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Sensitivity Analysis Without Assumptions

Peng Ding, Tyler VanderWeele

arXiv:1507.03984v1math.STstat.AP

TL;DR

Unmeasured confounding threatens causal inference from observational studies, and existing sensitivity analyses often depend on restrictive, untestable assumptions. The paper derives an assumption-free bounding factor and sharp inequality using two sensitivity parameters, finding that it is no more conservative than many assumption-based methods while strengthening Cornfield-style conditions. The authors qualify that some statements still require assumptions about parameter specifications or the potential-outcomes framework.

  • Problem

    Unmeasured confounding may undermine causal inference, while prior sensitivity analyses often impose restrictive assumptions about confounder structure and interaction.

  • Method

    The paper introduces a joint bounding factor and sharp inequality that assess whether unmeasured confounding could explain away or reduce an observed effect without assumptions about the confounder.

  • Results

    The bounding factor is no more conservative than many assumption-based techniques, makes the no-interaction assumption unnecessary in the relevant setting, and strengthens classical Cornfield conditions.

  • Takeaways & Limitations

    The bounding factor provides an implementable measure of confounding strength using sensitivity statements based on inequalities between two parameters.

  • Takeaways & Limitations

    Statements that specify sensitivity-parameter values, or interpret results within a potential-outcomes framework, still require corresponding assumptions.

Abstract

from arXiv · show

Unmeasured confounding may undermine the validity of causal inference with observational studies. Sensitivity analysis provides an attractive way to partially circumvent this issue by assessing the potential influence of unmeasured confounding on the causal conclusions. However, previous sensitivity analysis approaches often make strong and untestable assumptions such as having a confounder that is binary, or having no interaction between the effects of the exposure and the confounder on the outcome, or having only one confounder. Without imposing any assumptions on the confounder or confounders, we derive a bounding factor and a sharp inequality such that the sensitivity analysis parameters must satisfy the inequality if an unmeasured confounder is to explain away the observed effect estimate or reduce it to a particular level. Our approach is easy to implement and involves only two sensitivity parameters. Surprisingly, our bounding factor, which makes no simplifying assumptions, is no more conservative than a number of previous sensitivity analysis techniques that do make assumptions. Our new bounding factor implies not only the traditional Cornfield conditions that both the relative risk of the exposure on the confounder and that of the confounder on the outcome must satisfy, but also a high threshold that the maximum of these relative risks must satisfy. Furthermore, this new bounding factor can be viewed as a measure of the strength of confounding between the exposure and the outcome induced by a confounder.

1 Introduction

Unmeasured confounding can threaten causal conclusions from observational studies, while existing sensitivity analyses often rely on restrictive assumptions and may not assess alternative hypotheses. The paper introduces an assumption-free bounding factor intended to provide broader sensitivity assessments while remaining easy to implement.

  • 1 Introduction: Existing sensitivity analyses often assume a single binary confounder, no exposure–confounder interaction, or only one confounder.Some methods assess only the confounding strength needed to explain away an observed effect and do not evaluate weaker confounding scenarios or alternative hypotheses.
  • 1 Introduction: The paper proposes a new bounding factor and sensitivity analysis technique without assumptions about the unmeasured confounder or confounders.The approach is designed for binary, time-to-event, non-negative count, and continuous outcomes, using both ratio and difference scales.
  • 1 Introduction: The assumption-free bounding factor is no more conservative than many prior techniques that make simplifying assumptions and is easy to implement.The paper also presents the bounding factor as a measure of confounding strength between exposure and outcome induced by a confounder.
  • 1 Introduction: The new bounding factor strengthens the classical Cornfield conditions by requiring a threshold on the maximum of the relevant relative risks.The classical conditions concern the relative risk of exposure on the confounder and the relative risk of confounder on the outcome.
  • 1 Introduction: The paper qualifies “without assumptions”: specifying sensitivity parameters can itself be viewed as an assumption, and potential-outcomes interpretations require framework assumptions.Thus, some sensitivity statements are assumption-free, whereas others depend on parameter specifications or assumptions implicit in the potential-outcomes framework.
  • 1 Introduction: The bounding factor yields inequalities specifying when unmeasured confounding could explain away an observed association or reduce it to a given level.These statements use sensitivity-analysis parameters without imposing a specific structure on the confounder or confounders.

