10  Adjustment Methods

10.1 From identification to estimation

The previous chapter settled a question of principle: under consistency (Theorem 9.2), ignorability (Definition 9.7), and positivity (Definition 9.8), the average treatment effect (Definition 9.4) is a functional of the observable law of \((X,T,Y)\). The identification formula (Theorem 9.3), \[ \mathbb{E}[Y(t)] \;=\; \mathbb{E}_X\bigl[\,\mathbb{E}[Y \mid T=t,\,X]\,\bigr], \qquad t\in\{0,1\}, \] tells us what to compute but not how to compute it well. Two practical obstacles remain. First, the covariate vector \(X\in\mathcal{X}\) is usually high-dimensional, so stratifying on \(X\) and averaging within strata is infeasible: most strata contain a single unit, or none. Second, an estimator that plugs an estimated regression or an estimated treatment model into the identification formula inherits the error of that model, and different plug-ins fail in different ways. This chapter is about turning the identification formula into estimators that are both feasible and robust.

At bottom there are only two things one can model, and every method in this chapter is built from one or both of them. A surface here means a function defined over the covariate space \(\mathcal{X}\): hand it a patient’s covariate value \(x\) and it returns a number. The first surface is the treatment mechanism \(e(x)=P(T=1\mid X=x)\), the propensity score, the probability that a patient with covariates \(x\) received the active treatment; modeling it and reweighting the sample so that treated and control groups look alike is inverse probability weighting (IPW). The second surface is the outcome surface \(m_t(x)=\mathbb{E}[Y\mid T=t,X=x]\), the average outcome among patients with covariates \(x\) who received treatment \(t\); modeling it and averaging its predictions over the covariate distribution is standardization, also called outcome regression or the parametric g-formula. Matching, stratification, and machine-learning adjustments are not a third kind of object: each of them ultimately estimates the treatment mechanism, the outcome surface, or both. The augmented IPW (AIPW) estimator combines the two surfaces and enjoys a remarkable insurance property, made precise in Section 10.6: it returns the right answer if either the treatment model or the outcome model is correct, so getting one of the two models wrong does not by itself wreck the estimate. Finally, among all estimators that use no information beyond the identifying assumptions, there is a sharp lower bound on the asymptotic variance, encoded in a single function called the efficient influence function (EIF); AIPW attains it.

These four objects, the propensity score, IPW, standardization, and AIPW, are the within-study engine of every population-adjusted indirect comparison studied later in this book. Matching-adjusted indirect comparison (MAIC) is a weighting method in the IPW family; simulated treatment comparison (STC) is a standardization method in the outcome-regression family; and multilevel network meta-regression (ML-NMR) plays the role of the combined, doubly robust estimator. Mastering the four estimators here is therefore mastering the mathematical core of population adjustment. The development follows Rosenbaum and Rubin (1983) for the propensity score and Hernán and Robins (2020) for the estimators, and corresponds to results B.6 through B.12 in the proof catalogue.

One point of vocabulary bridges this chapter to the rest of the book. In an ordinary observational study the covariate distribution we average over is the study’s own. In an indirect treatment comparison the target covariate distribution is usually someone else’s: the covariate profile of another trial, or of the decision population that a health-technology-assessment (HTA) agency cares about. The very same reweighting and standardization machinery then does double duty. Within one study it removes confounding; across studies it transports a treatment effect from the population where it was measured to the population where it is needed. That is why the four estimators below are the engine of MAIC, STC, and ML-NMR, and why the recurring question of which covariate distribution to average over is not a technicality but the whole point of population adjustment. We flag that choice at the end of every section.

Three things this chapter deliberately does not do, so that its scope is honest. It does not handle unmeasured confounding: every result below assumes ignorability, and if some variable that affects both treatment and outcome is missing from \(X\), all four estimators are biased and no amount of weighting repairs it. That failure mode is the subject of the quantitative bias analysis of Chapter 28. It does not address the scale on which effects fail to be collapsible: the whole chapter works on the difference (linear) scale, where the marginal and conditional contrasts coincide, and the odds-ratio and hazard-ratio complications are deferred to Chapter 11 (Definition 7.7). And it does not yet use network structure: here there is a single study with two arms, and the step to a network of trials is Chapter 15.

10.1.1 A running clinical example

Before any theorem, here is a deliberately tiny clinical scenario that we return to at every step and compute in full in Section 10.8. It is small enough to check by hand and rich enough to show why adjustment is needed at all.

Example 10.1 (Two diabetes drugs) A hospital compares two treatments for type 2 diabetes: standard care, the control \(T=0\), and a new drug, the active treatment \(T=1\). The outcome \(Y\) is the reduction in a symptom score after six months, larger being better. The one baseline covariate that matters is disease severity \(X\in\{0,1\}\), with \(x=0\) “mild” and \(x=1\) “severe”; half the patients are severe. Two facts about the data generate the entire lesson of the chapter.

• Clinicians tend to give the new drug to the sicker patients: a mild patient receives it with probability \(e(0)=0.2\), a severe patient with probability \(e(1)=0.6\). This is the treatment mechanism.
• Severe patients improve more no matter which drug they take. The average outcomes are as follows.

Severity \(X\)	Share of patients	Chance of new drug \(e(x)\)	Mean \(Y\) on new drug \(m_1(x)\)	Mean \(Y\) on standard care \(m_0(x)\)
mild (\(x=0\))	\(50\%\)	\(0.2\)	\(1\)	\(0\)
severe (\(x=1\))	\(50\%\)	\(0.6\)	\(3\)	\(2\)

Within each severity group the new drug adds exactly \(1\) point, \(m_1(x)-m_0(x)=1\) for both \(x\), so the true average benefit is \(\tau=1\). \(\blacktriangleleft\)

Why this example is the whole chapter in miniature. Severity is a confounder: it pushes patients toward the new drug and it independently raises the outcome. If we simply compare the average outcome of everyone who got the new drug against everyone who got standard care, the new-drug group is crowded with severe, high-outcome patients, and the comparison flatters the new drug. The short calculation repeated in Section 10.8 gives a naive contrast of \(1.83\) against the true \(\tau=1\): an overstatement of \(0.83\) that is pure confounding. Every method in this chapter is a disciplined way to undo that crowding, either by reweighting the groups so their severity mix matches (inverse probability weighting, Section 10.4), by predicting each patient’s outcome under each drug and averaging (standardization, Section 10.5), or by combining the two (augmented IPW, Section 10.6). Keep the two numbers \(e(0)=0.2\) and \(e(1)=0.6\) in view; the small \(0.2\) is where all the later trouble with variance and overlap comes from.

10.1.2 Standing assumptions and notation

Throughout we follow the Notation. We observe \((X,T,Y)\) with \(X\in\mathcal{X}\) a baseline covariate vector, \(T\in\{0,1\}\) a binary treatment, and \(Y\in\mathbb{R}\) the outcome. The potential outcomes \(Y(0),Y(1)\) (Definition 9.1) satisfy SUTVA (Definition 9.3), so that the observed outcome is \(Y=TY(1)+(1-T)Y(0)\) (Theorem 9.2). We maintain the three identifying assumptions of the previous chapter as a standing hypothesis for the whole chapter.

Remark. Standing assumptions (A). Unless stated otherwise:

• (Consistency) \(Y=TY(1)+(1-T)Y(0)\) (Theorem 9.2).
• (Strong ignorability) \((Y(0),Y(1))\perp T \mid X\) (Definition 9.7).
• (Strict positivity) there is \(\varepsilon\in(0,\tfrac12)\) with \(\varepsilon \le e(X) \le 1-\varepsilon\) almost surely (Definition 9.8), where \(e(x)=P(T=1\mid X=x)\).
• (Integrability) \(\mathbb{E}[Y(t)^2]<\infty\) for \(t\in\{0,1\}\).

The conditional-independence symbol \(\perp\), the conditional-expectation tower property, and the graphoid axioms are those of Chapter 4 (Definition 4.6, Theorem 4.4, Theorem 4.5).

Remark. Plain-English translation of the assumptions. In the language of a trial:

• Consistency says the outcome we recorded for a patient really is that patient’s outcome under the treatment they actually received, with no interference between patients (Definition 9.3): no hidden versions of the treatment.
• Ignorability says that once we condition on the measured covariates \(X\), the treatment a patient received carries no further information about how they would fare under either treatment. In one phrase: no unmeasured confounding given \(X\). This is the one assumption that cannot be checked from data, and the one Chapter 28 revisits.
• Strict positivity says every patient had a genuine chance, bounded away from \(0\) and \(1\), of receiving either treatment: no patient was a foregone conclusion. In Example 10.1 the smallest chance is \(e(0)=0.2\), safely positive.
• Integrability says outcomes have finite variance, so the averages and variances below all exist. Every bounded outcome, and every realistic clinical score, satisfies it.

Remark. A few symbols and terms, gathered in one place. A novice can read the estimator sections knowing only the plain-language meanings; the precise measure-theoretic versions are collected here for reference and are used mainly inside proofs.

• \(P_X\) is the marginal law of \(X\), the distribution of covariates in the population. “\(P_X\)-almost every \(x\)” means “for every covariate value except possibly a set of probability zero.”
• \(\sigma(W)\) is the information carried by a random object \(W\) (formally the \(\sigma\)-algebra it generates), and \(\sigma(W)\vee\mathcal{G}\) is the join, the smallest information set containing both \(\sigma(W)\) and \(\mathcal{G}\); conditioning on it means conditioning on \(W\) and on \(\mathcal{G}\) at once.
• \(\operatorname*{ess\,inf} e(X)\) is the essential infimum, the largest number below which \(e(X)\) almost never falls; it ignores probability-zero exceptions, which is exactly the care the positivity margin needs.
• The graphoid axioms (Theorem 4.5) are bookkeeping rules for the symbol \(\perp\). Two are used repeatedly: decomposition (G2), if \(A\perp(B,C)\mid D\) then \(A\perp B\mid D\), so dropping part of the right block is legal; and contraction (G4), if \(A\perp B\mid D\) and \(A\perp C\mid(B,D)\) then \(A\perp(B,C)\mid D\), so two conditional independences combine.
• A nuisance function is a part of an estimator we must estimate but do not ultimately care about, here the propensity score \(\hat e\) and the outcome regression \(\hat m_t\). The word does not mean “confounder.”
• The symbol \(\blacktriangleleft\) marks the end of a definition, example, or remark.

For orientation, the four estimators and their optimal-variance summand are collected in Table 10.1. You do not need the symbols yet; use the table as a map of where each object is defined.

Table 10.1: Estimators and functionals used in this chapter.

Symbol	Name	What it targets	Defined in
\(\hat\psi_1^{\mathrm{IPW}}\)	IPW estimator of the treated mean	\(\psi_1=\mathbb{E}[Y(1)]\), by reweighting	Section 10.4
\(\hat\tau^{\mathrm{IPW}}\)	IPW estimator of the ATE	\(\tau=\psi_1-\psi_0\)	Section 10.4
\(\hat\tau^{\mathrm{OR}}\)	g-computation (outcome-regression) estimator	\(\tau\), by standardizing predictions	Section 10.5
\(\hat\psi_1^{\mathrm{AIPW}},\ \hat\tau^{\mathrm{AIPW}}\)	augmented IPW (doubly robust)	\(\psi_1,\ \tau\), combining both surfaces	Section 10.6
\(\varphi\)	efficient influence function	the variance-optimal estimating summand	Section 10.7

It is convenient to name two regression surfaces and to keep the counterfactual and observable versions notationally distinct, because the equality between them is the identification step and we do not want to assume it silently. Recall from the Notation that \(\mu_t(x)=\mathbb{E}[Y(t)\mid X=x]\) is the counterfactual conditional mean. We write \[ m_t(x) \;:=\; \mathbb{E}[Y \mid T=t,\,X=x] \qquad (t\in\{0,1\}) \tag{10.1}\] for the observable outcome regression. The next lemma records that, under the standing assumptions, the two coincide; this is the engine that lets every estimator below target a counterfactual mean while only ever touching observable quantities.

