28  Quantitative Bias Analysis for Indirect Comparisons

28.1 Introduction

When a health technology assessment (HTA) committee, for example a NICE appraisal panel or an FDA advisory committee, must decide whether to reimburse a new treatment against an established comparator, the pivotal evidence is frequently an indirect comparison rather than a head-to-head trial. The committee’s sharpest question is seldom “what is the point estimate?” but “how confident are you that the estimate survives a covariate you were never able to adjust for?” This chapter develops the mathematics of answering that question honestly, so that a reimbursement dossier can report not merely an effect but a defensible statement about how fragile that effect is.

Every method in this book up to now has removed bias by measuring the variable that causes it. The anchored Bucher comparison (Theorem 14.1) is exact when every effect modifier is shared and common; matching-adjusted indirect comparison (Chapter 18) reweights the individual patient data (IPD) until the imbalanced effect modifiers are balanced; simulated treatment comparison (Chapter 19) and multilevel network meta-regression (Chapter 20) model the same covariates inside an outcome regression. All of them adjust for a covariate by using it. What none of them can do is adjust for a covariate that was never measured, or, in the most common and most dangerous case for an unanchored indirect comparison, one that the comparator publication never reported.

The distinction is worth a table, because it says precisely where this chapter begins. Each earlier method neutralizes a covariate only by having access to it in some form; QBA is what remains when even that access is gone.

Method	How it removes the covariate’s bias	What it needs the covariate to be
Bucher (Chapter 14)	cancels it through a common comparator arm	shared and balanced across trials
MAIC (Chapter 18)	reweights the IPD until it is balanced	measured in the IPD, summarized by the comparator
STC (Chapter 19)	enters it in an outcome regression	measured in the IPD, summarized by the comparator
ML-NMR (Chapter 20)	integrates the outcome model over it	measured in the IPD, summarized by the comparator
QBA (this chapter)	quantifies what it could be doing	neither measured nor reported

The threat is sharpest in the unanchored setting. Chapter 17 showed that an unanchored population-adjusted indirect comparison (PAIC) is consistent only under conditional constancy of absolute effects and balance of every imbalanced prognostic variable, not merely the effect modifiers (Theorem 17.2, Definition 17.5), and Chapter 12 showed that the shared-effect-modifier assumption is by itself insufficient to identify the target-population contrast (Theorem 12.6). The single covariate that escapes the analysis, an unmeasured effect modifier (Definition 11.3) whose distribution differs between the two trial populations, is residual bias that no reweighting and no outcome model can touch, because the quantity it depends on, the covariate’s distribution in the comparator population, is simply unknown. The honest response is not to pretend the bias is zero but to quantify it: to ask how large the unmeasured mechanism would have to be to matter, and to report the answer alongside the point estimate. This is quantitative bias analysis (QBA), the subject of this chapter.

Remark. Prognostic variable versus effect modifier, in one line. Two kinds of covariate recur throughout the book, and the difference decides which comparisons they threaten. A prognostic variable shifts a patient’s outcome risk regardless of which treatment they take; an effect modifier changes the size of the treatment effect itself, so the benefit of \(B\) over \(A\) differs across its levels (Definition 11.3). An anchored comparison, which subtracts a within-trial contrast, is undisturbed by a purely prognostic variable and needs only its effect modifiers balanced; an unanchored comparison has no such protection and needs balance of every imbalanced prognostic variable as well (Theorem 17.2). That is why the unreported covariate is the sharpest threat exactly in the unanchored setting, and why the running example below places its unreported covariate there. \(\blacktriangleleft\)

QBA originates in epidemiology, where the same three threats to validity recur in every observational study: uncontrolled confounding, selection of the study sample on exposure and outcome, and mismeasurement of variables. The canonical reference is the monograph of Lash et al. (2021), whose deterministic and probabilistic machinery we adapt. We develop four instruments, in increasing sophistication. First, the E-value of VanderWeele and Ding (2017): a single number, derived from the point estimate alone, giving the minimum strength of unmeasured confounding or effect modification, on the risk-ratio scale, that could explain the observed effect away. We give its full derivation, resting on a sharp bound for the confounding bias factor that we prove from scratch. Second, an indirect-comparison-specific E-value that exploits a structural feature unique to the unanchored ITC: the unmeasured covariate is observed in the analyst’s own IPD, so half of the sensitivity problem is estimable rather than assumed, and the resulting E-value is sharper and can certify a result as provably robust. Third, the NORTA (normal-to-anything) construction, which imputes the unreported joint distribution of the comparator’s covariates by gluing the reported margins to a dependence structure borrowed from the IPD through a Gaussian copula (Definition 22.5, Theorem 22.4); we prove that the construction reproduces the margins exactly and induces precisely the intended copula. Fourth, the tipping-point analysis and its probabilistic upgrade, which locate the assumed value of the unreported covariate at which the conclusion reverses and place a prior distribution over that value to propagate the uncertainty honestly.

This material is new to the formal theory of indirect comparison; it is not in the proof catalogue, and it has no machine-checked counterpart in the companion Coq development. We therefore build it from first principles, proving every bound in full and grounding each tool in a fully worked, hand-checkable example at the end of the chapter.

Throughout, \(g\) is the link function on whose scale effects are reported and modeled, and \(g^{-1}\) its inverse; for binary and count outcomes the analysis scale is the log odds ratio or log rate ratio, so the effects of interest live on the risk-ratio scale where the E-value is defined. We write \(\mathrm{RR}\), \(\mathrm{OR}\), \(\mathrm{HR}\) for risk, odds, and hazard ratios as in Notation. The symbol \(U\) denotes the unmeasured (in the confounding sense) or unreported (in the ITC sense) covariate, distinguished from the measured covariate vector \(\mathbf{x}\); \(\mathbf{b}\) denotes a vector of bias parameters; and we reserve the plain italic \(E\) for the E-value, distinct from the expectation operator \(\mathbb{E}[\cdot]\).

This chapter builds five instruments. They are not competitors but a toolkit: some answer how robust is the estimate?, one answers what value would overturn it?, one answers how do I propagate my uncertainty about that value?, and one is the machinery that builds the comparator population the others need. Keep the following map in view.

Instrument	Question it answers	Main input	Output
E-value (Section 28.3)	how strong must unmeasured confounding be to explain the estimate away?	the point estimate on the risk-ratio scale	one number \(E\)
ITC E-value (Section 28.4)	the same, using the outcome association measured in the IPD	point estimate plus \(\mathrm{RR}_{UD}\) from the IPD	\(E_{\mathrm{ITC}}\), possibly \(+\infty\)
NORTA (1)	how do I build a comparator population to recompute the effect?	reported margins plus IPD dependence	a simulated covariate population
Tipping point (Section 28.6)	what comparator value flips the conclusion?	a sensitivity curve and a threshold	a value \(m^{\dagger}\) in covariate units
Probabilistic bias analysis (Section 28.7)	how does uncertainty about that value propagate?	a prior on the bias parameter plus the sampling error	systematic and total intervals

We anchor all five to one concrete comparison, introduced now and computed in full in Section 28.8.

Example 28.1 (A running example: an unanchored COPD comparison) A manufacturer seeks reimbursement for a new inhaled therapy \(B\) for chronic obstructive pulmonary disease (COPD). Only a single-arm trial of \(B\) is available, with full individual patient data (IPD); the established comparator \(A\) has only a published summary. The outcome is a severe exacerbation within 12 months, a bad event, so a risk ratio below \(1\) favors \(B\). Two baseline covariates matter. The first, \(x_R\), is age (standardized), which the \(A\) publication reports as a mean and standard deviation. The second, \(u\), is baseline lung function \(\mathrm{FEV}_1\) (forced expiratory volume in one second, in liters; higher is healthier, so it lowers exacerbation risk): the \(A\) publication does not report it at all. An unanchored analysis that plugs the IPD value of \(\mathrm{FEV}_1\) into the comparator population returns an adjusted risk ratio of about \(\mathrm{RR}=0.50\) (\(95\%\) confidence interval \(0.30\) to \(0.81\)), so \(B\) appears to roughly halve the exacerbation risk. The committee’s question is the subject of this chapter: how much of that apparent benefit could be an artifact of the unreported \(\mathrm{FEV}_1\)?

We carry this trial through every instrument below, previewing each result where the instrument is defined and computing all of it by hand in Section 28.8. The model fitted to the IPD, with a log link on the risk, is \(\log P(Y=1\mid x_R,u)=\beta_0+\beta_R x_R+\beta_U u\) with \(\beta_0=-1.0\), \(\beta_R=0.4\) (older age raises risk), and \(\beta_U=-0.5\) (higher \(\mathrm{FEV}_1\) lowers risk); the comparator reports age mean \(m_R=1.0\), standard deviation \(\sigma_R=0.5\), and its own event risk \(\mu_0=0.47\), and in the IPD the covariate \(\mathrm{FEV}_1\) has mean \(\bar{x}_U=2.0\) and standard deviation \(\sigma_U=1.0\). \(\blacktriangleleft\)

28.2 The framework of bias analysis

Bias analysis begins by writing down, explicitly, the mechanism that links what we observe to what is true. An estimate is valid when the estimator converges to the estimand; it is biased by a systematic error when a structural feature of the data, distinct from sampling variability, drives a wedge between the two. Quantitative bias analysis makes that wedge a function of named, interpretable parameters and then explores it.

A natural first reaction to a worrying covariate is to fit a richer model that includes it. The reason that does not work here is worth stating plainly: in an unanchored indirect comparison the covariate is unreported by the comparator, so there is no column for it in the comparator data and no likelihood can condition on it. Bayesian model expansion is the right tool when you hold the data and merely distrust a functional form; quantitative bias analysis is the right tool when the data themselves are missing and the best you can do is ask how bad the omission could be. QBA does not recover the missing information; it bounds the damage.

Definition 28.1 (Quantitative bias analysis) Let \(\theta\) be a target estimand and \(\hat\theta_{\mathrm{obs}}\) an estimator that is consistent for a biased surrogate \(\theta_{\mathrm{obs}}=\theta\cdot B(\mathbf{b})\) (multiplicative scale) or \(\theta_{\mathrm{obs}}=\theta + B(\mathbf{b})\) (additive scale), where the bias function \(B\) depends on a vector of bias parameters \(\mathbf{b}\in\mathcal{B}\) that index the unverified structural assumptions (the strength and imbalance of an unmeasured confounder, the selection probabilities of each cell, the sensitivity and specificity of a mismeasured variable, or the comparator distribution of an unreported covariate). A quantitative bias analysis is the map \[ \mathbf{b}\;\longmapsto\;\hat\theta_{\mathrm{adj}}(\mathbf{b}) \;=\;h\big(\hat\theta_{\mathrm{obs}},\mathbf{b}\big), \tag{28.1}\] where the correction \(h\) inverts the bias model, returning the value of the target consistent with the observed estimate under the assumed \(\mathbf{b}\). The analysis is deterministic when \(\mathbf{b}\) ranges over a fixed grid or a single value, and probabilistic when \(\mathbf{b}\) carries a prior distribution \(\pi\) that is propagated to a distribution over \(\hat\theta_{\mathrm{adj}}\). The three classical threats, uncontrolled confounding, selection bias, and misclassification, are the corners of the bias triangle and each supplies its own \(B\) and \(h\). \(\blacktriangleleft\)

Remark. Reader’s translation. Read Definition 28.1 as a recipe with three moves: (i) name the unverified quantities as a parameter vector \(\mathbf{b}\) (how imbalanced the missing covariate is, how strongly it acts, and so on); (ii) write the bias as an explicit function \(B(\mathbf{b})\) of them; (iii) invert it to see what the honest estimate would have been. On the multiplicative scale the correction divides: \(\hat\theta_{\mathrm{adj}}=\hat\theta_{\mathrm{obs}}/B(\mathbf{b})\), as in the risk-ratio confounding factor below. On the additive scale it subtracts: for a risk difference distorted by a selection or misclassification shift \(B(\mathbf{b})\), the corrected estimate is \(\hat\theta_{\mathrm{obs}}-B(\mathbf{b})\). The three corners of the bias triangle just supply three different shapes of \(B\). \(\blacktriangleleft\)

28.2.1 A practical workflow

Before the derivations, here is the applied procedure the rest of the chapter makes precise, written as a health researcher would run it for an unanchored indirect comparison.

1. Identify the threat. Name the covariate that is imbalanced and unreported by the comparator, and decide whether it is prognostic, an effect modifier, or both.
2. Estimate what you can. From the IPD, read off the covariate’s association with the outcome, \(\mathrm{RR}_{UD}=\exp(|\beta_U|)\) on a ratio scale, and its own IPD distribution.
3. Screen with an E-value. Compute the E-value and, using \(\mathrm{RR}_{UD}\), the sharper indirect-comparison E-value (Section 28.4). If the latter is \(+\infty\), stop: the result is certified robust to this covariate.
4. Choose plausible comparator values. Fix a range or a prior for the covariate’s comparator marginal from external evidence (registries, related trials, elicitation); this is a sensitivity assumption, not new data.
5. Build the comparator population. If dependence with the reported covariates matters, use NORTA

1. to glue your assumed marginal to a dependence structure borrowed from the IPD; otherwise the marginal alone suffices.

