Add callouts for Fine and Technical Points mentioned without one

chapters/04-effect-modification.qmd

@@ -518,8 +518,19 @@ The explanation (leaving aside random variability) is **qualitative effect modif
 
 The lesson: always specify the population, or subset, to which an effect measure corresponds.
 
+::: {.callout-note title="Technical Point 4.3: Relation between marginal and conditional causal risk ratios"}
+Knowing the stratum-specific causal risk ratios is not enough to tell
+whether the marginal causal risk ratio lies above or below 1.
+The marginal ratio averages the stratum-specific ratios,
+with weights that depend on how large each stratum is
+and on how common the outcome would be in it without treatment.
+Solving for the weights turns the question into an inequality about those two ingredients
+[@hernan2020causal, p. 58].
+@prp-marginal-rr-condition below works this out for Table 4.3.
+:::
+
 ::: {#prp-marginal-rr-condition}
-## When is the marginal risk ratio below 1? (Technical Point 4.3)
+## When is the marginal risk ratio below 1?
 
 Let $\RR_l \eqdef \Pr[Z^{a=1} = 1 \mid L = l] / \Pr[Z^{a=0} = 1 \mid L = l]$, $p_l \eqdef \Pr[Z^{a=0} = 1 \mid L = l]$, and $\pi_l \eqdef \Pr[L = l]$.
 With $\RR_0 = 2$ and $\RR_1 = 0.5$, the marginal risk ratio is below 1 if and only if
@@ -539,14 +550,15 @@ $$
 \end{aligned}
 $$
 
-Substituting $\RR_0 = 2$, $\RR_1 = 0.5$, and $w(1) = 1 - w(0)$:
+Substituting $\RR_0 = 2$, $\RR_1 = 0.5$, and $w(1) = 1 - w(0)$,
+and using $\Pr[Z^{a=0} = 1] = p_0 \pi_0 + p_1 \pi_1$:
 
 $$
 \begin{aligned}
 2\, w(0) + 0.5\,(1 - w(0)) < 1
 &\iff 1.5\, w(0) < 0.5 \\
 &\iff w(0) < 1/3 \\
-&\iff 3 p_0 \pi_0 < p_0 \pi_0 + p_1 \pi_1 && \text{(definition of } w(0)\text{; } \Pr[Z^{a=0} = 1] = p_0 \pi_0 + p_1 \pi_1\text{)} \\
+&\iff 3 p_0 \pi_0 < p_0 \pi_0 + p_1 \pi_1 && \text{(definition of } w(0)\text{)} \\
 &\iff 2 p_0 \pi_0 < p_1 \pi_1 \\
 &\iff \frac{p_1}{p_0} > \frac{2 \pi_0}{\pi_1}.
 \end{aligned}

chapters/15-outcome-regression-propensity-scores.qmd

@@ -348,13 +348,23 @@ Two caveats from the book [@hernan2020causal, pp. 201-202]:
 :::
 :::
 
-::: {.notes}
+---
+
+::: {.callout-note title="Technical Point 15.1: Balancing Scores and Prognostic Scores"}
+The propensity score is the simplest **balancing score** (@def-balancing-score):
+once we fix its value, the treated and the untreated have the same distribution of $L$.
 Rosenbaum and Rubin proved that exchangeability and positivity given $L$ imply exchangeability and positivity given any balancing score
 [@rosenbaum1983central, Theorem 3; @hernan2020causal, p. 202].
+So if adjusting for $L$ suffices, adjusting for $b(L)$, and in particular for $\pi(L)$, suffices too.
+
 Graphically, $\pi(L)$ sits between $L$ and $A$, with a deterministic arrow $L \to \pi(L)$ (book Figure 15.2);
 conditioning on $\pi(L)$ blocks the backdoor paths from $A$ to $Y$ through $L$ ($A \leftarrow \pi(L) \leftarrow L \to Y$), so adjusting for it suffices.
 
