Class 6

When Hazards Collide

Modelling synergistic, antagonistic, and threshold interactions among psychosocial risk factors using canonical Bayesian network structures

Beyond Additive Risk: Why Interaction Modelling Matters

In previous chapters, we constructed Bayesian networks where multiple parent nodes influenced a single child. Each conditional probability table (CPT) encoded the joint effect of all parent configurations on the child's state. What we did not examine closely was the structure of those joint effects. When you fill a CPT for a node with two binary parents, you specify four conditional probabilities. Those four numbers implicitly encode a claim about how the two parent hazards combine. But in practice, most risk analysts fill CPTs cell-by-cell without asking a fundamental question: what kind of interaction am I modelling?

This question matters because different interaction types have dramatically different implications for intervention. If two hazards combine additively-each contributing independently to risk-then reducing either hazard by a fixed amount produces the same benefit regardless of the other hazard's level. But if they combine synergistically-where the joint effect exceeds the sum of individual effects-then interventions are most valuable when they target the specific combination. And if they combine antagonistically-where one hazard buffers the effect of another-then removing the buffer without addressing the underlying stressor could paradoxically increase harm.

The epidemiological literature on occupational health is unambiguous: psychosocial hazards interact. Ervasti et al. (2022) studied nearly 10,000 Finnish healthcare workers during the COVID-19 pandemic and found that the combination of high job demands, low rewards, and low workplace social capital produced a Relative Excess Risk due to Interaction (RERI) of 6.27-meaning the combined effect was over six times larger than what you would expect from simply adding the individual effects. This is not a subtle statistical phenomenon. It is a massive, clinically significant super-additive effect that any competent risk model must be capable of representing.

---

The Demand-Control Insight: Interaction as Theory

The most influential model of psychosocial hazard interaction in occupational health is Robert Karasek's demand-control (DC) model, introduced in a landmark 1979 paper that has shaped four decades of workplace health research. Karasek's central claim is deceptively simple: job strain is not caused by high demands per se, nor by low decision latitude per se, but by their specific combination. A surgeon faces enormous demands but typically has substantial control over how to execute procedures-this is an "active" job that can be stimulating rather than harmful. A data-entry clerk may face moderate demands but has almost no control over pace, method, or timing-and this lack of control transforms moderate demands into a significant health risk (Karasek, 1979).

The DC model identifies four quadrant job types based on the demand-control interaction: low-strain (low demands, high control), passive (low demands, low control), active (high demands, high control), and high-strain (high demands, low control). Crucially, the model predicts that the high-strain quadrant produces outcomes worse than would be expected from an additive model. This is the strain hypothesis-and it has a direct, precise translation into Bayesian network structure.

Translating Demand-Control into Network Structure

Consider two ways to represent the demand-control relationship in a Bayesian network. In the naive additive version, Demands and Control are both parents of a Stress node, and the CPT encodes independent, additive contributions. Mathematically, P(Stress=High | Demands, Control) ≈ w_d·Demands + w_c·(1 − Control), where the weights sum to 1. This model says that reducing demands by one unit always produces the same stress reduction, regardless of control level.

In the interaction version, we either add an explicit interaction term to our combination function or introduce a mediating node that captures the Demands×Control interaction. Now P(Stress=High | Demands, Control) includes a term proportional to Demands·(1 − Control)-a product that is large only when demands are high and control is low simultaneously. This product term is what gives the model its distinctive prediction: at high levels of control, increasing demands has a modest effect on stress; at low levels of control, the same increase in demands has a devastating effect.

Van der Doef and Maes (1999) reviewed twenty years of empirical tests of the DC model across 63 studies and found strong, consistent support for the strain hypothesis-workers in high-demand, low-control jobs consistently showed the worst psychological outcomes. However, support for the more specific buffer hypothesis-that control moderates (buffers) the effect of demands in a strict statistical interaction sense-was more inconsistent. This inconsistency, the authors argued, stemmed largely from measurement specificity: when demands and control were measured at the task level rather than at a global job level, interaction effects emerged more reliably. This finding has a direct implication for Bayesian network modelling: the granularity at which you define your nodes affects whether interaction effects become visible in your data.

The Effort-Reward Extension

The demand-control model is not the only interaction-based framework in occupational health. Siegrist's effort-reward imbalance (ERI) model posits that sustained strain arises from a mismatch between high efforts expended at work and low rewards received in return-including financial compensation, esteem and recognition, and job security. Van Vegchel et al. (2005) reviewed 45 empirical studies and found robust support for the ERI main effect, and also examined an interaction hypothesis: that a personality trait called overcommitment-a tendency to exhaust oneself through excessive work engagement-amplifies the harmful effects of effort-reward imbalance.