6. Compute the sensitivity curve and its tipping point. Marginalize the IPD outcome model over each assumed value to get \(\Delta(m)\), and locate the value \(m^{\dagger}\) at which the conclusion reverses (Section 28.6).
7. Propagate the uncertainty. Place a prior on the comparator marginal and report the systematic and total intervals and the probability of benefit (Section 28.7).
8. Report honestly. State the E-value, the tipping point in the covariate’s own units, and whether a believable comparator value would overturn the conclusion.

The sections below justify each step; the running example runs all eight. A handful of risk-ratio labels recur across those steps, so the following dictionary fixes them once.

Symbol	How to read it
\(\mathrm{RR}_{\mathrm{crude}}\), \(\mathrm{RR}_{\mathrm{obs}}\)	the confounded thing the data show, before removing bias (used interchangeably)
\(\mathrm{RR}_{\mathrm{adj}}\), \(\mathrm{RR}_{\mathrm{true}}\)	the de-biased target you would see with the confounder controlled (likewise interchangeable)
\(B(\mathbf{b})\)	the general bias function of Definition 28.1
\(B\) (no argument)	the specific confounding bias factor of Lemma 28.1, one instance of \(B(\mathbf{b})\)
\(\mathrm{RR}_{XY}\)	the risk ratio of \(Y\) across levels of \(X\) (so \(\mathrm{RR}_{EU}\), \(\mathrm{RR}_{UD}\) read off the roles \(E\) = exposure, \(U\) = unmeasured covariate, \(D\) = disease)
\(E\) (italic), \(\mathbb{E}\)	the E-value Equation 28.9 and the expectation operator: never the exposure, which we write \(A\)

The simplest instance, and the one that motivates the E-value, is a single binary unmeasured confounder acting on a risk ratio. We derive its correction now, both because it is the prototype of Equation 28.1 and because the E-value is exactly the question “how large must \(\mathbf{b}\) be?” asked of this model.

Lemma 28.1 (Confounding bias factor for a binary confounder) Consider a binary exposure (treatment) \(A\in\{0,1\}\), a binary outcome \(D\), and a binary unmeasured confounder \(U\in\{0,1\}\) with prevalence \(p_1=P(U=1\mid A=1)\) among the exposed and \(p_0=P(U=1\mid A=0)\) among the unexposed. Suppose \(U\) is the only confounder, so that within strata of \(U\) the exposure is unconfounded, and suppose \(U\) acts on the outcome with a constant risk ratio \(\mathrm{RR}_{UD}=P(D=1\mid U=1,A=a)/P(D=1\mid U=0,A=a)\) that does not depend on \(a\). Then the crude (confounded) risk ratio and the \(U\)-adjusted risk ratio differ by the multiplicative bias factor \[ B \;=\; \frac{p_1\,(\mathrm{RR}_{UD}-1)+1}{p_0\,(\mathrm{RR}_{UD}-1)+1}, \qquad \mathrm{RR}_{\mathrm{adj}}=\frac{\mathrm{RR}_{\mathrm{crude}}}{B}. \tag{28.2}\] In particular \(B=1\) when \(p_1=p_0\) (the confounder is balanced) or \(\mathrm{RR}_{UD}=1\) (the confounder is inert).

Proof. Write \(r_a=P(D=1\mid A=a)\) for the marginal risk in arm \(a\) and \(q_a(u)=P(D=1\mid A=a,U=u)\) for the stratum risks. The lemma’s constant-risk-ratio hypothesis says exactly that the within-arm outcome risk ratio across \(U\) is \(\mathrm{RR}_{UD}\), that is \(q_a(1)/q_a(0)=\mathrm{RR}_{UD}\), so \(q_a(1)=\mathrm{RR}_{UD}\,q_a(0)\). By the law of total probability, \(r_a=q_a(0)\,(1-p_a)+q_a(1)\,p_a=q_a(0)\big[(1-p_a)+p_a\,\mathrm{RR}_{UD}\big]=q_a(0)\big[1+p_a(\mathrm{RR}_{UD}-1)\big]\). The crude risk ratio is therefore \[ \mathrm{RR}_{\mathrm{crude}}=\frac{r_1}{r_0} =\frac{q_1(0)}{q_0(0)}\cdot\frac{1+p_1(\mathrm{RR}_{UD}-1)}{1+p_0(\mathrm{RR}_{UD}-1)}. \] Because \(U\) is the only confounder, the ratio \(q_1(0)/q_0(0)\) of the stratum-\(U{=}0\) risks is the unconfounded, \(U\)-adjusted risk ratio \(\mathrm{RR}_{\mathrm{adj}}\) (the within-stratum exposure effect, assumed constant across \(U\)). The remaining factor is exactly \(B\) in Equation 28.2, so \(\mathrm{RR}_{\mathrm{crude}}=\mathrm{RR}_{\mathrm{adj}}\cdot B\), which rearranges to the stated correction. Setting \(p_1=p_0\) makes numerator and denominator equal, \(B=1\); setting \(\mathrm{RR}_{UD}=1\) makes both equal \(1\), again \(B=1\). \(\square\)

Remark. What the bias factor says. In words, the crude estimate you see equals the true effect times a factor that measures how differently the confounder is spread across the two arms. That factor is a product of two intuitions the formula makes exact: a confounder equally common in both arms (\(p_1=p_0\)) cannot bias anything, and a confounder unrelated to the outcome (\(\mathrm{RR}_{UD}=1\)) cannot either. Bias needs both an imbalance and an outcome link, which is exactly why an unreported covariate is dangerous only when it is both differently distributed across the trials and prognostic. \(\blacktriangleleft\)

Remark. The deterministic correction needs three numbers you rarely have. Formula Equation 28.2 is the correction \(h\) of Definition 28.1 with bias parameters \(\mathbf{b}=(p_1,p_0,\mathrm{RR}_{UD})\). It is honest but demanding: it presumes the analyst can name the prevalence of the unmeasured confounder in each arm and its outcome association. Selection bias and misclassification have analogous corrections (the selection odds ratio divides out of the crude odds ratio; the classification correction inverts the sensitivity-specificity map cell by cell), each with its own \(\mathbf{b}\); see Lash et al. (2021). The E-value of the next section sidesteps the need to specify \(\mathbf{b}\) by turning the question around. \(\blacktriangleleft\)

28.2.2 The other two corners: selection and misclassification

The bias triangle has three vertices, and the rest of this chapter develops only the confounding vertex, because it is the one that maps directly onto the unreported effect modifier of an unanchored comparison. For completeness we record the other two corrections, both standard and both taken from Lash et al. (2021) rather than re-derived here, so the reader can see the common shape: each names bias parameters, writes a \(B\), and inverts it.

Selection bias. When inclusion in the analysis depends on both exposure and outcome, let \(s_{ad}\) be the probability that a subject with exposure \(a\in\{0,1\}\) and outcome \(d\in\{0,1\}\) enters the sample. The observed (selected-sample) odds ratio and the target-population odds ratio differ by the multiplicative selection bias factor \[ B_{\mathrm{sel}}=\frac{s_{11}\,s_{00}}{s_{10}\,s_{01}}, \qquad \mathrm{OR}_{\mathrm{adj}}=\frac{\mathrm{OR}_{\mathrm{obs}}}{B_{\mathrm{sel}}}, \tag{28.3}\] so selection is harmless exactly when the four selection probabilities factor into an exposure term times an outcome term (no exposure-by-outcome interaction in who gets in).

Misclassification. When a binary exposure is measured with sensitivity \(\mathrm{Se}=P(A^{*}=1\mid A=1)\) and specificity \(\mathrm{Sp}=P(A^{*}=0\mid A=0)\), where \(A^{*}\) is the recorded and \(A\) the true exposure, the true exposed count \(N_1\) in a stratum of size \(n\) with observed exposed count \(N_1^{*}\) is recovered by inverting the classification map, \[ N_1=\frac{N_1^{*}-(1-\mathrm{Sp})\,n}{\mathrm{Se}+\mathrm{Sp}-1}, \tag{28.4}\] valid whenever \(\mathrm{Se}+\mathrm{Sp}>1\) (the measurement is better than a coin flip); one applies it within outcome strata and recomputes the effect from the corrected counts. Both corrections have probabilistic upgrades, priors on \((s_{ad})\) or on \((\mathrm{Se},\mathrm{Sp})\), exactly as in Section 28.7. In an indirect comparison the confounding corner dominates, so from here on bias means residual imbalance in an unreported covariate unless stated otherwise.

28.3 The E-value

The E-value asks not “what is the bias, given \(\mathbf{b}\)?” but “how strong must \(\mathbf{b}\) be to overturn the conclusion?” To answer it we need a bound that holds for any unmeasured confounder, not just a binary one with a constant association, in terms of two scalar sensitivity parameters. The bound of VanderWeele and Ding (2017), building on the bias-factor analysis of Ding and VanderWeele, is sharp, and we prove it in full.

Fix a binary exposure \(A\) and binary outcome \(D\), with an unmeasured confounder \(U\) taking values in a finite set \(\mathcal{U}\). Write \(f_a(u)=P(U=u\mid A=a)\) for the confounder distribution in arm \(a\) and \(p_a(u)=P(D=1\mid A=a,U=u)\) for the stratum risks. Define the two sensitivity parameters \[ \mathrm{RR}_{EU}\;=\;\max_{u\in\mathcal{U}}\frac{f_1(u)}{f_0(u)}, \qquad \mathrm{RR}_{UD}\;=\;\frac{\max_{u}p_0(u)}{\min_{u}p_0(u)}, \tag{28.5}\] both at least \(1\). A word on the subscripts, which follow the E-value literature and are unrelated to the italic E-value \(E\) of Equation 28.9 or to the expectation \(\mathbb{E}\): in \(\mathrm{RR}_{XY}\) the pair \(XY\) reads “the risk ratio of \(Y\) across levels of \(X\)”, with \(E\) naming the exposure role (the treatment variable we write \(A\)), \(U\) the unmeasured covariate, and \(D\) the disease outcome. So \(\mathrm{RR}_{EU}\) measures how exposure shifts the confounder distribution and \(\mathrm{RR}_{UD}\) how the confounder shifts outcome risk. In words, \(\mathrm{RR}_{EU}\) is the largest factor by which any confounder level is over-represented in the exposed relative to the unexposed arm, and \(\mathrm{RR}_{UD}\) is the widest spread of outcome risk across confounder levels within the unexposed arm.

Remark. A number to hold onto. Suppose a biomarker is twice as common in the exposed arm as in the unexposed arm, \(\mathrm{RR}_{EU}=2\), and carries a threefold spread in outcome risk, \(\mathrm{RR}_{UD}=3\). The bound below says the most these two facts can bias a risk ratio is \(\mathrm{BF}_U=\dfrac{2\times 3}{2+3-1}=\dfrac{6}{4}=1.5\): no configuration of that confounder can inflate an observed risk ratio by more than a factor of \(1.5\). The next lemma proves this formula and, crucially, that \(1.5\) is attained, not merely an over-cautious ceiling. \(\blacktriangleleft\)

Lemma 28.2 (Sharp bound on the confounding bias factor) With the notation above, suppose \(U\) carries all the confounding, so that within strata of \(U\) the exposure is unconfounded. Then the confounding bias factor \(B\) relating the observed risk ratio to the \(U\)-adjusted (causal) risk ratio standardized to the exposed (that is, averaging the stratum risks over the exposed arm’s confounder distribution) satisfies \[ B\;\le\;\mathrm{BF}_U\;:=\;\frac{\mathrm{RR}_{EU}\,\mathrm{RR}_{UD}}{\mathrm{RR}_{EU}+\mathrm{RR}_{UD}-1}, \tag{28.6}\] and the bound is sharp: for every pair \((\mathrm{RR}_{EU},\mathrm{RR}_{UD})\) with both values at least \(1\) there is a confounder configuration attaining equality.

Proof. The bias factor as a ratio of weighted averages. Because \(U\) carries all the confounding, the counterfactual risk had the exposed arm been left unexposed is, by standardization over the exposed confounder distribution, \(\sum_u p_0(u)f_1(u)\). Hence the causal risk ratio for the effect on the exposed is \(\big(\sum_u p_1(u)f_1(u)\big)\big/\big(\sum_u p_0(u)f_1(u)\big)\), while the observed risk ratio is \(\big(\sum_u p_1(u)f_1(u)\big)\big/\big(\sum_u p_0(u)f_0(u)\big)\). Dividing, the common numerator cancels and \[ B=\frac{\sum_{u}p_0(u)f_1(u)}{\sum_{u}p_0(u)f_0(u)} =\frac{\mathbb{E}_{f_1}[\,p_0(U)\,]}{\mathbb{E}_{f_0}[\,p_0(U)\,]}. \tag{28.7}\] The bias factor is the ratio of the unexposed-risk function averaged over the two confounder distributions.

