11 Effect Modification, Interaction, and Collapsibility
11.1 Why a constant effect can still move
Two randomized trials study the same active drug against the same placebo. They report the relative effect on the same outcome, fit the same model, and yet their headline numbers disagree. There are exactly two pure reasons this can happen, and they are completely different in kind.
The first reason is effect modification. The drug genuinely helps younger patients more than older ones, the two trials enrolled different age mixes, and so the population-averaged effect differs because the populations differ. Here the per-patient story is heterogeneous, and the disagreement is a faithful reflection of that heterogeneity. Adjusting for age would reconcile the trials.
The second reason is non-collapsibility. The drug helps every patient by exactly the same amount on the scale the model uses, with no heterogeneity whatsoever, and yet the population-averaged effect still differs between the trials because the reporting scale does not commute with averaging. To say a scale does not commute with averaging is to say that the order of two operations matters: averaging the patient risks first and then converting to an odds ratio gives a different number than converting each patient to an odds ratio first and then averaging. Here the per-patient story is perfectly homogeneous, and the disagreement is an arithmetic artifact of the chosen effect measure. No covariate adjustment can remove it, because there is nothing in the data generating mechanism to adjust away; the gap is built into the odds ratio and the hazard ratio themselves.
These two phenomena are routinely confused, partly because both are sometimes called “heterogeneity” and partly because both make a marginal number depend on the covariate distribution. This chapter separates them with full proofs. It is the conceptual hinge of Part II: every later method, Bucher’s indirect comparison (Theorem 14.1), the transportability calculus of Chapter 12, matching and simulated comparison, rests on knowing precisely which covariates must be balanced, on which scale, and which apparent discrepancies are not about balance at all.
The plan is as follows. Section 11.2 defines the two population estimands that the rest of the chapter contrasts: the conditional effect and the marginal effect. Section 11.3 defines an effect modifier, separates it from the looser notion of a statistical interaction, and proves that effect modification is scale dependent. Section 11.4 defines a prognostic variable and proves, with a machine-checked counterexample, that effect modifiers and prognostic variables are logically distinct; this is the algebraic engine behind the anchored-versus-unanchored divide. Section 11.5 defines collapsibility and direct collapsibility, proves the risk difference directly collapsible, and proves the odds ratio and the hazard ratio non-collapsible, with closed-form examples. Section 11.6 ties the two axes together, and Section 11.7 carries out one fully numerical example in which every number is reproducible by hand. Before all of this, Section 11.1.1 fixes one small trial and a table of effect-measure scales, and we return to that trial at every transition.
Throughout we adopt the potential-outcome framework of Definition 9.1 under Definition 9.3, so that the observed outcome obeys the consistency identity Theorem 9.2. Write \(\mu_t(x)=\mathbb E[Y(t)\mid X=x]\) for the stratum-specific mean potential outcome under treatment \(t\), as in Notation. In words, \(\mu_0(x)\) is the average outcome (for a binary outcome, the risk) we would see under control among patients whose covariate value is \(x\), and \(\mu_1(x)\) is the same under treatment; these two functions carry everything the population knows about the treatment. When several populations are in play we write \(\mu_t^{\mathcal P}(x)\) to name the population; inside a single fixed population we drop the superscript and write \(\mu_t(x)\). Inside a single randomized trial the no-confounding identity \[ \mathbb E[Y(t)] = \mathbb E_X[\mu_t(X)] \tag{11.1}\] holds by Definition 9.7 and Definition 9.8; it is the within-trial g-formula of Theorem 9.3 and Theorem 10.4. This is a restatement of the g-formula, not a new assumption: because randomization makes treatment independent of the potential outcomes, the population mean under treatment \(t\) is obtained by averaging the stratum means \(\mu_t(x)\) over the covariate distribution. We invoke Equation 11.1 freely. The single most important thing to keep in mind is that everything in this chapter happens inside a randomized experiment, where treatment is assigned by coin flip and no confounding is possible. Every discrepancy we exhibit below is therefore not bias: it is not something a better-designed study or a cleverer adjustment could remove.
Remark (Reader’s map: five ideas, kept apart). A novice can hold this whole chapter together with two questions and one warning.
Two questions to ask of any covariate and any effect measure.
- Effect modification. Does the treatment work differently in different patients, measured on the scale you plan to report? If the per-stratum contrast changes across patients, the covariate is an effect modifier on that scale.
- Collapsibility. If you imagine the treatment working equally in everyone, does the population-average effect still equal that common value after you average over the patient mix? If yes the measure is collapsible; if no it is non-collapsible.
These are separate checks. The first is about real variation across patients; the second is about the arithmetic of the effect measure itself.
One warning. Non-collapsibility is not bias. It happens inside a perfect randomized trial, with no confounding to adjust away; the boxed remark in Section 11.5 states this precisely.
The five terms the chapter keeps apart:
| term | plain-language question it answers |
|---|---|
| effect modification | Does the treatment effect vary across patients, on your reporting scale? |
| statistical interaction | Does a fitted model need a treatment-by-covariate product term, on its scale? |
| prognostic variable | Does the covariate predict the outcome within an arm, ignoring treatment? |
| collapsibility | If the effect is the same in everyone, does the average effect equal it? |
| non-collapsibility | The failure of that: the average effect differs even with no variation. |
Four pairs never to conflate: effect modification is not the same as having a prognostic variable; a statistical interaction is not automatically a causal effect modifier; non-collapsibility is not confounding; and a marginal odds ratio is not the average of the stratum odds ratios. Each is proved below. \(\blacktriangleleft\)
11.1.1 A running example and the effect-measure scales
Why this matters (ITC). Every indirect comparison reports its answer on some scale: a risk difference, a risk ratio, an odds ratio, or a hazard ratio. The word scale in this chapter means the mathematical form of that effect measure, not the units of the outcome. A link \(g\) (Definition 7.2) is the strictly increasing function that turns a mean outcome into the number the model treats as additive, and the effect on scale \(g\) is a difference on the \(g\)-transformed outcomes. The four measures a health researcher meets correspond to four links.
| effect measure | link \(g\) | contrast on the \(g\)-scale | outcome range of the mean \(\mu\) |
|---|---|---|---|
| risk difference | identity, \(g(\mu)=\mu\) | subtract the risks | any real number |
| risk ratio | log, \(g(\mu)=\log\mu\) | difference of log risks \(=\log\) risk ratio | positive, \(\mu\in(0,\infty)\) |
| odds ratio | logit, \(g(\mu)=\log\tfrac{\mu}{1-\mu}\) | difference of logits \(=\log\) odds ratio | probability, \(\mu\in(0,1)\) |
| hazard ratio | log-hazard | difference of log hazards \(=\log\) hazard ratio | positive rate, over time |
Reading the table left to right, choosing an effect measure is choosing a link, and the effect on scale \(g\) is always a plain subtraction once the outcomes have been pushed through \(g\). Keep this table nearby: the whole chapter is about how the choice of \(g\) changes the answers. We fix one small trial now and return to it at every transition.
11.2 Conditional and marginal effects
Why this matters (ITC). Bucher’s indirect comparison, network meta-analysis, MAIC, and STC each estimate one of two different numbers: the average of the per-patient effects, or the effect on the average patient. This section defines both. Confusing them is the root of most disagreements between an individual-level model and a trial’s headline result.
The rest of the chapter contrasts two ways to turn the stratum means \(\mu_a(x),\mu_b(x)\) into a single effect. They differ only in the order of two operations, transform (apply the link \(g\)) and average (over the covariate \(X\)).
| step | transform, then average (conditional) | average, then transform (marginal) |
|---|---|---|
| 1 | convert each stratum’s risks to the \(g\)-scale | average the arm risks over patients |
| 2 | average the stratum contrasts over patients | convert the two averaged risks to the \(g\)-scale |
| result | \(\Delta^{\mathrm{Cond},g}_{ab}\) (defined below) | \(\Delta^{\mathrm{Marg},g}_{ab}\) (defined below) |
When \(g\) is the identity these two orders always agree; when \(g\) bends (the log, the logit, the log-hazard) they generally do not. This is the same commutative-square idea as in Example 11.1, and everything below is a consequence of that one asymmetry. With the picture in mind, here are the two estimands.
Fix two treatments, a control \(a\) and an active treatment \(b\) (for binary problems we also write \(0\) and \(1\)). Fix a population \(\mathcal P\) with covariate density \(f_{\mathcal P}\) on \(\mathcal X\), and a strictly increasing, twice differentiable link \(g\) (Definition 7.2) on the range of the mean outcome, for example the identity for a mean difference, \(\log\) for a risk or rate ratio, the logit for an odds ratio. The conditional mean functions \(\mu_a^{\mathcal P}(x)\) and \(\mu_b^{\mathcal P}(x)\) carry all the information about the population; everything else is a contrast built from them. Two contrasts are natural, and they are not the same.
Definition 11.1 (Conditional effect) The conditional effect of \(b\) versus \(a\) on scale \(g\) at covariate value \(x\) in population \(\mathcal P\) is the stratum-specific contrast \[ \tau^{g}_{ab}(x) \;=\; g\bigl(\mu_b^{\mathcal P}(x)\bigr) - g\bigl(\mu_a^{\mathcal P}(x)\bigr). \tag{11.2}\] Its average over the population is the population-average conditional effect \[ \Delta^{\mathrm{Cond},g}_{ab}(\mathcal P) \;=\; \mathbb E_{X\sim f_{\mathcal P}}\bigl[\tau^{g}_{ab}(X)\bigr] \;=\; \int \bigl[g(\mu_b(x)) - g(\mu_a(x))\bigr]\, f_{\mathcal P}(x)\,dx . \tag{11.3}\] For binary treatments with \(g\) fixed we abbreviate \(\tau^{g}_{01}(x)=\psi_g(x)\), matching the notation of 1. \(\blacktriangleleft\)1
Reader translation. \(\tau^g_{ab}(x)\) is the treatment effect for one kind of patient, those with covariate value \(x\), expressed on your reporting scale \(g\). Averaging it over the patients, \(\Delta^{\mathrm{Cond},g}_{ab}\), answers the question: what is the typical per-patient effect? For the running trial Example 11.1 on the odds-ratio scale, \(\tau^{\mathrm{logit}}_{01}(x)=\log 6\) in both strata, so the average is \(\log 6\): the conditional odds ratio is \(6\).