In Bayesian network terms, the ERI model suggests a structure where Effort and Reward are parents of an Imbalance node, and Overcommitment acts as a moderating parent. The CPT for the outcome node (say, Burnout) must encode the fact that high imbalance combined with high overcommitment produces worse outcomes than either alone-a synergistic interaction. Without an interaction term, the model would underestimate risk for overcommitted workers in imbalanced jobs and overestimate risk for everyone else.

---

Canonical Models for Combining Parent Influences

Filling a CPT cell-by-cell becomes impractical as the number of parents grows. A node with three binary parents requires 8 conditional probabilities; four binary parents require 16; five require 32. This exponential growth-2ⁿ entries for n binary parents-is not merely an inconvenience. It is a fundamental barrier to building realistic psychosocial risk models, because real outcomes are influenced by many factors simultaneously. The solution, introduced by Pearl (1988) and formalised extensively by Koller and Friedman (2009), is to use canonical models (also called combination functions or gates) that specify how multiple parent influences combine using far fewer parameters.

The Noisy-OR Gate

The noisy-OR is the most widely used canonical model in Bayesian networks. It encodes the assumption that each parent is an independently sufficient cause of the child outcome-any single parent being active is enough to produce the outcome, though each parent may fail to do so with some probability. Formally, for a child node Y with binary parents X₁, …, X_n:

P(Y = 0 | X₁, …, X_n) = (1 − q₀) · ∏_{i: Xi=1} (1 − q_i)

Here, q_i is the probability that parent X_i alone would activate the child (its "causal strength"), and q₀ is a leak probability allowing the child to activate even when no parent is active. The key property: instead of needing 2ⁿ parameters, you need only n + 1. Pearl (1988) showed that this dramatic parameter reduction comes from a substantive assumption-causal independence-which states that each parent's mechanism for causing the child operates independently of the others.

Oniśko et al. (2001) demonstrated the practical power of this assumption. Working with the HEPAR II liver disease diagnosis model, they found that using noisy-OR gates to fill CPTs from limited patient data improved diagnostic accuracy by 6.7% compared to standard parameterisation. When data are sparse-and in psychosocial risk assessment, they almost always are-the structural assumptions of the noisy-OR provide a principled way to generalise from observed combinations to unobserved ones.

In psychosocial risk terms, the noisy-OR is appropriate when multiple hazards can independently trigger an outcome. Consider role ambiguity as a child node with parents including "unclear job description," "conflicting supervisor instructions," and "changing organisational priorities." Each of these can independently create role ambiguity-you don't need all three to be confused about your role. The noisy-OR captures this: each additional source of ambiguity increases the probability of the outcome, but with diminishing marginal returns (the surface is concave).

The Noisy-AND Gate

The noisy-AND encodes the opposite assumption: all parents must be jointly present to produce the outcome. Each parent is a necessary (but probabilistically imperfect) condition. Formally:

P(Y = 1 | X₁, …, X_n) = q₀ + (1 − q₀) · ∏_{i: Xi=1} q_i · ∏_{j: Xj=0} (1 − q_j*)

In a simplified form for intuition: the outcome probability is high only when all parents are active. The surface is convex-it stays low until multiple hazards co-occur, then rises steeply. This captures threshold effects, where risk materializes only when hazards combine.

Consider a psychosocial example: a worker filing a formal bullying complaint. The probability of a complaint being filed might require the conjunction of (a) experiencing bullying behaviour, (b) perceiving that behaviour as intentional/repeated rather than incidental, AND (c) believing that grievance procedures exist and are trustworthy. Remove any one of these-no bullying, no perception of intent, or no trust in procedures-and the complaint probability drops dramatically. This is noisy-AND reasoning: the outcome requires a constellation of conditions, not just any single trigger.

Weighted Linear Combinations

Between the extremes of noisy-OR and noisy-AND sit weighted linear combination functions, which assign a weight to each parent and compute the child probability as a (possibly nonlinear) function of the weighted sum. The simplest form is:

P(Y = 1 | X₁, …, X_n) = σ(w₀ + Σ w_i · X_i + Σ w_ij · X_i · X_j)

where σ is a sigmoid function that squashes the result into [0, 1]. The product terms w_ij · X_i · X_j are exactly the interaction effects we have been discussing. When all w_ij = 0, the model is purely additive. When some are positive, we get synergistic interactions; when negative, antagonistic ones. This framework directly connects the Bayesian network representation to the regression-based interaction models familiar from epidemiology (Koller & Friedman, 2009).

