9 Sensitivity Analysis and Partial Identification
9.1 Why Sensitivity Analysis?
Chapters 5–8 developed a sequence of identification strategies: back-door adjustment, propensity-score re-weighting, instrumental variables, and front-door and mediation analysis. Each strategy identifies a causal parameter as a functional of the observed-data distribution under a corresponding set of assumptions. Each set of assumptions is, however, stated in terms of unobserved quantities — potential outcomes, hypothetical interventions, or unmeasured variables — and cannot be verified from the data alone.
The chapter ahead (Chapter 10) opens by noting that identification does not by itself provide a statistically reliable estimator: once a parameter is identified, a separate theory of estimation and inference is still needed. The present chapter makes the complementary point, which closes out Part II:
Statistical reliability does not by itself validate identification.
A confidence interval reports sampling variation conditional on a maintained identification assumption. It is silent about whether that assumption is correct. A causal analysis is credible only if both components — identification and estimation — are credible. Sensitivity analysis is the tool for reporting how much the conclusion depends on the first.
9.1.1 Sampling Uncertainty vs. Identification Uncertainty
Consider an analyst who reports \(\hat\tau = 2.0\), 95% CI \(= [1.2,\; 2.8]\). This interval answers one question and only one:
If the identifying assumptions are correct, what range of values of \(\tau\) is consistent with the observed sampling variation?
As \(n \to \infty\) the interval will shrink around \(\tau\), provided the identifying assumptions hold. If the assumptions do not hold, the interval will instead shrink around a biased target. More data drawn from the same observational regime do not remove this bias.
Formally, there are three distinct sources of error:
- Sampling uncertainty: \(\hat P_n \neq P\). The empirical distribution differs from the population. Vanishes as \(n \to \infty\).
- Model misspecification: the working parametric or semiparametric model for a nuisance function is incorrect. Partly addressable by flexible modeling, doubly robust construction, and cross-fitting (Chapters 10–12).
- Identification uncertainty: the causal parameter \(\psi\) is not in fact identified by the functional \(\Psi(P)\) assumed by the analysis. Does not vanish merely by increasing the sample size from the same observational regime; can be reduced only by adding new assumptions, new design information, or new measurements.
The usual confidence interval addresses only the first; the estimation chapters to come address primarily the second; this chapter addresses the third.
9.1.2 The Role of Sensitivity Analysis
Sensitivity analysis is a structured way to vary the strength of violations of identifying assumptions and to report how the causal conclusion responds. It is not a test of the assumptions — they involve unobserved quantities and are therefore not testable. It is a statement of conditional robustness: how far can the assumptions be violated before the substantive conclusion changes?
Three reporting objects:
- Sensitivity curve. A plot of the estimand against a single sensitivity parameter \(\lambda\), with \(\lambda=0\) recovering the baseline identifying assumption.
- Sensitivity bounds. The range of estimand values consistent with any \(\lambda\) in a specified plausibility set \(\Lambda\).
- Tipping point. The smallest violation magnitude sufficient to change the sign, magnitude, or statistical significance of the conclusion.
9.2 A General Framework for Sensitivity Analysis
The three views of sensitivity analysis — curves, bounds, tipping points — all arise from a single construction. Let \(\psi\) be the causal estimand and \(A_0\) the baseline identifying assumption with \(\psi = \Psi(P)\). A sensitivity model is a one-parameter relaxation \(\{A_\lambda : \lambda \in \Lambda\}\) of \(A_0\), with \(A_0\) recovered at \(\lambda = 0\). Under \(A_\lambda\): \[\psi(\lambda) = \Psi(P;\, \lambda), \qquad \Psi(P;\, 0) = \Psi(P). \tag{9.1}\]
The key reporting object is the set \(\{\psi(\lambda) : \lambda \in \Lambda\}\).
Sensitivity parameters used in this chapter:
- \(\lambda\) = strength of unmeasured confounding (odds-ratio bound, E-value, or likelihood-ratio bound).
- \(\delta\) = magnitude of a structural outcome-effect coefficient treated as zero under the baseline assumption (used both for \(U\)-outcome effect and for IV exclusion violations).
- \(\alpha\) = propensity-score trimming threshold.
- \(\rho\) = residual correlation between mediator and outcome disturbances.