Lemma 10.1 (Regression identity) Under the standing assumptions (A), for \(P_X\)-almost every \(x\) and each \(t\in\{0,1\}\), \[ m_t(x) \;=\; \mu_t(x) \;=\; \mathbb{E}[Y(t)\mid X=x]. \]

Proof. Fix \(t\). Strict positivity guarantees \(P(T=t\mid X=x)>0\) for \(P_X\)-almost every \(x\), so the conditioning event \(\{T=t, X=x\}\) has positive probability and \(m_t(x)\) is well defined. On the event \(\{T=t\}\) consistency gives \(Y=Y(t)\) pointwise, hence \[ m_t(x)=\mathbb{E}[Y\mid T=t,X=x]=\mathbb{E}[Y(t)\mid T=t,X=x]. \] Strong ignorability \((Y(0),Y(1))\perp T\mid X\) implies the weak form \(Y(t)\perp T\mid X\) by decomposition (G2 of Theorem 4.5), and weak ignorability means precisely that conditioning further on \(\{T=t\}\) does not change the conditional mean: \(\mathbb{E}[Y(t)\mid T=t,X=x]=\mathbb{E}[Y(t)\mid X=x]=\mu_t(x)\). Combining the two displays gives \(m_t(x)=\mu_t(x)\). \(\square\)

Why this matters. Lemma 10.1 is the hinge on which the whole chapter turns. Every estimator below is built to recover a counterfactual mean \(\mu_t(x)=\mathbb{E}[Y(t)\mid X=x]\), a quantity about the outcome a patient would have had under a treatment they may not have received; yet every formula we can actually compute touches only the observable regression \(m_t(x)=\mathbb{E}[Y\mid T=t,X=x]\), a plain conditional average in the data. The lemma says the two are equal under the standing assumptions, so from here on we may write \(m_t\) and \(\mu_t\) interchangeably and read any observable regression as if it were the counterfactual one. If ignorability failed, this equality would break, and with it every identification result in the chapter; that is precisely why unmeasured confounding is fatal and why Chapter 28 exists. For the first few theorems we occasionally write “\(m_t=\mu_t\) under (A)” as a reminder that the coincidence is earned, not assumed.

With \(m_t=\mu_t\) in hand, the estimands of interest are the mean potential outcomes \(\psi_t:=\mathbb{E}[Y(t)]\) and their contrast, the average treatment effect \[ \tau \;=\; \psi_1-\psi_0 \;=\; \mathbb{E}[Y(1)-Y(0)] \;=\; \mathbb{E}_X[\tau(X)], \qquad \tau(x)=\mu_1(x)-\mu_0(x). \tag{10.2}\] Every theorem below is stated for the treated mean \(\psi_1\); the control mean \(\psi_0\) follows by the symmetric argument \(T\mapsto 1-T\), \(e\mapsto 1-e\), and the contrast \(\tau\) follows by linearity of expectation (Theorem 4.3).

10.2 Balancing scores

The identification formula Theorem 9.3 tells us to average treatment-specific outcomes within each covariate value \(x\). With a high-dimensional \(X\), say thirty baseline variables, this is hopeless in practice: split patients into cells defined by all thirty variables and almost every cell holds a single patient, or none, so there is nothing to average within it. Matching patients on the full \(X\) fails for the same reason, the so-called curse of dimensionality. So we look for a low-dimensional summary \(b(X)\), ideally a single number, that we can match or stratify on while still killing confounding. The precise requirement on such a summary is that, given its value, the covariates carry no further information about treatment assignment: conditional independence of covariates and treatment given the summary.

Definition 10.1 (Balancing score) A measurable function \(b:\mathcal{X}\to\mathcal{B}\) is a balancing score if \[ X \perp T \mid b(X), \] that is, within each level set \(\{x:b(x)=c\}\) the conditional law of \(X\) is the same for treated and control units.

[machine-checked: theories/BalancingScore.v, Definition is_balancing_score]

Reader translation. A balancing score is any summary of the covariates with this property: among patients who share the same score value, the treated and the untreated have the same distribution of measured baseline characteristics. Picture two overlaid histograms of severity within one score level, one for treated patients and one for untreated; balance means the two histograms coincide. Before conditioning on the score the treated are sicker on average; after conditioning on it, severity lines up, so a difference in outcomes within a score level can no longer be blamed on severity. This is the formal version of the familiar applied habit of “checking covariate balance after matching.”

Two trivial balancing scores bracket the interesting ones. The identity \(b(X)=X\) is always balancing, since \(X\perp T\mid X\) holds vacuously; it achieves no dimension reduction. At the other extreme, a constant \(b(X)\equiv c\) is balancing if and only if \(T\perp X\) outright, that is, in a completely randomized experiment. The propensity score, defined in the next section, is the optimal point between these extremes.

Remark. Reader guide: a short technical interlude. The next two lemmas are proof tools, not new causal ideas, and a reader may skim them now and return when they are first used, in the proof of Theorem 10.1 and in Step 3 of Theorem 10.3. Their entire job is bookkeeping for the symbol \(\perp\) when the treatment is binary. The one causal idea underneath is simple: a binary treatment is completely described by a single number, its conditional probability of being \(1\), so any statement about “the conditional law of \(T\)” reduces to a statement about that one regression function.

Because everything that follows manipulates conditional independence with a binary treatment, we isolate two elementary measure-theoretic facts. The first turns a statement about the conditional law of \(T\) into a statement about a single regression function. The second is the factorization of a product under conditional independence, and it is stated for integrable rather than merely bounded factors because the product we later feed into it, \(\mathbf{1}_{\{T=1\}} Y(1)\), involves the outcome \(Y(1)\), which is square-integrable under assumption (A) but need not be bounded; the extra work in its proof, the truncation argument, is there only to cover that unbounded case, and if every outcome in your problem is bounded you may ignore it. Both facts rest only on the definition Equation 4.9 and the tower property.

Lemma 10.2 (Conditional-independence criterion for a binary variable) Let \(T\in\{0,1\}\), let \(W\) be a random element, and let \(\mathcal{G}\) be a sub-\(\sigma\)-algebra. Then \(W \perp T \mid \mathcal{G}\) if and only if \[ \mathbb{E}\bigl[\mathbf{1}_{\{T=1\}}\,\big|\,\sigma(W)\vee\mathcal{G}\bigr] \;=\; \mathbb{E}\bigl[\mathbf{1}_{\{T=1\}}\,\big|\,\mathcal{G}\bigr] \qquad\text{almost surely.} \]

Proof. Write \(\pi:=\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid\mathcal{G}]\) and \(\pi_W:=\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid\sigma(W)\vee\mathcal{G}]\).

(\(\Rightarrow\)) Conditional independence (Definition 4.6) applied with the bounded test function \(f(T)=\mathbf{1}_{\{T=1\}}\) and \(g\equiv 1\) gives \(\pi_W=\pi\) immediately, since Equation 4.9 with \(g\equiv1\) states \(\mathbb{E}[f(T)\mid\sigma(W)\vee\mathcal{G}]=\mathbb{E}[f(T)\mid\mathcal{G}]\).

(\(\Leftarrow\)) Suppose \(\pi_W=\pi\). Any function of the binary variable \(T\) is affine in \(\mathbf{1}_{\{T=1\}}\): for \(h:\{0,1\}\to\mathbb{R}\), \(h(T)=h(0)+\bigl(h(1)-h(0)\bigr)\mathbf{1}_{\{T=1\}}\). Taking conditional expectations and using \(\pi_W=\pi\) shows \[ \mathbb{E}[h(T)\mid\sigma(W)\vee\mathcal{G}] =\mathbb{E}[h(T)\mid\mathcal{G}] \qquad\text{for every }h. \tag{$\ast$} \] Now let \(f\) be any bounded function of \(W\) and \(h\) any function of \(T\). By the tower property (Theorem 4.4), since \(f(W)\) is \(\sigma(W)\vee\mathcal{G}\)-measurable, \[ \mathbb{E}[f(W)\,h(T)\mid\mathcal{G}] =\mathbb{E}\bigl[f(W)\,\mathbb{E}[h(T)\mid\sigma(W)\vee\mathcal{G}]\bigm|\mathcal{G}\bigr] \overset{(\ast)}{=} \mathbb{E}\bigl[f(W)\,\mathbb{E}[h(T)\mid\mathcal{G}]\bigm|\mathcal{G}\bigr] =\mathbb{E}[h(T)\mid\mathcal{G}]\,\mathbb{E}[f(W)\mid\mathcal{G}], \] where the last step pulls out the \(\mathcal{G}\)-measurable factor \(\mathbb{E}[h(T)\mid\mathcal{G}]\). This is exactly the factorization Equation 4.9 defining \(W\perp T\mid\mathcal{G}\). \(\square\)

Reader translation, and a warning. The lemma says: to check that covariates and a binary treatment are conditionally independent, it is enough to check that adding \(W\) to what we condition on does not change the one regression \(\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid\cdot]\). What makes this work is that every function of a binary \(T\) is affine in \(\mathbf{1}_{\{T=1\}}\), so pinning down that single function pins down all of them. This is the only place we use that \(T\) is binary, and it is essential: for a treatment with three or more levels the lemma is false, and the whole shortcut of summarizing treatment by one propensity number collapses. That is why this chapter, and anchored two-arm indirect comparison generally, lives in the binary-treatment world.

Lemma 10.3 (Factorization of an integrable product) Let \(T\in\{0,1\}\) and let \(V\) be integrable with \(T\perp V\mid\mathcal{G}\) for a sub-\(\sigma\)-algebra \(\mathcal{G}\). Then \[ \mathbb{E}[\mathbf{1}_{\{T=1\}}\,V \mid \mathcal{G}] \;=\; \mathbb{E}[\mathbf{1}_{\{T=1\}}\mid\mathcal{G}]\;\mathbb{E}[V\mid\mathcal{G}] \qquad\text{almost surely.} \]

Proof. For bounded \(V\) this is the definition Equation 4.9 with \(f(T)=\mathbf{1}_{\{T=1\}}\) and \(g=V\). For general integrable \(V\), apply the bounded case to the truncations \(V_n:=(V\wedge n)\vee(-n)\), which satisfy \(T\perp V_n\mid\mathcal{G}\) because \(V_n\) is a measurable function of \(V\) (conditional independence is preserved under measurable maps of either argument, a special case of decomposition). As \(n\to\infty\), \(V_n\to V\) with \(|V_n|\le|V|\), so conditional dominated convergence (Theorem 4.2) gives \(\mathbb{E}[\mathbf{1}_{\{T=1\}}V_n\mid\mathcal{G}]\to\mathbb{E}[\mathbf{1}_{\{T=1\}}V\mid\mathcal{G}]\) and \(\mathbb{E}[V_n\mid\mathcal{G}]\to\mathbb{E}[V\mid\mathcal{G}]\) almost surely. Passing to the limit in the bounded identity yields the claim. \(\square\)

Reader translation. For bounded \(V\) the identity is immediate from the definition of conditional independence. The truncations \(V_n=(V\wedge n)\vee(-n)\) simply chop \(V\) down to the range \([-n,n]\) so the bounded case applies; then “conditional dominated convergence” is the conditional-expectation version of the familiar fact that if \(V_n\to V\) and all of them are dominated by the integrable \(|V|\), their (conditional) means converge too. Nothing causal happens here; the truncation is the price of allowing an unbounded outcome.

10.3 The propensity score and the Rosenbaum-Rubin theorem

Definition 10.2 (Propensity score) The propensity score is the conditional probability of treatment given covariates, \[ e(x) \;=\; P(T=1\mid X=x) \;=\; \mathbb{E}[\mathbf{1}_{\{T=1\}}\mid X=x], \] a measurable map \(e:\mathcal{X}\to[0,1]\). As a random variable we write \(e(X)\). \(\blacktriangleleft\)

Reader translation. The propensity score of a patient is one number in \([0,1]\): the probability, given everything we measured about them, that they received the active treatment. In Example 10.1 it takes just two values, \(e(0)=0.2\) for mild patients and \(e(1)=0.6\) for severe patients. In practice we do not know \(e\) and must estimate it, almost always by a logistic regression of the treatment indicator on the covariates (Definition 7.2, Theorem 7.3); the fitted value \(\hat e(X_i)\) is patient \(i\)’s estimated chance of treatment. Keep the true \(e\) and the fitted \(\hat e\) mentally separate: the theorems in this section are about the true \(e\), and plugging in \(\hat e\) is a further step, handled when we reach the estimators.

The propensity score collapses the entire covariate vector to a single number in \([0,1]\), regardless of the dimension of \(X\). The central theorem of Rosenbaum and Rubin (1983) is that this drastic compression loses nothing relevant to confounding: \(e(X)\) is a balancing score, and it is the coarsest one, so no balancing score can compress further.