-Methods based on prognostic scores need stronger assumptions and do not extend readily to time-varying treatments;
+A **prognostic score** $s(L)$ is the outcome-side counterpart:
+given $s(L)$, $L$ carries no further information about $Y^{a=0}$,
+just as, given $b(L)$, it carries none about $A$ (@def-balancing-score).
+Adjustment methods exist for both kinds of score,
+but those based on prognostic scores need stronger assumptions and do not extend readily to time-varying treatments;
 see Hansen (2008) and Abadie et al. (2013), as cited in @hernan2020causal [p. 202].
 :::
 

chapters/16-instrumental-variable-estimation.qmd

@@ -301,10 +301,19 @@ Versions 3 and 4 are special cases: they say "$e(U)$ is constant" and "$t(U)$ is
 
 ---
 
-::: {.proof}
-## Proof (Technical Point 16.5)
+::: {.callout-note title="Technical Point 16.5: Proof of the General Homogeneity Condition"}
+The proof of @thm-general-homogeneity rewrites the denominator and the numerator of the usual IV estimand
+as averages over the unmeasured confounders $U$:
+the denominator becomes $\E{t(U)}$ and the numerator $\E{e(U)\, t(U)}$.
+Zero covariance then lets the numerator factor as $\E{e(U)}\, \E{t(U)}$,
+and $\E{e(U)}$ is the average causal effect
+[@hernan2020causal, p. 217].
+The proof below fills in each step.
+:::
 
+::: {.proof}
 In Figure 16.1, $Y^a \ind (A, Z) \mid U$ for each $a$ and $U \ind Z$.
+Sums over $u$ become integrals when $U$ is continuous.
 
 **Step 1: the denominator is $\E{t(U)}$.**
 Because $U \ind Z$, $f(u \mid z) = f(u)$, so
@@ -324,13 +333,17 @@ For each $z$ and $u$,
 $$
 \begin{aligned}
 \E{A (Y^{a=1} - Y^{a=0}) \mid Z = z, U = u}
-&= \Pr(A = 1 \mid Z = z, U = u)\, \E{Y^{a=1} - Y^{a=0} \mid A = 1, Z = z, U = u} \\
-&= \Pr(A = 1 \mid Z = z, U = u)\, \E{Y^{a=1} - Y^{a=0} \mid U = u} && (Y^a \ind (A, Z) \mid U) \\
+&= \Pr(A = 1 \mid Z = z, U = u) \\
+&\qquad \times \E{Y^{a=1} - Y^{a=0} \mid A = 1, Z = z, U = u} \\
+&= \Pr(A = 1 \mid Z = z, U = u)\, \E{Y^{a=1} - Y^{a=0} \mid U = u} \\
 &= \Pr(A = 1 \mid Z = z, U = u)\, e(u), \\
-\E{Y^{a=0} \mid Z = z, U = u} &= \E{Y^{a=0} \mid U = u}. && (Y^{a=0} \ind Z \mid U)
+\E{Y^{a=0} \mid Z = z, U = u} &= \E{Y^{a=0} \mid U = u}.
 \end{aligned}
 $$
 
+The second equality uses $Y^a \ind (A, Z) \mid U$,
+and the last line uses $Y^{a=0} \ind Z \mid U$.
+
 Averaging over $U$ (with $f(u \mid z) = f(u)$):
 
 $$
@@ -514,9 +527,21 @@ Greenland calls compliers "cooperative" and defiers "non-cooperative" to avoid c
 