Definition 11.2 (Marginal effect) The marginal effect of \(b\) versus \(a\) on scale \(g\) (equivalently, the marginal effect on the effect-measure scale \(h=g\)) in population \(\mathcal P\) is the contrast of the transformed population-mean outcomes, \[ \Delta^{\mathrm{Marg},g}_{ab}(\mathcal P) \;=\; g\bigl(\mathbb E_{f_{\mathcal P}}[\mu_b(X)]\bigr) - g\bigl(\mathbb E_{f_{\mathcal P}}[\mu_a(X)]\bigr). \tag{11.4}\] For binary treatments we write \(\psi_g^{\mathrm M}\) for \(\Delta^{\mathrm{Marg},g}_{01}(\mathcal P)\), again matching 1. \(\blacktriangleleft\)
Reader translation. \(\Delta^{\mathrm{Marg},g}_{ab}\) first averages the outcomes as if the whole population took \(a\), then as if it took \(b\), and only then forms the contrast on scale \(g\). It answers the population question a health technology assessment usually asks: if everyone switched from \(a\) to \(b\), how far would the population-average outcome move on scale \(g\)? For the running trial on the odds-ratio scale this is \(\log 4.5\), not \(\log 6\). The two definitions in one table:
| notation | plain meaning |
|---|---|
| \(\mu_0(x)\) | expected outcome (risk) under control for patients with covariate value \(x\) |
| \(\mu_1(x)\) | expected outcome (risk) under treatment for those patients |
| \(\tau^g_{01}(x)\) | treatment effect within that stratum, on scale \(g\) |
| \(\Delta^{\mathrm{Cond},g}\) | average of the stratum effects: transform, then average |
| \(\Delta^{\mathrm{Marg},g}\) | effect of the averaged outcomes: average, then transform |
The distinction is the order of two operations. The conditional effect transforms first, by \(g\), then averages over \(X\); the marginal effect averages first, on the outcome scale, then transforms. The two quantities answer different scientific questions. The marginal effect asks: if the whole population received \(b\) rather than \(a\), by how much would the population-mean outcome move, measured on scale \(g\)? The population-average conditional effect asks: what is the average of the per-stratum effects, each measured on scale \(g\)? Health technology assessment usually wants the former; reasoning about an individual patient usually wants the latter. The catalogue records both as the two target estimands (corresponds to E.0 and E.12 in the proof catalogue).
A note on notation, since two conventions run in parallel from here on. For binary treatments the conditional contrast \(\tau^g_{01}(x)\) is also written \(\psi_g(x)\), and the marginal contrast \(\Delta^{\mathrm{Marg},g}_{01}(\mathcal P)\) is also written \(\psi_g^{\mathrm M}\), matching 1; we use the \(\psi\) letters in the collapsibility discussion of Section 11.5 and the \(\tau,\Delta\) letters when the treatment pair \((a,b)\) is general. They name the same objects; only the notation changes with context.
The next lemma is the whole reason the chapter exists. It quantifies exactly when the two estimands agree. It rests on one inequality, which is worth recalling first.
Remark (Jensen’s inequality, in one line). For a convex link \(g\) and any random mean \(Z\), transforming the average never exceeds averaging the transform: \[ g\bigl(\mathbb E[Z]\bigr)\;\le\;\mathbb E\bigl[g(Z)\bigr], \] with strict inequality when \(g\) is strictly convex and \(Z\) is non-degenerate (genuinely varies). For a concave \(g\) the inequality reverses. This is Theorem 4.6. The logit and the log are the nonlinear links that concern us, and the difference between the two sides is exactly the gap the next lemma names. \(\blacktriangleleft\)
Lemma 11.1 (The marginal-conditional gap) Fix a population \(\mathcal P\) and a link \(g\). Then \[ \Delta^{\mathrm{Marg},g}_{ab}(\mathcal P) - \Delta^{\mathrm{Cond},g}_{ab}(\mathcal P) \;=\; J_g(\mu_b) - J_g(\mu_a), \qquad J_g(\mu_t) \;:=\; g\bigl(\mathbb E_{f_{\mathcal P}}[\mu_t(X)]\bigr) - \mathbb E_{f_{\mathcal P}}\bigl[g(\mu_t(X))\bigr]. \tag{11.5}\] If \(g\) is affine (that is, linear plus a constant), \(g(u)=cu+d\) with \(c\neq0\), then \(J_g\equiv 0\) and the two effects coincide for every population and every pair of mean functions. If \(g\) is strictly convex (respectively strictly concave) and \(\mu_t(X)\) is non-degenerate, then \(J_g(\mu_t)<0\) (respectively \(>0\)) strictly.
The quantity \(J_g(\mu_t)\) is the Jensen gap (or nonlinearity gap) of the mean \(\mu_t(X)\) under \(g\): the amount by which transforming the average differs from averaging the transform.
Proof. Write out the two estimands. By Equation 11.4 and Equation 11.3, \[ \Delta^{\mathrm{Marg},g}_{ab}=g(\mathbb E[\mu_b])-g(\mathbb E[\mu_a]), \qquad \Delta^{\mathrm{Cond},g}_{ab}=\mathbb E[g(\mu_b)]-\mathbb E[g(\mu_a)], \] where in the second equality we used linearity of expectation to split \(\mathbb E[g(\mu_b)-g(\mu_a)]=\mathbb E[g(\mu_b)]-\mathbb E[g(\mu_a)]\), and every expectation is over \(X\sim f_{\mathcal P}\). Subtract the second from the first and regroup by treatment: \[ \Delta^{\mathrm{Marg},g}_{ab} - \Delta^{\mathrm{Cond},g}_{ab} = \underbrace{\Bigl[g(\mathbb E[\mu_b]) - \mathbb E[g(\mu_b)]\Bigr]}_{J_g(\mu_b)} - \underbrace{\Bigl[g(\mathbb E[\mu_a]) - \mathbb E[g(\mu_a)]\Bigr]}_{J_g(\mu_a)} = J_g(\mu_b) - J_g(\mu_a), \] which is Equation 11.5. If \(g(u)=cu+d\) is affine then \(g(\mathbb E[\mu_t])=c\,\mathbb E[\mu_t]+d =\mathbb E[c\mu_t+d]=\mathbb E[g(\mu_t)]\), so each \(J_g(\mu_t)=0\) and the gap vanishes identically. If \(g\) is strictly convex and \(\mu_t(X)\) is non-degenerate, the strict form of Jensen’s inequality (Theorem 4.6) gives \(g(\mathbb E[\mu_t(X)]) < \mathbb E[g(\mu_t(X))]\), that is \(J_g(\mu_t)<0\); the concave case is identical with the inequality reversed. \(\square\)
Two consequences are worth stating immediately. First, on the identity scale the marginal and the population-average conditional effect are always equal, for every population and every pair of mean functions, because \(J_{\mathrm{id}}\equiv 0\). The mean difference therefore never produces the second kind of discrepancy described in Section 11.1. Second, the gap Equation 11.5 is generally nonzero on a nonlinear scale even when the two treatments share the same conditional mean function up to a constant; the two terms \(J_g(\mu_a)\) and \(J_g(\mu_b)\) need not cancel because they apply the same nonlinear \(g\) to two different mean functions. This non-cancellation is non-collapsibility, made precise in Section 11.5.
Why this matters. The gap Equation 11.5 is the exact reason two correct trials, or two correct systematic reviews, can report different marginal effects on a nonlinear scale even when every patient responds identically. For the running trial Example 11.1 on the logit scale the marginal effect is \(\log 4.5\) and the conditional effect is \(\log 6\), so the gap is \[ J_{\mathrm{logit}}(\mu_1)-J_{\mathrm{logit}}(\mu_0) =\Delta^{\mathrm{Marg},\mathrm{logit}}_{01}-\Delta^{\mathrm{Cond},\mathrm{logit}}_{01} =\log 4.5-\log 6=-0.288 : \] transforming after averaging shaved the log odds ratio down by \(0.288\). On the identity scale the same computation gives \(0\), which is why the risk difference never shows this effect. Bucher, NMA, MAIC, and STC all inherit this gap the moment they report an odds ratio or a hazard ratio; Section 11.5 makes it precise and Example 11.3 carries the numbers through to an indirect comparison.
11.3 Effect modification and statistical interaction
Why this matters (ITC). Anchored indirect comparisons (Bucher, NMA) are unbiased only when the trials are balanced on the effect modifiers of the reported scale (Theorem 14.1). This section says exactly what an effect modifier is, why it is not the same as a regression interaction term, and why the answer depends on the scale you report.
We now define an effect modifier and contrast it with the looser, model-bound notion of an interaction.
Definition 11.3 (Effect modifier) A covariate \(X\) is an effect modifier of treatment \(b\) versus treatment \(a\) on scale \(g\) in population \(\mathcal P\) if the conditional effect Equation 11.2 depends non-trivially on \(x\), that is, if there exist \(x,x'\) in the support of \(f_{\mathcal P}\) with \[ \tau^{g}_{ab}(x) \;\neq\; \tau^{g}_{ab}(x'). \tag{11.6}\] Otherwise \(\tau^{g}_{ab}(\cdot)\) is constant on the support and \(X\) is not an effect modifier on scale \(g\) for the pair \((a,b)\) in \(\mathcal P\). \(\blacktriangleleft\)
Reader translation. \(X\) is an effect modifier when the treatment genuinely helps (or harms) one kind of patient more than another, measured on scale \(g\). In the running trial Example 11.1 the drug adds \(0.3\) to the risk in the low-risk stratum and \(0.4\) in the high-risk stratum, so on the risk-difference scale \(X\) is an effect modifier; but the odds ratio is \(6\) in both strata, so on the odds-ratio scale it is not. Same drug, same patients, opposite verdicts, because the scale differs.
The definition is parameterized by three things, each of which can change the answer (corresponds to E.1 in the proof catalogue). It depends on the treatment pair \((a,b)\): a covariate can modify the effect of \(b\) versus \(a\) but not of \(c\) versus \(a\). It depends on the scale \(g\): the central Proposition 11.1 below shows a covariate that modifies the effect on one scale and not on another. And it depends on the population \(\mathcal P\), through the conditional mean functions, since \(\mu_t^{\mathcal P}\) is itself population specific. Informal usage that says “\(X\) is an effect modifier” without naming all three is incomplete, and the omission is the source of much confusion in applied indirect comparison.
Example 11.2 (Effect-modifier status depends on the population) Keep the stratum means of the running trial, \(\mu_0(0)=0.1,\mu_1(0)=0.4,\mu_0(1)=0.4,\mu_1(1)=0.8\), and change only which patients are enrolled. In a broad trial \(\mathcal P\) that enrolls both strata, \(X\) takes both values, the risk-difference contrast is \(0.3\) at \(x=0\) and \(0.4\) at \(x=1\), and these differ, so \(X\) is an effect modifier on the identity scale. In a narrow trial \(\mathcal P'\) that enrolls only lower-risk patients (\(X\equiv 0\)), the covariate does not vary at all, so Definition 11.3 is vacuous: with no \(x\neq x'\) in the support, \(X\) cannot be an effect modifier in \(\mathcal P'\). The mean functions are identical; only the support of \(X\) changed, and with it the verdict. This is why “\(X\) is an effect modifier” is a statement about a population, not about a drug alone. \(\blacktriangleleft\)
In a regression model the notion is operationalized by an interaction term. Consider the conditional mean model, linear on the \(g\)-scale, that the network meta-regression of Dias et al. (2018) uses, \[ g\bigl(\mu_t(x)\bigr) \;=\; \alpha + \boldsymbol\beta_1^\top x + \bigl(\gamma + \boldsymbol\beta_2^\top x\bigr)\,t, \qquad t\in\{0,1\}. \tag{11.7}\] Here \(\boldsymbol\beta_1\) are the prognostic main effects, \(\gamma\) is the treatment main effect, and \(\boldsymbol\beta_2\) are the treatment-by-covariate interactions, the parameter labeled \(\boldsymbol\beta_{2,k}\) in Notation. The next lemma makes the link between the estimand notion (effect modification) and the parametric notion (a nonzero interaction coefficient) exact.
