Mechanistic Evaluations: Causal Pathways

• Mechanistic evaluations are systematic approaches that reveal and quantify internal causal processes using deterministic and probabilistic frameworks.
• They employ dichotomization and risk difference tests (e.g., R11 – R10 – R01 > 0) to identify interactions beyond mere statistical correlation.
• Causal graphical models, such as ADAGs, support these evaluations by linking interventional and observational data under conditional independence.

Mechanistic evaluations are systematic approaches for elucidating, validating, and quantifying the internal causal processes underlying the predictions or behaviors of complex models or systems. In the context of scientific modeling, statistics, epidemiology, and machine learning, mechanistic evaluation refers to methods that move beyond correlational or surface-level interaction analysis—aiming to uncover and test whether specific explanatory factors interact within the system via shared, structured, and, in some cases, deterministic pathways.

1. Formalization of Mechanistic Interactions

Mechanistic interaction is formalized through the specification of a deterministic or probabilistic functional relationship that governs how multiple causal variables combine to yield an outcome. The "deep determinism" assumption posits that there exists a deterministic function

$Y = f(A, B, V)$

where $Y$ is a binary response (e.g., disease presence), $A$ and $B$ are causal factors (which can be categorical, binary, or continuous), and $V$ is a set of possibly unobserved context variables. For any fixed $V$, setting $A$ and $B$ fixes the outcome $Y$ with certainty under this framework (Berzuini et al., 2010).

Mechanistic interaction is then defined not merely as statistical dependence between effects, but as "interference" or "coaction": variable $B$ interferes with $A$ if, for some context, setting $A$ to a particular value guarantees $Y=0$ regardless of $B$ (and vice versa). This asymmetry allows for nuanced mechanistic statements (e.g., $A$ blocks $B$ but not vice versa) and forms the basis for distinguishing mechanistic from statistical interaction, especially where variable types and data collection designs differ.

2. Testing Mechanistic Interactions: Probabilistic and Statistical Approaches

To enable empirical evaluation, mechanistic interaction criteria are linked to observable quantities. The key test involves dichotomizing continuous or ordinal predictors and defining indicator sets:

• For $A$, let threshold $\tau_A$ define $\alpha = \{a: a > \tau_A\}$ and its complement $\bar{\alpha}$, analogously for $B$.
• Risk functions are calculated as

$$R_{ijc} = P(Y = 1 \mid A \in \{\alpha\, \text{if}\ i = 1,\ \bar{\alpha}\ \text{if}\ i = 0\},\ B \in \{\beta\, \text{if}\ j = 1,\ \bar{\beta}\ \text{if}\ j = 0\},\ C = c)$$

for covariate strata $C = c$.

The central test (under core conditional independence conditions and a monotonicity assumption) is: $R_{11c} - R_{10c} - R_{01c} > 0$ for some $c$ (and appropriately chosen dichotomizations), supplemented by "insensitivity" properties on the dichotomization process (Berzuini et al., 2010, Berzuini et al., 2013). This positive difference is interpreted as direct evidence of mechanistic coaction—i.e., combined presence of $A$ and $B$ conferring risk beyond their individual effects.

In fully probabilistic frameworks, mechanistic interaction is equated with a deviation from a generalized "noisy OR" model for the negative (failure) outcome: $\pi_w(a, b) = \lambda_w(a)\, \mu_w(b)$ (with $\pi_w(a, b)$ the probability of $Y=0$ for context $w$ under intervention) (Berzuini et al., 2013). Observed risk-based inequalities (as above) correspond to this factorization holding or failing, regardless of variable type or scale.

3. Causal Frameworks: Augmented Directed Graphs and Conditional Independence

Mechanistic evaluations require careful mapping between interventional and observational data regimes, which is encoded in causal graphical models:

• Augmented Directed Acyclic Graphs (ADAGs): Nodes represent variables $A$, $B$, $Y$, and possible context $V$ (latent or observed), supplemented by regime indicators $\sigma_A$, $\sigma_B$ denoting manipulation or observation of $A$, $B$ values.
• Key independence is: $Y \perp (\sigma_A,\sigma_B)\,|\, (A,B)$, expressing "no unmeasured confounding" once $A$, $B$ are known.
• Conditional Independence: Core conditions stipulate specific conditional independencies (e.g., independence of unobserved context given observed covariates), supporting valid translation of observed associations to mechanistic interpretations.

