Skip to content

Non-Collapsibility

Non-collapsibility is a fundamental statistical concept that describes a situation where a measure of association between two variables changes when a third variable is introduced into the analysis, even when that third variable is not a confounder. This phenomenon is particularly relevant in the interpretation of regression models, especially logistic regression, and has significant implications for drawing accurate causal inferences and understanding epidemiological studies.

At its heart, the concept of collapsibility deals with the invariance of an association measure. A measure is considered collapsible if its value remains unchanged when conditioning on or marginalizing over a third variable that is not a confounder. Conversely, non-collapsibility occurs when this measure does change despite the absence of confounding. In essence, it's about whether the relationship between two variables "collapses" back to its original form when a third variable is accounted for, or if it remains altered.

The odds ratio (OR), a widely used measure of association, is a prime example of a non-collapsible measure, particularly within the context of logistic regression. This means that the odds ratio calculated from a simple, unadjusted model can differ numerically from the odds ratio calculated in a model that includes a covariate, even if that covariate does not confound the relationship. This difference is not due to random sampling error but is an inherent mathematical property of the odds ratio and the logistic regression model itself.

Origin and Key Developments

The theoretical underpinnings of collapsibility can be traced back to the analysis of contingency tables. The term "collapsibility" itself was formally introduced and popularized by scholars like Bishop, Fienberg, and Holland in their seminal 1975 book, "Discrete Multivariate Analysis: Theory and Practice."

Early discussions often drew parallels between non-collapsibility and Simpson's paradox, a statistical phenomenon where trends observed in different groups of data can disappear or even reverse when these groups are combined. However, the modern understanding, particularly within causal inference and epidemiology, carefully distinguishes non-collapsibility from confounding. While confounding also leads to changes in effect estimates when adjusting for a variable, non-collapsibility can occur even in the absence of any confounding bias.

Key milestones in the understanding and formalization of collapsibility include:

  • Graphical Models and Collapsibility: Judea Pearl, a leading figure in causal inference, has made significant contributions by defining collapsibility in terms of functional relationships with probability distributions, often visualized and analyzed using graphical models. This work clarified the conditions under which measures are collapsible or non-collapsible. For instance, a functional \(g[P(x,y)]\) of the joint distribution of \(X\) and \(Y\) is collapsible on \(Z\) if the expected value of \(g[P(x,y|z)]\) over \(Z\) equals \(g[P(x,y)]\). A special case, "simple collapsibility," occurs when \(g[P(x,y|z)]\) is constant across all values of \(Z\).1
  • Addressing Non-Collapsibility: Researchers like Greenland and Pearl (2011) provided a robust framework for understanding collapsibility using graphical models, offering clear distinctions between confounding and non-collapsibility. More recently, work by Zhang (2008) and Daniel et al. (2020) has focused on developing methods to "marginalize" conditional odds ratios, effectively providing ways to estimate a marginal odds ratio from conditional odds ratios, thereby acknowledging and addressing the inherent issue of non-collapsibility.2
  • Ongoing Research: Contemporary research continues to explore the subtle nuances of non-collapsibility, its impact on various statistical models (including time-to-event models), and its intricate relationship with confounding bias.

How It Works: The Odds Ratio and Logistic Regression

The non-collapsibility of the odds ratio in logistic regression can be understood by considering the nature of the logistic function. Logistic regression models the probability of an outcome \(Y\) given an exposure \(X\) using the logit link function:

\[ \text{logit}(P(Y=1|X, Z)) = \log\left(\frac{P(Y=1|X, Z)}{P(Y=0|X, Z)}\right) = \alpha + \beta_X X + \beta_Z Z \]

Here, \(P(Y=1|X, Z)\) is the probability of the outcome occurring given exposure \(X\) and covariate \(Z\). The odds ratio for the exposure \(X\), when adjusting for \(Z\), is \(e^{\beta_X}\).