Sensitivity curve: \(\lambda \mapsto \hat\psi(\lambda)\). Useful when the reader wants to see the effect deform as the assumption is relaxed.
Sensitivity bounds: \(\psi_L = \inf_\Lambda \psi(\lambda)\), \(\psi_U = \sup_\Lambda \psi(\lambda)\).
Tipping point: \(\lambda^\star = \inf\{\lambda \in \Lambda : \psi(\lambda) = 0\}\), or \(\lambda^\star_{\mathrm{CI}} = \inf\{\lambda : 0 \in \widehat{\mathrm{CI}}(\lambda)\}\).
9.3 Sensitivity to Unmeasured Confounding
The most common application is to unmeasured confounding under the back-door framework. The baseline assumption is conditional exchangeability \(Y(t) \indep T \mid X\) together with consistency and positivity. Sensitivity analysis contemplates a world with an unobserved \(U\) such that \(Y(t) \indep T \mid X, U\) but \(Y(t) \nindep T \mid X\).
9.3.1 The Bias Decomposition
Let \(\Delta_{\mathrm{obs}} = \E\{\E(Y \mid T=1, X) - \E(Y \mid T=0, X)\}\) denote the observed adjusted contrast. Under ignorability given \(X\) alone, \(\Delta_{\mathrm{obs}} = \tau\). Under the relaxed condition, the contrast departs from \(\tau\) by a bias \(B\).
9.3.2 A Simple Linear Sensitivity Model
In the fully linear model \(Y = \alpha + \tau T + \gamma^\top X + \delta U + \varepsilon\) with \(T = h(X, U, \eta)\):
Proof. In the linear model, \(m_t(x, u) = \alpha + \tau t + \gamma^\top x + \delta u\). The components of \(m_t\) not depending on \(u\) multiply integrals of \(g_t(X, u)\), which integrate to zero. The \(\delta u\) term gives Equation 9.4. \(\square\)
This gives the canonical teaching decomposition: \[\text{bias} \approx \underbrace{\delta}_{U\text{-outcome effect}} \;\times\; \underbrace{\E\{\E(U \mid T=1, X) - \E(U \mid T=0, X)\}}_{U\text{-treatment imbalance}}. \tag{9.5}\]
Both factors are sensitivity parameters because \(U\) is unobserved.
9.3.3 Binary Unmeasured Confounder
When \(U \in \{0,1\}\), define \(p_t(x) = P(U=1 \mid T=t, X=x)\).
Proof. With \(U \in \{0,1\}\), \(m_t(x,u) = m_t(x,0) + u\,\delta\). Substituting into Equation 9.3: \(B = \delta\,\E\{P(U{=}1 \mid T{=}1, X) - P(U{=}1 \mid T{=}0, X)\}\). \(\square\)
Formula Equation 9.6 is the workhorse of applied sensitivity analysis. A sensitivity analysis consists of computing \(\hat\tau_{\mathrm{sens}} = \hat\Delta_{\mathrm{obs}} - \hat B\) over a grid of \((\delta,\, \E\{p_1-p_0\})\) values.
9.3.4 A Sensitivity Table
For reporting, a table tabulates the adjusted effect over a grid. Here for observed adjusted effect \(\hat\Delta_{\mathrm{obs}} = 2.0\) (cells show \(\hat\tau_{\mathrm{sens}} = \hat\Delta_{\mathrm{obs}} - \delta(p_1-p_0)\)):
| \(U\)-outcome effect \(\delta\) | \(p_1 - p_0 = 0.10\) | \(p_1 - p_0 = 0.30\) | \(p_1 - p_0 = 0.50\) |
|---|---|---|---|
| weak (\(\delta = 2\)) | 1.80 | 1.40 | 1.00 |
| moderate (\(\delta = 4\)) | 1.60 | 0.80 | 0.00 |
| strong (\(\delta = 8\)) | 1.20 | −0.40 | −2.00 |
The tipping point for the sign is the diagonal cell \((\delta=4,\, p_1-p_0=0.5)\).
9.4 Three Canonical Sensitivity Models
The bias decomposition Equation 9.3 is a general identity. Three canonical models specialize it by imposing a structural restriction on either \(g_t\) or on the induced observed-data relationships. They are presented in order of increasing generality of the estimator they support.