2 Main Result: A New Bounding Factor

The paper defines two relative-risk sensitivity parameters for unmeasured confounding and derives a sharp joint bounding factor without assumptions about the confounder’s structure. This factor bounds how much the observed relative risk can be reduced and supports sensitivity analysis using the pair (RREU, RRUD).

  • Sensitivity parameters: RREU measures the maximal relative risk between exposure E and unmeasured confounder U within measured-confounder strata.For multiple unmeasured confounders, U may be a vector and the maximum compares any two categories of that vector.
  • Sensitivity parameters: RRUD measures the maximal relative risk between unmeasured confounder U and outcome D, allowing different effects among exposed and unexposed groups.RRUD is defined as the maximum of the corresponding relative risks with and without exposure.
  • Main result: The joint bounding factor is RREU × RRUD/(RREU + RRUD −1), and dividing the observed relative risk by it yields a lower bound on the true causal relative risk.The inequality is sharp: a confounder model can attain equality.
  • Implications: If confounding reduces an observed estimate to a target level, RREU and RRUD must be sufficiently large to satisfy the joint inequality.The pair (RREU, RRUD) measures the strength of confounding induced between exposure E and outcome D.
  • Main result: The bound applies without assumptions about the nature of the unmeasured confounder, including whether it is binary, categorical, continuous, mixed, or multiple.The approach also does not impose the no-interaction assumption and can assess reduction to a prespecified true causal relative risk.
  • Illustration: With (RREU, RRUD) = (2,2), an observed relative risk of 2.1 is corrected to 1.58 with a 95% confidence interval of [1.05,2.33].The confounder cannot explain away either the point estimate or its lower confidence limit of 1.4.
  • Illustration: With (RREU, RRUD) = (2.5,3.5), the corrected estimate is 1.20 with a 95% confidence interval of [0.8,1.77].This confounder cannot explain away the point estimate but reduces the lower confidence limit below one.
  • Implications: For RREU = RRUD = 2.5, each parameter must be at least 2.72 to reduce an observed relative risk of 3, while at least one must reach 4.44 to explain it away.These thresholds illustrate the high-threshold implication of the joint bound.

3 Relation with Cornfield Conditions

The new joint bound contains the classical Cornfield conditions as special cases and adds a high-threshold condition on the larger confounding association. The joint condition is more informative than either threshold heuristic for evaluating whether confounding can explain an observed effect.

  • Cornfield conditions: The classical Cornfield conditions require both the exposure–confounder and confounder–outcome relative risks to exceed specified thresholds.The paper derives these conditions as special cases of the joint result when one sensitivity parameter tends to infinity.
  • High threshold: The main result additionally requires the maximum of RREU and RRUD to satisfy a high-threshold condition.This applies to general unmeasured confounders, not only binary confounders.
  • Comparison: The high-threshold conditions are weaker than the joint bounding factor because some confounding scenarios pass the thresholds but fail the joint inequality.The paper characterizes the thresholds as useful heuristics rather than substitutes for the joint bound.
  • Threshold example: For an observed relative risk of 3, the low threshold is 3 for both parameters, whereas the high threshold is 5.45 for at least one parameter.These conditions are useful heuristics when one marginal association or their relative magnitudes are specified.
  • Threshold example: For (RREU, RRUD) = (5.5,3.1), the bounding factor is 2.24 < 3, so confounding cannot explain away an observed relative risk of 3.Both classical and high-threshold conditions would be satisfied, but the joint condition rules out the explanation.

4 Illustration

A cigarette-smoking and lung-cancer example illustrates how the joint bounding factor evaluates extreme unmeasured confounding. Even very strong equal associations leave a causal effect above one, and Figure 1 and Table 2 provide visual and tabular sensitivity analyses.