This graphical approach underpins the rationale for applying the risk-based tests above to observational data, as the core independencies are justified in either randomized-interventional or well-stratified observational studies.

4. Deep Determinism Assumption: Role, Justification, and Limitations

The deep determinism assumption ($Y=f(A,B,V)$, exact for all relevant $V$) is central. It posits that, for given context, the outcome is a deterministic function of causes—justifying the test for mechanistic interaction as revealing strict causal relationships.

• Justification: In molecular biology and genetics, many processes (e.g., gene–environment disease mechanisms) plausibly operate via deterministic biological pathways conditional on unmodeled context.
• Caveats: In practical settings, not all relevant context $V$ may be observable; stochastic effects may persist due to measurement error, model misspecification, or inherent biological randomness. Thus, deep determinism may at times be an idealization, but provides a tractable working assumption whose appropriateness requires domain evaluation.

5. Case Studies: Coronary Artery Disease Examples

Mechanistic evaluation methodology is illustrated in two coronary artery disease case studies:

Study Context	Variables & Model	Mechanistic Interaction Test & Interpretation
Post-infarction, genetic marker & statin	$G$ = rs1333040 (dichotomized); $S$ = statin use; Linear-risk Bernoulli, covariates $Z$, $T$, $I$	$G$, $S$ interaction term significant; under assumptions, test claims G and S "strongly coact" for reinfarction
Early MI, smoking habit & alternate SNP	$G$ = rs4620585 (dichotomized); Smoking (bin.); Linear-odds regression	$R_{11}-R_{10}-R_{01}$ significantly $>0$, interpreted as genetic variant "blocking" smoking risk, or vice versa

These studies support the claim that mechanistic coaction can be observed in real-world data, provided the assumptions hold and the causal structure is appropriately modeled.

6. Scientific Relevance, Limitations, and Practical Implications

Mechanistic evaluations grounded in the above frameworks move the analysis from mere correlation or statistical interaction (e.g., nonadditivity or nonmultiplicativity) to statements about shared mechanisms and interference.

• Advantages:
  • Applicable to arbitrary mixtures of categorical and continuous variables (post-dichotomization);
  • Integrates directly with both randomized and observational data by leveraging explicit conditional independence and causal graph formalism;
  • Guides clinical or mechanistic inference, aiding in the identification of contexts where interventions are effective or ineffective depending on genotype or exposure.
• Limitations:
  • Relies on strong assumptions (deep determinism, monotonicity, correct model specification, and sufficient control for confounding);
  • Requires careful justification of dichotomization choices and context covariate selection to avoid loss of sensitivity or introduction of bias;
  • May not be robust to residual stochasticity, model violations, or insufficient context measurement.
• Broader Implications:
  • Provides a template for mechanistic inference in fields ranging from genetics and epidemiology to psychology, wherever hypotheses about shared causal mechanisms are of interest.
  • Encourages mechanistic thinking in epidemiological design, policy evaluation, and biomarker discovery by focusing attention on causal pathways rather than observed associations alone.

7. Summary Table: Mechanistic Evaluation Test Structure

Step	Mechanistic Principle	Critical Formula / Condition
Dichotomize predictors	Seek thresholds $\tau_A$, $\tau_B$ for $A$, $B$	$\alpha = \{a : a > \tau_A\}$, etc.
Compute stratified risks	Probabilistic assessment in stratum $C = c$	$R_{ijc} = P(Y=1 | A \in \ldots, B \in \ldots, C=c)$
Test for coaction	Mechanistic interference or coaction inequality	$R_{11c} - R_{10c} - R_{01c} > 0$
Justify observational generalization	Conditional independence via ADAG	$Y \perp (\sigma_A,\sigma_B) | (A,B)$
Interpret finding	Map result to coaction, shared pathway, or blocking	Narrative domain-specific mechanistic claim

In summary, mechanistic evaluations provide a principled approach to identifying and testing for interaction at the pathway or mechanism level in complex systems, uniting formal causal modeling, rigorous statistical testing, and domain knowledge in a coherent framework (Berzuini et al., 2010, Berzuini et al., 2013).