9.4.1 Rosenbaum’s \(\Gamma\)-Sensitivity Model
The practical outcome is a tipping \(\Gamma^\star\): the smallest \(\Gamma\) at which the test fails to reject. Small \(\Gamma^\star\) (near 1) indicates a fragile conclusion; large \(\Gamma^\star\) indicates a robust one.
9.4.2 The E-Value and the VanderWeele–Ding Bound
Ding and VanderWeele (2016) and VanderWeele and Ding (2017) proposed a sensitivity summary that requires no matched design and takes the form of a single closed-form number. It has become the most widely reported sensitivity summary in the applied literature.
Define two risk-ratio sensitivity parameters: \(\mathrm{RR}_{TU} = \max_x P(U=1 \mid T=1, X=x)/P(U=1 \mid T=0, X=x)\) (maximum RR of \(U\) with \(T\)), and \(\mathrm{RR}_{UY} = \max_{t,x} P(Y=1 \mid T=t, X=x, U=1)/P(Y=1 \mid T=t, X=x, U=0)\) (maximum RR of \(U\) with \(Y\)). The bias factor is: \[B(\mathrm{RR}_{TU}, \mathrm{RR}_{UY}) = \frac{\mathrm{RR}_{TU}\, \mathrm{RR}_{UY}}{\mathrm{RR}_{TU} + \mathrm{RR}_{UY} - 1}. \tag{9.8}\]
9.4.3 The Marginal Sensitivity Model
Proof sketch. The true IPW weights equal the nominal weights multiplied by a perturbation factor \(\phi_i \in [\Lambda^{-1}, \Lambda]\). Because the target is linear in \(\phi\), the extrema are attained at \(\phi_i \in \{\Lambda^{-1}, \Lambda\}\). Zhao et al. (2019) give closed-form percentile expressions. \(\square\)
9.4.4 Comparing the Three Models
| Rosenbaum \(\Gamma\) | E-value | MSM (\(\Lambda\)) | |
|---|---|---|---|
| Parameter bounds | odds ratio of treatment given \(X, U\) | pair of risk-ratio associations \(\mathrm{RR}_{TU}, \mathrm{RR}_{UY}\) | odds ratio of nominal to true propensity |
| Natural estimator | matched-pair and weighted rank tests | any relative-effect summary | IPW, AIPW |
| One-number summary | tipping \(\Gamma^\star\) | \(\mathrm{EV}\) (symmetric diagonal) | bounds \([\tau_L, \tau_U]\) or tipping \(\Lambda^\star\) |
| Primary reference | Rosenbaum (2002) | VanderWeele and Ding (2017) | Tan (2006); Zhao et al. (2019) |
9.5 Benchmarking Sensitivity Parameters
Every sensitivity model has parameters that are not identified by the data. A sensitivity curve of the form “at \(\lambda = 0.2\) the effect is halved” is mathematically precise but scientifically empty until \(\lambda = 0.2\) has been anchored to something concrete. Benchmarking gives sensitivity parameters an empirical referent by comparing them to the analogous quantities computed for the observed covariates (Cinelli and Hazlett 2020).
9.5.1 Benchmarking Against Observed Covariates
For the binary-confounder setup, compute for each observed covariate \(X_j\): the coefficient on \(X_j\) in a regression of \(Y\) on \((T, X_1, \ldots)\) (the role of \(\delta\)), and the difference in conditional means of \(X_j\) across treatment arms (the role of the imbalance). Plot these against the tipping contour.
For the E-value, for each \(X_j\) compute the bias factor: \[B_j = \frac{\mathrm{RR}_{TX_j}\,\mathrm{RR}_{X_jY}}{\mathrm{RR}_{TX_j} + \mathrm{RR}_{X_jY} - 1}.\] If \(B_j < \mathrm{EV}\) for every observed covariate, an unmeasured confounder inducing the observed effect would have to be at least as strong as some observed covariate. Cinelli and Hazlett (2020) develop partial-\(R^2\) benchmarks for the linear regression setting.
9.6 Partial Identification and Bounds
The sensitivity models of Section 9.4 each restrict the magnitude of a violation by a single parameter. A more radical approach imposes no parametric restriction at all — asking what the observed data can say about the causal parameter without any untestable identifying assumption. The answer is a set of values: the theory of partial identification (Manski 1990, 2003).