Reduction to a box. Both \(B\) and the constraints are invariant to multiplying \(p_0(\cdot)\) by a positive constant, so normalize \(\min_u p_0(u)=1\); then \(p_0(u)\in[1,R]\) for all \(u\), where \(R:=\mathrm{RR}_{UD}\). Write \(S:=\mathrm{RR}_{EU}\), so the constraint \(\max_u f_1(u)/f_0(u)\le S\) reads \(f_1(u)\le S f_0(u)\) for every \(u\). We maximize Equation 28.7 over \(p_0\in[1,R]^{\mathcal{U}}\) and over probability vectors \(f_1,f_0\) obeying this pointwise constraint.

Maximizing over \(p_0\): the optimum is at a vertex. Fix \(f_1,f_0\) and vary a single coordinate \(t=p_0(u)\in[1,R]\), holding the others fixed. Writing \(a,b>0\) for the partial sums over the other coordinates, \(B(t)=\big(a+f_1(u)\,t\big)\big/\big(b+f_0(u)\,t\big)\), whose derivative \(B'(t)=\big(f_1(u)\,b-f_0(u)\,a\big)\big/\big(b+f_0(u)\,t\big)^2\) has a sign independent of \(t\): a ratio of two affine functions of \(t\) is monotone across the whole interval, since the numerator of \(B'(t)\) never changes sign. A function monotone in each coordinate separately attains its maximum over the box \([1,R]^{\mathcal{U}}\) at a vertex, because moving any single coordinate toward its better endpoint never lowers \(B\), so no interior point can be optimal; hence the optimum has every \(p_0(u)\in\{1,R\}\), and coordinate ascent reaches it.

The two-cell reduction. At such a vertex, partition \(\mathcal{U}=\mathcal{U}_R\sqcup\mathcal{U}_1\) with \(p_0=R\) on \(\mathcal{U}_R\) and \(p_0=1\) on \(\mathcal{U}_1\). Put \(a=\sum_{u\in\mathcal{U}_R}f_1(u)\) and \(b=\sum_{u\in\mathcal{U}_R}f_0(u)\), the masses the two confounder distributions place on the high-risk set. Then \(\sum_u p_0(u)f_1(u)=R\,a+(1-a)=1+(R-1)a\) and likewise the denominator is \(1+(R-1)b\), so \[ B=\frac{1+(R-1)a}{1+(R-1)b}. \tag{28.8}\] This is increasing in \(a\) and decreasing in \(b\). The pointwise constraint gives \(a=\sum_{\mathcal{U}_R}f_1\le S\sum_{\mathcal{U}_R}f_0=S\,b\), and of course \(a\le 1\); so for fixed \(b\) the largest feasible \(a\) is \(\min(1,S b)\).

Optimizing the two cells. If \(Sb\le 1\), set \(a=Sb\); then \(B=\big(1+(R-1)Sb\big)\big/\big(1+(R-1)b\big)=:g(b)\), and the numerator of \(g'(b)\) is \[ (R-1)S\big(1+(R-1)b\big)-\big(1+(R-1)Sb\big)(R-1) =(R-1)\big[S+S(R-1)b-1-(R-1)Sb\big]=(R-1)(S-1)\ge 0, \] where the two middle terms \(S(R-1)b\) and \((R-1)Sb\) cancel; so \(g\) increases and we push \(b\) up to the regime boundary \(Sb=1\), that is \(b=1/S\) with \(a=1\). If instead \(Sb>1\), then \(a=1\) and \(B=R\big/\big(1+(R-1)b\big)\) decreases in \(b\), so we push \(b\) down to the same boundary \(b=1/S\). Either way the maximum is at \(a=1\), \(b=1/S\), giving \[ B^{\star}=\frac{1+(R-1)}{1+(R-1)/S}=\frac{R}{(S+R-1)/S}=\frac{RS}{R+S-1} =\frac{\mathrm{RR}_{EU}\,\mathrm{RR}_{UD}}{\mathrm{RR}_{EU}+\mathrm{RR}_{UD}-1}=\mathrm{BF}_U, \] which is Equation 28.6.

Sharpness. The optimizing configuration is realizable with a two-point confounder \(\mathcal{U}=\{0,1\}\): take \(p_0(0)=1\), \(p_0(1)=R\), put all of the exposed mass on the high-risk level, \(f_1(1)=1\), and split the unexposed mass as \(f_0(1)=1/S\), \(f_0(0)=1-1/S\), which satisfies \(f_1(1)/f_0(1)=S\) and \(f_1(0)/f_0(0)=0\le S\). Substituting into Equation 28.7 returns \(B=\mathrm{BF}_U\) exactly, so no smaller bound is possible. \(\square\)

Remark. Why sharpness, not just validity, matters. Many sensitivity bounds are merely valid: the true bias is no larger than the bound. Sharpness says more, that the bound is attained by an actual confounder configuration, so it cannot be improved. This is what lets the E-value be an exact threshold rather than a conservative guess. A valid but loose bound would overstate how much confounding is needed and so overstate robustness; a bound that is attained means “an association of exactly the stated size on both legs is enough, and nothing weaker is”, which is the precise claim a reader can act on. \(\blacktriangleleft\)

The bound depends on the data only through the two sensitivity parameters, and it is symmetric in them: \(\mathrm{BF}_U\) is unchanged under \(\mathrm{RR}_{EU}\leftrightarrow\mathrm{RR}_{UD}\). The E-value is the point on the curve \(\mathrm{BF}_U=\mathrm{RR}_{\mathrm{obs}}\) where the two parameters are equal.

Theorem 28.1 (The E-value and its minimality) Let \(\mathrm{RR}_{\mathrm{obs}}>1\) be an observed risk ratio (for a protective effect \(\mathrm{RR}_{\mathrm{obs}}<1\), replace it by \(1/\mathrm{RR}_{\mathrm{obs}}\) throughout). Define the E-value \[ E\;=\;\mathrm{RR}_{\mathrm{obs}}+\sqrt{\mathrm{RR}_{\mathrm{obs}}\,(\mathrm{RR}_{\mathrm{obs}}-1)}. \tag{28.9}\] Then:

1. Unmeasured confounding can fully explain the observed association away (reduce the true risk ratio to the null) only if the two sensitivity parameters satisfy \(\mathrm{BF}_U=\mathrm{RR}_{EU}\mathrm{RR}_{UD}/(\mathrm{RR}_{EU}+\mathrm{RR}_{UD}-1)\ge\mathrm{RR}_{\mathrm{obs}}\).

2. Setting \(\mathrm{RR}_{EU}=\mathrm{RR}_{UD}=E\) achieves equality \(\mathrm{BF}_U=\mathrm{RR}_{\mathrm{obs}}\), and \(E\) is the smallest common value with this property.

3. \(E\) is minimal in the strong sense that if both sensitivity parameters are strictly below \(E\) then \(\mathrm{BF}_U<\mathrm{RR}_{\mathrm{obs}}\), so the result cannot be explained away: at least one of \(\mathrm{RR}_{EU},\mathrm{RR}_{UD}\) must be at least \(E\).

Moreover \(E\ge 1\), with \(E=1\) iff \(\mathrm{RR}_{\mathrm{obs}}=1\), and \(E\) is strictly increasing in \(\mathrm{RR}_{\mathrm{obs}}\), so a larger observed effect is harder to explain away.

Proof. Part 1. Write \(\mathrm{RR}_{\mathrm{obs}}=\mathrm{RR}_{\mathrm{true}}\cdot B\) with \(B\) the confounding bias factor. “Explaining away” means \(\mathrm{RR}_{\mathrm{true}}\le 1\), i.e. \(B\ge\mathrm{RR}_{\mathrm{obs}}\). By Lemma 28.2 the largest bias factor attainable with sensitivity parameters \((\mathrm{RR}_{EU},\mathrm{RR}_{UD})\) is \(\mathrm{BF}_U\), so such a \(B\) exists only if \(\mathrm{BF}_U\ge\mathrm{RR}_{\mathrm{obs}}\).

Part 2. On the diagonal \(\mathrm{RR}_{EU}=\mathrm{RR}_{UD}=v\ge 1\), \(\mathrm{BF}_U(v,v)=v^2/(2v-1)\). Setting this equal to \(\mathrm{RR}_{\mathrm{obs}}=:\rho\) gives \(v^2=\rho(2v-1)\), i.e. the quadratic \(v^2-2\rho v+\rho=0\), whose roots are \(v=\rho\pm\sqrt{\rho^2-\rho}=\rho\pm\sqrt{\rho(\rho-1)}\). The smaller root is below \(1\) for \(\rho>1\) (at \(\rho=2\) it equals \(2-\sqrt2\approx0.586\)), hence infeasible: \(v\) is a common value of the two sensitivity parameters, which are risk ratios of associations and so are at least \(1\) by construction (Equation 28.5), forcing \(v\ge1\); the admissible root is the larger one, \(E=\rho+\sqrt{\rho(\rho-1)}\), which is Equation 28.9. To see \(E\) is the smallest common value giving \(\mathrm{BF}_U\ge\rho\), note \(\psi(v):=v^2/(2v-1)\) has derivative \(\psi'(v)=\big(2v(2v-1)-2v^2\big)/(2v-1)^2=2v(v-1)/(2v-1)^2\ge 0\) for \(v\ge1\), so \(\psi\) is increasing; thus \(\mathrm{BF}_U(v,v)=\psi(v)\ge\rho\) holds exactly for \(v\ge E\).

Part 3. The bias factor is increasing in each argument: with \(x=\mathrm{RR}_{EU}\), \(y=\mathrm{RR}_{UD}\), \[ \frac{\partial}{\partial x}\frac{xy}{x+y-1}=\frac{y(x+y-1)-xy}{(x+y-1)^2}=\frac{y(y-1)}{(x+y-1)^2}\ge0, \] and symmetrically in \(y\). Hence if \(x<E\) and \(y<E\) then \(\mathrm{BF}_U(x,y)<\mathrm{BF}_U(E,E)=\rho=\mathrm{RR}_{\mathrm{obs}}\), contradicting Part 1; so at least one parameter is \(\ge E\). Finally \(E(\rho)=\rho+\sqrt{\rho(\rho-1)}\) has \(E(1)=1\) and \(E'(\rho)=1+\frac{2\rho-1}{2\sqrt{\rho(\rho-1)}}>0\) for \(\rho>1\), so \(E\ge1\) with equality iff \(\rho=1\), and \(E\) strictly increases in \(\rho\). \(\square\)

Remark. Why this matters, and a feel for the formula. The E-value converts an estimate into a robustness score on the same scale a clinician already uses for risk ratios, which is why it travels well into an HTA report and why it now appears routinely alongside NMA and ITC estimates. A quick numerical anchor: for an observed \(\mathrm{RR}_{\mathrm{obs}}=2\), \(E=2+\sqrt{2\cdot 1}=2+\sqrt2\approx 3.41\), so confounding would need a risk-ratio strength above \(3.4\) on both legs to erase a doubling of risk. The shape \(E=\rho+\sqrt{\rho(\rho-1)}\) repays reading: near the null (\(\rho\to 1^{+}\)) it behaves like \(1+\sqrt{\rho-1}\), rising as the square root of the excess, so a barely significant estimate has an E-value only a little above \(1\) and is easy to explain away; for large \(\rho\) it grows like \(2\rho\), so strong estimates demand proportionally strong confounding. For our COPD running example the plug-in effect is \(\mathrm{RR}=0.50\), giving \(E\approx 3.44\) (computed in Section 28.8); an unreported prognostic difference weaker than a \(3.4\)-fold association on each leg cannot overturn it. \(\blacktriangleleft\)

Remark. Reading and reporting the E-value. The E-value translates a point estimate into a single interpretable number: an unmeasured confounder (or, in the ITC setting below, an unmeasured effect modifier) would have to be associated with both the treatment assignment and the outcome by a risk ratio of at least \(E\), beyond the measured covariates, to explain the observed effect away; anything weaker cannot. A large \(E\) signals a robust finding, a small one a fragile finding. One also reports the E-value of the confidence-interval limit nearest the null, computed by Equation 28.9 with \(\mathrm{RR}_{\mathrm{obs}}\) replaced by that limit (and equal to \(1\) when the interval already includes the null): it is the strength of confounding needed to shift the interval to touch the null. The E-value is a property of the estimate alone, requiring no specification of the bias parameters; that economy is its appeal and also its limitation, since it is silent on whether confounding of strength \(E\) is plausible. The general bound was established by Ding and VanderWeele and popularized as the E-value by VanderWeele and Ding (2017); the same \(\mathrm{BF}_U\) bounds the bias for total-population standardization, not only the effect-on-treated standardization we proved. \(\blacktriangleleft\)

Remark. Mind the scale: the E-value lives on the risk-ratio scale. Formula Equation 28.9 takes a risk ratio. Feeding it a log odds ratio, or an odds ratio for a common outcome, is a common and consequential misuse. Convert first. The practical rules, with the approximate conversions of VanderWeele and Ding (2017), are collected here.