The issue arises because the odds ratio is defined on the odds scale, which is a non-linear transformation of probability. When you adjust for a covariate \(Z\), even if \(Z\) is not a confounder, the odds of the outcome for a given exposure level can change across the strata of \(Z\) in a way that makes the overall (marginal) odds ratio different from the stratum-specific (conditional) odds ratios.

Consider a scenario where you analyze the relationship between an exposure (X) and an outcome (Y) without considering a third variable (Z). You then adjust for Z (e.g., by stratifying the analysis or including it in a regression model). If Z is not a confounder, ideally, the measure of association (like an odds ratio) should remain the same. However, with the odds ratio in logistic regression, it often does not. This change is not due to Z biasing the estimate, but because the odds ratio itself is non-collapsible.

An extreme example cited in research illustrates this starkly: the odds ratio for an exposure can be approximately 2 in both strata of a population (conditional ORs), but approximately 1 in a sample composed of equal numbers from each stratum (marginal OR). This dramatic difference highlights how the marginal odds ratio can diverge significantly from the conditional odds ratio due to the inherent property of non-collapsibility.

Real-World Examples and Case Studies

A classic illustration of non-collapsibility involves the odds ratio in logistic regression, particularly in epidemiological studies.

Example Scenario:

Imagine a study investigating the association between a new drug (Exposure, X) and recovery from a disease (Outcome, Y). A third variable, the severity of the illness (Covariate, Z - e.g., Mild/Severe), is also recorded.

  • Scenario 1: Z is a Confounder: Suppose the new drug is more likely to be prescribed to patients with milder illness (Z is associated with X). Furthermore, patients with milder illness are more likely to recover regardless of whether they receive the drug (Z is associated with Y, even after accounting for the drug's effect). In this case, Z is a confounder, and adjusting for it is necessary to obtain an unbiased estimate of the drug's true effect. The change in the odds ratio upon adjustment is due to confounding.

  • Scenario 2: Z is NOT a Confounder, but Non-Collapsibility Occurs: Now, consider a situation where Z is not a confounder. For instance, Z might be a strong predictor of the outcome (prognostic) but is not associated with the exposure (the drug is prescribed randomly with respect to illness severity). Even in this scenario, if you adjust for Z in a logistic regression model, the estimated odds ratio for the drug's effect on recovery can still change. This change is not because Z is confounding the association but because the odds ratio itself is non-collapsible with respect to Z. The adjustment for a strong prognostic factor, even without confounding, can alter the odds ratio.

This phenomenon means that simply observing a change in an odds ratio after adjusting for a covariate does not automatically imply confounding. It could be, at least in part, a result of non-collapsibility.

Current Applications and Implications

The understanding of non-collapsibility is crucial across several disciplines:

  • Epidemiology and Public Health: Researchers must be aware that changes in effect estimates when adjusting for covariates in epidemiological studies might not solely be due to confounding. Misinterpreting non-collapsibility as confounding can lead to erroneous conclusions about the efficacy of interventions or the risks associated with environmental exposures.
  • Clinical Trials and Medical Research: In randomized controlled trials (RCTs), confounding is minimized by design. However, even in RCTs, adjusting for baseline covariates in logistic regression or Cox proportional hazards models can lead to changes in effect estimates due to non-collapsibility. This impacts the interpretation of treatment effects and the validity of using the "change-in-estimate" criterion to identify confounding.
  • Causal Inference: Non-collapsibility underscores the critical need to clearly define the estimand of interest—whether it's a marginal (population-averaged) effect or a conditional (stratum-specific) effect. It highlights that "adjusted" and "marginal" are not interchangeable, especially when dealing with non-collapsible measures.
  • Risk Prediction Modeling: In developing and updating risk prediction models, understanding non-collapsibility is vital. It can influence how predicted risks are adjusted when models are applied to new populations or settings, potentially leading to under-correction of predicted risks if not properly considered.

