When Cross-Layer Edges Support Causal Claims

From a Directed Edge to a Causal Claim

A directed edge from a molecular feature in one omics layer to a feature in another may look like a mechanistic statement: changing the first feature changes the second. But a directed edge estimated from observational data does not establish that statement by itself. It may encode a conditional association, reflect a modeling constraint such as a prespecified layer order, or represent a causal relationship. These interpretations are not interchangeable.

For example, suppose \(X\) is a transcript abundance and \(Y\) is a protein abundance measured in the same samples. A fitted network containing \(X\to Y\) is evidence that the chosen model represents a directed relationship under its assumptions. To interpret it causally, we need assumptions that rule out alternative explanations, including an unmeasured common cause, the reverse direction, and inaccurate measurements.

Definition: In a causal directed acyclic graph (DAG), an edge \(X\to Y\) means that \(X\) is a direct cause of \(Y\) in the represented causal model, conditional on the other relevant causes of \(Y\). A causal effect compares the distribution of \(Y\) under interventions that set \(X\) to different values, written \(P(Y\mid do(X=x))\). An edge in an observationally estimated network is not automatically a causal edge.

The distinction matters because observational data record variables as they occurred; they do not directly show what would happen if one feature were deliberately changed while other causes were held to the appropriate conditions. A directed edge can support a causal claim only when the graph, data, and study design justify that counterfactual interpretation.

Assumptions Behind a Causal Edge

A causal DAG describes a proposed causal process, not merely a pattern of correlations. Its arrows must be acyclic, and each arrow must have a defensible direction. In multi-omics studies, a biological layer order can rule out some directions in the fitted graph, but an imposed order is not independent evidence that the arrows are biologically causal.

For an observational edge \(X\to Y\) to support a causal interpretation, several conditions are especially important:

Correct causal structure and direction. The graph must include relevant causes and represent the direction of influence correctly. Acyclicity is appropriate for the variables and time scale being modeled; unresolved feedback may require time-indexed variables.
No unmeasured confounding for the relationship. After conditioning on suitable pre-exposure variables, there must be no unmeasured common cause of \(X\) and \(Y\). A common cause can make them associated even when \(X\) does not cause \(Y\).
Appropriate conditioning. Adjustment variables should be chosen from the causal structure. Conditioning on a variable caused by both \(X\) and \(Y\), for example, can itself induce an association. Conditioning on a mediator also changes a total-effect question into a direct-effect question.
Valid measurement and sampling. The measured features should adequately represent the causal variables, and selection into the analyzed sample should not create a misleading relationship. Batch effects, cell mixture, and detection limits can violate this condition.
Consistency and positivity for effect estimation. The observed outcome for a sample with exposure \(X=x\) must correspond to the outcome under that exposure level, and each relevant covariate group must have a positive probability of the exposure levels being compared.

These assumptions are substantive, not merely computational. In particular, a method that selects a sparse DAG does not verify that unmeasured confounding is absent or that a layer order is biologically correct. Earlier material on D-separation and Markov equivalence established that distinct DAGs can encode the same observational conditional independences; observational data alone may therefore leave causal directions unresolved.

Key distinction: A conditional association is a property of the observed joint distribution. A causal effect is a contrast between distributions under interventions. Causal interpretation requires assumptions or design information connecting the two.

Adjustment: The Logic and Its Conditions

A useful justification for adjustment is the backdoor formula. Let \(U\) denote measured pre-exposure covariates that block all noncausal paths from \(X\) to \(Y\). If, conditional on \(U\), exposure assignment is independent of the potential outcome under each exposure level, and consistency and positivity hold, then the mean outcome under an intervention can be recovered by standardizing within covariate groups:

E[Y\mid do(X=x)] = \sum_u E[Y\mid X=x,U=u]P(U=u).

The formula averages the observed conditional mean at \(X=x\) over the covariate distribution in the target population. Its justification is that conditional exchangeability makes the exposed and unexposed samples comparable within \(U=u\), while consistency equates the observed outcome at \(X=x\) with the potential outcome under that exposure. If an important common cause is unmeasured, the conditional means need not identify the intervention means.