Reported effect	Convert to a risk ratio by	Then
risk ratio	nothing, it already is one	apply Equation 28.9
odds ratio, rare outcome	\(\mathrm{RR}\approx\mathrm{OR}\)	apply Equation 28.9 to the OR
odds ratio, common outcome	\(\mathrm{RR}\approx\sqrt{\mathrm{OR}}\)	apply Equation 28.9 to \(\sqrt{\mathrm{OR}}\)
hazard ratio, rare outcome	\(\mathrm{RR}\approx\mathrm{HR}\)	apply Equation 28.9 to the HR
hazard ratio, common outcome	approximate conversion of VanderWeele and Ding (2017)	apply Equation 28.9 to the converted value
risk difference, mean difference, RMST difference	no ratio-scale E-value exists	use the tipping point (Section 28.6) instead

For a protective effect (\(\mathrm{RR}<1\)) convert to \(1/\mathrm{RR}\) first, as in the theorem. The running example works on the log-risk-ratio scale, so no conversion is needed there; a survival endpoint reported as a hazard ratio would use one of the hazard-ratio rows. \(\blacktriangleleft\)

28.4 The indirect-comparison E-value

In a classical observational study both sensitivity parameters in Equation 28.5 are unknown, and the symmetric E-value is the natural summary. An unanchored indirect comparison is different in a way that can be exploited. The covariate that threatens the comparison, an effect modifier (Definition 11.3) that is imbalanced between the IPD and comparator populations and is unreported by the comparator, is nonetheless observed in the analyst’s own IPD. Its association with the outcome is therefore estimable from the IPD outcome model, even though its distribution in the comparator population is not. One of the two arms of the bias factor becomes known data rather than an assumed sensitivity parameter, and the E-value sharpens accordingly.

The map from the classical confounding picture to the indirect-comparison picture is worth pinning down, because every symbol in the E-value derivation keeps its algebra but changes its clinical referent.

Classical confounding (Section 28.3)	Unanchored ITC (this section)
exposure \(A\in\{0,1\}\)	trial-population membership: IPD population versus comparator population
the exposed arm	the IPD (index-treatment) population
the unexposed arm	the comparator population
unmeasured confounder \(U\)	the covariate the comparator did not report
\(\mathrm{RR}_{EU}\), imbalance of \(U\) across exposure	imbalance of the covariate across the two trial populations
\(\mathrm{RR}_{UD}\), association of \(U\) with the outcome	covariate-outcome association, estimable from the IPD model
both arms of the bias factor unknown	one arm (\(\mathrm{RR}_{UD}\)) is known data; only the imbalance is assumed

The last row is the whole point: half of the sensitivity problem is estimated rather than guessed.

We set the stage with the unanchored estimand of Chapter 17. The IPD study fits an outcome model \(g\big(\mathbb{E}[Y\mid \mathbf{x},u]\big)=\beta_0+\boldsymbol{\beta}_R^{\top}\mathbf{x}+\beta_U u\) on the reported covariates \(\mathbf{x}\) and the unreported covariate \(U\). The comparator-population effect of the index treatment against the comparator, on the analysis scale, is the g-computation contrast \[ \Delta=g\!\left(\int g^{-1}\!\big(\beta_0+\boldsymbol{\beta}_R^{\top}\mathbf{x}+\beta_U u\big)\, dF_0(\mathbf{x},u)\right)-g(\mu_0), \tag{28.10}\] where \(F_0\) is the comparator covariate distribution and \(\mu_0\) its reported outcome under the comparator treatment. In words, reading Equation 28.10 from the inside out: take the IPD outcome model, predict each comparator patient’s risk as if they had received the index treatment from their covariates \((\mathbf{x},u)\), average those predicted risks over the comparator population \(F_0\), transform that average back to the analysis scale with \(g\), and subtract the comparator’s own reported outcome \(g(\mu_0)\). The result is what the index-versus-comparator contrast would be in the comparator population. The bias from omitting \(U\) is the gap between \(\Delta\) computed with \(U\) held at the analyst’s best guess (its IPD value) and \(\Delta\) computed at \(U\)’s true, unknown comparator value. On the risk-ratio scale this bias is a bias factor, and the question of the E-value is how large the across-arm imbalance of \(U\) must be to produce it, given that \(U\)’s outcome association is pinned down by the IPD.

Two things need saying before the definition. First, what “\(\mathrm{BF}_{\mathrm{obs}}\)” means: it is not the treatment effect but the amount of bias that would have to be present to nullify it, read on the ratio scale. If the adjusted risk ratio is \(0.50\), explaining it away means multiplying it back up to \(1\), a factor of \(2\); so \(\mathrm{BF}_{\mathrm{obs}}=1/0.50=2.0\), and in general \(\mathrm{BF}_{\mathrm{obs}}=\exp(|\Delta_{\mathrm{IPD}}|)\), where \(\Delta_{\mathrm{IPD}}\) is the log-scale adjusted effect with the unreported covariate held at its IPD value. “Held at its IPD value” means the comparator population is given the covariate’s IPD distribution as a best guess, so \(\Delta_{\mathrm{IPD}}\) is the plug-in adjusted effect, not the covariate’s raw mean. Second, why the formula has three cases: with the outcome association fixed at \(\mathrm{RR}_{UD}\), the bias factor can never exceed \(\mathrm{RR}_{UD}\) no matter how extreme the imbalance (proved as Proposition 28.1 below), so the equation \(\mathrm{BF}_U=\mathrm{BF}_{\mathrm{obs}}\) has a finite solution only when \(\mathrm{BF}_{\mathrm{obs}}<\mathrm{RR}_{UD}\); that ceiling is what splits the definition into its three branches.

Definition 28.2 (The indirect-comparison E-value) Consider an unanchored indirect comparison on a ratio scale (binary or count outcome). Let \(\mathrm{BF}_{\mathrm{obs}}\ge 1\) be the observed bias factor, the ratio-scale magnitude \(\exp(|\Delta_{\mathrm{IPD}}|)\) of the adjusted effect Equation 28.10 evaluated with the unreported covariate held at its IPD value, and let \(\mathrm{RR}_{UD}=\exp(|\beta_U|)\ge 1\) be the IPD-estimated association of the unreported covariate \(U\) with the outcome, conditional on the reported covariates. The indirect-comparison E-value is the minimum across-arm association \(\mathrm{RR}_{EU}\) that the unreported covariate must have, given \(\mathrm{RR}_{UD}\), to produce a bias factor as large as \(\mathrm{BF}_{\mathrm{obs}}\): \[ E_{\mathrm{ITC}}= \begin{cases} 1, & \mathrm{BF}_{\mathrm{obs}}\le 1,\\[2pt] \dfrac{\mathrm{BF}_{\mathrm{obs}}\,(\mathrm{RR}_{UD}-1)}{\mathrm{RR}_{UD}-\mathrm{BF}_{\mathrm{obs}}}, & 1<\mathrm{BF}_{\mathrm{obs}}<\mathrm{RR}_{UD},\\[8pt] +\infty, & \mathrm{RR}_{UD}\le\mathrm{BF}_{\mathrm{obs}}. \end{cases} \tag{28.11}\] \(\blacktriangleleft\)

Remark. Reading the three cases. The indirect-comparison E-value answers one question, “how imbalanced would the unreported covariate have to be, across the two trial populations, to overturn the result, given how strongly the IPD says it acts on the outcome?”, and its three cases are three honest answers. If \(\mathrm{BF}_{\mathrm{obs}}\le 1\) the estimate already sits at or past the null and no bias is needed, so \(E_{\mathrm{ITC}}=1\). If \(1<\mathrm{BF}_{\mathrm{obs}}<\mathrm{RR}_{UD}\) a finite imbalance suffices, and the formula returns exactly how much. If \(\mathrm{RR}_{UD}\le\mathrm{BF}_{\mathrm{obs}}\) the covariate is too weakly prognostic to matter: no imbalance, however extreme, can manufacture enough bias, and \(E_{\mathrm{ITC}}=+\infty\) certifies the result robust to this covariate. The word “infinite” here does not mean “no bias exists”; it means “no across-arm imbalance in this one covariate can explain the result away, under the single-covariate bias-factor model, with \(\mathrm{RR}_{UD}\) treated as fixed.” That narrower claim is exactly what a dossier can defend. For the running example the IPD gives \(\mathrm{RR}_{UD}=1.65\), below the needed \(\mathrm{BF}_{\mathrm{obs}}=2.01\), so we land in the third case and \(E_{\mathrm{ITC}}=+\infty\) (computed in Section 28.8). \(\blacktriangleleft\)

The three cases, and in particular the infinite case that the symmetric E-value can never deliver, follow from the same sharp bound.

Proposition 28.1 (Derivation and the robustness certificate) With \(\mathrm{BF}_{\mathrm{obs}}\) and \(\mathrm{RR}_{UD}\) as in Definition 28.2:

1. If \(1<\mathrm{BF}_{\mathrm{obs}}<\mathrm{RR}_{UD}\), the value Equation 28.11 is the unique \(\mathrm{RR}_{EU}\) solving \(\mathrm{BF}_U(\mathrm{RR}_{EU},\mathrm{RR}_{UD})=\mathrm{BF}_{\mathrm{obs}}\), and it exceeds \(1\).

2. The maximum bias factor achievable by any finite across-arm imbalance, with the outcome association fixed at \(\mathrm{RR}_{UD}\), is \(\mathrm{RR}_{UD}\) itself: \(\lim_{\mathrm{RR}_{EU}\to\infty}\mathrm{BF}_U(\mathrm{RR}_{EU},\mathrm{RR}_{UD})=\mathrm{RR}_{UD}\), the limit being approached from below and monotonically. Consequently, if \(\mathrm{RR}_{UD}\le\mathrm{BF}_{\mathrm{obs}}\), no across-arm imbalance can overturn the result under the single-covariate bias model, and \(E_{\mathrm{ITC}}=+\infty\) is a provable robustness certificate.

3. When the IPD-estimated outcome association happens to equal the symmetric E-value, \(\mathrm{RR}_{UD}=E\) with \(E\) from Equation 28.9 applied to \(\mathrm{BF}_{\mathrm{obs}}\), the indirect-comparison E-value coincides with it, \(E_{\mathrm{ITC}}=E\).

Proof. Part 1. Solve \(\mathrm{BF}_U(x,y)=\dfrac{xy}{x+y-1}=\mathrm{BF}_{\mathrm{obs}}\) for \(x=\mathrm{RR}_{EU}\) with \(y=\mathrm{RR}_{UD}\) fixed. Cross-multiplying, \(xy=\mathrm{BF}_{\mathrm{obs}}(x+y-1)\), hence \(x\,(y-\mathrm{BF}_{\mathrm{obs}})=\mathrm{BF}_{\mathrm{obs}}(y-1)\) and \(x=\mathrm{BF}_{\mathrm{obs}}(y-1)/(y-\mathrm{BF}_{\mathrm{obs}})\), which is Equation 28.11. When \(1<\mathrm{BF}_{\mathrm{obs}}<y\) the denominator is positive and \(x>0\); moreover \(x-1=\big(\mathrm{BF}_{\mathrm{obs}}(y-1)-(y-\mathrm{BF}_{\mathrm{obs}})\big)/(y-\mathrm{BF}_{\mathrm{obs}}) =(y-1)(\mathrm{BF}_{\mathrm{obs}}-1)/(y-\mathrm{BF}_{\mathrm{obs}})>0\), so \(x>1\). Uniqueness is immediate since \(\mathrm{BF}_U(\cdot,y)\) is strictly increasing in its first argument (Part 3 of 2).

Part 2. For fixed \(y>1\), \(\mathrm{BF}_U(x,y)=\dfrac{xy}{x+y-1}=\dfrac{y}{1+(y-1)/x}\uparrow y\) as \(x\to\infty\), monotonically since \((y-1)/x\) decreases. Thus the achievable bias factor is bounded above by \(\mathrm{RR}_{UD}\). If \(\mathrm{RR}_{UD}\le\mathrm{BF}_{\mathrm{obs}}\) the target is unreachable for every finite imbalance, so the minimal sufficient \(\mathrm{RR}_{EU}\) is \(+\infty\) by convention, and the result cannot be explained away by imbalance in \(U\) alone no matter how extreme; this is the certificate. (Algebraically, the formula Equation 28.11 has a nonpositive denominator here, confirming no finite solution.)

Part 3. If \(y=\mathrm{RR}_{UD}=E\) then, by Part 2 of 2, \(\mathrm{BF}_U(E,E)=\mathrm{BF}_{\mathrm{obs}}\), so \(x=E\) already solves the equation; by uniqueness \(E_{\mathrm{ITC}}=E\). \(\square\)

Corollary 28.1 (The indirect-comparison E-value against the symmetric one) Fix an observed bias factor \(\mathrm{BF}_{\mathrm{obs}}=\rho>1\) and regard \(E_{\mathrm{ITC}}\) as a function of the IPD-estimated outcome association \(\mathrm{RR}_{UD}=y\). On the finite range \(y>\rho\) the map \(y\mapsto E_{\mathrm{ITC}}(y)=\rho(y-1)/(y-\rho)\) is strictly decreasing, with \(E_{\mathrm{ITC}}(y)\to+\infty\) as \(y\downarrow\rho\) and \(E_{\mathrm{ITC}}(y)\to\rho\) as \(y\to\infty\). It crosses the symmetric E-value \(E=\rho+\sqrt{\rho(\rho-1)}\) exactly at \(y=E\): for \(\rho<y<E\) it exceeds \(E\), and for \(y>E\) it is smaller than \(E\).