Non-collapsibility is intricately linked with several other statistical and causal inference concepts:

  • Confounding: While both lead to changes in effect estimates upon adjustment, confounding is a bias caused by an extraneous variable associated with both the exposure and the outcome. Non-collapsibility, on the other hand, is an intrinsic property of the measure of association (like the OR) and the modeling framework (like logistic regression).
  • Simpson's Paradox: Non-collapsibility can be viewed as a statistical mechanism that contributes to or explains instances of Simpson's paradox, where aggregate-level associations differ from subgroup-level associations.
  • Marginal vs. Conditional Effects: This is perhaps the most direct implication. For non-collapsible measures, marginal and conditional effects are numerically distinct even in the absence of confounding. The choice of analysis technique and the interpretation of the resulting estimate depend on whether one aims to estimate a marginal or a conditional effect.
  • Causal Inference: The concept is foundational in causal inference, emphasizing that the interpretation of an effect measure is dependent on the conditioning set. It guides the selection of appropriate statistical models and methods to estimate specific causal quantities.
  • Link Functions: The non-collapsibility of measures like the odds ratio is directly tied to the non-linear nature of the logit link function in logistic regression. In contrast, measures like the risk ratio or risk difference, often associated with linear link functions (e.g., identity or log link), tend to be collapsible.

Common Misconceptions and Debates

Several common misunderstandings surround non-collapsibility:

  • Equating Non-Collapsibility with Bias: A frequent misconception is to treat non-collapsibility as synonymous with bias, particularly omitted variable bias. While adjusting for a non-collapsible measure can alter an estimate, this change is not necessarily a bias if the objective is to estimate a specific conditional effect. The problem arises when marginal and conditional interpretations are conflated.
  • Odds Ratio vs. Risk Ratio: Due to its non-collapsible nature, the odds ratio is sometimes viewed with caution in causal inference compared to collapsible measures like the risk ratio (RR) or risk difference (RD). However, the OR possesses other desirable statistical properties, such as remaining constant across different baseline risks in certain scenarios. The choice between these measures depends critically on the research question and the underlying data-generating process.
  • "Confounding without Non-collapsibility": Some discussions explore theoretical scenarios where the effects of confounding and non-collapsibility might coincidentally cancel each other out, leading to an apparent absence of non-collapsibility. However, the underlying non-collapsibility effect may still be present, even if masked.

Practical Implications and Why It Matters

Understanding non-collapsibility is crucial for several practical reasons:

  1. Accurate Interpretation of Results: Researchers must recognize that changes in effect estimates after covariate adjustment in models like logistic regression can stem from non-collapsibility, not solely from confounding. This awareness prevents the misattribution of changes to bias when they are inherent properties of the chosen measure and model.
  2. Sound Causal Inference: For drawing valid causal conclusions, it is essential to differentiate between marginal and conditional effects. Non-collapsibility highlights that these effects can be numerically distinct, and the analytical choices made (e.g., including or excluding covariates) precisely define the causal or associational quantity being estimated.
  3. Appropriate Model Selection: Awareness of collapsibility properties can guide the selection of statistical models and effect measures. If a marginal effect is the primary interest, methods like marginal structural models, which are designed to estimate marginal effects and account for non-collapsibility, may be necessary.
  4. Avoiding Misleading Conclusions: Failing to account for non-collapsibility can lead to incorrect conclusions about the presence and magnitude of confounding, potentially impacting public health recommendations, clinical practice, and policy decisions.

In summary, non-collapsibility is a fundamental statistical property that demands careful consideration in data analysis and interpretation, particularly when utilizing regression models for association and causal inference. It reminds us that the way we model relationships and the measures we choose have inherent mathematical consequences that must be understood for accurate scientific conclusions.

  1. Pearl, J. (2009). Causality: Models, Reasoning, and Inference. Cambridge University Press. (While not explicitly cited in the research, Pearl's foundational work on causal inference extensively covers collapsibility.) ↩

  2. Daniel, R. M., Williamson, E. J., & Patel, A. (2020). Making apples from oranges: Comparing noncollapsible effect estimators and their standard errors after adjustment for different covariate sets. Statistical Methods in Medical Research, 29(9), 2697-2711. ↩