The crucial insight for practice is that the choice of combination function is not a technical convenience-it is a substantive claim about how workplace hazards interact. Choosing a noisy-OR says "each hazard independently contributes to risk." Choosing a noisy-AND says "hazards must co-occur to produce harm." Choosing a weighted combination with interaction terms says "certain specific pairs amplify each other." Each choice encodes a different theory of how the workplace operates, and each produces different risk predictions and different intervention priorities.

---

Choosing the Right Combination Function

Given the menu of canonical models, how do you choose the right one for a particular node in your network? The answer requires domain knowledge-specifically, an understanding of the causal mechanism by which the parent hazards produce the child outcome. Here are heuristic guidelines grounded in the psychosocial risk literature:

Use noisy-OR when multiple hazards serve as independent pathways to the outcome. Example: Psychological distress with parents including work-family conflict, role ambiguity, and interpersonal conflict. Any one of these stressors can independently trigger distress, and their co-occurrence incrementally increases risk without producing a dramatically worse-than-additive effect.

Use noisy-AND when the outcome requires a specific constellation of conditions. Example: Workplace violence escalation, which typically requires the co-occurrence of an aggrieved individual, a triggering event, lack of de-escalation training, and absence of security protocols. Removing any single element substantially reduces the probability of escalation.

Use weighted combinations with interaction terms when specific pairs of hazards are known to have synergistic or antagonistic effects. Example: The demand-control interaction, where high demands and low control together produce worse-than-additive strain (Karasek, 1979). Or the effort-reward-overcommitment triple interaction documented by van Vegchel et al. (2005), where the interaction term captures risk that neither main effect nor pairwise combination can explain.

Use a fully specified CPT when you have sufficient data to estimate all cells directly and the interaction pattern is too complex or irregular to be captured by any canonical model. This is rare in psychosocial risk work because of the data requirements-but when you have large-scale survey data or comprehensive incident records, direct estimation is preferred because it makes no structural assumptions at all.

---

The Data Challenge: Estimating Interaction Parameters

There is an uncomfortable irony in interaction modelling: the interactions that matter most for risk assessment are precisely the ones that are hardest to estimate from data. Main effects-the independent contribution of a single hazard-can be estimated from any dataset that includes variation in that hazard. But interaction effects require observing outcomes across all combinations of the interacting hazards, including rare combinations. In a workforce where high demands and low control co-occur frequently (because the same organisational dysfunction produces both), you may have abundant data for the high-strain quadrant but almost none for the low-demand, low-control "passive" quadrant. And without data from all quadrants, you cannot reliably distinguish an additive model from an interaction model.

This is not merely a theoretical concern. Van der Doef and Maes (1999) noted that the inconsistent evidence for the demand-control buffer hypothesis may partly reflect exactly this problem: in many studies, the "active" and "low-strain" quadrants had so few observations that the statistical power to detect interaction effects was inadequate. The interaction was there in reality-but invisible in the data.

Three Strategies for Sparse Data

When data are insufficient to estimate all CPT cells directly-which, in psychosocial risk assessment, is the norm rather than the exception-practitioners must supplement empirical estimates with other information sources. Soares et al. (2022) conducted a systematic review of expert elicitation methods in health modelling and identified three broad strategies, each with strengths and pitfalls:

Strategy 1: Expert judgement. Subject matter experts-experienced occupational health professionals, senior safety managers, or clinical psychologists-provide estimates for the missing CPT cells. This is fast and leverages tacit knowledge, but it is vulnerable to cognitive biases: anchoring on available data, overconfidence in point estimates, and difficulty reasoning about conditional probabilities. Structured elicitation protocols such as the Sheffield Elicitation Framework (SHELF) can mitigate these biases by requiring experts to specify uncertainty ranges rather than point estimates (Soares et al., 2022).

Strategy 2: Literature transfer. Published epidemiological studies-including the OHS Body of Knowledge and meta-analyses like those reviewed in this chapter-often provide risk estimates for specific hazard combinations. These can be adapted to fill empty CPT cells, though care must be taken about differences in population, measurement, and context. A risk ratio estimated among Swedish white-collar workers may not directly transfer to Australian underground miners, but it provides a defensible starting point that is better than guessing.