  • Historical example: With RREU = RRUD = 10.73, the joint bounding factor is 5.63, leaving a corrected causal relative risk of 1.91.The corrected 95% confidence interval is [1.42,2.55].
  • Historical example: Even relative risks of 10.73 for both exposure–confounder and confounder–outcome associations cannot explain away the point estimate or lower confidence limit.The lower confidence limit remains above one after correction.
  • Sensitivity thresholds: If RREU = RRUD, explaining away the point estimate requires each confounding relative risk to be at least 20.95, versus 15.52 for the lower confidence limit.These values come from the joint threshold condition.
  • Sensitivity thresholds: Figure 1 plots the joint values of (RREU, RRUD) required to explain away the point estimate and the lower confidence limit.The solid line corresponds to the point estimate of 10.73, and the dotted line to the lower confidence limit of 8.02.
  • Sensitivity analysis: Table 2 organizes corrected point estimates and confidence limits by RRUD columns and RREU rows across confounding scenarios.The paper notes that such tables are most informative when considering extreme sensitivity-parameter values.

5 Discussion

The paper introduces a joint bounding factor and sensitivity-analysis inequality that require no assumptions about the structure of unmeasured confounders. It provides stronger conclusions than classical Cornfield conditions while remaining no more conservative than several assumption-based methods.

  • Contribution: The approach uses an inequality to determine when two sensitivity parameters could explain away an observed effect or reduce it to a specified level.The method avoids imposing a specific structure on the unmeasured confounder or confounders.
  • Comparison with prior methods: The new bounding factor requires only two sensitivity parameters rather than three and does not assume a single binary confounder or no exposure–confounder interaction.Earlier approaches often relied on those assumptions and, in some cases, required specifying several confounder prevalences.
  • Comparison with prior methods: The no-interaction assumption is unnecessary in the setting where the earlier formula yields the same bounding factor.In that setting, imposing no interaction does not strengthen the bounds.
  • Risk-difference extensions: For risk-difference analyses, sensitivity parameters expressed on the relative-risk scale yield a bounding factor that is unchanged by the number of confounder categories.By contrast, risk-difference-scale conditions become weaker as the number of categories increases.
  • Interpretation and use: The bounding factor can be interpreted as a measure of the strength of confounding between the exposure and outcome induced by an unmeasured confounder.The paper also describes the approach as a way to assess how much confounding would be needed to reduce an estimate or confidence interval toward the null.
  • Cornfield conditions: The bounding factor implies both classical Cornfield conditions and a high-threshold condition on the maximum of the two relative risks.These conditions provide conclusions stronger than the original Cornfield conditions.

Appendix 1: SAS Code

The appendix provides SAS code that computes bias factors, corrected estimates, and corrected confidence intervals across grids of confounding strengths. Users can adapt the example by changing the observed estimate, interval, and sensitivity-parameter ranges.

  • Code operation: Researchers can adapt the example by changing the observed relative risk and its lower and upper confidence limits.The relevant inputs are the estimate adjusted for measured covariates and its confidence interval.
  • Sensitivity grid: The minimum and maximum confounding strengths can be changed through the RREU and RRUD inputs.The appendix recommends including values at least as high as 5 to examine fairly severe confounding.
  • Sensitivity grid: The example evaluates relative-risk and confounding-strength values from 1.2 through 10 and calculates a high-threshold quantity from the observed relative risk.The supplied example uses RR = 10.73, with confidence limits 8.02 and 14.36.
  • Code operation: The code computes a bias factor and corrected relative-risk estimates and confidence intervals for each pair of sensitivity parameters.The rows represent increasing exposure–confounder relative risks, while the columns represent increasing confounder–outcome relative risks.

Appendix 2: Conditions for the Risk Difference Using Sensitivity Parameters on the Relative Risk Scale

This appendix extends the bounding-factor analysis to causal risk differences using sensitivity parameters on the relative-risk scale. It derives lower and upper bounds, illustrative calculations, and conditions for confounding to reduce an observed risk difference to a target level.