Lemma 11.2 (Effect modification equals a nonzero interaction on the model scale) Under the model Equation 11.7, the conditional effect on scale \(g\) is \[ \tau^{g}_{01}(x) \;=\; \gamma + \boldsymbol\beta_2^\top x . \] Consequently, the covariate component \(X_j\) is an effect modifier on scale \(g\) if and only if its interaction coefficient is nonzero, \(\beta_{2,j}\neq 0\), and \(X_j\) has nondegenerate support. If \(\boldsymbol\beta_2=\mathbf 0\), no component is an effect modifier on scale \(g\) and the conditional effect is the constant \(\gamma\).
Proof. Set \(t=1\) in Equation 11.7: \(g(\mu_1(x)) = \alpha + \boldsymbol\beta_1^\top x + \gamma + \boldsymbol\beta_2^\top x\). Set \(t=0\): the treatment terms drop and \(g(\mu_0(x)) = \alpha + \boldsymbol\beta_1^\top x\). Here \(\boldsymbol\beta_2^\top x=\sum_j\beta_{2,j}x_j\) is the dot product, the sum of the interaction coefficients times the covariate values. Subtract the second from the first. The intercept \(\alpha\) sits in both and cancels; the prognostic term \(\boldsymbol\beta_1^\top x\) sits in both and cancels; what survives is exactly the treatment terms, \[ \tau^{g}_{01}(x)=\bigl[\alpha+\boldsymbol\beta_1^\top x+\gamma+\boldsymbol\beta_2^\top x\bigr] -\bigl[\alpha+\boldsymbol\beta_1^\top x\bigr]=\gamma+\boldsymbol\beta_2^\top x . \] Now hold every covariate but \(X_j\) fixed and vary \(X_j\): the contrast \(\gamma+\boldsymbol\beta_2^\top x\) moves with \(X_j\) if and only if its coefficient \(\beta_{2,j}\) is nonzero (and \(X_j\) genuinely varies). So Equation 11.6 holds for \(X_j\) exactly when \(\beta_{2,j}\neq 0\). If \(\boldsymbol\beta_2=\mathbf 0\) the contrast is the constant \(\gamma\), so Equation 11.6 fails for every component and there is no effect modification. \(\square\)
Why this matters. In the network meta-regression Equation 11.7 that NMA and ML-NMR fit, effect modification is not a vague notion: it is exactly a nonzero interaction coefficient \(\beta_{2,j}\) on the model’s scale. When such a model is correctly specified you can read effect-modifier status straight off the fitted coefficients, which is what makes the interaction term the workhorse of population adjustment.
This equivalence tempts one to treat “effect modifier” and “statistical interaction” as synonyms. They are not, and the difference matters. We record the model-bound notion as a definition for contrast.
Definition 11.4 (Statistical interaction) A statistical interaction between treatment \(T\) and covariate \(X_j\) on scale \(g\), relative to a specified model, is a departure of the model from additivity on the \(g\)-scale in \(T\) and \(X_j\); in the parameterization Equation 11.7 it is the statement \(\beta_{2,j}\neq 0\). \(\blacktriangleleft\)
Three differences separate the two notions; see Hernán and Robins (2020), Chapters 4 and 5, for the epidemiological development.
First, symmetry. A statistical interaction is symmetric in its two arguments: the product term \(\beta_{2,j}\,t\,x_j\) treats treatment and covariate interchangeably, and one may equally say “the effect of \(X_j\) is modified by \(T\).” Effect modification is asymmetric by construction: it singles out the treatment \(T\) as the manipulated cause whose effect is being studied, and asks how that effect varies across levels of the second variable. The arithmetic is the same; the causal reading is not.
Second, scale dependence with a model. A statistical interaction is a property of a chosen model on a chosen scale; it can be created or destroyed by re-expressing the same conditional means through a different link. Effect modification is likewise scale dependent (Proposition 11.1), but it is a property of the conditional means themselves, not of a fitted parameterization, so it is well defined even when no parametric model has been written down.
Third, manipulability and surrogates. The product term in Equation 11.7 can be nonzero for a covariate that is a mere marker, correlated with the true modifier but with no causal role; conversely a genuine biological interaction can be masked in a marginal analysis. Effect modification, defined through potential-outcome means, is agnostic about the mechanism: it asserts only that the contrast varies across strata, whether because \(X\) acts causally or because it is a surrogate for something that does. For the indirect-comparison mathematics of this book the estimand notion in Definition 11.3 is the operative one, and we use “effect modifier” in that sense throughout. The equivalence Lemma 11.2 lets us read effect modification off a correctly specified model when one is available.
In applied terms, suppose you fit a treatment-by-covariate interaction and its coefficient is not zero. Should you call it effect modification? On the model’s scale, and if the covariate has real variation and the coefficient is estimated with reasonable precision, yes: Lemma 11.2 makes the two equivalent. Be cautious in three cases. When the covariate is a surrogate for the true modifier, the interaction is real but the mechanism is not the covariate you named. When the model’s scale is not the scale you will report, a nonzero interaction on the logit scale need not be one on the risk-difference scale, by the next proposition. And when the interaction is a small, noisy estimate, statistical significance is not the same as a modeling decision. The estimand notion in Definition 11.3 is the one this book uses; the regression interaction is how you detect it when a correct model is available.
11.3.1 Effect modification is scale dependent
The single most consequential fact about effect modification is that it cannot be discussed without naming a scale. A covariate can be an effect modifier on the logit scale and not on the identity scale, and the choice of scale for an analysis is therefore a substantive modeling decision, not a convention.
Proposition 11.1 (Scale dependence of effect-modifier status) There exist a binary covariate \(X\in\{0,1\}\) and binary potential outcomes \(Y(0),Y(1)\in\{0,1\}\) such that \(X\) is not an effect modifier on the identity (risk-difference) scale but is an effect modifier on the logit (odds-ratio) scale.
Proof. Imagine a lower-risk stratum \(x=0\) and a higher-risk stratum \(x=1\), with a drug that adds a constant \(0.1\) of absolute risk in each. Take the stratum risks \[ \mu_0(0)=0.1,\quad \mu_1(0)=0.2,\qquad \mu_0(1)=0.5,\quad \mu_1(1)=0.6 . \] On the identity scale the conditional effects are \[ \tau^{\mathrm{id}}_{01}(0)=0.2-0.1=0.1,\qquad \tau^{\mathrm{id}}_{01}(1)=0.6-0.5=0.1, \] equal in the two strata, so by Definition 11.3 \(X\) is not an effect modifier on the identity scale. On the logit scale, \[ \tau^{\mathrm{logit}}_{01}(0)=\operatorname{logit}(0.2)-\operatorname{logit}(0.1) =\log\frac{0.2/0.8}{0.1/0.9}=\log\frac{0.25}{0.1111\ldots}=\log 2.25=0.8109\ldots, \] \[ \tau^{\mathrm{logit}}_{01}(1)=\operatorname{logit}(0.6)-\operatorname{logit}(0.5) =\log\frac{0.6/0.4}{0.5/0.5}=\log 1.5=0.4055\ldots . \] These differ, so by Definition 11.3 \(X\) is an effect modifier on the logit scale. \(\square\)
The proposition (corresponds to E.10 in the proof catalogue) has an immediate methodological corollary: the assumption “\(X\) is not an effect modifier”, which underwrites every anchored indirect comparison through the shared-effect-modifier assumption of Theorem 12.6, is meaningful only relative to a named scale. An analyst who balances covariates on the risk-difference scale and reports an odds ratio, or vice versa, has not balanced the quantity that governs the reported estimand. The reason the two scales cannot be reconciled by a constant is exactly non-collapsibility, which we prove next; the present example is its shadow at the level of effect modification.
Reader translation. A health researcher might ask: if age modifies the risk difference but not the odds ratio, is it “wrong” to call age an effect modifier? It is not wrong, but it is incomplete. Effect-modifier status has no scale-free meaning (Proposition 11.1), so the honest statement always names the scale, as in “age modifies the risk difference but not the odds ratio here.” An analyst who balances covariates on one scale and reports an effect on another has not balanced the quantity that governs the reported number.
11.4 Prognostic variables, and why they differ from effect modifiers
Why this matters (ITC). Here is the single most important practical rule this chapter proves: anchored indirect comparisons need to balance only effect modifiers, while unanchored ones must balance all prognostic variables. The reason is that a covariate can influence each treatment arm yet leave the between-arm contrast untouched; such a purely prognostic covariate cancels when two arms are compared against a shared comparator, but not when an absolute arm mean is transported on its own. This section proves that effect modifiers and prognostic variables are genuinely different, and isolates the cancellation. In the transportability literature the first rule is the shared effect modifier assumption (SEMA) and the second the shared prognostic factor assumption (SPFA); both names, listed in Notation, return in Chapter 12.
Effect modification concerns the contrast \(\tau^g_{ab}\). A different and weaker condition concerns each arm separately.
Definition 11.5 (Prognostic variable) A covariate \(X\) is a prognostic variable for treatment \(t\) in population \(\mathcal P\) if the conditional mean \(\mu_t^{\mathcal P}(x)\) depends non-trivially on \(x\), that is, if there exist \(x,x'\) in the support with \(\mu_t^{\mathcal P}(x)\neq \mu_t^{\mathcal P}(x')\). A covariate is prognostic but not an effect modifier for the pair \((a,b)\) on scale \(g\) if both \(\mu_a\) and \(\mu_b\) depend on \(x\) while the contrast \(\tau^g_{ab}\) does not. \(\blacktriangleleft\)
Reader translation. \(X\) is a prognostic variable for an arm if it predicts the outcome within that arm, treatment aside: sicker patients do worse whether or not they are treated. Baseline severity, age, and biomarker level are typical prognostic variables. A prognostic variable becomes an effect modifier only if it changes the contrast between arms, not merely the level of each arm. In the running trial the higher-risk stratum has worse outcomes in both arms, so \(X\) is prognostic for both; it also changes the risk-difference contrast, so on that scale it is additionally an effect modifier; on the odds-ratio scale it is prognostic but not an effect modifier.