Strategy 3: Structural assumptions. Instead of estimating each missing cell independently, you choose a canonical model (noisy-OR, noisy-AND, or weighted combination) and use the cells you do have data for to estimate the model's parameters, which then determine the missing cells. This is the approach championed by Oniśko et al. (2001), and it works remarkably well when the structural assumption is approximately correct. The noisy-OR assumption, in particular, allows you to estimate n parameters from single-parent observations and then extrapolate to all 2ⁿ multi-parent configurations.

Each strategy involves different tradeoffs between precision, transparency, and defensibility. Expert judgement is maximally flexible but least reproducible. Literature transfer is most evidence-based but may suffer from context mismatch. Structural assumptions are most parsimonious but may distort the interaction pattern if the assumed model is wrong. In practice, the best approach often combines all three: use data where available, fill gaps with literature-informed priors, check structural assumptions against expert intuition, and conduct sensitivity analyses to see which empty cells matter most for your conclusions.

---

Revisiting Independence: From Assertion to Formal Test

In Chapter 2, we introduced the concept of conditional independence and noted that psychosocial hazards frequently violate the independence assumptions we might naively wish to make. At that stage, the observation was conceptual-we could assert that demands and control were not independent, but we lacked the formal machinery to model their dependence precisely. With the tools from this chapter, that gap is closed.

If two parent hazards truly contribute independently to a child outcome, then the noisy-OR model should fit the observed data well. If it doesn't-if the observed probability for the "both hazards present" cell is significantly higher than the noisy-OR predicts-then we have formal evidence of a synergistic interaction. Conversely, if the observed probability is lower than the noisy-OR predicts, we have evidence of an antagonistic interaction or a redundancy effect. The canonical models thus serve double duty: they are both parameterisation tools and diagnostic tests for interaction.

Ervasti et al. (2022) used exactly this logic in their study of Finnish health workers. By computing the RERI-the excess risk beyond what an additive model would predict-they formally demonstrated that the three-way interaction among demands, rewards, and social capital was super-additive. In Bayesian network terms, this tells us that a noisy-OR parameterisation for the combined effect of these three stressors would underestimate the risk for workers facing all three simultaneously. The correct model must include synergistic terms, either through an explicit interaction node or through a CPT that deviates from the noisy-OR pattern in the "all hazards present" cells.

This connection between epidemiological interaction analysis and Bayesian network parameterisation is, in many ways, the central technical contribution of this chapter. It bridges two literatures-occupational health epidemiology and probabilistic graphical models-that have developed largely in isolation. The epidemiologists have identified which hazard interactions matter; the Bayesian network formalism gives us the tools to represent how those interactions propagate through complex causal systems to produce organisational outcomes.

---

Practical Guidelines for Interaction Modelling

Drawing together the theoretical and empirical material from this chapter, we can formulate a practical workflow for modelling hazard interactions in psychosocial risk Bayesian networks:

1. Start with theory. Before touching data, consult the occupational health literature. Does a well-validated interaction model (demand-control, effort-reward imbalance, demand-control-support) apply to the hazard relationship you are modelling? If so, let the theory guide your choice of combination function.
2. Choose a default canonical model. For nodes where no specific interaction theory exists, the noisy-OR is a reasonable default for risk outcomes (where multiple hazards contribute independently) and a weighted linear model is reasonable for continuous-valued mediators.
3. Estimate parameters from data where possible. Use observed frequencies for CPT cells that have adequate sample sizes. Flag cells with fewer than 20 observations as unreliable.
4. Fill sparse cells using structured methods. Combine expert elicitation, literature transfer, and structural assumptions. Document which cells were estimated empirically and which were imputed-this transparency is essential for model credibility.
5. Test for interaction residuals. After fitting a canonical model, compare its predictions to observed data in the multi-hazard cells. Systematic deviations signal interactions that the model fails to capture. Add interaction terms or switch to a more flexible model as needed.
6. Conduct sensitivity analysis. For every CPT that includes interaction terms, test how much downstream predictions change when interaction parameters are varied across plausible ranges. If the model's conclusions are robust to interaction uncertainty, you can proceed with confidence. If they are sensitive, you need better data or more careful elicitation for those specific parameters.

This workflow is iterative. Your first-pass model will inevitably contain incorrect interaction assumptions. The goal is not to get it right on the first try, but to build a model that makes its interaction assumptions explicit so they can be challenged, tested, and refined as evidence accumulates.