 ---
 
-::: {.proof}
-## Proof (Technical Point 16.7)
+::: {.callout-note title="Technical Point 16.7: Monotonicity and the Effect in the Compliers"}
+Imbens and Angrist (1994) showed that, for a dichotomous causal instrument $Z$
+and no defiers (monotonicity, condition (iv)),
+the usual IV estimand is the average causal effect in the compliers (@thm-late);
+a closely related argument for binary outcomes is due to Baker and Lindeman (1994)
+[@hernan2020causal, p. 224].
+The proof below assumes a causal instrument (Figure 16.1).
+
+For a surrogate instrument $Z$, Hernán and Robins (2006b) showed that the usual IV estimator still identifies the effect in the compliers (defined by $U_Z$),
+but only if $Z$ is independent of $A$ and $Y$ given $U_Z$ and $U_Z$ is binary, and that independence is rarely plausible unless $U_Z$ is continuous [@hernan2020causal, p. 224].
+So when an observational study has at best a surrogate for $U_Z$,
+interpreting the IV estimand as a complier effect is often doubtful.
+:::
 
+::: {.proof}
 Write the intention-to-treat effect as a weighted average over the strata (AT: always-takers, NT: never-takers, C: compliers, D: defiers):
 
 $$
@@ -547,12 +572,6 @@ $$
 $$
 :::
 
-::: {.notes}
-The result is due to Imbens and Angrist (1994);
-Baker and Lindeman (1994) had a related proof for a binary outcome [@hernan2020causal, p. 224].
-For a surrogate instrument $Z$, Hernán and Robins (2006b) showed that the usual IV estimator still identifies the effect in the compliers (defined by $U_Z$),
-but only if $Z$ is independent of $A$ and $Y$ given $U_Z$ and $U_Z$ is binary, and that independence is rarely plausible unless $U_Z$ is continuous [@hernan2020causal, p. 224].
-:::
 
 ---
 

chapters/21-g-methods-time-varying.qmd

@@ -578,6 +578,30 @@ so the plug-in estimate stays in $[0, 1]$;
 the estimator in the main text is an instance of this construction.
 :::
 
+::: {.callout-note title="Technical Point 21.5: A Plug-In K+2 Robust Estimator"}
+With a binary $Y$, the sample average of $\hat{U}_{TR}$ from Technical Point 21.4 can fall outside $[0, 1]$.
+The ICE estimator, the sample average of $\hat{b}_0(L_0)$, cannot,
+as long as both outcome regressions are logistic:
+it is a **plug-in** estimator.
+To make it triply robust as well,
+fit the outcome models so that both correction terms of $\hat{U}_{TR}$ have sample mean exactly 0;
+the average of $\hat{U}_{TR}$ then coincides with the average of $\hat{b}_0(L_0)$.
+
+- Fit the logistic model for $\hat{b}_1(L_0, L_1)$ by maximum likelihood among individuals with $A_0 = A_1 = 1$,
+  adding the single covariate $A_0 A_1 / (\hat{\pi}_0 \hat{\pi}_1)$ with its own coefficient.
+  The score equation for that coefficient is
+  $\sum_i \frac{A_{0i} A_{1i}}{\hat{\pi}_{0i} \hat{\pi}_{1i}} \paren{Y_i - \hat{b}_1(L_{0i}, L_{1i})} = 0$,
+  which is the first correction term
+  (individuals outside the fitting sample contribute 0 because $A_0 A_1 = 0$).
+- Fit the logistic model for $\hat{b}_0(L_0)$ among individuals with $A_0 = 1$,
+  with response $\hat{b}_1(L_0, L_1)$ and the extra covariate $A_0 / \hat{\pi}_0$;
+  the same argument makes the second correction term sum to 0.
+
+The doubly robust estimator of @alg-dr-time-varying, which uses the time-varying weight as a covariate,
+is an instance of this plug-in estimator
+[@hernan2020causal, p. 289].
+:::
+
 ::: {.notes}
 
 ::: {.callout-warning title="Fact-Check: A Term in the Book's First Display of the Estimator"}
@@ -742,8 +766,19 @@ Sequential exchangeability then says $H_k(\psi)$ is independent of $A_k$ given t
 