This is the catalogue’s E.2. The two concepts partition into a four-cell table, for the treatment pair \((a,b)\) on a fixed scale \(g\) in a fixed population. Read “not prognostic” strictly: it means the covariate predicts the outcome in neither arm.
| prognostic for both \(a\) and \(b\) | prognostic for neither arm | |
|---|---|---|
| effect modifier on \(g\) | both arms vary and so does the contrast. Running trial, identity scale: \(\mu_a=(0.1,0.4)\), \(\mu_b=(0.4,0.8)\), contrasts \(0.3\) then \(0.4\) | impossible: a varying contrast forces at least one arm to vary |
| not an effect modifier on \(g\) | both arms vary but the contrast is constant (purely prognostic): \(\mu_a=(0.3,0.7)\), \(\mu_b=(0.5,0.9)\), contrast \(0.2\) in both | neither arm varies: \(\mu_a\equiv 0.4\), \(\mu_b\equiv 0.6\), contrast \(0.2\) |
Each cell lists \((\text{value at }x=0,\ \text{value at }x=1)\) for the two arms. The lower-left cell, purely prognostic, is the one that powers anchored comparison: each arm depends on the covariate, but in the same way on the \(g\)-scale, so the contrast is free of it. The upper-right cell is empty, which is the content of the following proposition; it is the one forced implication between the two concepts.
One subtlety before the proofs. The two columns above, “prognostic for both” and “prognostic for neither arm,” do not exhaust the possibilities: a covariate can be prognostic for exactly one arm. Those one-arm cases are exactly where an effect modifier can be “not prognostic for a specified treatment,” which is Part 2 of Theorem 11.1 below. There is no contradiction with the “impossible” cell: an effect modifier must be prognostic for at least one arm (that is Proposition 11.2), but it need not be prognostic for both, and in particular need not be prognostic for the one arm you happen to name.
Proposition 11.2 (An effect modifier is prognostic for at least one treatment) If \(X\) is an effect modifier of \(b\) versus \(a\) on scale \(g\), then \(X\) is a prognostic variable for \(a\) or for \(b\) (or both).
Proof. Contrapositive. Suppose \(X\) is prognostic for neither \(a\) nor \(b\): then \(\mu_a(x)=\mu_a(x')\) and \(\mu_b(x)=\mu_b(x')\) for all \(x,x'\) in the support. Apply the link \(g\) to all four values. Because \(\mu_a(x)=\mu_a(x')\) we get \(g(\mu_a(x))=g(\mu_a(x'))\), and because \(\mu_b(x)=\mu_b(x')\) we get \(g(\mu_b(x))=g(\mu_b(x'))\). Subtracting the two control values from the two treated values, \[ \tau^g_{ab}(x)=g(\mu_b(x))-g(\mu_a(x))=g(\mu_b(x'))-g(\mu_a(x'))=\tau^g_{ab}(x'), \] so the contrast is constant and \(X\) is not an effect modifier. Contrapositively, an effect modifier must be prognostic for \(a\) or for \(b\). \(\square\)
The reverse implications all fail, and this is the crucial separation. A prognostic variable need not be an effect modifier (the purely prognostic case), and an effect modifier need not be prognostic for a given treatment (it may be prognostic only for the other arm). The first failure is the deeper one, because it is what distinguishes anchored from unanchored adjustment, and it is verified in the Coq development.
Theorem 11.1 (Effect modifiers and prognostic variables are logically distinct) Both of the following hold.
- There exist mean functions \(\mu_a,\mu_b\) and a covariate \(X\) such that \(X\) is prognostic for both \(a\) and \(b\) yet is not an effect modifier of \(b\) versus \(a\) on the identity scale. Hence “prognostic variable” does not imply “effect modifier.”
- There exist mean functions \(\mu_a,\mu_b\) and a covariate \(X\) such that \(X\) is an effect modifier of \(b\) versus \(a\) yet is not prognostic for \(a\). Hence “effect modifier” does not imply “prognostic for a specified treatment.”
[machine-checked: theories/EffectModifiers.v, Theorem not_EM_does_not_imply_not_PV]
Proof. Part 1. Take \(X\in\{0,1\}\) with both values in the support, and set the identity-scale means equal across arms, \(\mu_a(0)=\mu_b(0)=0.3\) and \(\mu_a(1)=\mu_b(1)=0.7\). Each arm depends on \(x\) (rising from \(0.3\) to \(0.7\)), so \(X\) is prognostic for both \(a\) and \(b\). The contrast is \(\tau^{\mathrm{id}}_{ab}(x)=\mu_b(x)-\mu_a(x)=0\) for every \(x\), constant, so \(X\) is not an effect modifier. This is exactly the purely prognostic configuration: the two arms move together and their difference is free of \(x\). (The Coq file proves the same fact with the two arms written as \(\mu_a(x,w)=\mu_b(x,w)=\mathbf 1\{w\}\) over a binary unmeasured covariate \(w\): counterexample_not_EM shows the contrast is identically zero and counterexample_not_PV_fails shows each arm genuinely depends on \(w\), and together they give not_EM_does_not_imply_not_PV.)
Part 2. Let \(X\in\{0,1\}\), set \(\mu_a(x)=0.4\) constant in \(x\), and \(\mu_b(0)=0.4\), \(\mu_b(1)=0.7\). Then \(X\) is not prognostic for \(a\), since \(\mu_a\) is constant. The identity-scale contrast is \(\tau^{\mathrm{id}}_{ab}(0)=0\) and \(\tau^{\mathrm{id}}_{ab}(1)=0.3\), which differ, so \(X\) is an effect modifier of \(b\) versus \(a\). Thus a covariate can modify the contrast while being prognostic only for the active arm, never for the control arm. \(\square\)
Part 1 is the workhorse. It says the purely prognostic configuration genuinely exists: covariates whose influence on each arm is real but whose influence on the link-scale contrast is nil. The next lemma turns this into the cancellation that anchored comparisons exploit.
Picture the simplest anchored network. Trial 1 compares \(b\) against a common comparator \(C\), trial 2 compares \(a\) against the same \(C\), and the indirect estimate of \(b\) versus \(a\) subtracts the two contrasts, \((b\text{ versus }C)-(a\text{ versus }C)\). Anything that sits identically in both contrasts, in particular a purely prognostic covariate that shifts every arm the same way on the \(g\)-scale, is removed by that subtraction. The lemma proves this at the level of the conditional contrast.
Lemma 11.3 (Anchored cancellation of purely prognostic variables) Suppose the covariate vector splits as \((X,W)\), with \(X\) measured and \(W\) possibly unmeasured, and suppose \(W\) is not an effect modifier of \(b\) versus \(a\) on the model scale \(g\) given \(X\), meaning \[ g\bigl(\mu_b(x,w)\bigr) - g\bigl(\mu_a(x,w)\bigr) \;=\; g\bigl(\mu_b(x,w')\bigr) - g\bigl(\mu_a(x,w')\bigr) \qquad\text{for all } x,w,w'. \tag{11.8}\] Then the conditional contrast \(\tau^g_{ab}(x,w)\) does not depend on \(w\), and for any two populations \(\mathcal P_1,\mathcal P_2\) with the same marginal law of \(X\) but different conditional laws of \(W\) given \(X\), the population-average conditional contrast is the same: \[ \Delta^{\mathrm{Cond},g}_{ab}(\mathcal P_1) \;=\; \Delta^{\mathrm{Cond},g}_{ab}(\mathcal P_2). \]
[machine-checked: theories/EffectModifiers.v, Theorem anchored_PV_cancellation]
Proof. By Equation 11.8 the map \(w\mapsto \tau^g_{ab}(x,w)\) is constant for each fixed \(x\); write its common value as \(\bar\tau^g_{ab}(x)\). Any population \(\mathcal P\) has a joint covariate density that factors as \(f(x,w)=f(w\mid x)\,f_X(x)\), the conditional density of \(W\) given \(X\) times the marginal density of \(X\). Then \[
\Delta^{\mathrm{Cond},g}_{ab}(\mathcal P)
= \int\!\!\int \tau^g_{ab}(x,w)\, f(w\mid x)\,dw\, f_X(x)\,dx
= \int \bar\tau^g_{ab}(x)\Bigl(\int f(w\mid x)\,dw\Bigr) f_X(x)\,dx .
\] The inner integral \(\int f(w\mid x)\,dw = 1\), because \(f(w\mid x)\) is a conditional probability density and integrates to one over \(w\) for each fixed \(x\). Hence \[
\Delta^{\mathrm{Cond},g}_{ab}(\mathcal P) = \int \bar\tau^g_{ab}(x)\, f_X(x)\,dx,
\] which depends on \(\mathcal P\) only through the marginal law \(f_X\) of \(X\), not through the conditional law \(f(w\mid x)\) of \(W\). Two populations with the same \(f_X\) therefore give the same value, however differently \(W\) is distributed given \(X\). The Coq theorem anchored_PV_cancellation proves the pointwise step, that the contrast is independent of \(w\) under the not-an-effect-modifier hypothesis. \(\square\)
The lemma is the formal basis of the slogan “prognostic variables cancel in anchored comparisons but not in unanchored ones,” and it explains the asymmetry in covariate requirements between the two designs. In an anchored comparison the estimator combines two contrasts that each share the comparator \(C\), and any covariate that is purely prognostic on the model scale, even an unmeasured one, drops out of each contrast by Lemma 11.3; only effect modifiers, whose distribution genuinely alters the contrast, must be balanced. In an unanchored comparison the estimator works with the absolute arm means \(\mu_t\) directly, with no comparator to subtract, so Theorem 11.1 part 1 bites: a purely prognostic \(W\) that is imbalanced between source and target changes \(\mu_t\) and is not cancelled. This is why Theorem 14.1 requires balancing only effect modifiers, while the unanchored case of Chapter 12 requires balancing all imbalanced prognostic variables as well. The precise estimand-level consequences are developed there; here we have isolated the algebraic mechanism and tied it to the machine-checked counterexample. In the language of the section opener, Lemma 11.3 is the mathematical content of the shared effect modifier assumption (SEMA): balance the effect modifiers and the anchored contrast transports. The shared prognostic factor assumption (SPFA) is the stronger requirement the unanchored case needs, developed in Chapter 12.
11.5 Collapsibility, direct collapsibility, and non-collapsibility
Why this matters (ITC). Whether your reported effect measure is even transportable in principle across trials with different patient mixes depends on whether it is collapsible. The risk difference and risk ratio are collapsible; the odds ratio and hazard ratio are not, which is why marginal odds ratios and hazard ratios are treated throughout this book as population-specific quantities that cannot simply be pooled.
We turn to the second axis. Section 11.3 was about whether the conditional effect varies across strata. Collapsibility is about whether, when the conditional effect does not vary, the marginal effect equals it. Chapter 7 (1) defined collapsibility of an effect measure under a constant conditional effect. We now refine that definition to separate two strengths, because the risk ratio and the risk difference behave differently even though both are collapsible in the Chapter 7 sense.