Worked Example: A Common Cause Creates an Apparent RNA-to-Protein Effect

Suppose \(X\) indicates high expression of a transcript and \(Y\) is protein abundance in arbitrary units. Let \(U\) indicate a cell-composition state. In a hypothetical population, \(P(U=1)=P(U=0)=0.5\), and the transcript is more often highly expressed when \(U=1\): \(P(X=1\mid U=1)=0.8\) and \(P(X=1\mid U=0)=0.2\). The protein follows \(Y=10+4U+\varepsilon\), where \(E[\varepsilon\mid X,U]=0\). There is no \(X\) term in this outcome equation, so changing \(X\) does not change \(Y\) in this model.

First calculate the distribution of \(U\) among samples at each value of \(X\). Bayes' rule gives \(P(X=1)=0.8(0.5)+0.2(0.5)=0.5\), so \(P(U=1\mid X=1)=0.8(0.5)/0.5=0.8\). Similarly, \(P(X=0)=0.5\), and \(P(U=1\mid X=0)=0.2(0.5)/0.5=0.2\).

Using the stated conditional-mean assumption, the observed means are \(E[Y\mid X=1]=10+4(0.8)=13.2\) units and \(E[Y\mid X=0]=10+4(0.2)=10.8\) units. The observed mean difference is \(13.2-10.8=2.4\) units. Yet under intervention, the distribution of \(U\) remains one-half in each state, and \(X\) does not enter the equation for \(Y\). Thus \(E[Y\mid do(X=1)]=E[Y\mid do(X=0)]=10+4(0.5)=12\) units: the causal effect is zero.

The association arises because cell composition predicts both transcript expression and protein abundance. The condition \(E[\varepsilon\mid X,U]=0\) is needed for the conditional means used above; merely assuming that \(\varepsilon\) has an overall mean of zero would not justify them.

Reverse Causation and Measurement Error

A plausible biological ordering does not guarantee that the measured direction is correct. If two features are measured at one time point, a downstream response may influence an upstream-looking marker, or both may participate in feedback. Time-indexed measurements can help distinguish such processes, but temporal precedence alone does not eliminate confounding.

Worked Example: A Reverse Direction Produces a Positive Edge

Let \(Y\) be a protein measured in arbitrary units and \(X\) a transcript measured at the same observation time. Consider the hypothetical system \(Y=2+V\) and \(X=1+0.5V+\eta\), where \(V\) and \(\eta\) are independent, have mean zero, and have variances 4 and 1, respectively. This model has \(Y\to X\), not \(X\to Y\).

The covariance is \(\operatorname{Cov}(X,Y)=0.5\operatorname{Var}(V)=0.5(4)=2\). The variance of \(X\) is \(0.5^2(4)+1=2\). Therefore the slope from regressing \(Y\) on \(X\) is \(2/2=1\) protein unit per transcript unit. This is the slope of the linear regression of \(Y\) on \(X\); without additional assumptions, the conditional mean \(E[Y\mid X]\) need not be linear in \(X\).

Despite that positive observational slope, intervening on \(X\) does not change \(Y\): its equation is \(Y=2+V\) and contains no \(X\). The positive relationship reflects their shared dependence on \(V\) and the direction \(Y\to X\). A cross-sectional edge oriented as \(X\to Y\) would therefore misstate the causal direction.

Measurement error creates a different problem: the observed feature may not be the causal quantity of interest. Classical error in an exposure measurement often weakens a relationship, while correlated assay errors or batch effects can create associations. Thus an estimated edge can be distorted even if the causal direction is otherwise known.

Worked Example: Classical Error Attenuates a True Cross-Layer Effect

Suppose the true transcript level \(X\) has mean zero and variance 1, and the protein obeys \(Y=2X+\varepsilon\), where \(\varepsilon\) has mean zero and variance 1. The true causal change in mean protein abundance is 2 units for a one-unit intervention increase in \(X\). However, the assay records \(X_{\mathrm{obs}}=X+\delta\), where independent measurement error \(\delta\) has mean zero and variance 3. Assume \(X,\delta,\varepsilon\) are mutually independent and normally distributed.