 ---
 
+::: {.callout-note title="Fine Point 21.3: G-Estimation with a Saturated Structural Nested Model"}
+With a saturated SNMM, g-estimation needs no search:
+at the true $\psi$, sequential exchangeability forces the mean of $H_k(\psi)$
+to agree between those treated and those untreated at time $k$
+within every stratum of the past.
+Each such equality is one linear equation in the parameters,
+and solving them from the last time point backwards recovers $\psi$
+[@hernan2020causal, p. 293].
+@exm-g-estimation-saturated below carries this out for @tbl-seq-rand.
+:::
+
 ::: {#exm-g-estimation-saturated}
-## G-Estimation in the Sequentially Randomized Experiment (Fine Point 21.3)
+## G-Estimation in the Sequentially Randomized Experiment
 
 **Time 1.**
 Within each stratum of $(A_0, L_1)$, the mean of $H_1(\psi)$ must not depend on $A_1$.
@@ -888,6 +923,53 @@ but for an SNMM linear in $\beta$ the estimating equation has a closed-form solu
 :::
 :::
 
+::: {.callout-note title="Technical Point 21.8: A Closed Form Estimator for Linear Structural Nested Mean Models"}
+Suppose the blip is linear in $\beta$,
+$\gamma_k(\bar{a}_{k-1}, \bar{l}_k, \beta) = \tp{\beta} R_k$,
+where $R_k = r_k(\bar{L}_k, \bar{A}_{k-1})$ is a vector of known functions,
+and the treatment model is $\logit \Pr[A_k = 1 \mid \bar{L}_k, \bar{A}_{k-1}] = \tp{\alpha} W_k$.
+Then $H_k(\beta) = Y - \tp{\beta} S_k$ with $S_k \eqdef \sum_{j=k}^{K} A_j R_j$,
+so the g-estimating equation
+
+$$\sum_{i=1}^{N} \sum_{k=0}^{K} X_{i,k}(\hat{\alpha})\, Q_{i,k}\, \paren{Y_i - \tp{S_{i,k}} \beta} = 0,
+\qquad X_{i,k}(\hat{\alpha}) \eqdef A_{i,k} - \expit\paren{\tp{\hat{\alpha}} W_{i,k}},$$
+
+is linear in $\beta$ and, when the matrix below is invertible, has the explicit solution
+
+$$\hb = \paren{\sum_{i=1}^{N} \sum_{k=0}^{K} X_{i,k}(\hat{\alpha})\, Q_{i,k}\, \tp{S_{i,k}}}^{-1}
+\sum_{i=1}^{N} \sum_{k=0}^{K} Y_i\, X_{i,k}(\hat{\alpha})\, Q_{i,k}.$$
+
+Here $Q_{i,k} = q_k(\bar{L}_{i,k}, \bar{A}_{i,k-1})$ has the dimension of $\beta$;
+its choice changes the efficiency of $\hb$ but not its consistency
+(Robins 1994 gives the optimal choice).
+
+A **multiply robust** version adds a working model $\tp{\varsigma} D_k$, with $D_k = d_k(\bar{L}_k, \bar{A}_{k-1})$,
+for $\E{H_k(\beta) \mid \bar{L}_k, \bar{A}_{k-1}} = \E{Y^{\bar{A}_{k-1}, \underline{0}_k} \mid \bar{L}_k, \bar{A}_{k-1}}$,
+and solves jointly for $(\tilde{\beta}, \tilde{\varsigma})$
+
+$$\sum_{i,k} X_{i,k}(\hat{\alpha})\, Q_{i,k}\, \paren{H_{i,k}(\beta) - \tp{\varsigma} D_{i,k}} = 0,
+\qquad
+\sum_{i,k} D_{i,k}\, \paren{H_{i,k}(\beta) - \tp{\varsigma} D_{i,k}} = 0,$$
+
+which are again linear, so the solution is in closed form.
+$\tilde{\beta}$ is consistent and asymptotically normal if, at each $k$,
+either the working outcome model or the treatment model is correct,
+which makes it $2^{K+1}$ multiply robust
+[@hernan2020causal, p. 296].
+:::
+
+::: {.notes}
+
+::: {.callout-warning title="Fact-Check: The Book's Closed-Form Display"}
+*Fact-check note*: the book's display [@hernan2020causal, p. 296] multiplies by $A_{i,k}$ outside the sum
+and defines $S_{i,k}$ as a sum of $R_{i,j}$ without the treatment indicators $A_{i,j}$.
+Since $H_k(\beta) = Y - \sum_{j \ge k} A_j \gamma_j$ (@def-candidate-counterfactuals-snmm),
+substituting the linear blip gives $\tp{\beta} \sum_{j \ge k} A_j R_j$,
+which is the form used above;
+the two agree when $K = 0$ but not in general.
+:::
+:::
+
 ### From $\hb$ to Counterfactual Means
 