Definition 11.6 (Collapsibility and direct collapsibility) Let an effect measure be defined by a strictly increasing link \(g\), with conditional and marginal effects \(\psi_g(x)=g(\mu_1(x))-g(\mu_0(x))\) and \(\psi_g^{\mathrm M}=g(\mathbb E_X[\mu_1(X)])-g(\mathbb E_X[\mu_0(X)])\) as in Definition 11.1 and Definition 11.2. The measure is:
- collapsible over \(X\) if, whenever \(\psi_g(\cdot)\equiv\psi_g\) is constant in \(x\), the marginal effect equals that common value, \(\psi_g^{\mathrm M}=\psi_g\), for every distribution of \(X\) and every pair of stratum means consistent with it; it is non-collapsible otherwise. This is the notion of
- directly collapsible over \(X\) if the marginal effect is always a weighting-respecting average of the conditional effects, namely \[ \psi_g^{\mathrm M} \;=\; \mathbb E_X\bigl[\psi_g(X)\bigr] \;=\; \Delta^{\mathrm{Cond},g}_{01}(\mathcal P), \tag{11.9}\] for every distribution of \(X\) and every pair of stratum means, whether or not the conditional effect is constant.
Direct collapsibility is the statement that the marginal-conditional gap Equation 11.5 vanishes identically; collapsibility is the weaker statement that it vanishes when the conditional effect is constant. \(\blacktriangleleft\)
Reader translation. Two nested questions. Collapsibility asks: if the effect is the same constant in every stratum, does that constant survive averaging over the patient mix? Direct collapsibility asks more: even when the effect varies across strata, is the marginal effect exactly the patient-mix-weighted average of those stratum effects? Directly collapsible is the stronger, friendlier property, because the marginal number is then always the honest average of the per-patient numbers. Merely collapsible guarantees only that a constant effect is reported faithfully; if the effect varies, the marginal number can drift away from the average. The risk difference is directly collapsible; the risk ratio is collapsible but not directly; the odds ratio and hazard ratio are neither.
Direct collapsibility implies collapsibility: if \(\psi_g^{\mathrm M}=\mathbb E_X[\psi_g(X)]\) always, then in particular when \(\psi_g(X)\equiv\psi_g\) is constant the average is \(\psi_g\), giving \(\psi_g^{\mathrm M}=\psi_g\). The converse fails, and the risk ratio is the witness. The terminology and the distinction follow Chandler and Ishak (2026), who make direct collapsibility the pivotal property for transporting a marginal effect across populations.
Proposition 11.3 (The risk difference is directly collapsible) On the identity scale, \(g(\mu)=\mu\), the effect measure is directly collapsible over every covariate \(X\). In particular the risk difference is collapsible, and the marginal risk difference equals the population-average conditional risk difference for every distribution of \(X\), even when the conditional risk difference is itself an effect modifier.
Proof. With \(g\) the identity, Equation 11.5 gives \(J_{\mathrm{id}}(\mu_t)=\mathbb E_X[\mu_t(X)]-\mathbb E_X[\mu_t(X)]=0\), so by Lemma 11.1 \[ \psi_{\mathrm{id}}^{\mathrm M} = \mathbb E_X[\mu_1(X)] - \mathbb E_X[\mu_0(X)] = \mathbb E_X\bigl[\mu_1(X)-\mu_0(X)\bigr] = \mathbb E_X[\psi_{\mathrm{id}}(X)], \] which is Equation 11.9. Direct collapsibility holds with no constancy assumption, hence the parenthetical remark about effect modification. \(\square\)
Why this matters. On the risk-difference scale the population effect is always the genuine average of the per-patient effects, so a trial’s headline risk difference is a faithful summary even when the drug helps some patients more than others. This is the cleanest possible behavior, and it is why absolute-scale reasoning is so robust in anchored comparisons.
This is the strongest possible good behavior: on the identity scale the marginal effect is the genuine population average of the per-stratum effects, so the two questions of Section 11.2 have the same answer, and a varying conditional effect is faithfully summarized by its average. Chapter 7 (1) established the collapsibility half of this for the risk difference and the risk ratio; Proposition 11.3 sharpens it to direct collapsibility for the risk difference. The risk ratio is collapsible but not directly collapsible, which is the next proposition and the reason the two strengths are worth distinguishing.
Proposition 11.4 (The risk ratio is collapsible but not directly collapsible) On the log scale, \(g(\mu)=\log\mu\) (the risk ratio): (i) the measure is collapsible over every \(X\); (ii) it is not directly collapsible, that is, there is a covariate \(X\) and stratum means for which \(\psi_{\log}^{\mathrm M}\neq\mathbb E_X[\psi_{\log}(X)]\).
Proof. (i) Collapsibility. Suppose the conditional log risk ratio is constant, \(\psi_{\log}(x)=\log\mu_1(x)-\log\mu_0(x)\equiv\rho\), equivalently \(\mu_1(x)=e^\rho\mu_0(x)\) for all \(x\). Taking expectations over \(X\), \(\mathbb E_X[\mu_1(X)]=e^\rho\,\mathbb E_X[\mu_0(X)]\), so \[ \psi_{\log}^{\mathrm M}=\log\frac{\mathbb E_X[\mu_1(X)]}{\mathbb E_X[\mu_0(X)]}=\log e^\rho=\rho =\psi_{\log}, \] for every distribution of \(X\); the measure is collapsible.
(ii) Not directly collapsible. Take \(X\in\{0,1\}\) with \(P(X=0)=P(X=1)=\tfrac12\) and stratum means \(\mu_0(0)=0.1,\ \mu_1(0)=0.4,\ \mu_0(1)=0.4,\ \mu_1(1)=0.8\). The conditional log risk ratios are \(\psi_{\log}(0)=\log(0.4/0.1)=\log 4\) and \(\psi_{\log}(1)=\log(0.8/0.4)=\log 2\), with average \(\mathbb E_X[\psi_{\log}(X)]=\tfrac12(\log 4+\log 2)=\tfrac12\log 8=\log 2\sqrt2=1.0397\ldots\). The marginal risks are \(\mathbb E[\mu_0]=\tfrac12(0.1+0.4)=0.25\) and \(\mathbb E[\mu_1]=\tfrac12(0.4+0.8)=0.60\), so \(\psi_{\log}^{\mathrm M}=\log(0.60/0.25)=\log 2.4=0.8755\ldots\). Since \(0.8755\ldots\neq1.0397\ldots\), Equation 11.9 fails and the risk ratio is not directly collapsible. \(\square\)
Reader translation. Why is the risk ratio collapsible but not directly collapsible? Collapsibility only has to handle a constant log risk ratio, and a constant \(\rho\) with \(\mu_1(x)=e^\rho\mu_0(x)\) scales both averaged risks by the same \(e^\rho\), so the constant survives (part i). Direct collapsibility must handle a varying log risk ratio, and there the log of a ratio of averages is not the average of the logs of the ratios: the strata with larger control risk \(\mu_0(x)\) pull more weight into the marginal ratio than they carry in the plain average of the log ratios, so the two disagree (part ii). Collapsibility asks “does the constant survive averaging?”; direct collapsibility asks “does a varying effect average correctly?” The risk ratio passes the first test and fails the second.
The gap in part (ii) appears only because the conditional risk ratio varies across strata; whenever it is constant, part (i) guarantees agreement. This is precisely the difference between the two strengths: a collapsible measure summarizes a constant conditional effect correctly, while a directly collapsible measure summarizes a varying one correctly too. The risk difference manages both; the risk ratio only the former. For an anchored indirect comparison reporting a log risk ratio under the shared-effect-modifier assumption, collapsibility suffices and the comparison transports (Theorem 12.5, Theorem 12.4); the failure of direct collapsibility becomes relevant only when the conditional effect is allowed to vary, which is the effect-modification axis again.
11.5.1 The odds ratio and the hazard ratio are non-collapsible
The odds ratio and the hazard ratio fail even collapsibility. They can differ between the conditional and the marginal scale with no effect modification at all, purely because their links are nonlinear and the gap Equation 11.5 does not cancel between the two arms. This is the catalogue’s B.13, and it is the formal reason the rest of the book treats marginal odds ratios and hazard ratios as population-specific quantities.
We first prove a clean general statement for the odds ratio, with strict attenuation toward the null, using a change-of-measure argument; then the hazard ratio, in closed form, with the additional feature that its marginal version varies with time.
Remark (The odds-ratio phenomenon before the proof). The running trial Example 11.1 already shows it. The conditional odds ratio is exactly \(6\) in both strata, so there is no effect modification on the odds-ratio scale and nothing to adjust for. Yet the odds ratio of the population-averaged risks is \[ \frac{0.60/0.40}{0.25/0.75}=4.5 \; < \; 6 . \] A perfectly homogeneous per-patient odds ratio of \(6\) became a marginal odds ratio of \(4.5\), pulled toward the null value \(1\), purely by averaging. The theorem below shows this attenuation is universal (it always happens when the baseline risk varies) and unidirectional (always toward \(1\)); the change-of-measure proof is the machinery, but \(6\to 4.5\) is the fact. \(\blacktriangleleft\)
Theorem 11.2 (Non-collapsibility of the odds ratio and the hazard ratio) Work under the no-confounding identity Equation 11.1, so that no confounding is present.
(a) Odds ratio. Let \(X\) be a covariate, let \(\mu_t(X)\in(0,1)\) be the stratum risks, and suppose the conditional odds ratio is constant across strata, \[ \omega \;=\; \frac{\mu_1(x)/(1-\mu_1(x))}{\mu_0(x)/(1-\mu_0(x))}\qquad\text{for all } x, \] so that \(X\) is not an effect modifier on the logit scale. If \(\omega\neq 1\) and the baseline odds \(W:=\mu_0(X)/(1-\mu_0(X))\) are non-degenerate, then the marginal odds ratio \[ \omega^{\mathrm M} \;=\; \frac{\mathbb E[\mu_1(X)]/(1-\mathbb E[\mu_1(X)])}{\mathbb E[\mu_0(X)]/(1-\mathbb E[\mu_0(X)])} \] satisfies \(1<\omega^{\mathrm M}<\omega\) when \(\omega>1\), and \(\omega<\omega^{\mathrm M}<1\) when \(\omega<1\). In particular \(\omega^{\mathrm M}\neq\omega\): the odds ratio is non-collapsible, and the marginal odds ratio is strictly attenuated toward the null value \(1\).
(b) Hazard ratio. For time-to-event outcomes with a covariate \(X\) and constant conditional hazard ratio \(r=h_1(x)/h_0(x)\) across strata (no effect modification on the log-hazard scale), the marginal hazard ratio, formed from the population-averaged survival functions \(\bar S_t(u)=\mathbb E_X[\exp(-\int_0^u h_t(X)\,ds)]\), is in general not equal to \(r\) and varies with the time \(u\). Hence the hazard ratio is non-collapsible.