Proof. Differentiating, \[ \frac{d}{dy}\,\frac{\rho(y-1)}{y-\rho} =\rho\,\frac{(y-\rho)-(y-1)}{(y-\rho)^2} =\frac{\rho(1-\rho)}{(y-\rho)^2}<0 \quad\text{for }\rho>1, \] so \(E_{\mathrm{ITC}}\) strictly decreases on \(y>\rho\). The two limits are immediate from the rational form: as \(y\downarrow\rho\) the denominator \(\to 0^{+}\) with positive numerator, and as \(y\to\infty\) the ratio \(\rho(y-1)/(y-\rho)\to\rho\). By Proposition 28.1(3), \(E_{\mathrm{ITC}}(E)=E\); strict monotonicity then forces \(E_{\mathrm{ITC}}(y)>E\) for \(\rho<y<E\) and \(E_{\mathrm{ITC}}(y)<E\) for \(y>E\). \(\square\)

Remark. What “sharper” does and does not mean. It is tempting to say the indirect-comparison E-value is “always larger because it uses more information”; Corollary 28.1 shows that is false. The ITC E-value is better calibrated, not uniformly larger: by plugging in the known \(\mathrm{RR}_{UD}\) it can land above or below the symmetric E-value. A weakly prognostic unreported covariate (\(y<E\), and especially \(y\le\rho\)) is harder or impossible to weaponize, so more imbalance would be required and \(E_{\mathrm{ITC}}>E\); a strongly prognostic one (\(y>E\)) needs less imbalance to do damage, so \(E_{\mathrm{ITC}}<E\). Using the data can cut either way, and that honest calibration, not a guaranteed increase, is the gain. \(\blacktriangleleft\)

Remark. Why this is the honest summary for an unmeasured effect modifier. The infinite case has no analogue in the classical E-value, where both arms are unknown and the bias factor can be pushed arbitrarily high by sending both to infinity. Here one arm is data. If the IPD says \(U\) is only weakly prognostic (\(\mathrm{RR}_{UD}\) small), then even a wildly imbalanced \(U\) cannot manufacture a large bias, and the unanchored comparison is robust to it whatever the comparator distribution turns out to be. This is exactly the residual-bias question raised by Theorem 12.6, now answered quantitatively for one covariate at a time. Three caveats must travel with the number, all visible in the derivation: it treats the IPD-estimated \(\mathrm{RR}_{UD}=\exp(|\beta_U|)\) as fixed, propagating none of that estimate’s sampling uncertainty; it is a single-covariate bias-factor argument, so several jointly imbalanced covariates require the multivariate analysis of 1; and it is a ratio-scale quantity, defined for binary and count outcomes, not for continuous or time-to-event effects, for which the tipping-point analysis of Section 28.6 is the appropriate instrument. \(\blacktriangleleft\)

28.5 NORTA: imputing the unreported covariate distribution

Before any formulas, the practical problem NORTA solves. The comparator publication gives you a few margins: the distribution of each covariate on its own, usually just a mean and a standard deviation (age is \(62\pm 9\) years, say) or a prevalence (\(30\%\) are current smokers). A margin says nothing about how two covariates move together in that population, their dependence (do older patients tend to have worse lung function?). To recompute the adjusted effect you need the full joint distribution of all covariates in the comparator population, and you are missing two things: the margin of the unreported covariate, which you must assume, and all of the dependence, which the comparator never reports. NORTA fills both gaps in a disciplined way. You supply the assumed margins, and you borrow the dependence from the one place it is observed, your own IPD. The name is literal, normal to anything: start from correlated normal draws, which carry the borrowed dependence, then reshape each coordinate into the margin you want. The rest of this section makes “reshape” and “borrow the dependence” precise and flags that borrowing as the method’s key assumption.

The E-value summarizes the threat of a single unmeasured covariate without ever simulating it. To go further, to actually recompute the adjusted effect Equation 28.10 under an assumption about the unreported covariate, we must construct a comparator-population covariate distribution that (i) honors every reported marginal summary, (ii) places a chosen marginal on the unreported covariate, and (iii) carries a credible dependence structure linking them. Aggregate comparator data fix the margins but say nothing about the dependence (Sklar’s theorem, Theorem 22.5, makes this precise: the margins and the copula are separately specifiable). The only place the dependence is observed is the IPD. The NORTA (normal-to-anything) construction supplies exactly this: it glues the target margins to a Gaussian copula (Definition 22.5) whose correlation is borrowed from the IPD.

Theorem 28.2 (The NORTA construction) Let \(F_1,\dots,F_d\) be target univariate distribution functions, with generalized inverses \(F_j^{-1}\), and let \(\boldsymbol{\Omega}\succeq 0\) be a correlation matrix (symmetric, unit diagonal). Draw \(\mathbf{Z}=(Z_1,\dots,Z_d)\sim\mathcal{N}_d(\mathbf{0},\boldsymbol{\Omega})\) and set \[ X_j\;=\;F_j^{-1}\!\big(\Phi(Z_j)\big),\qquad j=1,\dots,d, \tag{28.12}\] where \(\Phi\) is the standard normal distribution function. Then:

1. (Exact margins.) For every target \(F_j\), discrete or continuous, \(X_j\sim F_j\).

2. (Induced copula.) If in addition every \(F_j\) is continuous, the joint distribution function of \(\mathbf{X}=(X_1,\dots,X_d)\) is \[ F_{\mathbf{X}}(\mathbf{x})=C_{\boldsymbol{\Omega}}\big(F_1(x_1),\dots,F_d(x_d)\big), \tag{28.13}\] so \(\mathbf{X}\) has margins \(F_1,\dots,F_d\) and Gaussian copula \(C_{\boldsymbol{\Omega}}\); by uniqueness in Sklar’s theorem \(C_{\boldsymbol{\Omega}}\) is the unique copula of \(\mathbf{X}\).

Proof. Part 1. Since \(\boldsymbol{\Omega}\) has unit diagonal, each \(Z_j\sim\mathcal{N}(0,1)\), and \(\Phi\) is continuous, so by the probability integral transform (Theorem 22.4, part 2) the variable \(U_j:=\Phi(Z_j)\) is uniform on \((0,1)\). By the inverse-transform property (Theorem 22.4, part 1), \(F_j^{-1}(U_j)\sim F_j\) for any distribution function \(F_j\). Hence \(X_j=F_j^{-1}(\Phi(Z_j))\sim F_j\), with no continuity required.

Part 2. Let \(U_j=\Phi(Z_j)\) as above; by Proposition 22.1 the vector \((U_1,\dots,U_d)\) has joint distribution function exactly the Gaussian copula \(C_{\boldsymbol{\Omega}}\). The defining equivalence of a generalized inverse, \(\{F_j^{-1}(u)\le x\}=\{u\le F_j(x)\}\) (the Galois inequality Equation 22.12, used in Theorem 22.4), gives for each coordinate \(\{X_j\le x_j\}=\{F_j^{-1}(U_j)\le x_j\}=\{U_j\le F_j(x_j)\}\). Intersecting over \(j\), \[ F_{\mathbf{X}}(\mathbf{x})=P\Big(\bigcap_{j=1}^{d}\{X_j\le x_j\}\Big) =P\Big(\bigcap_{j=1}^{d}\{U_j\le F_j(x_j)\}\Big) =C_{\boldsymbol{\Omega}}\big(F_1(x_1),\dots,F_d(x_d)\big), \] which is Equation 28.13. With continuous margins, Sklar’s theorem (Theorem 22.5) says the copula in the factorization \(F_{\mathbf{X}}=C(F_1,\dots,F_d)\) is unique, and we have exhibited it as \(C_{\boldsymbol{\Omega}}\). \(\square\)

Remark. Reader’s translation. Theorem 28.2 is the guarantee that the recipe does what the introduction promised. Part 1 says the simulated covariates land on exactly the margins you asked for (the assumed prevalence or mean is honored, discrete or continuous). Part 2 says the dependence you wired in through \(\boldsymbol{\Omega}\) comes out as exactly a Gaussian copula, with nothing sneaking in from the marginal reshaping. So you may choose margins and dependence separately and trust that both emerge as specified, which is what makes NORTA a reliable comparator-population generator. \(\blacktriangleleft\)

The construction is exact for the margins but leaves one question open: how to choose \(\boldsymbol{\Omega}\) so that the simulated population reproduces the dependence seen in the IPD. The natural target is the rank correlation, because it is what survives the marginal transforms and because it is invariant to the choice of margins.

Lemma 28.3 (Spearman matching for the Gaussian copula) Let \((X_j,X_k)\) be constructed by Equation 28.12 with continuous margins and Gaussian-copula correlation \(\Omega_{jk}=\rho^{N}\). Then:

1. Spearman’s rank correlation \(\rho^{S}(X_j,X_k)\) depends only on the copula \(C_{\boldsymbol{\Omega}}\), not on the margins \(F_j,F_k\); it equals \(\rho^{S}(Z_j,Z_k)\).

2. For the bivariate Gaussian copula, \(\rho^{S}=\frac{6}{\pi}\arcsin(\rho^{N}/2)\). Hence choosing \[ \rho^{N}=2\sin\!\big(\pi\,\rho^{S}_{\mathrm{IPD}}/6\big) \tag{28.14}\] makes the simulated Spearman correlation equal the IPD Spearman correlation \(\rho^{S}_{\mathrm{IPD}}\) exactly.

Proof. Part 1. Spearman’s rho is the Pearson correlation of the probability-integral transforms, \(\rho^{S}(X_j,X_k)=\operatorname{corr}\!\big(F_j(X_j),F_k(X_k)\big)\). For continuous margins, \(F_j(X_j)=F_j\big(F_j^{-1}(\Phi(Z_j))\big)=\Phi(Z_j)\), since \(F_j\circ F_j^{-1}\) is the identity on the range of a continuous \(F_j\). The map \(\Phi\) is strictly increasing, so \(\operatorname{corr}(\Phi(Z_j),\Phi(Z_k))\) is a functional of the joint law of \((Z_j,Z_k)\) alone, hence of the copula; it does not involve \(F_j,F_k\). This proves the invariance and the identity \(\rho^{S}(X_j,X_k)=\rho^{S}(Z_j,Z_k)\).

Part 2. The identity \(\rho^{S}=\frac{6}{\pi}\arcsin(\rho^{N}/2)\) for a bivariate normal with correlation \(\rho^{N}\) is the classical “arcsine law” for the grade correlation of the Gaussian copula; we take it as a known property of the bivariate normal (see the Notes). Inverting, \(\rho^{N}=2\sin(\pi\rho^{S}/6)\), which is Equation 28.14; feeding the IPD value \(\rho^{S}_{\mathrm{IPD}}\) on the right reproduces it as the simulated Spearman correlation by Part 1. \(\square\)

Remark. Why rank correlation, and the discrete and definiteness caveats. The Pearson correlation of \((X_j,X_k)\) produced by Equation 28.12 is in general not \(\Omega_{jk}\): the nonlinear marginal transforms \(F_j^{-1}\circ\Phi\) distort linear correlation, and the distortion depends on the margins. Rank correlation is the right invariant precisely because, by Lemma 28.3, it passes through the transforms untouched, which is why the implementation borrows the IPD’s Spearman matrix and maps it by Equation 28.14. Two practical points. First, for a binary covariate (the unreported \(U\) is often binary) the margin is discrete, so Lemma 28.3 is approximate: the transform \(F_j(X_j)\) is no longer exactly uniform (Theorem 22.4, part 2 fails for discrete \(F\)), and the induced copula in Theorem 28.2 is one of the many copulas Sklar’s theorem allows rather than a unique one. The marginal-recovery guarantee (part 1) still holds exactly. Second, the element-wise transform Equation 28.14 applied to an empirical Spearman matrix need not return a positive semidefinite matrix, so before constructing the copula one projects it onto the nearest correlation matrix (symmetrize, floor the eigenvalues at a small positive value, and rescale to unit diagonal). Both adjustments are benign and standard. \(\blacktriangleleft\)

Remark. What actually changes when \(U\) is binary, and how much to trust the borrowed correlation. Two applied points, since the unreported covariate is often a binary flag (a biomarker-positive indicator, a smoking status). First, the discrete caveat above is narrow in its consequences. Marginal recovery stays exact: if you assume a comparator prevalence of \(30\%\), the simulated population has prevalence \(30\%\), full stop. What becomes approximate is only the dependence: the Spearman match Equation 28.14 no longer lands the simulated rank correlation exactly on the IPD value, and the copula is no longer unique. In practice this perturbs the induced covariate correlations slightly, and hence the adjusted effect slightly, but it never distorts the assumed prevalence, which is usually the very quantity the sensitivity analysis is sweeping. Second, borrowing the IPD dependence is a substantive assumption, not a technical convenience: it asserts that age and lung function (say) hang together in the comparator population the way they do in your trial. When that is doubtful, the honest move is to redo the analysis under a few alternative correlation matrices (including independence and a strong correlation of each sign) and report the range. If the conclusion is stable across them the borrowing did no harm; if it is not, the dependence itself has become a bias parameter deserving its own sensitivity sweep. \(\blacktriangleleft\)

With the comparator covariate population in hand, the deterministic ITC bias analysis is a sweep: choose a grid of assumed marginals for the unreported covariate, and for each one simulate the comparator population by Equation 28.12, marginalize the IPD outcome model over it, and record the adjusted effect Equation 28.10. The resulting curve is the input to the tipping-point and probabilistic analyses.