Before the proof, a word on “coarsest,” which points the opposite way from everyday usage. Think of a balancing score as a rule for grouping patients: a fine rule makes many small groups (in the extreme, \(b(X)=X\) puts every distinct covariate profile in its own group), while a coarse rule makes a few large groups (in the extreme, a constant puts everyone together). The theorem says the propensity score is the coarsest balancing score, the one making the fewest, largest groups you can get away with while still removing confounding. It uses the least covariate information of any balancing score, which is exactly what we wanted, because fewer, larger groups are what make matching and stratification feasible at all.

Example 10.2 (Two covariates collapsing to one number) Suppose \(X=(X_1,X_2)\) with \(X_1,X_2\in\{0,1\}\), so there are four covariate profiles, and suppose the treatment probability happens to depend on them only through their sum: \(e(x)=0.2\) if \(x_1+x_2=0\), \(e(x)=0.5\) if \(x_1+x_2=1\) (the two profiles \((1,0)\) and \((0,1)\)), and \(e(x)=0.8\) if \(x_1+x_2=2\). Then the four profiles collapse to three propensity values \(\{0.2,0.5,0.8\}\): the profiles \((1,0)\) and \((0,1)\) share the score \(0.5\) and are pooled. The finer score \(b(X)=X\) keeps all four groups; the propensity score \(e(X)\) keeps only three, yet by Theorem 10.1 both remove confounding equally well. Stratifying on the scalar \(e(X)\) does the same job with fewer groups, which is the whole practical payoff of the “coarsest” property. \(\blacktriangleleft\)

Theorem 10.1 (Rosenbaum-Rubin theorem) Let \(e\) be the propensity score of Definition 10.2.

1. (Balancing.) \(e(X)\) is a balancing score: \(X\perp T\mid e(X)\).
2. (Coarsest.) If \(b\) is any balancing score, then \(e(X)\) is a measurable function of \(b(X)\); equivalently \(\sigma(e(X))\subseteq\sigma(b(X))\) up to \(P\)-null sets, so every balancing score refines \(e(X)\).

[machine-checked: theories/PropensityScore.v, Theorem propensity_score_is_balancing; theories/BalancingScore.v, Lemma propensity_is_coarsest]

Proof. (1) Apply the criterion Lemma 10.2 with \(W=X\) and \(\mathcal{G}=\sigma(e(X))\). Because \(e(X)\) is a function of \(X\), we have \(\sigma(X)\vee\sigma(e(X))=\sigma(X)\), so the left-hand regression is \[ \mathbb{E}[\mathbf{1}_{\{T=1\}}\mid X,\,e(X)] =\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid X] =e(X). \] For the right-hand regression, the tower property (Theorem 4.4) with \(\sigma(e(X))\subseteq\sigma(X)\) gives \[ \mathbb{E}[\mathbf{1}_{\{T=1\}}\mid e(X)] =\mathbb{E}\bigl[\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid X]\bigm| e(X)\bigr] =\mathbb{E}[e(X)\mid e(X)] =e(X), \] the last equality because \(e(X)\) is \(\sigma(e(X))\)-measurable. The two regressions agree, so by Lemma 10.2, \(X\perp T\mid e(X)\).

(2) Let \(b\) be a balancing score, \(X\perp T\mid b(X)\). Apply Lemma 10.2 with \(W=X\) and \(\mathcal{G}=\sigma(b(X))\). As in part (1) the left regression equals \(\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid X]=e(X)\), so the criterion yields \[ e(X) =\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid b(X)] \qquad\text{almost surely.} \tag{10.3}\] The right-hand side is, by construction of conditional expectation, a \(\sigma(b(X))\)-measurable random variable (here we use the standard measure-theoretic fact that a conditional expectation given a \(\sigma\)-algebra is always measurable with respect to that \(\sigma\)-algebra), hence almost surely equal to a measurable function of \(b(X)\) (a \(\sigma(b(X))\)-measurable random variable is, almost surely, some function of \(b(X)\) itself). Therefore \(e(X)\) is a measurable function of \(b(X)\) up to a null set, which is the claim. \(\square\)

Remark. Identity Equation 10.3 is worth isolating: for any balancing score \(b\), the propensity score is the treatment regression on \(b(X)\), \(e(X)=P(T=1\mid b(X))\). We use it again in the sufficiency theorem. Part (2) is the precise sense of “coarsest”: among all balancing scores, \(e(X)\) generates the smallest \(\sigma\)-algebra, so stratifying or matching on \(e(X)\) uses the least possible covariate information while still removing confounding.

Why this matters. Theorem 10.1 is the license for every weighting and matching method in the book. It says you may replace a thirty-variable balancing problem with a one-variable one, matching or weighting on the single number \(e(X)\) instead of on all of \(X\), without giving up the removal of measured confounding. MAIC rests on exactly this idea: it never balances covariates one at a time; it builds weights from a single score and balances the summary. Part (2), “coarsest,” is what guarantees this scalar is enough rather than a lossy shortcut.

The balancing property concerns only \((X,T)\). The result that powers adjustment is that balancing transfers ignorability from \(X\) to the score: if treatment is unconfounded given the full \(X\), it remains unconfounded given the summary \(b(X)\), in particular given the scalar \(e(X)\).

Theorem 10.2 (Propensity-score sufficiency) Assume strong ignorability \((Y(0),Y(1))\perp T\mid X\). Then for every balancing score \(b\), \[ (Y(0),Y(1)) \perp T \mid b(X). \] In particular, taking \(b=e\), \[ (Y(0),Y(1)) \perp T \mid e(X). \]

[machine-checked: theories/PropensityScore.v, Theorem propensity_score_sufficiency_discrete; theories/BalancingScore.v, Theorem balancing_score_sufficiency]

Proof. Write \(W=(Y(0),Y(1))\). We verify the criterion Lemma 10.2 for \(W\perp T\mid b(X)\), that is, \(\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid W, b(X)] =\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid b(X)]\) almost surely.

For the left side, note \(\sigma(W)\vee\sigma(b(X))\subseteq\sigma(W)\vee\sigma(X)\) (because \(b(X)\) is a function of \(X\), it carries no more information than \(X\), so \(\sigma(b(X))\subseteq\sigma(X)\)), so by the tower property (Theorem 4.4), \[ \mathbb{E}[\mathbf{1}_{\{T=1\}}\mid W, b(X)] =\mathbb{E}\bigl[\,\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid W, X]\bigm| W, b(X)\bigr]. \] Strong ignorability gives \(W\perp T\mid X\), so by Lemma 10.2 (with the roles \(W\), \(\mathcal{G}=\sigma(X)\)) the inner regression is \(\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid W, X]=\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid X]=e(X)\). Hence \[ \mathbb{E}[\mathbf{1}_{\{T=1\}}\mid W, b(X)] =\mathbb{E}[e(X)\mid W, b(X)] =e(X), \] where the last equality holds because, by the coarsest property Equation 10.3, \(e(X)\) is \(\sigma(b(X))\)-measurable, hence \(\sigma(W)\vee\sigma(b(X))\)-measurable. For the right side, Equation 10.3 gives directly \(\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid b(X)]=e(X)\). The two sides agree, so Lemma 10.2 yields \(W\perp T\mid b(X)\). The case \(b=e\) is immediate, since \(e(X)\) is trivially a function of itself and \(\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid e(X)]=e(X)\) by Theorem 10.1. \(\square\)

Why this matters. This is the theorem that makes propensity-score adjustment legitimate, not merely convenient. Theorem 10.1 balanced the covariates; this theorem shows the balance is enough for causal inference: if all confounding is captured by the measured covariates \(X\), then removing confounding on the single score \(e(X)\) removes it entirely. So a health researcher may summarize a high-dimensional patient by one propensity number, match or weight on that number, and still estimate the treatment effect without bias, provided ignorability holds. Every population-adjustment method downstream leans on exactly this reduction; when it fails, it fails because ignorability fails, never because the scalar summary was too crude.

Remark. Theorem 10.2 can also be read off the graphoid algebra of Theorem 4.5: contraction (G4) combines \(X\perp T\mid b(X)\) with \(W\perp T\mid X\) (noting that conditioning \(W\perp T\mid X\) already includes \(b(X)\), which is redundant given \(X\)) to give \((X,W)\perp T\mid b(X)\), and decomposition (G2) drops \(X\). The direct proof above is preferred here because it is self-contained and makes explicit the role of the coarsest identity Equation 10.3. The practical payoff is decisive: confounding adjustment, which seemed to require the high-dimensional \(X\), can be carried out with the scalar \(e(X)\). The three estimators of the next sections are the three ways to cash this in, namely weighting, standardization, and their combination.

10.4 Inverse probability weighting

The idea of inverse probability weighting goes back to the survey-sampling estimator of Horvitz and Thompson: if a unit enters the sample with known probability \(\pi\), weighting its contribution by \(1/\pi\) recovers a population-level total in expectation. In the causal version the “sampling probability” into the treated arm is the propensity score \(e(X)\), and weighting treated units by \(1/e(X)\) reconstructs the mean of \(Y(1)\) as if everyone had been treated.

A first look on numbers. Return to Example 10.1. Among mild patients only a fraction \(e(0)=0.2\) received the new drug, so each treated mild patient must stand in for \(1/0.2=5\) mild patients; among severe patients the fraction is \(e(1)=0.6\), so each treated severe patient stands in for \(1/0.6\approx1.67\). Weighting the observed treated outcomes by these factors rebuilds the population as if everyone had taken the new drug: the arithmetic, carried out in full in Section 10.8, gives a reweighted treated mean of exactly \(\psi_1=2\), undoing the confounding that had inflated the naive figure to \(2.5\). The theorem next states this cancellation in general; the reader can track its four steps against these numbers.

Theorem 10.3 (Horvitz-Thompson identification) Under the standing assumptions (A), \[ \mathbb{E}\!\left[\frac{T\,Y}{e(X)}\right] \;=\; \mathbb{E}[Y(1)], \qquad \mathbb{E}\!\left[\frac{(1-T)\,Y}{1-e(X)}\right] \;=\; \mathbb{E}[Y(0)]. \tag{10.4}\]

[machine-checked: theories/IPWEstimator.v, Theorem mu1_IPW_unbiased, Corollary ATE_IPW_unbiased]

Proof. We prove the first identity; the second is the image under \(T\mapsto 1-T\), \(e\mapsto 1-e\). Strict positivity makes the weight \(1/e(X)\le 1/\varepsilon\) bounded, and since \(\mathbb{E}[Y(1)^2]<\infty\) the integrand is integrable, so all expectations below are finite.

Step 1 (consistency). The factor \(T\) vanishes off \(\{T=1\}\), and on \(\{T=1\}\) consistency gives \(Y=Y(1)\); therefore \(T\,Y=T\,Y(1)\) pointwise, and \[ \mathbb{E}\!\left[\frac{T\,Y}{e(X)}\right] =\mathbb{E}\!\left[\frac{T\,Y(1)}{e(X)}\right]. \]

Step 2 (tower and pull-out). Conditioning on \(X\) (Theorem 4.4) and pulling the \(\sigma(X)\)-measurable factor \(1/e(X)\) outside the inner expectation, \[ \mathbb{E}\!\left[\frac{T\,Y(1)}{e(X)}\right] =\mathbb{E}\!\left[\frac{1}{e(X)}\,\mathbb{E}\bigl[T\,Y(1)\mid X\bigr]\right]. \]

Step 3 (ignorability factorizes). Strong ignorability implies \(Y(1)\perp T\mid X\) (G2 of Theorem 4.5). Since \(Y(1)\) is integrable, Lemma 10.3 gives \[ \mathbb{E}[T\,Y(1)\mid X] =\mathbb{E}[\mathbf{1}_{\{T=1\}}\,Y(1)\mid X] =\mathbb{E}[\mathbf{1}_{\{T=1\}}\mid X]\,\mathbb{E}[Y(1)\mid X] =e(X)\,\mu_1(X). \]

Step 4 (cancel and average). Substituting, \[ \mathbb{E}\!\left[\frac{1}{e(X)}\,e(X)\,\mu_1(X)\right] =\mathbb{E}[\mu_1(X)] =\mathbb{E}\bigl[\mathbb{E}[Y(1)\mid X]\bigr] =\mathbb{E}[Y(1)], \] using the tower property once more. \(\square\)

The sample analogue is the IPW estimator \[ \hat\psi_1^{\mathrm{IPW}} =\frac1n\sum_{i=1}^n \frac{T_i Y_i}{\hat e(X_i)}, \qquad \hat\tau^{\mathrm{IPW}} =\frac1n\sum_{i=1}^n\!\left[\frac{T_i Y_i}{\hat e(X_i)} -\frac{(1-T_i)Y_i}{1-\hat e(X_i)}\right], \] where \(\hat e\) is an estimated propensity score, in practice a logistic regression fit (Definition 7.2, Theorem 7.3). By Theorem 10.3 the population functional equals the estimand, so \(\hat\tau^{\mathrm{IPW}}\) is consistent whenever \(\hat e\to e\) uniformly and the law of large numbers applies.