Why this matters (ITC). 2(a) is the reason you cannot pool marginal odds ratios across trials with different baseline risks, even when every trial is a flawless randomized experiment and the conditional odds ratio is identical in all of them. Two trials with the same drug and the same per-patient odds ratio will report different marginal odds ratios if they enrolled different risk mixes. For hazard ratios, part (b) is why almost every oncology health technology assessment, which reports a single Cox hazard ratio, is summarizing a quantity that in truth drifts with follow-up time. The consequence, not the proof, is the point; Example 11.3 Step 3 carries it into an indirect comparison.
Proof. (a) Odds ratio.
Step 1 (setup: reduce to the baseline odds). Because the conditional odds ratio is the constant \(\omega\), the treated odds equal \(\omega\) times the control odds in every stratum: \(\dfrac{\mu_1(x)}{1-\mu_1(x)} = \omega\,\dfrac{\mu_0(x)}{1-\mu_0(x)}\). Write the control odds as the random variable \(W=\mu_0(X)/(1-\mu_0(X))\ge 0\), non-degenerate by hypothesis. Then the stratum risks are \[ \mu_0(X)=\frac{W}{1+W},\qquad \mu_1(X)=\frac{\omega W}{1+\omega W}, \] since solving the constant-odds-ratio relation for the treated risk gives exactly the second expression. For any \(a>0\) define \[ A(a) \;=\; \frac{\mathbb E\!\left[\dfrac{aW}{1+aW}\right]}{\mathbb E\!\left[\dfrac{1}{1+aW}\right]} . \] Using \(1-\dfrac{aW}{1+aW}=\dfrac{1}{1+aW}\), the marginal control and treated odds are \[ \frac{\mathbb E[\mu_0]}{1-\mathbb E[\mu_0]}=\frac{\mathbb E[W/(1+W)]}{\mathbb E[1/(1+W)]}=A(1), \qquad \frac{\mathbb E[\mu_1]}{1-\mathbb E[\mu_1]}=\frac{\mathbb E[\omega W/(1+\omega W)]}{\mathbb E[1/(1+\omega W)]}=A(\omega), \] so that the marginal odds ratio is \(\omega^{\mathrm M}=A(\omega)/A(1)\). What we showed: the whole problem is now about the single function \(A\).
Step 2 (reweighting: a plain average with tilted weights). The idea is to rewrite \(A(a)\) as an ordinary average of \(W\), but with the patients reweighted. Give each patient a positive weight proportional to \(1/(1+aW)\) and renormalize so the weights sum to one; call the resulting reweighted distribution \(Q_a\). Formally, \(Q_a\) is the tilted probability measure whose density relative to the original law \(P\) of \(X\), its Radon-Nikodym derivative, is \[ \frac{dQ_a}{dP}=\frac{1/(1+aW)}{\mathbb E[1/(1+aW)]}; \] the weights are positive and integrate to one, so \(Q_a\) is a genuine probability distribution. Writing \(Q:=Q_1\) for the control tilt and keeping \(Q_\omega\) for the treated tilt, a direct computation gives \(A(1)=\mathbb E_Q[W]\) and \(A(\omega)=\omega\,\mathbb E_{Q_\omega}[W]\), because \[ \mathbb E_Q[W]=\frac{\mathbb E[W/(1+W)]}{\mathbb E[1/(1+W)]}=A(1),\qquad \mathbb E_{Q_\omega}[W]=\frac{\mathbb E[W/(1+\omega W)]}{\mathbb E[1/(1+\omega W)]}=\frac{A(\omega)}{\omega}. \] Therefore \[ \omega^{\mathrm M}=\frac{A(\omega)}{A(1)}=\omega\,\frac{\mathbb E_{Q_\omega}[W]}{\mathbb E_Q[W]} . \tag{11.10}\] What we showed: the marginal odds ratio is the conditional odds ratio \(\omega\) times a ratio of two reweighted averages of the baseline odds. If the two reweightings gave the same average, \(\omega^{\mathrm M}\) would equal \(\omega\); non-collapsibility is precisely that they do not.
Step 3 (compare the two reweighted means through a covariance). Compare \(\mathbb E_{Q_\omega}[W]\) with \(\mathbb E_Q[W]\) by writing one tilt as a reweighting of the other. The likelihood ratio (the ratio of the two weight functions) is, up to the positive constant \(c=\mathbb E[1/(1+W)]/\mathbb E[1/(1+\omega W)]\), \[ \ell(W)=\frac{dQ_\omega}{dQ}=c\,\frac{1+W}{1+\omega W}, \qquad \ell'(W)=c\,\frac{(1+\omega W)-\omega(1+W)}{(1+\omega W)^2}=c\,\frac{1-\omega}{(1+\omega W)^2}. \] For \(\omega>1\) this derivative is strictly negative, so \(\ell\) is a strictly decreasing function of \(W\), and by construction \(\mathbb E_Q[\ell(W)]=1\). Using the definition of covariance, \(\operatorname{Cov}_Q(U,V)=\mathbb E_Q[UV]-\mathbb E_Q[U]\,\mathbb E_Q[V]\), with \(U=W\), \(V=\ell(W)\), and \(\mathbb E_Q[\ell(W)]=1\), \[ \mathbb E_{Q_\omega}[W]-\mathbb E_Q[W]=\mathbb E_Q[W\,\ell(W)]-\mathbb E_Q[W]\,\mathbb E_Q[\ell(W)] =\operatorname{Cov}_Q\bigl(W,\ell(W)\bigr). \] Now \(W\) is increasing in itself while \(\ell(W)\) is decreasing in \(W\), and two oppositely monotone functions of the same random variable have non-positive covariance. Concretely, for an independent copy \(W'\) of \(W\) under \(Q\), \[ \operatorname{Cov}_Q(W,\ell(W))=\tfrac12\,\mathbb E_Q\bigl[(W-W')(\ell(W)-\ell(W'))\bigr]\le 0, \] because whenever \(W>W'\) we have \(\ell(W)<\ell(W')\) and vice versa, so each product \((W-W')(\ell(W)-\ell(W'))\le 0\); the inequality is strict because \(W\) is non-degenerate, so the product is strictly negative on an event of positive probability. Hence \(\mathbb E_{Q_\omega}[W]<\mathbb E_Q[W]\). What we showed: the treated reweighting shifts the average baseline odds strictly down, so by Equation 11.10, \(\omega^{\mathrm M}<\omega\).
Step 4 (the lower bound and the sign). For \(\omega>1\), the constant-odds-ratio relation gives \(\mu_1(x)=\omega W/(1+\omega W)>W/(1+W)=\mu_0(x)\) in every stratum, so \(\mathbb E[\mu_1]>\mathbb E[\mu_0]\); the marginal treated odds then exceed the marginal control odds and \(\omega^{\mathrm M}=A(\omega)/A(1)>1\). Combining with Step 3, \(1<\omega^{\mathrm M}<\omega\). The case \(\omega<1\) is identical with \(\ell\) strictly increasing and every inequality reversed, giving \(\omega<\omega^{\mathrm M}<1\).
(b) Hazard ratio. We exhibit a closed-form two-stratum example; a single counterexample proves both non-collapsibility and time-variation. Let \(X\in\{0,1\}\) with \(P(X=0)=P(X=1)=\tfrac12\); read \(x=0\) as a lower-risk subgroup and \(x=1\) as a higher-risk subgroup, for instance early versus advanced disease. Outcomes are exponential with constant stratum hazards \(h_0(0)=1\), \(h_0(1)=4\) under control, and a constant conditional hazard ratio \(r=\tfrac12\), so the treated hazards are \(h_1(0)=\tfrac12\), \(h_1(1)=2\). The conditional hazard ratio is \(\tfrac12\) in both strata, so \(X\) is not an effect modifier on the log-hazard scale. For an exponential stratum with hazard \(h\) the survival function is \(e^{-hu}\), so averaging over the two strata gives the population survival functions \[ \bar S_0(u)=\tfrac12 e^{-u}+\tfrac12 e^{-4u},\qquad \bar S_1(u)=\tfrac12 e^{-u/2}+\tfrac12 e^{-2u} . \] The marginal hazard is defined, as always, as minus the derivative of the log survival, \(\bar h_t(u)=-\frac{d}{du}\log\bar S_t(u)=-\bar S_t'(u)/\bar S_t(u)\); the minus sign is there because survival decreases, so the derivative of its logarithm is negative and the hazard is a positive rate. (Equivalently, the hazard is the instantaneous event rate among those still at risk, and for a mixture of exponentials it is a survival-weighted average of the stratum hazards, not a fixed average.) Differentiating, \[ \bar h_0(u)=\frac{e^{-u}+4e^{-4u}}{e^{-u}+e^{-4u}},\qquad \bar h_1(u)=\frac{\tfrac12 e^{-u/2}+2 e^{-2u}}{e^{-u/2}+e^{-2u}} . \] The marginal hazard ratio is \(\overline{\mathrm{HR}}(u)=\bar h_1(u)/\bar h_0(u)\). At \(u\to 0\) both averaged hazards reduce to the equal-weight averages, \(\bar h_0(0)=\tfrac12(1+4)=2.5\) and \(\bar h_1(0)=\tfrac12(\tfrac12+2)=1.25\), so \(\overline{\mathrm{HR}}(0)=0.5=r\). But at \(u=1\), using \(e^{-1}=0.36788\), \(e^{-4}=0.018316\), \(e^{-1/2}=0.60653\), \(e^{-2}=0.13534\), \[ \bar h_0(1)=\frac{0.36788+4(0.018316)}{0.36788+0.018316}=\frac{0.44114}{0.38620}=1.1423,\qquad \bar h_1(1)=\frac{\tfrac12(0.60653)+2(0.13534)}{0.60653+0.13534}=\frac{0.57394}{0.74187}=0.77364, \] so \(\overline{\mathrm{HR}}(1)=0.77364/1.1423=0.6773\neq 0.5=r\). The marginal hazard ratio therefore differs from the constant conditional hazard ratio and depends on the time \(u\), both of which establish non-collapsibility of the hazard ratio.
Why does the marginal hazard ratio move over time when nothing about any patient changed? Because the two arms deplete their high-hazard stratum at different rates. Early on (\(u\to 0\)) nobody has had the event yet, both arms are still the equal-weight mixtures, and \(\overline{\mathrm{HR}}(0)=r\). As time passes the high-hazard stratum \(x=1\) dies out faster, and it dies out fastest in the arm with the higher hazards, so the survivors in the two arms come to have different stratum mixes and the marginal hazard ratio drifts away from \(r\). This selection-of-the-frail is not confounding: it is a mechanical consequence of averaging survival curves, and it happens in a perfect randomized trial. \(\square\)
Corollary 11.1 (Attenuation of the marginal odds ratio toward the null) Under the hypotheses of 2(a), the marginal log odds ratio is strictly smaller in magnitude than the conditional log odds ratio: \(0<|\log\omega^{\mathrm M}|<|\log\omega|\) with the same sign. The conditional and marginal odds ratios agree, \(\omega^{\mathrm M}=\omega\), if and only if the baseline odds \(W\) are degenerate (a single stratum), in which case there is no prognostic variation to average over.