Example 28.2 (NORTA with a borrowed dependence, in closed form) The worked example of Section 28.8 takes the comparator covariates independent, which makes NORTA invisible; here we let NORTA do its job so the dependence is seen to bite. Keep the running COPD model \(\log P(Y=1\mid x_R,u)=\beta_0+\beta_R x_R+\beta_U u\) with \(\beta_R=0.4\), \(\beta_U=-0.5\), normal margins \(x_R\sim\mathcal{N}(1.0,0.5^2)\) (age) and \(u\sim\mathcal{N}(2.0,1.0^2)\) (\(\mathrm{FEV}_1\)), and comparator risk \(\mu_0=0.47\). Suppose the IPD reports a Spearman rank correlation between age and \(\mathrm{FEV}_1\) of \(\rho^{S}_{\mathrm{IPD}}=-0.3\) (older patients tend to have worse lung function). NORTA borrows it through Equation 28.14: the Gaussian-copula correlation is \[ \rho^{N}=2\sin\!\big(\pi\,(-0.3)/6\big)=2\sin(-0.1571)=-0.313. \] For normal margins the copula correlation is the Pearson correlation, so \(\operatorname{Cov}(x_R,u)=\rho^{N}\sigma_R\sigma_U=-0.313\times 0.5\times 1.0=-0.1564\). The linear predictor \(W=\beta_0+\beta_R x_R+\beta_U u\) is then normal with variance \[ \operatorname{Var}(W)=\beta_R^2\sigma_R^2+\beta_U^2\sigma_U^2+2\beta_R\beta_U\operatorname{Cov}(x_R,u) =0.04+0.25+2(0.4)(-0.5)(-0.1564)=0.29+0.0626=0.3526, \] against \(\operatorname{Var}(W)=0.29\) under independence. By the log-normal mean \(\mathbb{E}[e^{W}]=\exp\!\big(\mathbb{E}[W]+\tfrac12\operatorname{Var}(W)\big)\), the dependence adds \(\tfrac12(0.3526-0.29)=0.0313\) to the log risk, so the adjusted log-risk-ratio moves from \(-0.700\) (the independence value of Section 28.8) to \(-0.700+0.0313=-0.669\), a risk ratio of \(e^{-0.669}=0.512\) instead of \(0.497\). Borrowing a realistic negative age-lung-function dependence nudges the estimate about \(3\%\) toward the null; a positive borrowed correlation would push it the other way. That shift, invisible to the margins alone, is exactly the information NORTA imports from the IPD. \(\blacktriangleleft\)

28.6 Tipping-point analysis

A sensitivity curve answers “what if the unreported mean were such-and-such?” The tipping point distills the curve to its single decision-relevant value: the assumed marginal of the unreported covariate at which the conclusion changes.

Definition 28.3 (Tipping point) Let \(m\mapsto\Delta(m)\) be the sensitivity curve that maps an assumed comparator marginal \(m\) of the unreported covariate \(U\) (its mean for a continuous covariate, its prevalence for a binary one) to the adjusted effect Equation 28.10 on the analysis scale, all other inputs held fixed. Fix a decision threshold \(c\) (the null value, \(c=0\) on a log-ratio or difference scale, by default). A tipping point is a value \(m^{\dagger}\) with \[ \Delta(m^{\dagger})=c. \tag{28.15}\] The tipping gap is \(m^{\dagger}-\bar{x}_{U}\), the signed distance from the value \(\bar{x}_U\) the covariate takes in the IPD. Given a plausible range \([m_{\mathrm{lo}},m_{\mathrm{hi}}]\) for the comparator marginal, the conclusion is flagged fragile if \(m^{\dagger}\in[m_{\mathrm{lo}},m_{\mathrm{hi}}]\), since a believable comparator value would reverse it, and robust if \(m^{\dagger}\) lies outside the range. One may report the tipping point of the confidence limit nearest the threshold in place of the point estimate. \(\blacktriangleleft\)

Remark. Reader’s translation. The tipping point is the sensitivity analysis a clinician can argue about directly, because it speaks in the covariate’s own units. It says: the conclusion holds unless the comparator’s true mean \(\mathrm{FEV}_1\) (or prevalence, for a binary covariate) is as extreme as \(m^{\dagger}\). You then ask a subject-matter question, not a statistical one: is \(m^{\dagger}\) inside the range of values the comparator population could plausibly have? If yes, the result is fragile; if no, it is robust. The tipping gap \(m^{\dagger}-\bar{x}_U\) measures how far the comparator would have to differ from your IPD before the story changes. For the running example this comes out at \(m^{\dagger}=0.6\) liters, against an IPD mean of \(2.0\) (see Section 28.8). \(\blacktriangleleft\)

For the tipping point to be a well-defined single number we need the sensitivity curve to be monotone, so that it crosses the threshold at most once. The next proposition shows monotonicity holds under the natural structural assumptions, via a coupling argument that fixes the copula draws and moves only the unreported margin.

Proposition 28.2 (Monotonicity and existence of the tipping point) Suppose the unreported covariate enters the IPD outcome model through a single term \(\beta_U u\) with \(\beta_U\neq 0\), the link \(g\) is continuous and strictly monotone (so \(g^{-1}\) is monotone), and the assumed marginal family \(\{F_{U,m}\}\) is ordered by first-order stochastic dominance in \(m\) through a quantile function \(F_{U,m}^{-1}(t)\) that is nondecreasing in \(m\) for each \(t\in(0,1)\) (as for a location family, where \(F_{U,m}^{-1}(t)=m+\sigma\Phi^{-1}(t)\), or a Bernoulli family, where \(F_{U,m}^{-1}(t)=\mathbf{1}\{t>1-m\}\)). Construct the comparator population by NORTA Equation 28.12 with a fixed correlation matrix and fixed reported margins. Then:

1. The sensitivity curve \(\Delta(m)\) is monotone in \(m\), strictly so where \(g^{-1}\) is strictly monotone; its direction is the sign of \(\beta_U\) (composed with the monotonicity of \(g\)).

2. If the threshold \(c\) lies between \(\Delta(m_{\mathrm{lo}})\) and \(\Delta(m_{\mathrm{hi}})\), a tipping point \(m^{\dagger}\in[m_{\mathrm{lo}},m_{\mathrm{hi}}]\) exists and is unique. If \(c\) lies outside the interval spanned by \(\Delta\) over the plausible range, there is no tipping point inside the range and the conclusion is robust on that range.

The idea behind the proof is simpler than its machinery. If pushing the unreported covariate up always moves the predicted outcome the same way, because it enters the model through a single term \(\beta_U u\) with a fixed sign, then the sensitivity curve can only go one direction and so crosses any threshold at most once. The one technical device is a coupling: use the same normal draws \(\mathbf{Z}\) for every assumed value \(m\), changing only the quantile function applied to the last coordinate, so that raising \(m\) raises that covariate for every simulated patient at once and never touches the others. Monotonicity then survives the averaging over patients.

Proof. Part 1. Realize the comparator covariates by the common-\(\mathbf{Z}\) coupling of Equation 28.12: draw \(\mathbf{Z}\sim\mathcal{N}_d(\mathbf{0},\boldsymbol{\Omega})\) once, set the reported coordinates \(X_{R,j}=F_{R,j}^{-1}(\Phi(Z_j))\) (independent of \(m\)) and the unreported coordinate \(U(m)=F_{U,m}^{-1}(\Phi(Z_d))\). By hypothesis \(F_{U,m}^{-1}(t)\) is nondecreasing in \(m\) for each fixed \(t=\Phi(Z_d)\), so for \(m\le m'\) we have \(U(m)\le U(m')\) pointwise in \(\mathbf{Z}\), while \(\mathbf{X}_R\) is unchanged. The linear predictor \(\eta(m)=\beta_0+\boldsymbol{\beta}_R^{\top}\mathbf{X}_R+\beta_U U(m)\) is therefore monotone in \(m\) with the sign of \(\beta_U\), pointwise in \(\mathbf{Z}\), and since \(g^{-1}\) is monotone, the per-subject prediction \(g^{-1}(\eta(m))\) is monotone in \(m\) pointwise. Taking expectation over \(\mathbf{Z}\) preserves the inequality, so the marginal mean \(\mathbb{E}_{\mathbf{Z}}[g^{-1}(\eta(m))]\) is monotone in \(m\); applying the monotone \(g\) and subtracting the constant \(g(\mu_0)\) leaves \(\Delta(m)\) monotone, with strictness inherited wherever \(g^{-1}\) is strictly monotone and \(\beta_U\neq0\).

Part 2. The curve \(\Delta\) is continuous in \(m\) (the integrand is continuous in \(m\) and bounded by an integrable envelope, so dominated convergence makes the marginal mean continuous; \(g\) is continuous). A continuous monotone function on \([m_{\mathrm{lo}},m_{\mathrm{hi}}]\) attains the value \(c\) exactly once when \(c\) lies between its endpoint values (intermediate value theorem plus injectivity from strict monotonicity), giving a unique \(m^{\dagger}\) in the range; if \(c\) is outside the endpoint interval, \(\Delta-c\) keeps a constant sign on the range, so no crossing occurs and the sign of the conclusion is constant, i.e. robust. \(\square\)

Remark. Computation and the link to the E-value. In practice the curve is evaluated on a grid by the NORTA marginalization of 1, and the crossing is located by one-dimensional root-finding (interpolation plus a bracketed solver) on the grid. The tipping point and the indirect-comparison E-value are two windows onto the same residual bias: the tipping point speaks in the natural units of the covariate’s comparator marginal, which is the most transparent scale for a clinician to judge plausibility, while \(E_{\mathrm{ITC}}\) recasts the same threat as a ratio-scale association. They can give qualitatively different impressions, since a small tipping gap and a large \(E_{\mathrm{ITC}}\) both occur, and reporting both is the cautious practice. \(\blacktriangleleft\)

28.7 Probabilistic bias analysis

Deterministic analysis commits to a single bias parameter or sweeps a bare grid. Probabilistic bias analysis instead places a prior distribution over the bias parameters and propagates it, separating two kinds of uncertainty that a single confidence interval conflates: the systematic uncertainty from not knowing the bias parameter, and the ordinary random sampling uncertainty.

Definition 28.4 (Probabilistic bias analysis) Let \(\hat\theta_{\mathrm{adj}}(\mathbf{b})\) be the QBA correction Equation 28.1 and let \(\pi\) be a prior on the bias parameter \(\mathbf{b}\in\mathcal{B}\), encoding external evidence about it. The probabilistic bias analysis draws \(\mathbf{b}^{(s)}\stackrel{\text{iid}}{\sim}\pi\) for \(s=1,\dots,S\), computes the adjusted effects \(\theta^{(s)}=\hat\theta_{\mathrm{adj}}(\mathbf{b}^{(s)})\), and reports two intervals:

• the systematic interval, the central \(1-\alpha\) empirical quantiles of \(\{\theta^{(s)}\}_{s=1}^{S}\), which captures bias-parameter uncertainty alone;
• the total interval, the same quantiles of the error-perturbed draws \(\theta_{\mathrm{tot}}^{(s)}=\theta^{(s)}+\varepsilon^{(s)}\) with \(\varepsilon^{(s)}\sim\mathcal{N}(0,\mathrm{SE}^2)\) independent draws of the sampling error, which folds ordinary statistical uncertainty on top.

Draws that violate a structural constraint (a negative reconstructed cell count, an out-of-range prevalence) are discarded, and the retained fraction is reported. The probability of benefit is the share of draws on the favorable side of the threshold, \(\widehat{P}_{\mathrm{ben}}=S^{-1}\sum_s\mathbf{1}\{\theta^{(s)}\text{ favorable}\}\). \(\blacktriangleleft\)

Remark. Where the prior comes from, and which draws to throw away. The prior \(\pi\) is a sensitivity assumption, external evidence about the bias parameter, never new data from the trial, and it must be defensible in a report. Practical defaults: for a continuous comparator mean, center a normal or truncated normal prior at the IPD value and set its standard deviation to a fraction of the IPD standard deviation (a half or a third), widening it when the comparator population is known to differ; for a binary comparator prevalence, elicit a plausible interval from a clinician and fit a Beta prior whose central range matches it (for instance \(\mathrm{Beta}(2,8)\) has mean \(0.2\) and most of its mass in \([0.03,0.45]\)). Registries, baseline tables of related trials, and epidemiologic cohorts are the usual sources; when nothing is known, a deliberately wide prior is the conservative choice. Invalid draws are those a prior can produce but the world cannot: a normal prior on a prevalence can wander below \(0\) or above \(1\), a drawn covariance can be incompatible with the fixed margins, a reconstructed cell count can go negative. These are discarded and the retained fraction reported; a large discard rate is itself a warning that the prior is misspecified. In the running example the prior is \(\mathcal{N}(2.0,0.3^2)\), an IPD-centered normal whose standard deviation is about a third of the IPD’s (see Section 28.8). \(\blacktriangleleft\)

The two intervals estimate well-defined population quantities, and the Monte Carlo estimates converge to them; we record this so the procedure is not merely a heuristic.