Why this matters. Theorem 10.3 is the mathematical heart of MAIC and of every weighting-based indirect comparison. It says that if you know each patient’s chance of treatment, you can rebuild the population mean under a treatment from the patients who actually received it, by up-weighting the under-represented ones. The catch lives in the word “know.” The theorem is exact for the true \(e\); the estimator plugs in a fitted \(\hat e\), so its consistency needs \(\hat e\to e\), and uniformly rather than merely pointwise. The reason is the weight \(1/\hat e(X_i)\): a single patient whose \(\hat e(X_i)\) is badly underestimated produces an enormous weight that can dominate the whole average, and pointwise closeness would let such a spike survive, while uniform closeness rules it out. This sensitivity to small \(\hat e\) is the same phenomenon the overlap discussion below quantifies.

Remark. Recipe: the IPW estimator in practice.

1. Fit a model for treatment, usually a logistic regression of \(T\) on the covariates \(X\), and read off each patient’s fitted propensity \(\hat e(X_i)\).
2. Form the weights: \(1/\hat e(X_i)\) for treated patients and \(1/(1-\hat e(X_i))\) for control patients.
3. Check overlap before trusting anything: look for very small \(\hat e(X_i)\) or very large weights, and compute the effective sample size \(\mathrm{ESS}=(\sum_i w_i)^2/\sum_i w_i^2\) (Proposition 10.1). A small ESS is a red flag.
4. Compute the weighted mean outcome in each arm and take the difference; in practice use the normalized (Hájek) form of Section 10.4.1.

10.4.1 Stabilized weights

The raw Horvitz-Thompson estimator divides by \(\hat e(X_i)\), which can be tiny, producing a few enormous weights and a large variance. A first repair multiplies the weights by a constant, the marginal treatment probability, which leaves the estimand unchanged but rescales the weights to average one.

Lemma 10.4 (Stabilized and normalized weights) Under the standing assumptions (A):

1. The treated inverse-probability weight has unit mean, \(\displaystyle \mathbb{E}\!\left[\frac{T}{e(X)}\right]=1\).
2. Consequently the Hájek (normalized) estimand equals the treated mean, \[ \frac{\mathbb{E}[T\,Y/e(X)]}{\mathbb{E}[T/e(X)]} =\mathbb{E}[Y(1)]. \]
3. The stabilized weight \(sw=T\,p_1/e(X)+(1-T)\,p_0/(1-e(X))\), with \(p_1=P(T=1)\) and \(p_0=P(T=0)\), also has unit mean, \(\mathbb{E}[sw]=1\).

Proof. (1) By the tower property and the definition of \(e\), \(\mathbb{E}[T/e(X)] =\mathbb{E}\bigl[\mathbb{E}[T\mid X]/e(X)\bigr] =\mathbb{E}[e(X)/e(X)]=\mathbb{E}[1]=1\). (2) Divide the first identity of Equation 10.4, which equals \(\mathbb{E}[Y(1)]\), by the unit constant of part (1). (3) Linearity (Theorem 4.3) and the conditional computation \(\mathbb{E}[T\,p_1/e(X)]=p_1\,\mathbb{E}[T/e(X)]=p_1\) and likewise \(\mathbb{E}[(1-T)p_0/(1-e(X))]=p_0\) give \(\mathbb{E}[sw]=p_1+p_0=1\). \(\square\)

The Hájek estimator \(\sum_i T_iY_i/\hat e(X_i)\big/\sum_i T_i/\hat e(X_i)\) replaces the population denominator \(\mathbb{E}[T/e(X)]=1\) by its sample version, which self-corrects when weights happen to be large by chance; it is the form used in practice and is the population analogue of the matching weights in MAIC. The estimand is unchanged by part (2), only the finite-sample variance improves.

The three weight schemes are easy to conflate, so Table 10.2 lays them side by side.

Table 10.2: Raw, stabilized, and Hájek (normalized) weights. All three leave the IPW estimand unchanged; they differ only in finite-sample variance.

Scheme	Treated weight	Mean weight	Finite-sample variance	Used in practice?
Raw (Horvitz-Thompson)	\(T/\hat e(X)\)	\(1\) at truth	largest; a few tiny \(\hat e\) dominate	rarely alone
Stabilized	\(T\,\hat p_1/\hat e(X)\)	\(1\)	smaller; rescaled by \(\hat p_1=\widehat{P(T=1)}\)	in marginal structural models
Hájek (normalized)	\(T/\hat e(X)\), then divide the estimator by \(\sum_i T_i/\hat e(X_i)\)	\(1\) by construction	smallest; self-correcting denominator	yes, the default and the MAIC form

Reader translation. All three rows estimate the same quantity, the treated mean \(\psi_1\); they differ only in how noisy the estimate is in a finite sample. Raw weights can be dominated by one patient with a tiny \(\hat e\). Stabilized weights multiply by the marginal treatment probability, shrinking the spread without touching the target. The Hájek estimator divides by the sample sum of the weights, so if the weights happen to come out large by chance the denominator grows to match and cancels the accident. That self-correction is why the Hájek form is the practical default.

From IPW to MAIC. This is the exact point where the abstract machinery becomes the method a health researcher runs. MAIC is IPW with two specializations: the propensity model is reparametrized as \(w(x)=\exp(\alpha+\boldsymbol{\beta}^\top x)\), an unnormalized weight rather than a fitted logistic probability, and the weights are normalized to the Hájek form so they average one. Everything proved here, unbiasedness of the reweighted mean and the overlap penalty below, transfers verbatim to MAIC; the MAIC chapter only adds a recipe for choosing \(\alpha\) and \(\boldsymbol{\beta}\) by matching covariate moments to the target population.

10.4.2 Variance and the cost of poor overlap

Identification Equation 10.4 holds for any positivity margin \(\varepsilon>0\), but the variance of the IPW integrand is governed by how small \(e(X)\) can be. The following proposition makes the dependence explicit and connects it to the effective sample size of Notation.

Proposition 10.1 (Variance inflation under poor overlap) Let \(W=T/e(X)\) be the treated weight and let \(Z=T\,Y/e(X)\) be the treated IPW integrand. Under the standing assumptions (A):

1. \(\displaystyle \mathrm{Var}(Z) =\mathbb{E}\!\left[\frac{\mathbb{E}[Y^2\mid T=1,X]}{e(X)}\right]-\psi_1^2\), which increases without bound as \(\operatorname*{ess\,inf} e(X)\to 0\) whenever \(\mathbb{E}[Y^2\mid T=1,X]\) is bounded away from zero.
2. The population effective-sample-size fraction of the treated weights is \[ \frac{\bigl(\mathbb{E}[W]\bigr)^2}{\mathbb{E}[W^2]} =\frac{1}{\mathbb{E}[1/e(X)]}\;\le\; \mathbb{E}[e(X)] \;=\; P(T=1)\;\le\;1, \] with the first inequality strict unless \(e(X)\) is almost surely constant.

[machine-checked: theories/IPWEstimator.v, Lemma var_mu1_IPW_ge0, Lemma ESS_variance_bound]

Proof. (1) Since \(T\in\{0,1\}\) we have \(T^2=T\), so \(Z^2=T\,Y^2/e(X)^2\). Conditioning on \(X\) and using consistency (\(T\,Y^2=T\,Y(1)^2\), and on \(\{T=1\}\), \(\mathbb{E}[Y^2\mid T=1,X]\) is the second moment), \[ \mathbb{E}[Z^2\mid X] =\frac{1}{e(X)^2}\,\mathbb{E}[T\,Y^2\mid X] =\frac{1}{e(X)^2}\,e(X)\,\mathbb{E}[Y^2\mid T=1,X] =\frac{\mathbb{E}[Y^2\mid T=1,X]}{e(X)}, \] where \(\mathbb{E}[T\,Y^2\mid X]=e(X)\,\mathbb{E}[Y^2\mid T=1,X]\) follows by the same conditioning on \((T,X)\) used in Lemma 10.1. Taking expectations gives \(\mathbb{E}[Z^2]=\mathbb{E}[\mathbb{E}[Y^2\mid T=1,X]/e(X)]\), and subtracting \((\mathbb{E}Z)^2=\psi_1^2\) from Theorem 10.3 yields the formula. If \(\mathbb{E}[Y^2\mid T=1,X]\ge c>0\), the integrand is at least \(c/e(X)\), which \(\to\infty\) on the set where \(e(X)\to0\), so the variance diverges as the positivity margin vanishes.

(2) From part (1)’s computation with \(Y\equiv1\), \(\mathbb{E}[W^2]=\mathbb{E}[1/e(X)]\), and \(\mathbb{E}[W]=1\) by Lemma 10.4. Hence the ratio is \(1/\mathbb{E}[1/e(X)]\). The map \(u\mapsto 1/u\) is strictly convex on \((0,\infty)\), so Jensen’s inequality (Theorem 4.6) gives \(\mathbb{E}[1/e(X)]\ge 1/\mathbb{E}[e(X)]\), with equality iff \(e(X)\) is almost surely constant; inverting, \(1/\mathbb{E}[1/e(X)]\le \mathbb{E}[e(X)]=P(T=1)\le1\). \(\square\)

Part (2) is the population shadow of the finite-sample bound \(\mathrm{ESS}=(\sum_i w_i)^2/\sum_i w_i^2\le n\): extreme propensities drive the effective sample size far below the nominal one, which is exactly the failure mode that population-adjustment methods must monitor. The diagnostic recurs in the MAIC chapter as the headline warning sign of an unreliable comparison.

To make the cost of poor overlap concrete rather than symbolic, Table 10.3 tracks it through the two-stratum world of Example 10.1 as the smaller propensity \(e(0)=\varepsilon\) shrinks, holding \(e(1)=0.6\) and \(\mathrm{Var}(Y\mid T=1,X)=4\). As \(\varepsilon\) falls, the largest weight grows, the effective sample size collapses, and the efficiency bound \(V^\ast\) of Equation 10.11 (computed later) climbs.

Table 10.3: The cost of poor overlap in Example 10.1, with \(e(1)=0.6\) and conditional outcome variance \(4\) held fixed. Halving the smallest propensity roughly doubles the variance floor.

Smallest propensity \(e(0)=\varepsilon\)	Largest treated weight \(1/\varepsilon\)	Treated ESS fraction \(1/\mathbb{E}[1/e(X)]\)	Efficiency bound \(V^\ast\)
\(0.40\)	\(2.5\)	\(0.48\)	\(9.33\)
\(0.20\)	\(5.0\)	\(0.30\)	\(14.33\)
\(0.10\)	\(10.0\)	\(0.17\)	\(24.33\)
\(0.05\)	\(20.0\)	\(0.09\)	\(44.33\)

Positivity, overlap, and extrapolation, in health-research terms. Strict positivity is a yes-or-no mathematical condition (formally, the essential infimum \(\operatorname*{ess\,inf} e(X)\) stays above \(0\), meaning \(e(X)\) almost never touches zero). Its practical, graded cousin is overlap. Poor overlap means some kind of patient in the target population almost never receives one of the treatments, so the corresponding propensity sits near \(0\) or \(1\). Concretely: an eligibility criterion that all but excludes a subgroup from one arm, a prescribing habit that reserves the new drug for the sickest patients, or a sparsely populated covariate cell all produce near-zero propensities. When they do, IPW must lean on a handful of enormous weights (the table above), and an outcome-regression method must extrapolate, predicting outcomes for patient types it has barely seen under that treatment. Neither is a software bug; both are the data telling you the comparison you want is only weakly supported. This is why applied guidance insists on an overlap check, an effective-sample-size report, and honesty about which subgroups drive the result.

Why this matters. Proposition 10.1 is the theorem behind the single most-quoted diagnostic in population adjustment. When a MAIC reports an effective sample size of, say, thirty percent of the nominal one, that number is the finite-sample face of part (2): the weights have concentrated, and the comparison rests on far fewer patients than it appears to. A reviewer who understands this reads a small ESS not as a nuisance statistic but as a quantitative statement that the two trial populations barely overlap, and therefore that no method, MAIC included, can deliver a precise answer from these data.

10.5 Standardization and the g-formula

The complementary estimator models the outcome rather than the treatment. Having fitted \(m_1(x)=\mathbb{E}[Y\mid T=1,X=x]\), one predicts the treated outcome for every unit, treated or not, and averages. This is standardization, also called the parametric g-formula or outcome regression.

