Risk Regression Models in Clinical Epidemiology: Estimating Risk Difference and Risk Ratio

Introduction

When analyzing binary outcomes in clinical epidemiology—such as the presence or absence of a disease—choosing the appropriate regression model is essential for accurately estimating effect sizes like risk difference and risk ratio. These models underpin much of the decision-making in therapeutic, prognostic, and etiological studies. This article provides a comprehensive guide to the statistical frameworks used for risk regression, explains when each approach is appropriate, and highlights technical nuances that can influence interpretation and validity.

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1. Understanding Risk Regression

Risk regression refers to modeling strategies that estimate the absolute or relative probability of a binary event occurring. The two primary effect measures in this context are:

• Risk Difference (RD): The absolute difference in event probability between groups.

• Risk Ratio (RR): The relative probability (multiplicative scale) of event occurrence between groups.

While both are interpretable and clinically relevant, their statistical modeling frameworks differ in flexibility, assumptions, and susceptibility to issues such as convergence failure or misestimated confidence intervals.

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2. Risk Difference Regression

Characteristics and Methods

Risk difference models quantify the absolute change in outcome probability associated with a predictor. These models are naturally unconstrained because probabilities on the additive scale can potentially fall outside the 0–1 interval, especially when extrapolating.

Statistical implementations include:

• Ordinary Least Squares (OLS): Simple linear regression applied to binary outcomes:
  

  Regress outcome predictors
  

  While intuitive, OLS is inefficient for binary outcomes due to heteroscedasticity.
  

• Generalized Linear Model (GLM) with Identity Link and Binomial Family:

  
  

  glm outcome predictors, link(identity) family(binomial)
  

  This approach uses maximum likelihood estimation and is more robust than OLS.
  

• Truncated Predictions to Constrain Values: To avoid predicted probabilities falling outside [0,1], constrained models truncate fitted values:

  
  

  binreg outcome predictors, rd
  

  This explicitly estimates risk differences with bounds respected.
  

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3. Risk Ratio Regression

3.1. Theoretical Challenges

Unlike the risk difference, modeling the risk ratio introduces non-linearity and often results in convergence issues. A common concern is that predicted values can exceed 1, especially with high baseline risks or large effect sizes.

3.2. Log-Binomial Regression

This method models the log of the risk (not the odds):

• Unconstrained Approach:

  
  

  glm outcome predictors, link(log) family(binomial) eform
  

  This often fails to converge in datasets with extreme risks or sparse cells.
  

• Constrained (Truncated) Risk Estimation:

  binreg outcome predictors, rr
  

  This imposes constraints to ensure valid predicted probabilities.
  

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4. Modified Poisson Regression

To address convergence problems in log-binomial models, Poisson regression with robust error variance is often used to approximate the risk ratio:

Poisson Working Model Variants

• Unadjusted Poisson:
  

  glm outcome predictors, link(log) family(poisson) eform
  

• Chi-squared Adjusted:

  
  

  glm outcome predictors, link(log) family(poisson) eform scale(x2)
  

• Deviance Adjusted:

  
  

  glm outcome predictors, link(log) family(poisson) eform scale(deviance)
  

• Robust Error Variance:

  
  

  glm outcome predictors, link(log) family(poisson) eform robust
  

This approach is computationally stable and provides consistent estimates with corrected standard errors.

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5. Gaussian Working Model for Log-Risk

An alternative, though less common, method uses a Gaussian distribution with a log link:

• Unadjusted:

  
  

  glm outcome predictors, link(log) family(gaussian) eform
  

• Robust Adjustment:

  
  

  glm outcome predictors, link(log) family(gaussian) eform robust
  

This approach can produce valid estimates but requires careful justification since the Gaussian assumption on binary data is questionable.

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6. Logistic Regression: A Comparator

Though logistic regression models the log odds rather than risk, it remains a mainstay in clinical research. However, odds ratios overestimate risk ratios when the event is common (>10–20%).

Example:

logistic outcome predictors

In datasets with high event rates, logistic regression may produce misleading magnitudes when interpreted as risk measures.

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7. Practical Implications and Model Selection

When deciding between models, consider the following:

• Use risk difference regression for absolute effect communication, such as in therapeutic benefit estimation.

• Use modified Poisson regression when the risk ratio is the target and the outcome is not rare.

• Avoid unconstrained log-binomial models if convergence problems arise.

• Use logistic regression when interested in odds ratios or when data is case-control in design.

• Apply robust error variance adjustments to improve inference in models not naturally tailored to binary outcomes.
  

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Conclusion

Regression modeling of binary outcomes offers several paths to estimate clinically meaningful effect measures such as risk difference and risk ratio. Each model comes with trade-offs in interpretability, convergence, and robustness. Understanding their mathematical underpinnings and technical limitations ensures more accurate and appropriate application in clinical research. Analysts must match their model choice to the research aim, data structure, and practical constraints of inference to yield results that inform meaningful clinical decisions.