Theorem 28.3 (Consistency of the probabilistic-bias-analysis intervals) Let \(\theta=\hat\theta_{\mathrm{adj}}(\mathbf{b})\) with \(\mathbf{b}\sim\pi\) have distribution function \(G_{\mathrm{sys}}\) (the law of the correction under the prior, the systematic distribution), and let \(G_{\mathrm{tot}}\) be the law of \(\theta+\varepsilon\) with \(\varepsilon\sim\mathcal{N}(0,\mathrm{SE}^2)\) independent of \(\theta\), that is the convolution \(G_{\mathrm{tot}}(t)=\int\Phi\!\big((t-\hat\theta_{\mathrm{adj}}(\mathbf{b}))/\mathrm{SE}\big)\,d\pi(\mathbf{b})\). As the number of draws \(S\to\infty\):

1. The empirical quantiles of \(\{\theta^{(s)}\}\) converge almost surely to the quantiles of \(G_{\mathrm{sys}}\) at every continuity point of \(G_{\mathrm{sys}}^{-1}\); likewise the error-perturbed quantiles converge to those of \(G_{\mathrm{tot}}\).

2. \(\widehat{P}_{\mathrm{ben}}\to P(\theta\text{ favorable})\) almost surely.

3. \(G_{\mathrm{tot}}\) is continuous and strictly increasing wherever \(\mathrm{SE}>0\), so its quantiles are unique and the total interval is well defined for every \(\alpha\).

Proof. Part 1. The draws \(\theta^{(s)}=\hat\theta_{\mathrm{adj}}(\mathbf{b}^{(s)})\) are independent and identically distributed with distribution function \(G_{\mathrm{sys}}\), so by the Glivenko-Cantelli theorem the empirical distribution function \(\widehat{G}_{S}\) converges to \(G_{\mathrm{sys}}\) uniformly, almost surely. Uniform convergence of distribution functions implies convergence of the corresponding quantile (generalized inverse) functions at every point where \(G_{\mathrm{sys}}^{-1}\) is continuous, which is the standard quantile-convergence consequence of Glivenko-Cantelli. For the total interval, note \(\theta_{\mathrm{tot}}^{(s)}=\theta^{(s)}+\varepsilon^{(s)}\) are iid with distribution function \(G_{\mathrm{tot}}\) (the independent sum has the stated convolution law), and apply Glivenko-Cantelli again.

Part 2. The indicators \(\mathbf{1}\{\theta^{(s)}\text{ favorable}\}\) are iid Bernoulli with mean \(P(\theta\text{ favorable})\) and finite variance, so the strong law of large numbers gives \(\widehat{P}_{\mathrm{ben}}\to P(\theta\text{ favorable})\) almost surely.

Part 3. For \(\mathrm{SE}>0\) the inner map \(t\mapsto\Phi((t-\theta_0)/\mathrm{SE})\) is continuous and strictly increasing for each \(\theta_0\), and these properties are preserved under the mixture \(\int(\cdot)\,d\pi\): continuity by dominated convergence (the integrand is bounded by \(1\)), strict monotonicity because for \(t<t'\) the integrand strictly increases for every \(\theta_0\) and the prior integrates a strictly positive gap to a positive number. Hence \(G_{\mathrm{tot}}\) is a continuous, strictly increasing distribution function, with unique quantiles. \(\square\)

Remark. Interpretation. The systematic interval answers “how much does my conclusion move as the unknown bias parameter ranges over its prior?”; the total interval answers “and what does ordinary sampling error add on top?” Their difference is the price of the missing information, isolated from statistical noise; that separation is the entire point of probabilistic bias analysis (Lash et al. (2021)). In the unanchored ITC the bias parameter is the comparator marginal of the unreported covariate, the correction \(\hat\theta_{\mathrm{adj}}\) is the NORTA marginalization, and a normal or beta prior on that marginal yields the two intervals directly. When the sensitivity curve is monotone (Proposition 28.2) the favorable fraction \(\widehat{P}_{\mathrm{ben}}\) is a clean Monte Carlo estimate of the prior probability that the comparator marginal lies on the favorable side of the tipping point. In the language of the rest of the book this is the same random-effects logic as a Bayesian NMA (Chapter 15), except the “random” quantity is not a study effect drawn from a hierarchical prior but the unreported covariate value drawn from an external prior, and the total interval is its predictive spread folded together with sampling error. \(\blacktriangleleft\)

28.8 A worked E-value and tipping-point example

We now carry the running COPD comparison of Example 28.1 through every instrument of the chapter, choosing the numbers so that each quantity is exactly reproducible by hand. Recall the setup: the new inhaled therapy \(B\) is compared, unanchored, against comparator \(A\) on the binary outcome severe exacerbation within 12 months, analyzed on the log-risk-ratio scale. The IPD outcome model, fitted with a log link, is \[ \log P(Y=1\mid x_R,u)=\beta_0+\beta_R\,x_R+\beta_U\,u, \qquad \beta_0=-1.0,\ \beta_R=0.4,\ \beta_U=-0.5, \] where \(x_R\) is the one reported covariate (standardized age) and \(u\) the one unreported covariate (baseline \(\mathrm{FEV}_1\), a prognostic factor the comparator did not report). The comparator publication reports the marginal of \(x_R\) as mean \(m_R=1.0\), standard deviation \(\sigma_R=0.5\), and its own event risk \(\mu_0=0.47\); it does not report \(u\). In the IPD, \(u\) has mean \(\bar{x}_U=2.0\) and standard deviation \(\sigma_U=1.0\). For a hand-checkable marginalization we take \(x_R\) and \(u\) independent in the comparator population (Gaussian copula equal to the independence copula) and both normal.

The sensitivity curve in closed form. Under independence and normality, the comparator-population risk of the index treatment is a product of normal moment generating functions. With \(\mathbb{E}[e^{tX}]=e^{t\mu+t^2\sigma^2/2}\) for \(X\sim\mathcal{N}(\mu,\sigma^2)\), \[ r_B(m)=\mathbb{E}\!\big[e^{\beta_0+\beta_R X_R+\beta_U U}\big] =\exp\!\Big(\beta_0+\beta_R m_R+\tfrac12\beta_R^2\sigma_R^2+\beta_U m+\tfrac12\beta_U^2\sigma_U^2\Big), \] where \(m\) is the assumed comparator mean of \(U\). The unanchored log-risk-ratio against the comparator is \(\Delta(m)=\log r_B(m)-\log\mu_0=K+\beta_U m\) with \[ K=\beta_0+\beta_R m_R+\tfrac12\beta_R^2\sigma_R^2+\tfrac12\beta_U^2\sigma_U^2-\log\mu_0. \] Plugging in: \(\beta_R m_R=0.4\), \(\tfrac12\beta_R^2\sigma_R^2=\tfrac12(0.16)(0.25)=0.02\), \(\tfrac12\beta_U^2\sigma_U^2=\tfrac12(0.25)(1)=0.125\), and \(-\log(0.47)=0.755\), so \(K=-1.0+0.4+0.02+0.125+0.755=0.300\). The sensitivity curve is the line \[ \Delta(m)=0.300-0.5\,m, \] monotone decreasing in \(m\) exactly as Proposition 28.2 predicts (sign of \(\beta_U=-0.5\)).

(a) The standard E-value. The natural plug-in for the unknown comparator mean of \(U\) is the IPD value \(\bar{x}_U=2.0\), giving the adjusted effect \(\Delta(2.0)=0.300-1.0=-0.700\), that is a risk ratio \(\mathrm{RR}=e^{-0.700}=0.497\): treatment \(B\) roughly halves the risk. The corresponding bias factor to explain this away is \(\mathrm{BF}_{\mathrm{obs}}=e^{0.700}=2.014\). The symmetric E-value Equation 28.9 is \[ E=\mathrm{BF}_{\mathrm{obs}}+\sqrt{\mathrm{BF}_{\mathrm{obs}}(\mathrm{BF}_{\mathrm{obs}}-1)} =2.014+\sqrt{2.014\times1.014}=2.014+\sqrt{2.042}=2.014+1.429=3.443. \] An unmeasured effect modifier would need a risk-ratio association of at least \(3.44\) with both the treatment-population membership and the outcome to nullify the result; an association of, say, \(1.5\) on each arm could not. Had the analysis instead reported a confidence limit nearest the null of, for example, \(\mathrm{RR}=1/1.3\) (i.e. bias factor \(1.3\)), its E-value would be \(1.3+\sqrt{1.3\times0.3}=1.3+0.625=1.925\).

(b) The tipping point. Solving \(\Delta(m^{\dagger})=0\) gives \(m^{\dagger}=0.300/0.5=0.600\). The tipping gap is \(m^{\dagger}-\bar{x}_U=0.600-2.000=-1.400\): the comparator’s true mean of \(U\) would have to fall to \(0.6\), far below the IPD value of \(2.0\), before the conclusion reverses. If the plausible range for the comparator mean is, say, \([1.4,2.6]\) (the IPD value plus or minus \(0.6\)), then \(m^{\dagger}=0.6\) lies outside it, and the conclusion is robust to the unreported covariate over the plausible range.

The whole sensitivity analysis fits in one small table. Sweeping the assumed comparator mean \(m\) of \(\mathrm{FEV}_1\) through the affine curve \(\Delta(m)=0.300-0.5\,m\) gives:

Assumed comparator mean \(m\)	\(\Delta(m)\) (log RR)	Risk ratio \(e^{\Delta(m)}\)	Verdict
\(0.0\)	\(0.300\)	\(1.35\)	favors \(A\)
\(0.6\)	\(0.000\)	\(1.00\)	null (tipping point \(m^{\dagger}\))
\(1.0\)	\(-0.200\)	\(0.82\)	favors \(B\)
\(1.4\)	\(-0.400\)	\(0.67\)	favors \(B\) (edge of plausible range)
\(2.0\)	\(-0.700\)	\(0.50\)	favors \(B\) (IPD plug-in)
\(2.6\)	\(-1.000\)	\(0.37\)	favors \(B\) (edge of plausible range)
\(3.0\)	\(-1.200\)	\(0.30\)	favors \(B\)

The tipping point \(m^{\dagger}=0.6\) sits well below the plausible range \([1.4,2.6]\), so every plausible comparator value leaves \(B\) favored: a table restatement of the robustness verdict.

(c) The indirect-comparison E-value. The IPD-estimated outcome association of \(U\) is \(\mathrm{RR}_{UD}=e^{|\beta_U|}=e^{0.5}=1.649\). Compare it to the observed bias factor \(\mathrm{BF}_{\mathrm{obs}}=2.014\). Since \(\mathrm{RR}_{UD}=1.649<2.014=\mathrm{BF}_{\mathrm{obs}}\), we are in the third case of Definition 28.2: by Proposition 28.1 the maximal bias factor any imbalance in \(U\) can produce, with its outcome association fixed at \(1.649\), is \(1.649\) itself, which is smaller than the \(2.014\) needed. Hence \[ E_{\mathrm{ITC}}=+\infty: \] no across-arm imbalance in \(U\) can overturn the result, a robustness certificate the symmetric E-value of part (a) cannot provide. This is the sentence such a certificate puts in a dossier: the unreported \(\mathrm{FEV}_1\), estimated from the IPD to carry an outcome association of \(\mathrm{RR}_{UD}=1.65\), is too weakly prognostic to matter; since no across-arm imbalance can produce a bias factor exceeding \(\mathrm{RR}_{UD}\), while a factor of \(2.01\) would be needed to nullify the result, the comparison is robust to \(\mathrm{FEV}_1\) regardless of its comparator distribution. To exhibit the finite formula, suppose instead the IPD had estimated a stronger prognostic covariate, \(\mathrm{RR}_{UD}=3.0\). Then \[ E_{\mathrm{ITC}}=\frac{\mathrm{BF}_{\mathrm{obs}}(\mathrm{RR}_{UD}-1)}{\mathrm{RR}_{UD}-\mathrm{BF}_{\mathrm{obs}}} =\frac{2.014\times(3.0-1)}{3.0-2.014}=\frac{4.028}{0.986}=4.085, \] so the across-arm association would need to reach \(4.09\) to explain the result away. As a consistency check on Proposition 28.1(3), set \(\mathrm{RR}_{UD}=E=3.443\) from part (a); the formula returns \(2.014\times2.443/(3.443-2.014)=4.920/1.429=3.443=E\), recovering the symmetric E-value exactly, as it must when both arms are equal.