9.6.1 Point Identification vs. Partial Identification
Under point identification, \(\psi = \Psi(P)\). Under partial identification, the assumptions pin down a set: \[\psi \in \Psi_A(P) = \{\text{values of }\psi\text{ consistent with assumption set }A\text{ and distribution }P\}. \tag{9.11}\]
9.6.2 Manski’s No-Assumption Bound
The no-assumption bound is as wide as the support of the outcome — typically too wide to be informative. Productive partial identification therefore proceeds by adding shape restrictions that narrow the set.
Monotone treatment response (\(Y(1) \geq Y(0)\) a.s.) tightens the bounds. Monotone treatment selection (\(\E\{Y(t) \mid T=1\} \geq \E\{Y(t) \mid T=0\}\)) is another common restriction. Combining these produces informative bounds even when neither alone suffices (Manski 2003).
9.7 Sensitivity to Positivity and Overlap Violations
Chapter 6 introduced positivity as a structural requirement for the identification formulas underlying propensity-score methods. When \(\pi(x_0) = 0\), the counterfactual mean \(\E\{Y(1) \mid X=x_0\}\) is not identified, and the ATE may not be point-identified over the full covariate support. When positivity holds but weakly, IPW weights \(1/\pi(X)\) generate extreme values that destabilize estimates.
9.7.1 Trimming as a Sensitivity Analysis
Proof. Conditional on \(X \in S_\alpha\), positivity holds with slack \(\alpha\), so the back-door formula applies pointwise. Integrating and normalizing gives Equation 9.16. The weight bound follows directly from \(\pi(X) \geq \alpha\) on \(S_\alpha\). \(\square\)
\(\tau_\alpha\) and \(\tau_0\) are different estimands: trimming is a retargeting strategy, not a variance-reduction trick. A trimmed analysis answers “what is the causal effect on the subpopulation for which the treatment decision is not already essentially forced by covariates?”
9.8 Sensitivity to Invalid Instruments
Of the three IV assumptions, relevance is the most empirically diagnosable. Exogeneity and exclusion involve unobserved relationships and are not testable. A sensitivity analysis for IV concentrates on these two untestable assumptions.
Relevance is a statement about \((Z, T)\) given \(X\) — observable. Exogeneity and exclusion are statements about \((Z, Y)\) given \(X\) and the unobserved \(U\) — not observable. A complete robustness argument has two parts: (a) evaluation of first-stage strength and (b) sensitivity analysis for exogeneity and exclusion.
9.8.1 Direct-Effect Violation of Exclusion
Consider the scalar-IV model with a direct effect \(\delta\) of \(Z\) on \(Y\): \[Y = \alpha + \beta T + \delta Z + \varepsilon, \qquad \E(\varepsilon \mid Z) = 0. \tag{9.17}\]
Proof. Take covariance of Equation 9.17 with \(Z\): \(\mathrm{Cov}(Y,Z) = \beta\,\mathrm{Cov}(T,Z) + \delta\,\mathrm{Var}(Z) + \mathrm{Cov}(\varepsilon,Z)\). By orthogonality, \(\mathrm{Cov}(\varepsilon,Z)=0\). Dividing and rearranging gives Equation 9.18 and Equation 9.19. \(\square\)
Key pedagogical consequence: the bias is proportional to \(\mathrm{Var}(Z)/\mathrm{Cov}(T,Z)\), the inverse of the first-stage slope. A weak first stage amplifies the bias from any exclusion violation.
9.8.2 Connection to LATE and Monotonicity
Under heterogeneous treatment effects, sensitivity analysis can also address monotonicity violations, with the sensitivity parameter becoming the proportion of defiers (Angrist et al. 1996). The same conceptual framework — parameterize the violation, report a curve or bound — applies.
9.9 Sensitivity in Mediation Analysis
Sensitivity analysis is especially important in mediation analysis because the identifying assumptions are strictly stronger than those for total effects. Even in a randomized trial, Assumptions 3 and 4 of sequential ignorability cannot be guaranteed.