Theorem 10.4 (G-computation identification) Under the standing assumptions (A), \[ \mathbb{E}\bigl[m_1(X)\bigr]=\mathbb{E}[Y(1)], \qquad \mathbb{E}\bigl[m_0(X)\bigr]=\mathbb{E}[Y(0)], \tag{10.5}\] and therefore the standardized contrast equals the average treatment effect, \(\mathbb{E}[m_1(X)-m_0(X)]=\tau\).

[machine-checked: theories/OutcomeRegression.v, Theorem mu1_OR_unbiased, Corollary ATE_OR_unbiased]

Proof. By Lemma 10.1, \(m_1(x)=\mu_1(x)=\mathbb{E}[Y(1)\mid X=x]\) for \(P_X\)-almost every \(x\). Taking the expectation over \(X\) and applying the tower property (Theorem 4.4), \[ \mathbb{E}[m_1(X)] =\mathbb{E}[\mu_1(X)] =\mathbb{E}\bigl[\mathbb{E}[Y(1)\mid X]\bigr] =\mathbb{E}[Y(1)]. \] The control identity is symmetric, and the contrast follows by linearity of expectation (Theorem 4.3). This is exactly the \(t=1\) and \(t=0\) instances of the identification formula Theorem 9.3, now read as a statement about the regression surface \(m_t\). \(\square\)

The sample analogue is the g-computation estimator \[ \hat\tau^{\mathrm{OR}} =\frac1n\sum_{i=1}^n\bigl[\hat m_1(X_i)-\hat m_0(X_i)\bigr], \] where \(\hat m_t\) is a fitted regression. By Theorem 10.4 it is consistent whenever \(\hat m_t\to m_t\) uniformly, by the strong law of large numbers applied to the empirical average over \(X_i\).

Why this matters. Standardization is the logic of STC. Instead of reweighting patients, it fits the outcome surface once and then asks it a counterfactual question for every patient, “what would this patient’s outcome be under the other treatment?”, and averages the answers. For an indirect comparison the crucial freedom is which patients you average over: fit the regression where you have individual data, then average its predictions over the covariate distribution of the target trial, and you have transported the effect to that trial’s population.

Remark. Recipe: the g-computation (standardization) estimator in practice.

1. Fit an outcome model \(\hat m_t(x)\), a regression of \(Y\) on covariates \(X\) within each treatment arm (or one model with a treatment term).
2. For every patient \(i\), predict both \(\hat m_1(X_i)\) and \(\hat m_0(X_i)\), the outcome under each treatment, whether or not they received it.
3. Average each prediction over the target covariate distribution (the study sample for an ATE, or another trial’s covariates for a transported effect) and take the difference.
4. There is no weight and no effective sample size to check; the analogous danger is extrapolation, so confirm that the target covariates lie inside the range where the model was fit.

Remark. No positivity needed for the estimand, but extrapolation creeps in. Unlike IPW, the identity Equation 10.5 does not divide by \(e(X)\), so it does not require strict positivity for the population statement; only Lemma 10.1 needed positivity to define \(m_t\) on the support of \(X\). The price is that the estimator \(\hat m_t(X_i)\) is evaluated at covariate values \(X_i\) that may be far from where treatment-\(t\) data exist, so the fitted model silently extrapolates. IPW refuses to extrapolate but pays in variance; standardization extrapolates freely but pays in bias if the model is wrong. This is the complementarity that the next section exploits.

Remark. Matching as nonparametric standardization. Covariate or propensity-score matching is a third implementation of the same identification formula. Exact matching on \(X\) estimates \(m_t(x)\) nonparametrically by the within-cell treated and control averages, then averages over the empirical covariate distribution, which is Equation 10.5 with a histogram regression. Matching on the scalar \(e(X)\) is licensed by Theorem 10.2: pairs with equal propensity scores are exchangeable in treatment, so their outcome difference is an unbiased CATE estimate at that score. One-to-one propensity matching of treated to control units targets the treatment effect on the treated rather than the ATE, because it reweights toward the treated covariate distribution; this distinction returns in the anchored-versus-unanchored discussion of population adjustment.

10.5.1 The estimand depends on whom you average over: ATE versus ATT

The matching remark quietly raised the most consequential subtlety in the whole chapter, and it deserves to be stated plainly rather than in passing. Every identification result above computes the average treatment effect (ATE) by averaging \(\tau(x)=\mu_1(x)-\mu_0(x)\) over the whole covariate distribution \(P_X\). But nothing forces that choice. If instead we average over the covariate distribution of the treated patients, \(P(X\mid T=1)\), we obtain a different estimand, the average treatment effect on the treated (ATT), \[ \tau_{\mathrm{ATT}} =\mathbb{E}\bigl[\tau(X)\mid T=1\bigr] =\int \tau(x)\,dP(x\mid T=1). \tag{10.6}\] When the effect \(\tau(x)\) varies with \(x\), that is, under effect modification, the ATE and the ATT genuinely differ, because they weight the same per-patient effects by different covariate mixes. One-to-one propensity matching of each treated patient to a control targets \(\tau_{\mathrm{ATT}}\), not the ATE, precisely because it reweights everything toward the treated covariate distribution.

Reader translation. “The treatment effect” is not one number until you say in which population. Averaging the same conditional effects over a sicker population gives a different answer than averaging over a healthier one. In Example 10.1 the per-stratum effect is a constant \(1\), so the ATE and the ATT both equal \(1\) and the distinction is invisible; make the effect differ by severity and they part company.

Why this matters, and where it returns. This is the mathematical pivot of the indirect-comparison literature. Choosing between Equation 10.6 and Equation 10.2, that is, choosing which distribution \(dP(x)\) sits inside the average, is a choice of target population dressed in estimand language. When later chapters debate which population a MAIC or an STC estimates, and split anchored from unanchored comparisons, they are debating exactly this choice.

From standardization to STC. STC is this estimator with the outcome regression fit on the trial that has individual patient data and its predictions averaged over the aggregate covariate summary of the comparator trial; the ATE-versus-ATT choice above is what becomes the anchored-versus-unanchored choice there.

10.6 Augmented IPW and double robustness

We now combine the two estimators. The augmented IPW estimator starts from the IPW integrand and subtracts a regression-based correction that has mean zero when the propensity model is correct. The resulting functional has a startling property: it returns the right answer if either the propensity model or the outcome model is correct, a property called double robustness.

For a candidate propensity model \(e^\ast\) and a candidate outcome model \(m_1^\ast\), define the treated AIPW functional \[ \psi_1^{\mathrm{AIPW}}(e^\ast,m_1^\ast) :=\mathbb{E}\!\left[\frac{T\bigl(Y-m_1^\ast(X)\bigr)}{e^\ast(X)} +m_1^\ast(X)\right]. \tag{10.7}\] The sample analogue replaces \(\mathbb{E}\) by the empirical average and \((e^\ast,m_1^\ast)\) by fitted nuisances \((\hat e,\hat m_1)\). The key is an exact algebraic rewriting of the integrand, valid pointwise, that isolates a single “weighted residual” whose mean we can kill in two different ways.

Before the algebra, one reassurance. The identity below and its proof are written in terms of the unobserved potential outcome \(Y(1)\), which can look alarming for an estimator meant to run on real data. But \(Y(1)\) appears only inside the proof, as a device for showing the bias cancels; the estimator you actually compute, the sample version of Equation 10.7, uses only the observed \(X\), \(T\), \(Y\) and the fitted \(\hat e,\hat m_1\). No potential outcome is ever evaluated.

Lemma 10.5 (AIPW cancellation identity) Assume consistency \(Y=TY(1)+(1-T)Y(0)\). For any functions \(e^\ast,m_1^\ast\) with \(e^\ast(X)\neq 0\), \[ \frac{T\bigl(Y-m_1^\ast(X)\bigr)}{e^\ast(X)}+m_1^\ast(X) \;=\; \underbrace{\left(\frac{T}{e^\ast(X)}-1\right)\bigl(Y(1)-m_1^\ast(X)\bigr)}_{ =:\,R(e^\ast,m_1^\ast)} \;+\;Y(1) \qquad\text{pointwise.} \tag{10.8}\]

[machine-checked: theories/DoublyRobust.v, Lemma AIPW1_pointwise]

Proof. Since \(T\in\{0,1\}\) we have \(T(1-T)=0\), so consistency gives \(T\,Y=T\,Y(1)\). Hence the left-hand side equals \(\dfrac{T\,Y(1)-T\,m_1^\ast(X)}{e^\ast(X)}+m_1^\ast(X) =\dfrac{T\bigl(Y(1)-m_1^\ast(X)\bigr)}{e^\ast(X)}+m_1^\ast(X)\). Expanding the right-hand side, \[ \left(\frac{T}{e^\ast(X)}-1\right)\bigl(Y(1)-m_1^\ast(X)\bigr)+Y(1) =\frac{T\bigl(Y(1)-m_1^\ast(X)\bigr)}{e^\ast(X)} -\bigl(Y(1)-m_1^\ast(X)\bigr)+Y(1), \] and the last two terms collapse to \(m_1^\ast(X)\). The two sides agree. \(\square\)

Taking expectations in Equation 10.8 and using \(\mathbb{E}[Y(1)]=\psi_1\), \[ \psi_1^{\mathrm{AIPW}}(e^\ast,m_1^\ast) =\psi_1+\mathbb{E}\bigl[R(e^\ast,m_1^\ast)\bigr], \qquad R(e^\ast,m_1^\ast)=\left(\frac{T}{e^\ast(X)}-1\right)\bigl(Y(1)-m_1^\ast(X)\bigr). \tag{10.9}\] The population functional Equation 10.7 equals the target \(\psi_1\) exactly when the error term \(\mathbb{E}[R(e^\ast,m_1^\ast)]\) vanishes. (Two statements are worth keeping apart. That the functional is unbiased is one thing; that the finite-sample estimator is consistent is another, and it needs this and that the fitted \(\hat e,\hat m_1\) converge to their targets. We are checking the functional here, the population version, so “the error term vanishes” is the exact condition; the estimator-level story is Section 10.6.1.) Double robustness is the statement that this error term vanishes under either of two single-model conditions.

Theorem 10.5 (Double robustness of AIPW) Maintain the standing assumptions (A) (so that ignorability and consistency hold). Let \((e^\ast,m_1^\ast)\) be candidate nuisance functions with \(e^\ast(X)\ge\varepsilon\) almost surely. If either

1. the propensity model is correct, \(e^\ast=e\), or
2. the outcome model is correct, \(m_1^\ast=m_1\),

then the error term in Equation 10.9 vanishes and \[ \psi_1^{\mathrm{AIPW}}(e^\ast,m_1^\ast)=\mathbb{E}[Y(1)]. \] The same holds for \(\psi_0^{\mathrm{AIPW}}\) under \(T\mapsto1-T\), and hence for the AIPW contrast \(\tau^{\mathrm{AIPW}}\).

[machine-checked: theories/DoublyRobust.v, Theorem double_robustness, Corollary ATE_AIPW_double_robust]

Proof. By Equation 10.9 it suffices to show \(\mathbb{E}[R(e^\ast,m_1^\ast)]=0\) in each case. Recall \(R=\bigl(T/e^\ast(X)-1\bigr)\bigl(Y(1)-m_1^\ast(X)\bigr)\).

Case 1: \(e^\ast=e\) (propensity correct). Condition on the pair \((Y(1),X)\). The factor \(Y(1)-m_1^\ast(X)\) is \(\sigma(Y(1),X)\)-measurable, so it pulls out of the conditional expectation, leaving the conditional mean of the weight residual: \[ \mathbb{E}[R\mid Y(1),X] =\bigl(Y(1)-m_1^\ast(X)\bigr)\, \mathbb{E}\!\left[\frac{T}{e(X)}-1\,\Big|\,Y(1),X\right]. \] Strong ignorability gives \(T\perp Y(1)\mid X\), so \(\mathbb{E}[T\mid Y(1),X]=\mathbb{E}[T\mid X]=e(X)\), whence \(\mathbb{E}[T/e(X)-1\mid Y(1),X]=e(X)/e(X)-1=0\). Therefore \(\mathbb{E}[R\mid Y(1),X]=0\) and, taking expectations, \(\mathbb{E}[R]=0\). Note the outcome model \(m_1^\ast\) was arbitrary.

