Concept Architecture
Concept
Theoretically, an Interaction Effect occurs when the effect of one explanatory variable on an outcome depends on the value of another explanatory variable. Interaction effects are fundamental to regression modelling because they allow relationships between predictors and outcomes to vary across subgroups or levels of another variable rather than assuming purely additive effects. In health economics, interaction effects are used to investigate treatment effect heterogeneity, subgroup differences and effect modification in economic evaluations and clinical studies.
Mathematically, an interaction effect is represented by including a product term between two or more variables within a regression model. The coefficient of the interaction term quantifies the extent to which the effect of one predictor changes according to the value of another predictor. Statistical inference is based on estimation of the interaction coefficient and hypothesis testing to determine whether the interaction differs significantly from zero.
In practice, interaction effects are estimated using regression models such as linear regression, logistic regression, generalised linear models and Cox proportional hazards models. They are routinely evaluated in subgroup analyses, health technology assessments and economic evaluations to determine whether treatment effectiveness, costs or quality-of-life outcomes differ across patient characteristics.
Purpose
Used to determine whether the effect of one variable depends on the level of another variable, thereby identifying effect modification and treatment effect heterogeneity within health economic and clinical analyses.
Mathematical Formulae
Primary Formula
y = ?? + ??X? + ??X? + ??(X? ? X?) + �
where:
- ?? = interaction effect coefficient
Supporting Formulae
Null hypothesis:
H?: ?? = 0
Wald statistic:
W = (??? / SE(???))�
W ~ ?�?
Related Mathematical Methods
- Multiple Linear Regression
- Logistic Regression
- Generalised Linear Model
- Cox Proportional Hazards Model
- Wald Test
- Likelihood Ratio Test
- Treatment Effect Heterogeneity
Example
A health economic evaluation examines whether the effect of a new treatment differs according to patient age.
Estimated regression model:
Cost = ?? ? 1200(Treatment) + 35(Age) + 18(Treatment ? Age)
The estimated interaction coefficient is:
??? = 18
p = 0.02
The significant interaction indicates that the treatment effect on healthcare costs varies with patient age and should be considered when evaluating subgroup cost-effectiveness.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| PRODUCT | =B2*C2 | Create an interaction term between two predictors |
| SUMPRODUCT | =SUMPRODUCT(XRange,BetaRange) | Calculate predicted values including interaction effects |
| LINEST | =LINEST(YRange,XRange,TRUE,TRUE) | Estimate regression coefficients including interaction terms |
| IF | =IF(P2<0.05,"Interaction Present","No Interaction") | Interpret statistical significance |
| EXP | =EXP(B2) | Convert interaction coefficients to odds or hazard ratios when appropriate |
VBA (Optional)
A VBA procedure can automatically generate interaction terms, estimate regression models and summarise statistically significant effect modifications across candidate predictors.
Sources
- Hosmer DW, Lemeshow S, Sturdivant RX. Applied Logistic Regression.
- Kleinbaum DG, Klein M. Regression: A Self-Learning Text.
- Harrell FE. Regression Modeling Strategies.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is an interaction effect?
The portion of output variance attributable to two or more parameters varying together, beyond what each parameter's individual effect alone would explain.
Source: Saltelli et al. 2008
Why can interaction effects be missed by one-way analysis?
An interaction effect is the part of a model's output variation that appears only when two or more inputs change together, beyond the sum of what each does alone. A one-way analysis, moving inputs singly, holds the others fixed and so can never reveal it, potentially understating an input whose influence emerges mainly in combination. Capturing interactions requires varying inputs jointly, as global methods do. Ignoring them can give a misleading picture of what drives the result. Saltelli and colleagues (2008) describe interaction effects.
Source: Saltelli et al. 2008
How is an interaction effect measured?
An interaction effect is measured in variance-based sensitivity analysis as variance not explained by the individual first-order effects: the difference between an input's total-effect index, which includes its interactions, and its first-order index reflects its interaction with other inputs, and higher-order indices quantify the joint contribution of specific groups of inputs. Estimating these requires Monte Carlo procedures with structured samples. A gap between total and first-order effects, or the sum of first-order effects falling short of the total variance, signals the presence of interaction effects.
Source: Sobol 2001
How does an interaction effect differ from a main effect?
A main effect is the output variance explained by an input varying on its own, while an interaction effect is the additional variance explained only when two or more inputs vary together, beyond the sum of their individual main effects. So the main effect is the input's direct influence and the interaction effect its joint influence with others. When interaction effects are large, the model is non-additive, and an input's importance cannot be judged from its main effect alone, since much of its influence may act through interactions.
Source: Saltelli et al. 2008
Why do interaction effects matter?
Interaction effects matter because when inputs act jointly, an input with a small individual effect can still be influential through its interactions, so judging importance from main effects alone would understate it and could wrongly fix a consequential input as negligible. Large interaction effects also reveal that the model behaves non-additively, which affects interpretation and how uncertainty is analysed. Recognising interaction effects, through the gap between total and first-order sensitivity indices, ensures that inputs mattering mainly through joint action are properly identified and retained.
Source: Sobol 2001
What are the limitations of analysing interaction effects?
Analysing interaction effects, especially quantifying specific higher-order interactions, is computationally demanding, since estimating variance-based indices for combinations of inputs requires many model runs, and the number of possible interactions grows rapidly with the number of inputs. Total-effect indices capture all of an input's interactions together without separating them, while fully decomposing interactions is often impractical. Results depend on the input assumptions. These limitations mean interaction effects are usually assessed through total-effect indices, with detailed decomposition reserved for cases where specific joint influences must be understood.
Source: Saltelli et al. 2008
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 28 Oct 2025
Content version: 1.0.0
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- Term code
- HE-EM-UA-034
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