9.9.1 Residual-Correlation Sensitivity Parameter
Consider the linear mediation system \(M = a_0 + aT + a_X^\top X + \varepsilon_M\), \(Y = b_0 + \tau' T + bM + b_X^\top X + \varepsilon_Y\), with sensitivity parameter: \[\rho = \mathrm{Corr}(\varepsilon_M, \varepsilon_Y \mid T, X). \tag{9.20}\]
If sequential ignorability holds, \(\rho = 0\): disturbances are orthogonal because any shared source of variation has been conditioned out. A nonzero \(\rho\) represents unobserved \(M\)–\(Y\) confounding. Imai et al. (2010) derive the bias in the estimated NIE as a smooth function of \(\rho\), giving a sensitivity curve \(\rho \mapsto \widehat{\mathrm{NIE}}(\rho)\) with tipping point \(\rho^\star\).
A well-reported mediation sensitivity analysis states: the point estimate of the indirect effect (with sampling CI), the tipping value \(\rho^\star\), and a benchmark for what magnitude of \(\rho\) is plausible.
9.10 Sensitivity Analysis and Modern Estimators
Chapters 10–13 develop doubly robust estimation, orthogonal scores, cross-fitting, and semiparametrically efficient IV estimation. These tools make estimators more robust — but robust to a specific class of perturbations: nuisance-model misspecification. They are silent about identification.
What robust estimation does. Neyman orthogonality makes AIPW consistent if either the outcome model or propensity model is correctly specified (double robustness). With cross-fitting, nuisance estimators need only converge at rate \(n^{-1/4}\) to yield \(\sqrt{n}\)-asymptotics. Both consequences address estimation under correctly maintained identifying assumptions.
What robust estimation does not do: Orthogonality and cross-fitting do not solve unmeasured confounding, positivity failure, IV invalidity, or mediation assumption failure.
Neyman orthogonality and cross-fitting protect the estimator against nuisance-estimation error. They do not protect against violations of causal identification assumptions.
This is the reason sensitivity analysis belongs in Part II (identification) rather than Part III (estimation). Every method in Chapters 10–13 assumes identification and refines the estimation.
9.11 Lab: A Tipping-Point Analysis for an Observational ATE
DGP. Let \(X, U \overset{\mathrm{i.i.d.}}{\sim} N(0,1)\) independently. Treatment: \(P(T=1 \mid X, U) = \mathrm{expit}(-0.3 + 0.6X + 1.0U)\). Outcome: \(Y(t) = 1.5t + 0.8X + 2.0U + \varepsilon\), \(\varepsilon \sim N(0,1)\). The analyst observes \((Y, T, X)\) but not \(U\). True ATE: \(\tau = 1.5\).
Naive adjusted estimator. OLS coefficient on \(T\) in the regression of \(Y\) on \((1, T, X)\). Running 500 Monte Carlo replicates (\(n = 2000\)): mean of \(\hat\tau_X\) is 3.172, SD is 0.099. Implied 95% CI \(\approx [2.98, 3.37]\). The naive estimator is badly biased, but the CI is narrow and does not contain the truth. This is the identification-uncertainty failure mode in concrete form.
Sensitivity adjustment. Using the binary-\(U\) decomposition, bias is \(B(\alpha_U, \delta) = \delta \cdot \mathrm{imbalance}(\alpha_U)\). Sensitivity-adjusted estimate \(\hat\tau_{\mathrm{sens}} = \hat\tau_X - B\) over a grid (\(\hat\tau_X = 3.172\); the true configuration is \(\alpha_U = 1.00, \delta = 2.0\)):
| Posited \(\delta\) | \(\alpha_U=0.25\) (imb=0.232) | \(\alpha_U=0.50\) (imb=0.440) | \(\alpha_U=1.00\) (imb=0.787) | \(\alpha_U=1.50\) (imb=1.025) |
|---|---|---|---|---|
| 0.5 | 3.056 | 2.952 | 2.779 | 2.660 |
| 1.0 | 2.940 | 2.732 | 2.385 | 2.147 |
| 2.0 | 2.708 | 2.292 | 1.598 ✓ | 1.123 |
| 3.0 | 2.477 | 1.851 | 0.811 | 0.098 |
The true configuration \((\alpha_U=1.00, \delta=2.0)\) gives the sensitivity-adjusted value \(1.598 \approx \tau = 1.5\) (checkmark).
Tipping points. \(\delta^\star(\alpha_U) = \hat\tau_X / \mathrm{imbalance}(\alpha_U)\). At \(\alpha_U = 1.00\): \(\delta^\star = 3.172/0.787 \approx 4.03\).