 ::: {#rem-counterfactual-means-after-g-estimation}
@@ -1120,7 +1202,7 @@ which we can never know for certain in an observational study.
 ---
 
 ::: {#exm-front-door-big-g}
-## The Front Door Formula from the Big G-Formula (Technical Point 21.11)
+## The Front Door Formula from the Big G-Formula
 
 Under Figure 7.14 ($A \to M \to Y$, with $U$ a common cause of $A$ and $Y$),
 the big g-formula for $\Pr[Y^a = y]$ is
@@ -1147,17 +1229,69 @@ This proof does not require that counterfactuals $Y^m$ exist
 [@hernan2020causal, p. 301].
 :::
 
-::: {.notes}
-
-::: {#rem-front-door-other-proofs}
-## Other proofs of the front door formula
+---
 
-Technical Point 21.11 also gives a coupling argument,
-and Technical Point 21.12 a proof based on a SWIG property (Shpitser et al. 2022):
-if a fixed treatment node $a_m$ is d-separated from $B^a$ given $C^a$ on the SWIG,
-then $\Pr[B^a = b \mid C^a = c]$ does not depend on $a_m$
-[@hernan2020causal, pp. 301-302].
+::: {.callout-note title="Technical Point 21.11: A Big G-Formula Proof of the Front Door Formula"}
+Technical Point 7.4 derived the front door formula using counterfactuals $Y^m$.
+Starting instead from the big g-formula,
+the conditional independencies of Figure 7.14 alone reduce it to observed data
+(@exm-front-door-big-g),
+so the front door formula holds even if no well-defined $Y^m$ exists.
+
+A second route, also free of $Y^m$, is a **coupling** argument.
+Even if everyone agrees that $Y^m$ is not well defined,
+any law of the observed data that factorizes according to Figure 7.14
+can be generated by some FFRCISTG model (Technical Point 6.2)
+that is "as detailed as the data" in the sense of Robins and Richardson (2010),
+and such a model formally includes a variable $Y^m$.
+Within that model, Technical Point 7.4 shows that the two formulas coincide.
+A factorizing law for which they differed
+could therefore not be generated by any such model,
+so no such distribution exists
+[@hernan2020causal, p. 301].
 :::
+
+---
+
+::: {.callout-note title="Technical Point 21.12: A Front Door Formula Proof Using d-Separation of Treatment Nodes on SWIGs"}
+**A SWIG property** (Shpitser et al. 2022).
+Let $G(a)$ be the SWIG for strategy $a$,
+assume only treatment counterfactuals are well defined,
+and let $B^a$ and $C^a$ be disjoint sets of random (not fixed) observed nodes of $G(a)$,
+such as $Y^a$, $M^a$, or the natural value of treatment $A$.
+If, in $G(a)$, $B^a$ and the fixed node $a_m$ are d-separated given $C^a$,
+then $\Pr[B^a = b \mid C^a = c]$ does not depend on $a_m$.
+This compares distributions from different single worlds;
+it is not a cross-world statement.
+
+**Front door example.**
+In the SWIG for Figure 7.14, let $C^a = (M^a, A)$ and $B^a = Y^a$.
+The only path from the fixed node $a$ to $Y^a$ passes through the non-collider $M^a$, which is conditioned on,
+so $\E{Y^a \mid M^a, A} = \E{Y^{a'} \mid M^{a'}, A}$ for all $a, a'$.
+
+**Proof of the front door formula.**
+Following Technical Point 7.4, it remains to show $\E{Y^a \mid M^a = m} = \sum_{a'} \E{Y \mid M = m, A = a'} \Pr[A = a']$:
+
+$$
+\begin{aligned}
+\E{Y^a \mid M^a = m}
+&= \sum_{a'} \E{Y^a \mid M^a = m, A = a'} \Pr[A = a' \mid M^a = m] \\
+&= \sum_{a'} \E{Y^a \mid M^a = m, A = a'} \Pr[A = a'] \\
+&= \sum_{a'} \E{Y^{a'} \mid M^{a'} = m, A = a'} \Pr[A = a'] \\
+&= \sum_{a'} \E{Y \mid M = m, A = a'} \Pr[A = a'].
+\end{aligned}
+$$
+
+The four equalities use, in turn,
+the law of total expectation,
+the d-separation of $M^a$ from $A$ in the SWIG,
+the SWIG property just shown,
+and consistency.
+
+So $\E{Y^a \mid M^a}$ is the same for every $a$,
+yet it generally differs from $\E{Y \mid M} = \sum_{a'} \E{Y \mid M, A = a'} \Pr[A = a' \mid M]$,
+because the factual $M = M^A$, unlike $M^a$, is associated with $A$
+[@hernan2020causal, p. 302].
 :::
 