(d) Probabilistic bias analysis. Suppose external evidence supports a normal prior on the comparator mean of \(U\), \(m\sim\mathcal{N}(2.0,\,0.3^2)\), centered at the IPD value with a standard deviation of \(0.3\). Because the sensitivity curve is affine, \(\Delta(m)=0.300-0.5\,m\), the adjusted effect is itself normal: \[ \Delta\sim\mathcal{N}\!\big(0.300-0.5\times2.0,\ 0.5^2\times0.3^2\big) =\mathcal{N}(-0.700,\ 0.0225),\qquad \mathrm{sd}=0.150. \] The systematic \(95\%\) interval is \(-0.700\pm1.96\times0.150=(-0.994,\,-0.406)\), i.e. a risk ratio in \((0.370,\,0.666)\), entirely below \(1\). Folding in a trial sampling standard error of, say, \(\mathrm{SE}=0.25\), the total standard deviation is \(\sqrt{0.150^2+0.25^2}=\sqrt{0.085}=0.292\), and the total \(95\%\) interval is \(-0.700\pm1.96\times0.292=(-1.271,\,-0.129)\), a risk ratio in \((0.281,\,0.879)\), still below \(1\). The probability of benefit is \(\widehat{P}_{\mathrm{ben}}=P(\Delta<0)\): systematically \(\Phi(0.700/0.150)=\Phi(4.67)\approx1.000\), and in total \(\Phi(0.700/0.292)=\Phi(2.40)=0.992\). Every instrument agrees: the unreported effect modifier is a real but, here, a manageable threat, and the conclusion that \(B\) lowers risk survives it. The systematic interval, narrow because the prior is informative about the comparator mean, is exactly the “how much we do not know about \(U\)” component, cleanly separated from the wider total interval that adds sampling error.

28.9 Notes and references

The deterministic and probabilistic framework of Definition 28.1 and Definition 28.4 follows the monograph of Lash et al. (2021) (Fox, MacLehose, and Lash), which is the standard reference for the bias triangle, the confounding bias factor of Lemma 28.1, the selection and misclassification corrections mentioned in passing, and the systematic-versus-total interval decomposition. The bias factor Equation 28.2 for a binary confounder goes back to Schlesselman; we have reproduced only the confounding corner in full because it is the one the E-value rests on.

The E-value (2) and the sharp bias-factor bound (Lemma 28.2) are due to VanderWeele and Ding; we cite VanderWeele and Ding (2017), the article that introduced the E-value and its interpretation. The vertex argument and the two-cell optimization in our proof of Lemma 28.2 follow the structure of the Ding-VanderWeele “sensitivity analysis without assumptions” result; the derivation here standardizes to the exposed (the effect-on-treated bias factor), and the same bound is known to hold for total-population standardization. The indirect-comparison E-value of Definition 28.2 and the robustness certificate of Proposition 28.1 specialize the bound to the unanchored ITC by treating the \(U\)-outcome association as estimable from the IPD; this exploitation of the IPD is, to our knowledge, due to the recent ITC bias-analysis literature (Ren and colleagues, 2025) and is implemented in the uitc and mlumr packages, whose lessons informed the worked example. The connection to the insufficiency of the shared-effect-modifier assumption is Theorem 12.6, and the object being stress-tested is the effect modifier of Definition 11.3.

The NORTA construction (Theorem 28.2) is the normal-to-anything method of Cario and Nelson; it is the copula sampler of Proposition 22.1 composed with the inverse-CDF transform (Theorem 22.4), assembled into the Gaussian-copula joint law guaranteed by Sklar’s theorem (Theorem 22.5, Sklar (1959)). The same machinery underlies the ML-NMR integration of Chapter 22 and the ML-UMR pseudo-population of Chapter 26; here it is repurposed to impute an unreported comparator distribution rather than to integrate a reported one. The Spearman-matching transform Equation 28.14 rests on the arcsine law \(\rho^{S}=\frac{6}{\pi}\arcsin(\rho^{N}/2)\) for the grade correlation of the bivariate normal, a classical identity (Kruskal; see Nelsen’s copula text); we used it rather than reproved it, since a full derivation of the arcsine law is a self-contained exercise in bivariate-normal geometry. The positive-definite projection noted after Lemma 28.3 is the standard nearest-correlation-matrix repair.

This chapter has no counterpart in the proof catalogue and no machine-checked counterpart in the ITC_Coq development: quantitative bias analysis for indirect comparison is new material, built here from the cited primary sources. The tipping-point monotonicity proposition (Proposition 28.2) and the probabilistic-bias-analysis convergence theorem (Theorem 28.3) are, as far as we know, folklore made precise; their proofs use only the coupling of Equation 28.12 and the Glivenko-Cantelli theorem.

For the reader who wants to go deeper, the essential sources sort cleanly by instrument.

• Quantitative bias analysis in general: Lash et al. (2021) (Fox, MacLehose, and Lash), the standard monograph, including the selection and misclassification corners of Section 28.2.2 that this chapter only sketches.
• The E-value: VanderWeele and Ding (2017), for the definition, the interpretation, and the approximate scale conversions for odds and hazard ratios collected in Section 28.3.
• NORTA and copulas: Cario and Nelson for the normal-to-anything construction, and Nelsen’s copula text for Sklar’s theorem and the arcsine law used in Lemma 28.3.
• The ITC specialization: the recent indirect-comparison bias-analysis literature, implemented in the uitc and mlumr packages, for the exploitation of the IPD that gives Definition 28.2 its edge.

28.10 Exercises

A star (\(\star\)) marks an exercise with a full solution in Appendix F. The first three are warm-ups on reading and interpretation; the later ones build to the proofs.

Exercise 28.1 (Reading an E-value (warm-up)) A published unanchored comparison reports an adjusted hazard ratio of \(\mathrm{HR}=0.62\) (\(95\%\) confidence interval \(0.45\) to \(0.85\)) for a rare event, and states an E-value of \(2.6\) for the point estimate and \(1.7\) for the confidence limit nearest the null.

1. In one sentence, say what the number \(2.6\) means to a committee, using the phrase “risk-ratio association on both legs”.

2. A discussant claims an unmeasured prognostic difference between the trial populations could plausibly reach a risk-ratio association of \(1.5\) on each leg. Does that threaten the result? What association would?

3. Why is the confidence-limit E-value (\(1.7\)) the more conservative number to report?

Exercise 28.2 (Judging a tipping point (interpretation)) In the running COPD example the tipping point for the comparator mean of \(\mathrm{FEV}_1\) is \(m^{\dagger}=0.6\) liters, while the IPD mean is \(2.0\) liters and the plausible comparator range is \([1.4,2.6]\).

1. State, in clinical language, what would have to be true of the comparator population for the conclusion to reverse.

2. Is the conclusion fragile or robust on this range, and why? What single piece of external evidence would most change your verdict?

Exercise 28.3 (A dossier paragraph (interpretation)) Using the running example’s results (adjusted \(\mathrm{RR}=0.50\), symmetric E-value \(3.44\), indirect-comparison E-value \(+\infty\) because \(\mathrm{RR}_{UD}=1.65<2.01=\mathrm{BF}_{\mathrm{obs}}\), and tipping point \(m^{\dagger}=0.6\) outside the plausible range), write a single honest paragraph for an HTA committee stating the estimate, the robustness verdict, and the one caveat that still travels with it (the single-covariate, fixed-\(\mathrm{RR}_{UD}\) assumption).

Exercise 28.4 (Computing E-values) An unanchored comparison reports an adjusted risk ratio of \(\mathrm{RR}_{\mathrm{obs}}=1.8\) with \(95\%\) confidence interval \((1.2,\,2.7)\).

1. Compute the E-value of the point estimate using Equation 28.9.

2. Compute the E-value of the confidence limit nearest the null.

3. A reviewer notes that the measured covariates leave at most a residual association of risk ratio \(1.4\) on each arm. Can unmeasured confounding of that strength explain the result away? Justify using

Exercise 28.5 (Protective effects) Show that for a protective observed effect \(\mathrm{RR}_{\mathrm{obs}}<1\), applying Equation 28.9 to \(1/\mathrm{RR}_{\mathrm{obs}}\) gives the same E-value as the following symmetric construction: the smallest \(E\ge 1\) such that confounding with \(\mathrm{RR}_{EU}=\mathrm{RR}_{UD}=E\) can raise the true risk ratio back up to the null. Compute the E-value for \(\mathrm{RR}_{\mathrm{obs}}=0.5\).

Exercise 28.6 (The indirect-comparison E-value and its certificate (\(\star\))) Use the worked example’s model (same \(\beta_0,\beta_R,m_R,\sigma_R,\sigma_U\), comparator risk \(\mu_0=0.47\), IPD value \(\bar{x}_U=2.0\)), but now suppose the IPD estimates a stronger unreported covariate, \(\beta_U=-0.8\).

1. Recompute the constant \(K=\beta_0+\beta_R m_R+\tfrac12\beta_R^2\sigma_R^2+\tfrac12\beta_U^2\sigma_U^2 -\log\mu_0\), the sensitivity curve \(\Delta(m)=K+\beta_U m\), the adjusted effect \(\Delta(2.0)\), the observed bias factor \(\mathrm{BF}_{\mathrm{obs}}=\exp(|\Delta(2.0)|)\), and the outcome association \(\mathrm{RR}_{UD}=e^{|\beta_U|}\). (Both \(K\) and the slope change, since each depends on \(\beta_U\).)

2. Decide whether \(E_{\mathrm{ITC}}\) is finite or infinite using Proposition 28.1, and compute it if finite. Then verify the finite formula on the artificial pair \(\mathrm{BF}_{\mathrm{obs}}=1.5\), \(\mathrm{RR}_{UD}=2.5\).

3. Prove the general statement behind the certificate: for fixed \(\mathrm{RR}_{UD}>1\), the bias factor \(\mathrm{BF}_U(\mathrm{RR}_{EU},\mathrm{RR}_{UD})\) is strictly increasing in \(\mathrm{RR}_{EU}\) and bounded above by \(\mathrm{RR}_{UD}\), with the bound approached but never attained.

Exercise 28.7 (The sharp bound for a binary confounder) Specialize Lemma 28.2 to a binary confounder \(U\in\{0,1\}\) with unexposed-arm risks \(p_0(0),p_0(1)\) and confounder distributions \(f_1,f_0\). Starting from the exact bias factor Equation 28.7, verify directly (without the vertex argument) that \(B\le \mathrm{RR}_{EU}\mathrm{RR}_{UD}/(\mathrm{RR}_{EU}+\mathrm{RR}_{UD}-1)\), and identify the configuration of \((f_1,f_0,p_0)\) that attains equality. Confirm it matches the configuration in the proof of Lemma 28.2.

Exercise 28.8 (NORTA recovers any margin (\(\star\))) Let \(F\) be the Bernoulli\((\pi)\) distribution function and let \(Z\sim\mathcal{N}(0,1)\).

1. Write down \(F^{-1}\) (the generalized inverse) explicitly and show that \(X=F^{-1}(\Phi(Z))\) is Bernoulli\((\pi)\), confirming Theorem 28.2(1) for a discrete margin.

2. Show by counterexample that Theorem 28.2(2) can fail for this discrete margin: exhibit two distinct correlation parameters \(\rho^{N}\) that induce the same bivariate distribution of two such Bernoulli variables, so the copula is not unique. Explain why marginal recovery nonetheless still holds.

3. For a continuous margin, prove that the Pearson correlation of \((X_1,X_2)=\big(F_1^{-1}(\Phi(Z_1)), F_2^{-1}(\Phi(Z_2))\big)\) need not equal \(\rho^{N}=\operatorname{corr}(Z_1,Z_2)\), by taking \(F_1=F_2\) a lognormal margin and computing both correlations for \(\rho^{N}=1\).

Exercise 28.9 (Tipping points for a nonmonotone curve) Suppose an analyst balances two unreported covariates \(U_1,U_2\) jointly, with the adjusted effect a function \(\Delta(m_1,m_2)\) of both assumed means.

1. Give an example of an outcome model and margins for which the single-covariate sensitivity curve \(m_1\mapsto\Delta(m_1,m_2^{0})\) (with \(m_2\) fixed) is not monotone, so that Proposition 28.2 does not apply. (Hint: let \(U_1\) enter the outcome model nonlinearly, for example through \(u_1^2\).)

2. Explain what “the tipping point” should mean when the sensitivity surface crosses the threshold on a curve rather than at a point, and how a plausible region (rather than a range) for \((m_1,m_2)\) enters the robust-versus-fragile verdict.

Exercise 28.10 (Probabilistic analysis with a beta prior) The unreported covariate is binary with comparator prevalence \(m\in(0,1)\), and the sensitivity curve is \(\Delta(m)=0.2-1.5\,m\) on the log-odds-ratio scale (threshold \(c=0\)). Place a Beta\((2,8)\) prior on \(m\) (mean \(0.2\)).

1. Find the tipping prevalence \(m^{\dagger}\) and decide robust-versus-fragile against the plausible range \([0.05,0.40]\).

2. Using that \(\Delta\) is a decreasing affine function of \(m\), express the probability of benefit \(P(\Delta<0)=P(m>m^{\dagger})\) in terms of the Beta\((2,8)\) distribution function, and state whether the systematic interval for \(\Delta\) excludes the null. (You may leave the Beta integral in closed form via its polynomial density.)

Lash, Timothy L., Matthew P. Fox, and Richard F. MacLehose. 2021. Applying Quantitative Bias Analysis to Epidemiologic Data. 2nd ed. Springer.

Sklar, Abe. 1959. “Fonctions de répartition à \(n\) Dimensions Et Leurs Marges.” Publications de l’Institut de Statistique de l’Université de Paris 8: 229–31.

VanderWeele, Tyler J., and Peng Ding. 2017. “Sensitivity Analysis in Observational Research: Introducing the e-Value.” Annals of Internal Medicine 167 (4): 268–74.