Benchmarking against observed \(X\) (outcome coefficient \(\gamma_X = 0.8\)): at \(\alpha_U = 1.00\), the unmeasured confounder would need an outcome association about \(4.03/0.8 \approx 5\) times stronger than \(X\) to zero out the estimate. A confounder as strong as \(X\) (\(\delta = 0.8\), \(\alpha_U = 1.0\)) would adjust the estimate to \(\approx 2.54\) — still clearly positive.
Four lessons: (1) The naive CI \([2.98, 3.37]\) is narrow but misses the true ATE \(\tau = 1.5\). Sampling uncertainty \(\neq\) identification uncertainty. (2) Sensitivity adjustment at the correct configuration recovers the truth to within Monte Carlo error. (3) The tipping point depends jointly on both sensitivity parameters. (4) Benchmarking against observed \(X\) anchors the analysis: the conclusion “the effect is positive” is robust to unmeasured confounders up to about \(5\gamma_X\) in outcome association.
9.12 Practical Reporting Guidelines
At minimum, a report should include: the point estimate \(\hat\psi\) and its 95% CI; a sensitivity summary (curve, bound, E-value, or tipping point); a benchmarking statement; and a plain-language interpretation.
9.13 Chapter Summary
| Symbol | Meaning |
|---|---|
| \(\Delta_{\mathrm{obs}}\) | Observed adjusted contrast |
| \(B\) | Confounding bias Equation 9.3 |
| \(\Gamma\) | Rosenbaum odds-ratio bound |
| \(\mathrm{EV}\) | E-value: \(R + \sqrt{R(R-1)}\) Equation 9.9 |
| \(\Lambda\) | MSM odds-ratio bound |
| \(\tau_\alpha\) | Trimmed ATE Equation 9.15 |
| \(\rho\) | Residual correlation for mediation Equation 9.20 |
| \(\lambda^\star\) | Tipping point for sign |
| \(\lambda^\star_{\mathrm{CI}}\) | Tipping point for statistical significance |
- Sampling uncertainty, model misspecification, and identification uncertainty are distinct sources of error. A confidence interval addresses only the first, and identification uncertainty does not vanish merely by collecting more data from the same observational regime.
- A sensitivity analysis introduces a parameter \(\lambda\) that quantifies the strength of a violation, with \(\lambda = 0\) recovering the baseline assumption. The three reporting objects are the sensitivity curve, sensitivity bounds, and the tipping point.
- The master bias decomposition \(\Delta_{\mathrm{obs}} = \tau + B\) underwrites sensitivity analysis for unmeasured confounding, specializing cleanly in the linear and binary-\(U\) cases.
- Three canonical sensitivity models differ by what the sensitivity parameter bounds: Rosenbaum’s \(\Gamma\) bounds a treatment-odds ratio; the E-value \(\mathrm{EV} = R + \sqrt{R(R-1)}\) gives a closed-form symmetric threshold; and the MSM bounds an odds ratio of nominal to true propensity.
- Sensitivity parameters need benchmarking against observed covariates to be interpretable.
- Partial identification reports an identified set rather than a single value. Manski’s no-assumption bound is as wide as the support of the outcome; shape restrictions narrow it.
- Positivity violations are an identification failure, not a variance problem. Trimming changes the estimand to a region of covariate overlap; it is a retargeting strategy, not a correction for the original population ATE.
- Invalid instruments produce bias \(\delta \cdot \mathrm{Var}(Z)/\mathrm{Cov}(T,Z)\) amplified by weak first stages. Weak-instrument diagnostics do not address exclusion violations.
- Mediation sensitivity analysis proceeds by a residual-correlation parameter \(\rho\) that captures unmeasured \(M\)–\(Y\) confounding.
- Modern estimation methods (AIPW, TMLE, DML, efficient GMM) address nuisance estimation, not identification. Neyman orthogonality protects the estimator; it does not protect the causal claim.
9.14 Problems
1. Sampling vs. identification uncertainty. Explain in your own words the difference between a narrow sampling confidence interval and a robust causal conclusion. Construct an example data-generating process, with numerical parameters, in which the CI for a back-door-adjusted ATE is very narrow (SD below 0.05) yet the true ATE lies outside it. Identify the identifying assumption that is violated and the magnitude of the violation.