Case 2: \(m_1^\ast=m_1\) (outcome correct). Condition instead on \((T,X)\). Now the weight factor \(T/e^\ast(X)-1\) is \(\sigma(T,X)\)-measurable and pulls out: \[ \mathbb{E}[R\mid T,X] =\left(\frac{T}{e^\ast(X)}-1\right)\, \mathbb{E}\bigl[Y(1)-m_1(X)\mid T,X\bigr]. \] Strong ignorability gives the mean form \(\mathbb{E}[Y(1)\mid T,X]=\mathbb{E}[Y(1)\mid X]=\mu_1(X)\), and \(\mu_1(X)=m_1(X)\) by Lemma 10.1, so \(\mathbb{E}[Y(1)-m_1(X)\mid T,X]=\mu_1(X)-m_1(X)=0\). Therefore \(\mathbb{E}[R\mid T,X]=0\) and \(\mathbb{E}[R]=0\). Note the propensity model \(e^\ast\) was arbitrary (subject to \(e^\ast\ge\varepsilon\)).

In either case \(\mathbb{E}[R]=0\), so Equation 10.9 gives \(\psi_1^{\mathrm{AIPW}}=\psi_1=\mathbb{E}[Y(1)]\). The control and contrast statements follow by symmetry and linearity. \(\square\)

Why this matters. Theorem 10.5 is the reason AIPW, and its network descendant ML-NMR, are the estimators of choice when one is unsure which model to trust. In an indirect comparison you rarely know whether the treatment-selection model or the outcome model is the better specified; double robustness says you do not have to choose, because being right about either one suffices. That is the precise content of the “insurance property” promised in the introduction: one premium, two ways to collect.

Remark. The two cases are perfectly symmetric in structure: each conditions on the “other” variable so that one factor of \(R\) becomes a residual with conditional mean zero. In Case 1 it is the weight residual \(T/e-1\) that is centered, killing \(R\) whatever the outcome model; in Case 2 it is the outcome residual \(Y(1)-m_1\) that is centered, killing \(R\) whatever the propensity model. Two independent chances to be right, from one estimator. The standing assumptions (consistency, ignorability, positivity) are not what is being protected here; they are maintained throughout, and double robustness concerns only misspecification of the fitted nuisances \(\hat e,\hat m_1\).

Remark. Recipe: the AIPW (doubly robust) estimator in practice.

1. Fit both nuisance models: a propensity model \(\hat e(X)\) and an outcome model \(\hat m_t(X)\).
2. For each patient form the augmented summand \(\hat m_1(X_i)+T_i\bigl(Y_i-\hat m_1(X_i)\bigr)/\hat e(X_i)\): the outcome prediction, corrected by an inverse-probability-weighted residual.
3. Average over patients for \(\hat\psi_1^{\mathrm{AIPW}}\); do the mirror-image control calculation and subtract for \(\hat\tau^{\mathrm{AIPW}}\).
4. Read double robustness carefully: it buys two chances at the nuisance models, not immunity. If both \(\hat e\) and \(\hat m_t\) are wrong, the estimate is biased, as the worked example shows explicitly.

10.6.1 Neyman orthogonality and why machine learning can be plugged in

Double robustness is a statement about getting a nuisance model exactly right. The reason AIPW dominates modern practice is a subtler, quantitative cousin, and it is worth its own subsection because it is the single idea that lets a health researcher use flexible machine-learning fits for \(\hat e\) and \(\hat m_t\).

Remark. Neyman orthogonality, a preview. A stronger statement underlies the finite-sample behavior of AIPW: the functional Equation 10.7 is insensitive to small errors in the nuisances around the truth. Concretely, the directional derivative of \(\psi_1^{\mathrm{AIPW}}\) with respect to \(e^\ast\) at the true \((e,m_1)\) is zero, and likewise with respect to \(m_1^\ast\), so first-order nuisance error does not propagate to first-order error in the estimate. This “rate double robustness” lets one plug in slowly converging machine-learning fits for \(\hat e,\hat m_1\) and still obtain a \(\sqrt n\)-consistent, asymptotically normal \(\hat\tau^{\mathrm{AIPW}}\). Exercise Exercise 10.10 asks for the derivative computation.

Here is the idea in slower motion, because it carries a large practical payoff. Think of the estimation error as having two layers. A first-order error is one proportional to how far a nuisance estimate sits from the truth, say \(\hat e-e\). A second-order error is proportional to a product of two such gaps, for example \((\hat e-e)(\hat m_1-m_1)\), and is therefore far smaller when both gaps are small. For a plain IPW estimator the nuisance error enters at first order: a one-percent error in \(\hat e\) moves the estimate by about one percent. For AIPW the first-order term cancels, and only the second-order product survives. The picture to keep is: first-order error killed, second-order error survives. Its consequence is decisive: because only the product of the two nuisance errors matters, each nuisance may be estimated by a slow, flexible method (a random forest, a boosted tree, a Super Learner) that would be far too crude for a first-order-sensitive estimator, and the AIPW estimate is still \(\sqrt n\)-consistent and asymptotically normal as long as the product of the two convergence rates beats \(1/\sqrt n\).

A practitioner’s question, answered. Do MAIC, STC, and ML-NMR need cross-fitting (sample splitting) in practice? For the classical implementations, no: MAIC and STC fit low-dimensional parametric nuisances (a logistic weight model, a linear or generalized linear outcome model), for which the technical Donsker-type conditions hold and orthogonality already delivers valid inference without splitting. Cross-fitting becomes relevant only if one deliberately swaps in high-complexity machine-learning nuisances, the regime the orthogonality theory was built for; ML-NMR’s Bayesian fit sidesteps the issue by propagating nuisance uncertainty through the posterior.

From AIPW to ML-NMR (a preview). Read forward, AIPW is the template for ML-NMR. ML-NMR fits an outcome regression surface across a network of studies and integrates it over each target population’s covariate distribution; the sense in which it inherits AIPW’s doubly robust “insurance” is made precise only in the ML-NMR chapters, so take the analogy here as a signpost, not a finished theorem. The efficiency ceiling of the next section, however, is shared by all of them: MAIC, STC, and ML-NMR each face the same variance floor set by overlap.

10.7 The efficient influence function and the variance bound

Remark. Advanced material; the one-paragraph takeaway. This is the most technical section of the chapter, and a reader may absorb its conclusion without its proof. The conclusion: there is a best possible variance for estimating \(\psi_1\) from data satisfying only assumption (A), no estimator can beat it, and AIPW achieves it. That best variance, Equation 10.11, grows larger exactly when overlap is poor, so a wide confidence interval under poor overlap is not a defect of your method but a law of the problem. If you retain only that, you have the part that matters for practice.

What kind of object to expect (the Cramér-Rao analogy, up front). In the parametric world of Chapter 5, the Cramér-Rao bound (Theorem 5.7) sets a floor on the variance of any unbiased estimator, and the maximum likelihood estimator attains it. This section tells the same story one rung more abstract, for a model too rich to be indexed by finitely many parameters: there is again a variance floor, again a single object that encodes it, and again a specific estimator (now AIPW, not the MLE) that attains it. Read “efficient influence function” as “the semiparametric analogue of the Cramér-Rao information,” and much of the mystery lifts.

Three ideas, kept separate. Beginners conflate three things the section keeps distinct.

• An influence function \(\phi\) is a per-patient contribution describing how an estimator wiggles when the data wiggle: an estimator is asymptotically linear if it equals its target plus the average of such contributions plus negligible error (the display below). Many estimators have one.
• The efficient influence function is the particular influence function with the smallest variance among all of them. It belongs to the parameter and model, not to any one estimator.
• The semiparametric efficiency bound is that smallest variance, the number \(V^\ast=\mathbb{E}[\varphi^2]\).
• A regular estimator is one whose limiting behavior does not jump under vanishingly small changes to the data-generating law; the qualifier rules out pathological “super-efficient” estimators that beat the bound at one law and misbehave next to it. The bound applies to all regular estimators, which is every estimator used in practice.

Double robustness is about bias. The second virtue of AIPW is about variance: when both nuisances are correct, its asymptotic variance is the smallest achievable by any regular estimator that assumes no more than the identifying assumptions. The object that encodes this optimum is the efficient influence function.

An estimator \(\hat\psi_1\) is asymptotically linear with influence function \(\phi\) if \[ \sqrt n\,(\hat\psi_1-\psi_1)=\frac1{\sqrt n}\sum_{i=1}^n\phi(X_i,T_i,Y_i)+o_P(1), \qquad \mathbb{E}[\phi]=0,\ \mathbb{E}[\phi^2]<\infty, \] so that \(\sqrt n(\hat\psi_1-\psi_1)\xrightarrow{d}\mathcal{N}(0,\mathbb{E}[\phi^2])\) by the central limit theorem. The influence function determines the asymptotic variance; the question is how small \(\mathbb{E}[\phi^2]\) can be.

Lemma 10.6 (The AIPW influence function is centered) Define \[ \varphi(X,T,Y):=\frac{T\bigl(Y-m_1(X)\bigr)}{e(X)}+m_1(X)-\psi_1. \tag{10.10}\] Under the standing assumptions (A), \(\mathbb{E}[\varphi]=0\), and the sample-average AIPW estimator with the true nuisances satisfies \(\hat\psi_1^{\mathrm{AIPW}}-\psi_1=\frac1n\sum_i\varphi(X_i,T_i,Y_i)\) exactly.

Proof. Taking \(e^\ast=e\), \(m_1^\ast=m_1\) in Equation 10.7 shows \(\psi_1^{\mathrm{AIPW}}(e,m_1)=\mathbb{E}\bigl[T(Y-m_1(X))/e(X)+m_1(X)\bigr]\), which equals \(\psi_1\) by Theorem 10.5 (either case applies). Hence \(\mathbb{E}[\varphi]=\psi_1^{\mathrm{AIPW}}(e,m_1)-\psi_1=0\). The exact linearity is immediate because \(\varphi\) is, by construction, the centered AIPW summand: averaging Equation 10.10 over the sample gives \(\hat\psi_1^{\mathrm{AIPW}}-\psi_1\). \(\square\)

Theorem 10.6 (Efficient influence function and the semiparametric variance bound) Consider the nonparametric model for the law of \((X,T,Y)\) generated by the standing assumptions (A) with no further restriction on \(e\), on the regressions \(m_t\), or on the conditional outcome laws. The function \(\varphi\) of Equation 10.10 is the efficient influence function for the parameter \(\psi_1=\mathbb{E}[Y(1)]\), and the semiparametric efficiency bound is its variance, \[ V^\ast:=\mathbb{E}[\varphi^2] =\mathbb{E}\!\left[\frac{\mathrm{Var}(Y\mid T=1,X)}{e(X)}\right] +\mathrm{Var}\bigl(\mu_1(X)\bigr). \tag{10.11}\] Every regular asymptotically linear estimator \(\hat\psi_1\) has asymptotic variance at least \(V^\ast\), and the AIPW estimator with consistent nuisances attains it.

Proof. We prove the variance decomposition Equation 10.11 in full and the attainment by AIPW; we then state precisely which step is imported as an external theorem.

Variance computation. Write \(\varphi=A+B\) with \[ A=\frac{T\bigl(Y-m_1(X)\bigr)}{e(X)}, \qquad B=m_1(X)-\psi_1, \] so \(V^\ast=\mathbb{E}[A^2]+2\,\mathbb{E}[AB]+\mathbb{E}[B^2]\) by Lemma 10.6 (\(\mathbb{E}[\varphi]=0\)).

For \(\mathbb{E}[A^2]\), use \(T^2=T\) and condition on \(X\). On \(\{T=1\}\), \(\mathbb{E}\bigl[(Y-m_1(X))^2\mid T=1,X\bigr]=\mathrm{Var}(Y\mid T=1,X)\) by definition of \(m_1\), so, as in Proposition 10.1, \[ \mathbb{E}[A^2\mid X] =\frac{1}{e(X)^2}\,\mathbb{E}\bigl[T\,(Y-m_1(X))^2\mid X\bigr] =\frac{e(X)\,\mathrm{Var}(Y\mid T=1,X)}{e(X)^2} =\frac{\mathrm{Var}(Y\mid T=1,X)}{e(X)}, \] and therefore \(\mathbb{E}[A^2]=\mathbb{E}\bigl[\mathrm{Var}(Y\mid T=1,X)/e(X)\bigr]\).

