the mean and variance, selected quantiles, proportion of zeros, and a scientifically plausible upper range. If targets are estimated from a reference data set, acknowledge that they inherit its sampling uncertainty and any preprocessing choices. A target distribution should represent the measurement process of interest, not merely be convenient to simulate.

Choosing and Generating Marginals

Distribution choice begins with the measurement’s support and sampling mechanism. A positive, right-skewed continuous measurement may be described by a lognormal model. Counts require distributions on the nonnegative integers; a Poisson model has equal mean and variance, whereas a negative-binomial model can allow overdispersion. Compositional measurements lie on a simplex, so their components are constrained to sum to one. Zero-inflated counts require a model that distinguishes an additional zero-generating process from the count distribution among observations not in that process.

For a continuous target, this construction gives an exact model-based marginal in repeated simulation, up to numerical implementation. For a discrete target, the cumulative probability jumps create ties, as they should. A finite simulated block will not have exactly the target mean, variance, or quantiles: random variation remains. The purpose of diagnostics is to decide whether discrepancies are ordinary finite-sample variation or evidence of an incorrect model, parameter, transformation, or implementation.

Checking Marginal Fit and Diagnosing Failure

Choose diagnostics before generating the block, based on the target and the intended use of the simulation. A compact diagnostic set often includes the minimum and maximum, mean and standard deviation where meaningful, selected quantiles such as the median and 95th percentile, and a histogram or empirical distribution plot. For counts, add the zero fraction and integer/nonnegative checks. For compositions, verify that each row is nonnegative and sums to one within numerical tolerance, then inspect component distributions. Compare the diagnostics with the target values, not with vague expectations such as “the values look plausible.”

Interpret discrepancies in context. An empirical 95th percentile far above its target may indicate a tail that is too heavy, a parameterization error, or simply a rare finite-sample fluctuation; inspect repeated simulation blocks or a wider set of quantiles before deciding. A negative value from a model intended to produce positive measurements indicates a support error, not harmless sampling noise. An unexpectedly high count zero fraction may indicate an unsuitable count model or a coding error. For compositions, row sums that differ materially from one signal a generation or normalization problem.

One useful justification for the last step concerns strictly increasing transformations. If \(Z\sim N(0,1)\) and \(X=F^{-1}\{\Phi(Z)\}\), where \(\Phi\) is the standard normal cumulative distribution function and \(F\) is a continuous target distribution, then \(X\) has marginal distribution \(F\). Indeed, for any \(x\) in the support,

The last equality follows because \(Z\) is standard normal. This establishes the marginal result, not that every dependence measure is preserved. Monotone transformations preserve the ordering of values within each feature, but generally change Pearson correlations. Discrete targets introduce ties, which can further alter rank-based dependence. Thus, a procedure that matches marginals must still be checked against the intended joint structure.

Alternatives, Limitations, and Common Pitfalls

No method makes the target distribution objectively “realistic” without a defensible target. A parametric model may be useful for stress-testing tails, but that does not establish that its tail is empirically plausible. Empirical resampling avoids some distributional assumptions but is limited by the reference data’s range and size. Quantile matching can reproduce chosen marginals while leaving important joint behavior misspecified. The right choice depends on the benchmark question and should be reported with the distributional assumptions and diagnostics.

• Confusing a marginal with a joint model. Correct per-feature histograms do not imply realistic cross-feature relationships. Diagnose both separately.
• Matching only the mean. Equal means can hide different skewness, zero rates, or upper tails. Check quantities that matter for the method comparison.
• Using a distribution with the wrong support. A normal model can generate negative values for a strictly positive assay, and a continuous model cannot represent integer counts exactly.
• Forcing exact finite-sample agreement. A random sample will fluctuate around its target. Altering generated observations to hit every target summary can change the intended distribution and must be justified as part of the design.
• Ignoring transformations and constraints. Log transformations, normalization, and compositional closure affect the scale and dependence of the measurements. State whether targets apply before or after such processing.

The central practical principle is to define the marginal target on the scale that the simulated analysis will receive, then verify support, central summaries, zero behavior, and tails. Matching only one of these features can yield data that look plausible at a glance but behave unlike the intended measurements.