The covariance of the observed transcript with the protein is \(\operatorname{Cov}(X_{\mathrm{obs}},Y)=\operatorname{Cov}(X+\delta,2X+\varepsilon)=2\operatorname{Var}(X)=2\). The observed transcript variance is \(\operatorname{Var}(X_{\mathrm{obs}})=1+3=4\). Hence the regression slope using the measured transcript is \(2/4=0.5\) protein units per observed transcript unit, rather than the true causal slope of 2.

The discrepancy follows from treating a noisy proxy as if it were the true exposure. An edge based on \(X_{\mathrm{obs}}\) may consequently understate the effect or be missed by a selection procedure. If transcript and protein measurements also share a batch error, even a relationship absent between the true quantities can appear between the measured values.

Evidence, Diagnostics, and Alternatives

No single diagnostic can prove that a causal edge is correct, especially when unmeasured causes are possible. Diagnostics can, however, reveal inconsistencies and motivate sensitivity checks. Before making a causal claim, ask whether the proposed cause precedes the outcome at a biologically meaningful time scale; whether important common causes such as cell composition, treatment, and batch were measured; and whether the edge persists under defensible adjustment choices. Check whether adjustment has included post-exposure variables or colliders, whether missingness or sample selection differs across groups, and whether assay reliability is adequate for both layers.

Compare the causal interpretation with its main alternatives. An undirected or conditional-association network describes dependence without asserting an intervention effect. A DAG learned under a layer order can encode direction consistent with that constraint, but the order is an assumption, not a causal test. Longitudinal sampling, perturbation experiments, and credible external instruments can add information about direction or intervention effects, though each has its own assumptions. In particular, a perturbation is informative only if it changes the intended feature and does not affect the outcome through unintended pathways.

A practical claim should be calibrated to the evidence. “The fitted network contains a directed cross-layer edge” reports a model result. “The association is consistent with a causal effect under the stated assumptions” is a conditional interpretation. A stronger claim that changing the transcript changes the protein requires credible support for those assumptions and for the measurement model. None of these claims establishes causation merely because the edge is statistically selected.

Common mistake: “The graph has an arrow from transcript to protein, and the transcript is earlier in the layer order, so the transcript causes the protein.” The arrow may depend on the imposed order, and an unmeasured common cause, reverse process, or measurement error may explain the observed data. The correction is to state the causal assumptions explicitly and seek design or measurement evidence that makes the causal interpretation credible.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Why does a directed edge in an estimated observational DAG not, by itself, show that intervening on its parent changes its child?
In the confounding example, what assumption about \(\varepsilon\) justifies the reported conditional means?
In the reverse-direction example, what is the regression slope of \(Y\) on \(X\), and why is it not the causal effect of \(X\) on \(Y\)?
If a measured exposure has independent classical error, what happened to the slope in the measurement-error example, and what were its true and observed values?
Name two design or data features that can strengthen causal interpretation of a cross-layer edge, and state one limitation of each.

Solutions. (1) The edge may represent conditional association or reflect constraints used in learning; causal interpretation requires assumptions connecting observational data to intervention distributions. (2) The required condition is \(E[\varepsilon\mid X,U]=0\), so the residual contributes zero to each conditional mean. (3) The slope is \(\operatorname{Cov}(X,Y)/\operatorname{Var}(X)=2/2=1\) protein unit per transcript unit, but the structural equation for \(Y\) does not contain \(X\); the association reflects the reverse direction and shared \(V\). (4) The true causal slope is 2 protein units per true transcript unit, while measurement error gives an observed regression slope of \(0.5\) protein units per observed transcript unit. (5) Longitudinal sampling can establish temporal ordering but does not rule out confounding; perturbation can provide intervention evidence but may have off-target effects. Each strengthens evidence only when its own assumptions are credible.