  • Bounding factor: The joint bounding factor is defined as BFU = RREU × RRUD/(RREU + RRUD − 1).RREU measures the exposure–confounder association and RRUD measures the confounder–outcome association on the relative-risk scale.
  • Risk-difference bounds: The appendix derives lower bounds for causal risk differences in the exposed, unexposed, and whole populations using BFU.The bounds can be obtained without knowing the exposure prevalence for some population-level calculations.
  • Illustration: With p1 = 0.25, p0 = 0.1, and (RREU,RRUD) = (2,2), BFU = 1.33 and the whole-population causal risk difference is at least 0.09.The exposed and unexposed population lower bounds are 0.12 and 0.09, respectively.
  • Target reduction: For an observed risk difference, the sensitivity parameters must satisfy a lower-bound condition for confounding to reduce it to a specified target.The appendix also gives formulas for the bounding factor required to reduce an estimate or confidence interval to zero or another quantity.
  • Cornfield conditions: In the example, both confounding measures must be at least 1.74, and their maximum must satisfy an additional high-threshold condition.The appendix applies the corresponding Cornfield conditions to the risk-difference setting.
  • Preventive exposures: For preventive exposures, analogous upper bounds are obtained after modifying the exposure–confounder sensitivity measure.The appendix treats apparently causative and apparently preventive exposures separately.

Appendix 3: Conditions for the Risk Difference Using Sensitivity Parameters on the Risk Difference Scale

This appendix studies risk-difference sensitivity parameters directly, defining exposure–confounder and confounder–outcome associations across confounder categories. It shows that the resulting conditions depend on the number of categories and are less practically useful than relative-risk-scale parameters.

  • Parameter definitions: The appendix defines the observed and standardized risk differences for the risk-difference-scale analysis.The observed risk difference compares outcome probabilities with and without exposure, while the standardized risk difference averages conditional differences over confounder categories.
  • Parameter definitions: The appendix defines αk as the exposed–unexposed difference in the probability of confounder category k and uses its maximum absolute value as RDEU.RDEU measures the exposure–confounder association on the risk-difference scale.
  • Parameter definitions: RDUD is defined as the maximum confounder–outcome risk difference with and without exposure.For a categorical confounder, the measure takes the maximum across the two exposure conditions.
  • Categorical confounders: For categorical confounders, the risk-difference-scale conditions depend on the number of confounder categories, and no simple bounding-factor form is available.The appendix provides conditions for RDEU and RDUD instead.
  • Limitations: The risk-difference-scale conditions become less informative as the number of confounder categories increases, so a binary confounder is not the most conservative case.This category-count problem does not occur for the relative-risk Cornfield conditions.
  • Scale choice: Sensitivity parameters on the relative-risk scale are presented as more appropriate for risk-difference sensitivity analysis because the bounding factor is the same regardless of category count.On the relative-risk scale, a binary confounder is the most conservative case.

Appendix 3: Another bounding factor with the exposure-confounder relationship on the odds

This appendix addresses a bounding factor on the ratio scale and its implied Cornfield conditions, with proofs.

  • The appendix concerns a bounding factor on the ratio scale.
  • It derives implied Cornfield conditions from the ratio-scale result.
  • The appendix includes proofs of these conditions.

Appendix 4: Relations between the new bounding factor and some existing results including

This appendix discusses Schlesselman’s formula and Flanders and Khoury’s results in relation to the paper’s sensitivity-analysis results.

  • The appendix discusses Schlesselman’s formula.
  • It also discusses Flanders and Khoury’s results.
  • These are presented as existing results relevant to the appendix’s analysis.

Appendix 6: SAS code for the risk difference using sensitivity parameters on the relative risk

The appendix develops a general sensitivity-analysis framework for arbitrary unmeasured confounders, derives bounding factors and sharp Cornfield-type conditions, and extends the results across outcome scales and populations.

  • Technical framework: The framework allows the unmeasured confounder U to take arbitrary values, rather than restricting it to a binary or categorical variable.
  • Technical framework: The generalized relative risk of exposure on U is defined through the Radon–Nikodym derivative RREU(u) = F1(du)/F0(du).
  • Technical framework: The analysis is conducted conditional on, or within strata of, the measured confounders C.
  • Sensitivity parameters: The maximal relative risks of U on D are defined separately without and with exposure, then combined as RRUD = max(RRUD|E=0,RRUD|E=1).
  • Bounding factor: The bounding factor is BFU = RREU × MRUD / (RREU + MRUD −1), and it bounds the confounding relative risks.
  • Cornfield conditions: The framework implies joint and threshold Cornfield conditions for RREU and RRUD, including a low threshold based on min(RRUD,RREU).
  • Extensions: The results extend to apparently preventive exposures, risk differences, rare time-to-event outcomes, hazard ratios, and nonnegative outcomes.
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