For the cross term, condition on \(X\) and pull out the \(\sigma(X)\)-measurable factor \(B=m_1(X)-\psi_1\): \[ \mathbb{E}[AB\mid X] =\frac{m_1(X)-\psi_1}{e(X)}\,\mathbb{E}\bigl[T\,(Y-m_1(X))\mid X\bigr] =\frac{m_1(X)-\psi_1}{e(X)}\,e(X)\,\mathbb{E}\bigl[Y-m_1(X)\mid T=1,X\bigr]=0, \] because \(\mathbb{E}[Y\mid T=1,X]=m_1(X)\). Hence \(\mathbb{E}[AB]=0\).

For \(\mathbb{E}[B^2]\), note \(\mathbb{E}[m_1(X)]=\mathbb{E}[\mu_1(X)]=\psi_1\) by Theorem 10.4, so \(B=m_1(X)-\mathbb{E}[m_1(X)]\) is centered and \(\mathbb{E}[B^2]=\mathrm{Var}(m_1(X))=\mathrm{Var}(\mu_1(X))\), using \(m_1=\mu_1\) (Lemma 10.1). Summing the three pieces gives Equation 10.11.

Attainment. By Lemma 10.6 the AIPW estimator with true nuisances is exactly the empirical average of \(\varphi\), so its influence function is \(\varphi\) and its asymptotic variance is \(\mathbb{E}[\varphi^2]=V^\ast\). With consistent estimated nuisances the same limit holds, by the Neyman-orthogonality remark of the previous section (the nuisance-estimation error enters at second order); a rigorous statement requires sample splitting or a Donsker condition, for which we refer to Hahn (1998) and Tsiatis (2006).

The lower bound. That no regular asymptotically linear estimator can have asymptotic variance below \(V^\ast=\mathbb{E}[\varphi^2]\) is the semiparametric efficiency theorem. Its proof characterizes the tangent space of the model as the closed linear span of scores of regular parametric submodels, shows that the influence function of any regular estimator differs from \(\varphi\) by an element orthogonal to \(\varphi\), and concludes by the Pythagorean identity \(\mathbb{E}[\phi^2]=\mathbb{E}[\varphi^2]+\mathbb{E}[(\phi-\varphi)^2]\ge \mathbb{E}[\varphi^2]\). We take this characterization as a cited external result (Hahn (1998) for the ATE specifically; Vaart (1998), Chapter 25, and Tsiatis (2006) for the general theory), since the tangent-space machinery is beyond the scope of this chapter. Everything else above is proved here. \(\square\)

Remark. The decomposition Equation 10.11 is interpretable term by term. The first term, \(\mathbb{E}[\mathrm{Var}(Y\mid T=1,X)/e(X)]\), is the irreducible outcome noise in the treated arm, inflated by the inverse propensity: where treatment is rare, each treated observation must stand in for many untreated units, so its noise counts more. This is the same overlap penalty seen in Proposition 10.1. The second term, \(\mathrm{Var}(\mu_1(X))\), is the genuine spread of the conditional mean surface across the covariate distribution, present even with noiseless outcomes. The bound is the analogue, for this semiparametric problem, of the Cramér-Rao bound (Theorem 5.7) for parametric ones: a floor on variance set by the information in the model, here achieved by AIPW rather than by a maximum likelihood estimator.

Why this matters, for a health researcher. The bound \(V^\ast\) answers the question an HTA reviewer actually asks: is my MAIC variance trustworthy, and how small could it possibly be? Equation 10.11 says the achievable variance contains the inverse-propensity inflation term \(\mathbb{E}[\mathrm{Var}(Y\mid T=1,X)/e(X)]\), which blows up as overlap worsens (the \(\varepsilon\) column of Table 10.3). So when two trial populations barely overlap, every estimator, MAIC included, must have large variance; a wide confidence interval is then a faithful report of a hard problem, not a failure of the method. Conversely, a suspiciously narrow interval under poor overlap is a warning that some model is extrapolating and buying precision it has not earned.

Remark. There is no machine-checked counterpart to Theorem 10.6 in the companion Coq development; the files theories/IPWEstimator.v and theories/DoublyRobust.v formalize unbiasedness and double robustness but, as their comments note, do not formalize efficiency bounds. The bias-level results Theorem 10.1, Theorem 10.2, Theorem 10.3, Theorem 10.4, and Theorem 10.5 are tagged above with their verified counterparts.

10.8 A worked example

We now carry a single small population through all four estimators with concrete numbers, so the reader can check every claim by hand.

Example 10.3 (Two-stratum population) Let \(X\in\{0,1\}\) be a binary covariate with \(P(X=0)=P(X=1)=\tfrac12\). Let the propensity score and the (true, hence observable) conditional means be \[ e(0)=0.2,\quad e(1)=0.6;\qquad \begin{array}{c|cc} & x=0 & x=1\\\hline m_1(x)=\mu_1(x) & 1 & 3\\ m_0(x)=\mu_0(x) & 0 & 2 \end{array} \] with conditional outcome variance \(\mathrm{Var}(Y\mid T=1,X=x)=4\) in each stratum. Treatment is more likely in stratum \(x=1\), which also has higher outcomes, so \(X\) is a confounder. The conditional treatment effect is \(\tau(x)=\mu_1(x)-\mu_0(x)=1\) in both strata.

True estimands. Averaging the conditional means over \(P(X=x)=\tfrac12\), \[ \psi_1=\mathbb{E}[Y(1)]=\tfrac12(1)+\tfrac12(3)=2,\qquad \psi_0=\mathbb{E}[Y(0)]=\tfrac12(0)+\tfrac12(2)=1,\qquad \tau=\psi_1-\psi_0=1. \]

Marginal treatment probabilities. \(P(T=1)=\tfrac12 e(0)+\tfrac12 e(1)=\tfrac12(0.2)+\tfrac12(0.6)=0.4\), so \(P(T=0)=0.6\).

The naive (unadjusted) contrast is biased. Using Bayes’ rule for the covariate mix within each arm, \[ P(X=1\mid T=1)=\frac{\tfrac12 e(1)}{P(T=1)}=\frac{0.3}{0.4}=0.75, \qquad P(X=1\mid T=0)=\frac{\tfrac12(1-e(1))}{P(T=0)}=\frac{0.2}{0.6}=\tfrac13. \] Hence \[ \mathbb{E}[Y\mid T=1]=0.25(1)+0.75(3)=2.5,\qquad \mathbb{E}[Y\mid T=0]=\tfrac23(0)+\tfrac13(2)=\tfrac23\approx0.667, \] and the unadjusted contrast is \(2.5-0.667=1.833\), overstating the true \(\tau=1\) by \(0.833\). The overstatement is pure confounding: the treated arm is overweighted toward the high-outcome stratum.

IPW (exact, population level). By the conditional calculation in Theorem 10.3, \(\mathbb{E}[TY/e(X)\mid X=x]=m_1(x)\) exactly, since the weight cancels \(e(x)\). Numerically, stratum \(x=0\) contributes \(\tfrac{1}{0.2}\cdot e(0)\cdot m_1(0)=\tfrac{1}{0.2}(0.2)(1)=1\) and stratum \(x=1\) contributes \(\tfrac{1}{0.6}(0.6)(3)=3\), so \(\mathbb{E}[TY/e(X)]=\tfrac12(1)+\tfrac12(3)=2=\psi_1\). Likewise the control identity gives \(\psi_0=1\), and \(\hat\tau^{\mathrm{IPW}}=1\). The unit-mean check of Lemma 10.4 holds: \(\mathbb{E}[T/e(X)]=\tfrac12\frac{0.2}{0.2} +\tfrac12\frac{0.6}{0.6}=1\).

Overlap diagnostic. The treated weights have second moment \(\mathbb{E}[1/e(X)]=\tfrac12\frac1{0.2}+\tfrac12\frac1{0.6} =\tfrac12(5)+\tfrac12(1.667)=3.333\), so by Proposition 10.1 the population effective-sample-size fraction is \(1/3.333=0.30\). Only thirty percent of the nominal sample size is effectively used in the treated weighting, a direct consequence of the small \(e(0)=0.2\).

G-computation (exact). Standardizing the regression, \(\mathbb{E}[m_1(X)]=\tfrac12(1)+\tfrac12(3)=2=\psi_1\) and \(\mathbb{E}[m_0(X)]=\tfrac12(0)+\tfrac12(2)=1=\psi_0\), so \(\hat\tau^{\mathrm{OR}}=1\), matching IPW.

AIPW and double robustness. We test the treated functional Equation 10.7 under three misspecification scenarios, using \(\mathbb{E}[TY]=\sum_x \tfrac12 e(x)m_1(x)=\tfrac12(0.2)(1)+\tfrac12(0.6)(3)=1.0\) and \(\mathbb{E}[T/e(X)]=1\) throughout.

• Correct propensity, wrong outcome (\(e^\ast=e\), \(m_1^\ast\equiv5\)): \[ \mathbb{E}\!\left[\frac{T(Y-5)}{e(X)}+5\right] =\mathbb{E}\!\left[\frac{TY}{e(X)}\right]-5\,\mathbb{E}\!\left[\frac{T}{e(X)}\right]+5 =2-5(1)+5=2=\psi_1.\ \checkmark \]
• Wrong propensity, correct outcome (\(e^\ast\equiv0.5\), \(m_1^\ast=m_1\)): conditioning on \(X\), \(\mathbb{E}\bigl[T(Y-m_1(x))/0.5\mid X=x\bigr] =2\,e(x)\,\mathbb{E}[Y-m_1(x)\mid T=1,X=x]=0\), so the functional reduces to \(\mathbb{E}[m_1(X)]=2=\psi_1.\ \checkmark\)
• Both wrong (\(e^\ast\equiv0.5\), \(m_1^\ast\equiv5\)): \[ \mathbb{E}\!\left[\frac{T(Y-5)}{0.5}+5\right] =2\bigl(\mathbb{E}[TY]-5\,\mathbb{E}[T]\bigr)+5 =2\bigl(1-5(0.4)\bigr)+5=2(-1)+5=3\neq\psi_1. \] With both models wrong the estimate is biased to \(3\). Double robustness is two chances, not unconditional immunity.

The variance bound. With \(\mathrm{Var}(Y\mid T=1,X=x)=4\), \[ \mathbb{E}\!\left[\frac{\mathrm{Var}(Y\mid T=1,X)}{e(X)}\right] =\tfrac12\frac{4}{0.2}+\tfrac12\frac{4}{0.6} =\tfrac12(20)+\tfrac12(6.667)=13.333, \] and \(\mu_1(X)\in\{1,3\}\) each with probability \(\tfrac12\) has \(\mathrm{Var}(\mu_1(X))=\tfrac12(1-2)^2+\tfrac12(3-2)^2=1\), so by Equation 10.11 \[ V^\ast=13.333+1=14.333=\tfrac{43}{3}. \] An efficient estimator of \(\psi_1\) therefore obeys \(\sqrt n(\hat\psi_1-\psi_1)\xrightarrow{d}\mathcal{N}(0,\,14.333)\). Had the design been balanced with \(e\equiv0.4\), the first term would be \(4/0.4=10\) and the bound \(V^\ast=11\); the realized poor overlap at \(e(0)=0.2\) inflates the efficiency bound from \(11\) to \(14.33\), a quantitative statement of the cost of imbalance.

10.8.1 An observed-data version: ten patients

The calculation above is at the level of the population, where expectations are exact. In real life we hold a finite dataset. Example 10.4 shows the same two-stratum world as a table of ten patients, with the columns an analyst would actually compute: a fitted propensity, a weight, and model predictions. It is the concrete observed-data anchor promised in the introduction.

Example 10.4 (A ten-patient sample) Five mild (\(x=0\)) and five severe (\(x=1\)) patients are observed. A logistic regression of \(T\) on severity returns fitted propensities \(\hat e(0)=0.2\) and \(\hat e(1)=0.6\), matching the design of Example 10.1. The raw IPW weight is \(w_i=1/\hat e(X_i)\) for treated patients and \(w_i=1/(1-\hat e(X_i))\) for controls. The fitted outcome regression returns the stratum means as predictions, \(\hat m_1(0)=1,\ \hat m_1(1)=3,\ \hat m_0(0)=0,\ \hat m_0(1)=2\).