2. Binary unmeasured confounder. Suppose \(\hat\Delta_{\mathrm{obs}} = 2.0\). Assume a binary unmeasured confounder \(U\) with constant outcome contrast \(\delta = 4\) and imbalance \(p_1 - p_0 = 0.3\).
- Use the Binary-Confounder Bias Lemma to compute \(B\) and the sensitivity-adjusted estimate \(\hat\tau_{\mathrm{sens}}\).
- What outcome contrast \(\delta\) would zero out the estimate at the same imbalance level?
- What imbalance \(p_1 - p_0\) would zero out the estimate at \(\delta = 4\)?
3. Tipping point in a linear bias model. For \(\hat\Delta_{\mathrm{obs}} = 1.5\) and bias model \(B(\lambda) = 0.4\lambda\):
- Find the tipping point \(\lambda^\star\) for the sign of the adjusted estimate.
- Suppose the sampling standard error is 0.3, independent of \(\lambda\). Find the tipping point \(\lambda^\star_{\mathrm{CI}}\) for the lower confidence limit to reach zero (95% level). Compare to \(\lambda^\star\).
4. Benchmarking. A researcher reports a sensitivity analysis in which the tipping point for the \(U\)-outcome effect is \(\delta^\star = 3.0\). Observed covariate outcome coefficients are \((\hat\gamma_{\mathrm{age}}, \hat\gamma_{\mathrm{income}}, \hat\gamma_{\mathrm{education}}) = (0.3, 1.8, 2.4)\). Identify the strongest benchmark. Is the result robust to an unmeasured confounder as strong as the strongest observed covariate? Write two sentences of interpretation suitable for an applied report.
5. E-value calculation. An observational study reports an adjusted risk ratio of \(\mathrm{RR}_{TY \mid X}^{\mathrm{obs}} = 1.9\) with 95% CI \([1.3, 2.8]\).
- Compute the E-value for the point estimate using Equation 9.9.
- Compute the E-value for the lower confidence limit.
- Write one sentence of interpretation for each.
6. IV sensitivity curve. In a scalar-IV model with \(\mathrm{Cov}(Y,Z) = 3\), \(\mathrm{Cov}(T,Z) = 1.5\), \(\mathrm{Var}(Z) = 1\):
- Use Equation 9.19 to compute \(\hat\beta(\delta)\) for \(\delta \in \{0, 0.5, 1.0, 1.5, 2.0\}\).
- Find the tipping point \(\delta^\star\).
- Repeat for the weaker-instrument case \(\mathrm{Cov}(T,Z) = 0.5\) (keeping other quantities fixed). Explain why the same \(\delta\) produces a larger bias.
7. Manski bound calculation. Suppose \(Y \in [0, 10]\), \(P(T=1) = 0.3\), \(\E(Y \mid T=1) = 6.5\), \(\E(Y \mid T=0) = 4.0\).
- Compute Manski’s no-assumption bound using Equation 9.12–Equation 9.13.
- Compute the width and verify Equation 9.14.
- The observed association is 2.5. Does the no-assumption identified set include zero? Discuss what this implies about the informational content of the data alone.
8. Positivity sensitivity. Explain why the trimmed estimand \(\tau_\alpha\) of Equation 9.15 is generally not equal to the untrimmed ATE \(\tau\). In a dataset with \(\pi(X)\) distributed uniformly on \([0,1]\), what fraction of the population is retained at trimming levels \(\alpha \in \{0.01, 0.05, 0.10, 0.20\}\)? Under what scientific questions is \(\tau_\alpha\) preferable to \(\tau\) as a target?
9. Mediation sensitivity. In a mediation study of a behavioral intervention (\(T\)) on depression (\(Y\)) through sleep quality (\(M\)), explain why randomization of \(T\) alone does not eliminate the need for a sensitivity analysis for the indirect effect. Draw the DAG that describes the residual concern and identify which arrow corresponds to a nonzero \(\rho\) in Equation 9.20. Give a plausible scientific story in which the residual correlation could be large.
10. Modern estimators and identification. A colleague argues that because DML and AIPW are “doubly robust and orthogonalized,” they “automatically correct for unmeasured confounding provided the machine-learning models are good enough.” Explain why this is incorrect, referring to the bias decomposition of the Bias Decomposition Theorem. Identify which term in that decomposition machine-learning methods can estimate consistently, and which term they cannot estimate at all.