Proof. The strict inequalities \(1<\omega^{\mathrm M}<\omega\) (for \(\omega>1\)) and \(\omega<\omega^{\mathrm M}<1\) (for \(\omega<1\)) translate under the strictly increasing \(\log\) into \(0<\log\omega^{\mathrm M}<\log\omega\) and \(\log\omega<\log\omega^{\mathrm M}<0\) respectively, which is the magnitude-and-sign statement. If \(W\) is degenerate at a single value \(w_0\) (that is, the baseline odds are identical for every patient, so there is effectively one stratum of baseline risk), then \(X\) has one stratum and the covariance in the proof of 2 is zero, giving \(\mathbb E_{Q_\omega}[W]=\mathbb E_Q[W]=w_0\) and \(\omega^{\mathrm M}=\omega\) by Equation 11.10. Conversely, if \(W\) is non-degenerate the covariance is strictly negative and \(\omega^{\mathrm M}<\omega\) strictly, so equality forces degeneracy. \(\square\)
The agreement-iff-degenerate clause is the precise sense in which non-collapsibility of the odds ratio is driven by prognostic heterogeneity in the baseline risk, not by effect modification: in 2 the conditional effect is held constant, so there is no effect modification whatsoever, and yet the moment the baseline risk varies across strata, the marginal and conditional odds ratios part company. The example Example 7.5 of Chapter 7 is one numerical instance of this theorem; the present result shows the phenomenon is universal and unidirectional, and gives the change-of-measure mechanism behind it.
Remark (Non-collapsibility is not confounding). It is essential to read 2 correctly: the gap between the marginal and the conditional odds ratio is not bias, and adjusting for \(X\) does not remove it, because there is nothing to remove. The theorem is proved under the no-confounding identity Equation 11.1, the very identity that holds in a randomized trial where \(X\) is a pure prognostic factor balanced across arms by design. No backdoor path, no omitted common cause, no selection. The conditional odds ratio and the marginal odds ratio are simply two different functionals of the same confounding-free data generating mechanism, and they disagree because the logit does not commute with averaging. By Lemma 11.1 the gap is exactly the Jensen defect Equation 11.5, \(J_{\mathrm{logit}}(\mu_1)-J_{\mathrm{logit}}(\mu_0)\), which is generically nonzero for the strictly convex-then-concave logit, and which the risk difference escapes because its link is affine. Confounding is a statement about which expectation identifies the causal effect; non-collapsibility is a statement about how a correctly identified effect changes when you transform before or after averaging. The two are independent, and conflating them is a standard error. The catalogue records this under B.13. \(\blacktriangleleft\)
We summarize the collapsibility axis in a table; “collapsible” and “directly collapsible” are in the sense of Definition 11.6. The effect-modification axis is separate and is governed by Proposition 11.1: whenever a measure is non-collapsible, whether a covariate is an effect modifier on its scale can differ from the answer on the identity scale. The table below is only about collapsibility.
| effect measure | link \(g\) | collapsible | directly collapsible |
|---|---|---|---|
| risk difference | identity | yes | yes |
| risk ratio | \(\log\) | yes | no |
| odds ratio | logit | no | no |
| hazard ratio | log-hazard | no | no (also time-varying) |
11.6 Synthesis: two scale-dependent axes
Why this matters (ITC). An analyst planning an indirect comparison makes two independent decisions: which scale to report (this fixes whether the marginal estimand is transportable at all), and which covariates are effect modifiers on that scale (this fixes what must be balanced). This section shows the two decisions are genuinely independent, so neither can substitute for the other.
Effect modification and collapsibility are two independent reasons a stratified and a pooled analysis can disagree, and both are scale dependent, but they are not the same axis. It is worth stating the logical relationship cleanly, because the applied literature often collapses the two.
Effect modification (Definition 11.3) is a property of how the conditional effect \(\tau^g_{ab}(x)\) varies across strata of a covariate. Non-collapsibility (Definition 11.6) is a property of how the effect measure behaves under averaging within a population, when the conditional effect does not vary at all. The worked example below realizes all four combinations on one data set. The following proposition records that the two axes are genuinely orthogonal.
Proposition 11.5 (Effect modification and non-collapsibility are independent) Each of the four combinations occurs.
- No effect modification and collapsible: a covariate with constant conditional risk difference, on the risk-difference scale.
- Effect modification and collapsible: a covariate with non-constant conditional risk difference, on the risk-difference scale, where the marginal effect is still the exact average of the conditional effects by Proposition 11.3.
- No effect modification and non-collapsible: a covariate with constant conditional odds ratio and non-degenerate baseline risk, on the odds-ratio scale, where marginal and conditional disagree by
- Effect modification and non-collapsible: a covariate with non-constant conditional odds ratio, on the odds-ratio scale.
Proof. Combination 1 is any covariate whose conditional risk difference is constant, for instance the identity-scale reading of the risks in Proposition 11.1, where the risk difference is \(0.1\) in both strata; collapsibility holds by Proposition 11.3. Combination 2 is the worked example Example 11.3 below, whose conditional risk differences \(0.3\) and \(0.4\) differ yet whose marginal risk difference equals their average. Combination 3 is the worked example read on the odds-ratio scale, where the conditional odds ratio is constant at \(6\) and the marginal odds ratio is \(\dfrac{0.60/0.40}{0.25/0.75}=4.5\) (the arithmetic is in Example 11.3 Step 2), by 2. Combination 4 is the logit-scale reading of Proposition 11.1, whose conditional odds ratios \(2.25\) and \(1.5\) differ. All four are realized. \(\square\)
The practical upshot is a discipline for indirect comparison. An analyst must decide two things separately: on which scale to model and report (which fixes whether the measure is collapsible and hence whether the marginal estimand is even transportable in principle), and which covariates are effect modifiers on that scale (which fixes what must be balanced). Confusing the two leads to the classic errors: balancing effect modifiers on one scale while reporting a non-collapsible measure on another, or attributing a non-collapsibility gap to unmeasured confounding and “adjusting” it away. Chapter 12 turns this discipline into the transportability theorems Theorem 12.4 and Theorem 12.5, and Chapter 16 shows the failure mode Theorem 16.1 and the ecological bias Definition 16.2 that result when the two axes are mishandled.
11.7 A fully worked example
We now carry out one example end to end, with every number reproducible by hand. It exhibits effect modification and non-collapsibility simultaneously, on different scales, and then shows the downstream consequence for an indirect comparison. This completes the running trial of Example 11.1: the same four numbers, now carried through every scale and into an indirect comparison. Recall \(\mu_0(x)\) is the control risk and \(\mu_1(x)\) the treated risk among patients with covariate value \(x\).
Example 11.3 (Effect modification and non-collapsibility in one trial) A randomized trial compares an active treatment (\(t=1\)) against control (\(t=0\)) on a binary outcome, in a population stratified by a binary prognostic covariate \(X\in\{0,1\}\) with \(P(X=1)=\tfrac12\). The stratum-specific risks are \[ \mu_0(0)=0.1,\quad \mu_1(0)=0.4,\qquad \mu_0(1)=0.4,\quad \mu_1(1)=0.8 . \]
Step 1: effect-modifier status, three scales. The conditional contrasts are
- risk difference: \(\tau^{\mathrm{id}}(0)=0.4-0.1=0.3\) and \(\tau^{\mathrm{id}}(1)=0.8-0.4=0.4\). These differ, so \(X\) is an effect modifier on the risk-difference scale.
- risk ratio: \(\mathrm{RR}(0)=0.4/0.1=4\) and \(\mathrm{RR}(1)=0.8/0.4=2\). These differ, so \(X\) is an effect modifier on the risk-ratio scale.
- odds ratio: in stratum \(0\), \(\dfrac{0.4/0.6}{0.1/0.9}=\dfrac{2/3}{1/9}=6\); in stratum \(1\), \(\dfrac{0.8/0.2}{0.4/0.6}=\dfrac{4}{2/3}=6\). These are equal, so \(X\) is not an effect modifier on the odds-ratio scale; the conditional odds ratio is the constant \(\omega=6\).
So the same covariate modifies the effect on two scales and not on the third. This is Proposition 11.1 made concrete, and it already shows that “is \(X\) an effect modifier?” has no scale-free answer.
Step 2: the marginal effects. The marginal risks are the equal-weight stratum averages, \[ \mathbb E[\mu_0]=\tfrac12(0.1+0.4)=0.25,\qquad \mathbb E[\mu_1]=\tfrac12(0.4+0.8)=0.60 . \] On the three scales:
- marginal risk difference: \(0.60-0.25=0.35\). The average of the conditional risk differences is \(\tfrac12(0.3+0.4)=0.35\). They agree exactly: the risk difference is directly collapsible (Proposition 11.3), even though \(X\) is an effect modifier on this scale. This is combination 2 of
- marginal odds ratio: \(\dfrac{0.60/0.40}{0.25/0.75}=\dfrac{1.5}{1/3}=4.5\). The conditional odds ratio is \(\omega=6\). They disagree: \(\omega^{\mathrm M}=4.5\neq 6\), with \(1<4.5<6\), exactly the strict attenuation of 2 and Corollary 11.1. There is no effect modification on this scale, and yet the marginal and conditional measures differ. This is combination 3 of 3.
The two findings sit side by side: on the risk-difference scale the covariate is an effect modifier but the measure collapses perfectly; on the odds-ratio scale the covariate is not an effect modifier but the measure fails to collapse. Effect modification and non-collapsibility are different phenomena, demonstrated on one data set.
Step 3: consequence for an indirect comparison. Suppose a second trial, sharing the same conditional mean functions (conditional constancy of relative effects, Definition 14.2), enrolled a different covariate mix, \(P(X=1)=0.8\). Its marginal risks are \[ \mathbb E'[\mu_0]=0.1(0.2)+0.4(0.8)=0.34,\qquad \mathbb E'[\mu_1]=0.4(0.2)+0.8(0.8)=0.72 . \] On the risk-difference scale its marginal effect is \(0.72-0.34=0.38=0.3(0.2)+0.4(0.8)\), the population average of the same conditional risk differences. Because \(X\) is an effect modifier on this scale, the marginal effect moved from \(0.35\) to \(0.38\) purely through the change in covariate distribution; a naive unadjusted comparison of the two trials would be biased by \(0.03\) on the risk-difference scale. This is the failure-of-Bucher mechanism Theorem 16.1, and the cure is to reweight both trials to a common target distribution.