\(i\)	severity \(X_i\)	\(T_i\)	\(Y_i\)	\(\hat e(X_i)\)	weight \(w_i\)	\(\hat m_1(X_i)\)	\(\hat m_0(X_i)\)
1	mild	0	\(0\)	\(0.2\)	\(1.25\)	\(1\)	\(0\)
2	mild	0	\(1\)	\(0.2\)	\(1.25\)	\(1\)	\(0\)
3	mild	0	\(-1\)	\(0.2\)	\(1.25\)	\(1\)	\(0\)
4	mild	0	\(0\)	\(0.2\)	\(1.25\)	\(1\)	\(0\)
5	mild	1	\(1\)	\(0.2\)	\(5.00\)	\(1\)	\(0\)
6	severe	0	\(1\)	\(0.6\)	\(2.50\)	\(3\)	\(2\)
7	severe	0	\(3\)	\(0.6\)	\(2.50\)	\(3\)	\(2\)
8	severe	1	\(3\)	\(0.6\)	\(1.67\)	\(3\)	\(2\)
9	severe	1	\(2\)	\(0.6\)	\(1.67\)	\(3\)	\(2\)
10	severe	1	\(4\)	\(0.6\)	\(1.67\)	\(3\)	\(2\)

\(\blacktriangleleft\)

Run the four calculations on this table.

• Naive contrast. Treated mean \(=(1+3+2+4)/4=2.5\); control mean \(=(0+1-1+0+1+3)/6=0.667\); difference \(1.833\), the same confounded overstatement as in the population.
• IPW, Hájek form. The treated numerator is \(\sum_{T_i=1}w_iY_i =5(1)+1.67(3+2+4)=20\) and the treated denominator is \(\sum_{T_i=1}w_i =5+1.67(3)=10\), giving treated mean \(20/10=2.0\). The control side gives \(10/10=1.0\). Hence \(\hat\tau^{\mathrm{IPW}}=2.0-1.0=1.0\), recovering \(\tau\). The single mild treated patient, raw weight \(5\), does the heavy lifting: normalized to mean one its weight is \(2.0\), while each severe treated patient’s is \(0.67\).
• G-computation. For every patient \(\hat m_1(X_i)-\hat m_0(X_i)=1\), so the average is \(\hat\tau^{\mathrm{OR}}=1.0\).
• Effective sample size. The treated weights are \(\{5,1.67,1.67,1.67\}\), so \((\sum w)^2/\sum w^2=10^2/(25+3\cdot 1.67^2)\approx 100/33.3=3.0\): three patients’ worth of information out of the ten, a \(30\%\) effective fraction that matches the population \(1/\mathbb{E}[1/e(X)]=0.30\) computed earlier. The lone tiny propensity \(\hat e(0)=0.2\) is what shrinks it.

A caution the finite table makes vivid: these numbers landed on \(\tau=1\) exactly because the sample stratum means happened to equal the population ones. With real noise the three estimators would scatter around \(1\), and, despite AIPW’s asymptotic efficiency, at \(n=10\) its variance need not be the smallest of the three. Efficiency is a large-sample statement; Exercise 10.11 pushes the reader to feel the finite-sample behavior on harder numbers.

10.9 Notes and references

The propensity score and the two theorems of Section 10.3 are due to Rosenbaum and Rubin (1983); their Theorem 1 is the balancing property and their Theorem 2 is sufficiency, here Theorem 10.1 and Theorem 10.2. The estimators follow Hernán and Robins (2020), Part II: inverse probability weighting traces to the Horvitz-Thompson survey estimator (Horvitz and Thompson (1952)), standardization and the g-formula to Robins (1986), and the augmented, doubly robust estimator to Robins et al. (1994) with the influence-function form popularized by Bang and Robins (2005). The semiparametric efficiency bound for the ATE is Hahn (1998); the tangent-space theory underlying Theorem 10.6 is developed in Vaart (1998), Chapter 25, and Tsiatis (2006). The non-collapsibility caveat that governs which scale these marginal estimands live on is taken up in the next chapter (Definition 7.7 and its consequences); for the ATE on the difference scale, collapsibility makes the marginal and conditional contrasts agree, which is why Equation 10.2 and Equation 10.5 need no scale correction.

This chapter corresponds to results B.6 through B.12 of the proof catalogue (proofs/B_causal_foundations.md): B.6 (balancing-score theorem) and B.7 (Rosenbaum-Rubin) give Definition 10.1 and Theorem 10.1; B.8 gives Theorem 10.2; B.9 (Horvitz-Thompson) gives Theorem 10.3; B.10 (g-computation) gives Theorem 10.4; B.11 (AIPW double robustness) gives Lemma 10.5 and Theorem 10.5; and B.12 (efficient influence function) gives Theorem 10.6.

Machine-checked counterparts. The bias-level results are formalized in the companion Coq development under theories/. The tags in the theorem statements record the exact names: propensity_score_is_balancing and propensity_is_coarsest for Theorem 10.1; propensity_score_sufficiency_discrete and balancing_score_sufficiency for Theorem 10.2; mu1_IPW_unbiased and ATE_IPW_unbiased for Theorem 10.3; mu1_OR_unbiased and ATE_OR_unbiased for Theorem 10.4; AIPW1_pointwise, double_robustness, and ATE_AIPW_double_robust for Lemma 10.5 and Theorem 10.5; and var_mu1_IPW_ge0, ESS_variance_bound for Proposition 10.1. The Coq proofs are carried out in a discrete probability-mass model, with the probabilistic content of ignorability entering as named hypotheses; the pen-and-paper proofs here supply the measure-theoretic versions. The efficiency bound Theorem 10.6 has no Coq counterpart.

10.10 Exercises

A starred exercise, marked \(\star\), has a full worked solution in Appendix F. The first three exercises are accessible warm-ups and interpretation questions; the computation and proof exercises follow.

10.10.1 Warm-ups and interpretation

Exercise 10.1 (Warm-up) The confounded naive contrast. In Example 10.1, use the covariate mix within each arm to confirm that the average outcome among treated patients is \(2.5\) and among controls is \(0.667\), so the naive contrast is \(1.833\). Explain in one sentence, in terms of severity, why this overstates the true effect \(\tau=1\).

Exercise 10.2 (Warm-up) IPW and g-computation by hand. Using the ten-patient table of Example 10.4: (a) recompute the Hájek IPW treated and control means and their contrast; (b) recompute the g-computation contrast from the prediction columns; (c) recompute the treated effective sample size and say what it implies about how many patients really drive the treated mean.

Exercise 10.3 (Interpretation) Overlap and the target population. A MAIC comparing two oncology drugs reports an effective sample size of \(18\%\) of the nominal one. (a) In a sentence or two, say what this implies about how similar the two trial populations are and about the width of the resulting confidence interval. (b) A colleague proposes to fix the small effective sample size by discarding patients with extreme weights. Explain, referring to Section 10.5.1, why discarding patients changes the population the estimate refers to rather than merely reducing variance.

10.10.2 Computation and proof

Exercise 10.4 (Control-arm Horvitz-Thompson) Prove the second identity of Equation 10.4, \(\mathbb{E}[(1-T)Y/(1-e(X))]=\mathbb{E}[Y(0)]\), by repeating the four steps of the proof of Theorem 10.3 with \(T\) replaced by \(1-T\) and \(e\) by \(1-e\). State explicitly where strict positivity of \(1-e(X)\) is used.

Exercise 10.5 (Stabilized weights have unit mean (\(\star\))) Let \(sw=T\,p_1/e(X)+(1-T)\,p_0/(1-e(X))\) with \(p_1=P(T=1)\), \(p_0=P(T=0)\). (a) Show \(\mathbb{E}[sw\mid X]=p_1+p_0=1\) pointwise in \(X\), hence \(\mathbb{E}[sw]=1\). (b) In the population of Example 10.3, compute the four stabilized weight values (one per \((T,x)\) cell) and verify directly that their \(P\)-weighted average is \(1\). (c) Explain in one sentence why stabilized weights have smaller variance than the raw weights \(T/e(X)\) while leaving the IPW estimand unchanged.

Exercise 10.6 (Functions of a balancing score) Let \(b\) be a balancing score and let \(h\) be a measurable injection on the range of \(b\). (a) Show \(h\circ b\) is a balancing score. (b) Show that a constant function is a balancing score if and only if \(T\perp X\), i.e. the experiment is completely randomized. (c) Using Theorem 10.1, show that if \(b\) takes finitely many values then \(e(X)\) takes at most that many values; interpret this as a limit on how much a coarse stratification can balance.

Exercise 10.7 (Randomized experiments) Suppose \(e(X)\equiv \bar e\) is constant (a completely randomized trial). (a) Show that the IPW, g-computation, and naive difference-in-means estimands all equal \(\tau\). (b) Show that the treated IPW integrand reduces to \(TY/\bar e\) and that its population effective-sample-size fraction from Proposition 10.1 equals \(1\), so there is no overlap penalty. (c) Conclude that adjustment is unnecessary for identification under randomization, and say what it is still useful for.

Exercise 10.8 (Control-arm double robustness) State and prove the control-arm analogue of Theorem 10.5 for \(\psi_0^{\mathrm{AIPW}}(e^\ast,m_0^\ast) =\mathbb{E}[(1-T)(Y-m_0^\ast(X))/(1-e^\ast(X))+m_0^\ast(X)]\), using the control-arm cancellation identity obtained from Lemma 10.5 under \(T\mapsto1-T\). Deduce double robustness of the AIPW contrast \(\tau^{\mathrm{AIPW}}=\psi_1^{\mathrm{AIPW}}-\psi_0^{\mathrm{AIPW}}\).

Exercise 10.9 (Variance bound for the control mean (\(\star\))) Write down the efficient influence function and the bound \(V_0^\ast\) for \(\psi_0=\mathbb{E}[Y(0)]\) by symmetry with Theorem 10.6. Then, in the population of Example 10.3 with \(\mathrm{Var}(Y\mid T=0,X=x)=4\), compute \(V_0^\ast\) numerically and compare it with \(V^\ast=14.333\) for the treated mean. Explain, using the inverse-propensity inflation term, why the control bound is the smaller of the two.

Exercise 10.10 (Neyman orthogonality of AIPW) For a fixed direction \(\Delta_e\) with \(\|\Delta_e\|_\infty<\infty\), consider the path \(e_s^\ast=e+s\,\Delta_e\) and let \(g(s):=\psi_1^{\mathrm{AIPW}}(e_s^\ast,m_1)\) with the correct outcome model. (a) Differentiate under the expectation to show \(g'(0)=0\), i.e. the AIPW functional is insensitive to first-order propensity error when the outcome model is correct. (b) Symmetrically, fix the correct \(e\) and perturb \(m_1^\ast=m_1+s\Delta_m\); show the derivative in \(s\) at \(0\) is also zero. (c) Explain how (a) and (b) together justify plugging in slowly converging nuisance estimators, referencing the remark following Theorem 10.5.

Exercise 10.11 (Recomputation with new numbers) Repeat the worked example Example 10.3 with \(e(0)=0.1\), \(e(1)=0.5\), \(\mu_1(0)=2,\ \mu_1(1)=4,\ \mu_0(0)=1,\ \mu_0(1)=1\), and \(\mathrm{Var}(Y\mid T=1,X=x)=9\). (a) Compute \(\psi_1,\psi_0,\tau\) and the naive contrast, and report the confounding bias. (b) Verify the IPW and g-computation identities. (c) Compute the overlap fraction \(1/\mathbb{E}[1/e(X)]\) and the efficiency bound \(V^\ast\), and comment on how the worse overlap at \(e(0)=0.1\) changes both relative to the original example.

Bang, Heejung, and James M. Robins. 2005. “Doubly Robust Estimation in Missing Data and Causal Inference Models.” Biometrics 61 (4): 962–73.

Hahn, Jinyong. 1998. “On the Role of the Propensity Score in Efficient Semiparametric Estimation of Average Treatment Effects.” Econometrica 66 (2): 315–31.

Hernán, Miguel A., and James M. Robins. 2020. Causal Inference: What If. Chapman & Hall/CRC.

Horvitz, Daniel G., and Donovan J. Thompson. 1952. “A Generalization of Sampling Without Replacement from a Finite Universe.” Journal of the American Statistical Association 47 (260): 663–85.

Robins, James. 1986. “A New Approach to Causal Inference in Mortality Studies with a Sustained Exposure Period.” Mathematical Modelling 7 (9-12): 1393–512.

Robins, James M., Andrea Rotnitzky, and Lue Ping Zhao. 1994. “Estimation of Regression Coefficients When Some Regressors Are Not Always Observed.” Journal of the American Statistical Association 89 (427): 846–66.

Rosenbaum, Paul R., and Donald B. Rubin. 1983. “The Central Role of the Propensity Score in Observational Studies for Causal Effects.” Biometrika 70 (1): 41–55.

Tsiatis, Anastasios A. 2006. Semiparametric Theory and Missing Data. Springer.

Vaart, A. W. van der. 1998. Asymptotic Statistics. Cambridge University Press.