 ## Summary

chapters/23-causal-mediation.qmd

@@ -183,6 +183,20 @@ The book compresses the last two lines into one step
 
 ---
 
+::: {.callout-note title="Technical Point 23.1: Proof of the mediation formula"}
+The proof of @thm-mediation-formula makes three moves.
+It averages over the value the mediator would take without treatment;
+it drops the conditioning on that value by cross-world independence;
+and it swaps the remaining counterfactual quantities for observed ones
+by exchangeability and consistency.
+Only the middle move is controversial:
+the cross-world independence $Y^{a=1, m} \ind M^{a=0}$ holds if Figure 23.1 is read as an NPSEM-IE,
+but not if it is read as an FFRCISTG model
+[@hernan2020causal, p. 324].
+:::
+
+---
+
 ### Why the mediation formula is under attack
 
 - The mediation formula identifies a cross-world quantity
@@ -305,11 +319,26 @@ that the g-formula be a function of the observed data distribution,
 which the last line shows it is.
 :::
 
-::: {.notes}
+---
+
+::: {.callout-note title="Technical Point 23.2: When the mediation formula is the g-formula"}
+The separable components $N$ and $O$ are exchangeable (Figure 23.3),
+so the g-formula would identify $\E{Y^{n=0, o=1}}$,
+except that nobody in the trial has $(N = 0, O = 1)$, so positivity fails.
+Positivity, however, is sufficient rather than necessary:
+given exchangeability and consistency,
+the g-formula identifies the mean whenever it can be written as a function of the observed data.
+The deterministic arrows $A \to N$ and $A \to O$,
+together with assumptions (i) and (ii) of no direct effect,
+make that function exactly the mediation formula (@thm-mediation-g-formula).
+
 The book calls this derivation "somewhat heuristic"
 because determinism between $A$, $N$, and $O$ creates null sets;
 Robins et al. (2022) give a rigorous proof
-using the SWIG Markov property of Technical Point 21.12 [@hernan2020causal, p. 326].
+using the SWIG Markov property of Technical Point 21.12.
+In the same way, under the expanded diagram of Figure 23.7
+the g-formula equals the front door formula (Technical Point 23.3)
+[@hernan2020causal, p. 326].
 :::
 
 ---