On the odds-ratio scale the second trial’s marginal odds ratio is \[ \frac{0.72/0.28}{0.34/0.66}=\frac{18/7}{17/33}=\frac{18\cdot 33}{7\cdot 17}=\frac{594}{119}=4.992 , \] compared with \(4.5\) in the first trial, even though the conditional odds ratio is the constant \(6\) in both and \(X\) is not an effect modifier on this scale. Here the marginal estimand moved from \(4.5\) to \(4.992\) not because of effect modification but because of non-collapsibility: the marginal odds ratio depends on the baseline-risk distribution, which the covariate mix changed. No covariate balancing on the logit scale can equalize these two marginal odds ratios while preserving the trials’ own populations, because the discrepancy is not about an imbalanced effect modifier; it is the structural population-dependence of a non-collapsible marginal effect. This is exactly the caveat that makes a marginal odds ratio non-transportable across populations even under the shared-effect-modifier assumption (Theorem 12.6), and it is why Chapter 12 distinguishes conditional from marginal estimands so carefully. \(\blacktriangleleft\)
The example is small enough to verify with a calculator and rich enough to contain the entire chapter: scale-dependent effect modification (Step 1), the contrast between a directly collapsible and a non-collapsible measure (Step 2), and the two distinct ways a marginal effect can fail to transport across populations (Step 3). If the reader internalizes one numerical fact from this chapter, let it be that \(0.35\) is the honest average of \(0.3\) and \(0.4\), while \(4.5\) is not equal to \(6\) even though nothing varied.
11.8 Notes and references
The separation of effect modification from statistical interaction follows Hernán and Robins (2020), Chapters 4 and 5, whose treatment emphasizes the asymmetry of effect modification and the scale dependence of additive versus multiplicative interaction; the symmetry of the product term and the surrogate-modifier phenomenon are theirs. The scale-conditional definitions of effect modifier and prognostic variable, and the four-cell table, are the formulation of Phillippo (2019) and the catalogue results E.1 and E.2; the regression operationalization Equation 11.7 is the network meta-regression model of Dias et al. (2018), Chapter 8, in the notation of Notation. The conditional and marginal estimands of Definition 11.1 and Definition 11.2, and the marginal-conditional gap Lemma 11.1, are the estimand-scale framework of Chandler and Ishak (2026), who name the two estimands and make the collapsibility gap the organizing idea; these correspond to E.0 and E.12 in the proof catalogue.
Collapsibility and non-collapsibility were defined in Chapter 7 (1), where the risk difference and risk ratio were shown collapsible and the odds ratio non-collapsible with the numerical instance Example 7.5. The refinement to direct collapsibility (Definition 11.6), and the observation that the risk ratio is collapsible but not directly collapsible (Proposition 11.4), follow Chandler and Ishak (2026); direct collapsibility is the property their transportability theorems require, and it is sharper than the constant-effect collapsibility of the classical literature. The general non-collapsibility and strict attenuation of the odds ratio, 2(a) and Corollary 11.1, are the catalogue’s B.13; the monotone-attenuation phenomenon is classical in categorical data analysis and the change-of-measure proof given here packages the convexity argument of Lemma 11.1 into a covariance inequality. The non-collapsibility and time-variation of the marginal hazard ratio, 2(b), is the survival-analysis counterpart; the closed-form two-stratum frailty computation is standard, and the same mechanism reappears in the survival chapters of this book where the marginal hazard ratio of a conditionally proportional-hazards model is shown to drift with time.
The machine-checked content is in theories/EffectModifiers.v of the companion Coq development. Theorem 11.1 (part 1) is Theorem not_EM_does_not_imply_not_PV, with the supporting lemmas counterexample_not_EM and counterexample_not_PV_fails; it formalizes that a purely prognostic variable need not be an effect modifier, the algebraic heart of the anchored-versus-unanchored distinction. Lemma 11.3 is Theorem anchored_PV_cancellation, the pointwise statement that a non-effect-modifier cancels in the link-scale contrast. The Coq file deliberately does not formalize the scale-specific population-adjustment consistency theorems, which depend on estimand scale, overlap, and model correctness beyond the algebraic abstraction; those live in the paper proofs of Part E and in Chapter 12 of this book.
Two bibliographic additions are proposed in the chapter work log for the orchestrator: the original collapsibility analysis of Greenland, Robins, and Pearl (1999), to which the term non-collapsibility and the odds-ratio attenuation are usually attributed, and the early demonstration by Gail and colleagues (1984) that omitting a balanced prognostic covariate biases a nonlinear-model treatment coefficient. The present chapter cites the keys available in references.bib.
11.9 Exercises
The exercises progress from interpretation and warm-ups to computation and proof. A starred exercise, marked \(\star\), has a full worked solution in Appendix F.
Exercise 11.1 (Interpretation.) A colleague reports that two well-conducted randomized trials of the same drug, on the same outcome, gave different odds ratios, and concludes that one of them must be biased. Give the two distinct innocent explanations this chapter provides (Section 11.1), and say for each whether adjusting for a covariate would reconcile the trials. Which explanation can occur even if both trials enrolled patients identical on every measured and unmeasured covariate?
Exercise 11.2 (Warm-up.) For the running trial Example 11.1, with \(\mu_0(0)=0.1,\mu_1(0)=0.4,\mu_0(1)=0.4,\mu_1(1)=0.8\) and \(P(X=1)=\tfrac12\): (a) compute the odds ratio within each stratum and confirm both equal \(6\) (transform, then average); (b) compute the population-average risks \(\mathbb E[\mu_0],\mathbb E[\mu_1]\) and then their odds ratio (average, then transform), confirming \(4.5\); (c) do the same on the identity scale and confirm the two orders agree at \(0.35\). Say in one sentence which scale is collapsible and why.
Exercise 11.3 (Classification.) For each of the four two-stratum tables below (equal stratum sizes, \(P(X=1)=\tfrac12\)), state (i) whether \(X\) is an effect modifier on the risk-difference scale, (ii) whether \(X\) is an effect modifier on the odds-ratio scale, and (iii) if the conditional odds ratio is constant across strata, whether the marginal odds ratio equals it (and if the conditional odds ratio varies, say so).
| table | \(\mu_0(0),\ \mu_1(0)\) | \(\mu_0(1),\ \mu_1(1)\) |
|---|---|---|
| A | \(0.2,\ 0.3\) | \(0.2,\ 0.3\) |
| B | \(0.1,\ 0.2\) | \(0.5,\ 0.6\) |
| C | \(0.1,\ 0.4\) | \(0.4,\ 0.8\) |
| D | \(0.2,\ 0.5\) | \(0.5,\ 0.8\) |
Which table is the running example, and which tables show non-collapsibility of the odds ratio with no effect modification on the odds-ratio scale?
Exercise 11.4 Constructing a scale flip. Find stratum risks \(\mu_0(0),\mu_1(0),\mu_0(1),\mu_1(1)\in(0,1)\), different from those in Proposition 11.1, such that the covariate is an effect modifier on the risk-difference scale but not on the odds-ratio scale. Verify your construction by computing both conditional contrasts in both strata. Explain why your example is combination 2, not combination 4, of 3.
Exercise 11.5 When is the risk ratio directly collapsible? Prove that on the log scale, if the conditional risk ratio is constant in \(x\), then the marginal log risk ratio equals the (then constant) conditional log risk ratio, recovering collapsibility. Then show by a two-stratum example that if the conditional risk ratio varies, the marginal log risk ratio is not the population average of the conditional log risk ratios, and determine whether it is larger or smaller than that average for your example. Relate the direction to the convexity of \(\log\).
Exercise 11.6 (\(\star\)) General monotone attenuation of the odds ratio. Reprove 2(a) for an arbitrary, possibly continuous, distribution of the baseline odds \(W\) by the monotonicity route. (The theorem’s own proof already covers this case through the covariance inequality; here you establish the same conclusion more directly, which is often easier to remember.) Specifically, with \(A(a)=\mathbb E[aW/(1+aW)]/\mathbb E[1/(1+aW)]\) and \(\omega^{\mathrm M}=A(\omega)/A(1)\), show that (a) \(a\mapsto A(a)\) is strictly increasing, so \(\omega^{\mathrm M}>1\) for \(\omega>1\); and (b) \(a\mapsto A(a)/a\) is strictly decreasing, so \(\omega^{\mathrm M}<\omega\) for \(\omega>1\). For (b), use the change-of-measure identity Equation 11.10 and the covariance inequality for oppositely monotone functions. Deduce the agreement-iff-degenerate clause of Corollary 11.1.
Exercise 11.7 Endpoints of the marginal hazard ratio. In 2(b), prove that \(\overline{\mathrm{HR}}(u)\to r\) as both \(u\to 0\) and \(u\to\infty\), and that \(\overline{\mathrm{HR}}(u)\neq r\) for \(u\in(0,\infty)\) unless the baseline hazards are equal across strata. Interpret the \(u\to\infty\) limit as a frailty-selection effect: the survivors in both arms come to be dominated by the lowest-hazard stratum. Sketch \(\overline{\mathrm{HR}}(u)\) for the given numbers and identify the time at which it departs furthest from \(r\).
Exercise 11.8 (\(\star\)) Why unanchored adjustment needs prognostic variables. Consider treatments \(A\) and \(C\) with a measured covariate \(X\) and an unmeasured covariate \(W\), and conditional means \(\mu_A(x,w)\), \(\mu_C(x,w)\) on the identity scale. (a) Suppose \(W\) is purely prognostic, \(\mu_A(x,w)-\mu_C(x,w)\) free of \(w\). Using Lemma 11.3, show that the anchored contrast \(\Delta^{\mathrm{Cond,id}}_{AC}\) is unchanged when the conditional law of \(W\) given \(X\) differs between a source and a target population, so \(W\) need not be balanced. (b) Now consider the unanchored target \(\mathbb E_{f_*}[\mu_A(X,W)]\) alone. Give explicit \(\mu_A\) and two conditional laws of \(W\) given \(X\), agreeing in \(f_X\), for which this absolute mean differs between source and target. Conclude that an imbalanced purely prognostic variable biases the unanchored estimand even though it cancels in the anchored one, and connect this to Theorem 11.1 part 1.
Exercise 11.9 Interaction in a saturated model. Write the saturated logistic model for binary \(X\) and \(T\), \(\operatorname{logit}\mu_t(x)=\beta_0+\beta_1 x+\beta_2 t+\beta_3 (xt)\). Express the four stratum risks of Example 11.3 through \(\beta_0,\dots,\beta_3\), solve for the coefficients, and verify that \(\beta_3=0\) corresponds exactly to the absence of effect modification on the logit scale (Lemma 11.2). Then compute, from the same fitted model, the marginal odds ratio \(4.5\) by averaging the fitted risks, and confirm that no value of \((\beta_0,\beta_1,\beta_2,\beta_3)\) makes the marginal odds ratio equal the conditional odds ratio \(e^{\beta_2}\) unless \(\beta_1=0\). Interpret \(\beta_1=0\) as the degeneracy clause of Corollary 11.1.
Throughout, the symbol \(\blacktriangleleft\) marks the end of a definition, example, or remark, and \(\square\) marks the end of a proof; both are only visual bookends.↩︎