docs(causal-inference): replace undefined \vL and \vl with \vec{L}, \vec{l}

Addresses the finding carried unaddressed across three review rounds.
`\vL` and `\vl` are not defined anywhere in the vendored latex-macros
submodule -- confirmed by searching all four TeX definition mechanisms
(`\def`, `\newcommand`, `\renewcommand`, `\providecommand`), not just the
`\newcommand` family -- so they rendered as undefined control sequences.

31 occurrences across three subfiles:

    _sec_assumptions.qmd            7
    _sec_propensity_scores.qmd     10
    _sec_regression_adjustment.qmd 14

`\vec{L}` and `\vec{l}` match how the file's neighbouring vector macros
are themselves built: `\renewcommand{\vec}[1]{\tilde{#1}}` (macros.qmd:354)
and `\def\vX{\vecf{X}}` (:366) both resolve to the same tilde form, so the
replacement renders consistently with `\vX` and `\vx` alongside it.

No `\vL` or `\vl` remains anywhere in the repository.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>

Author: d-morrison

_subfiles/causal-inference/_sec_assumptions.qmd

@@ -32,9 +32,9 @@ can introduce bias.
 
 **Exchangeability** (also called *no unmeasured confounding*
 or *ignorability*) states that,
-conditional on observed covariates $\vL$,
+conditional on observed covariates $\vec{L}$,
 treatment assignment is independent of potential outcomes:
-$$A \perp\!\!\!\perp (Y(0), Y(1)) \mid \vL$$
+$$A \perp\!\!\!\perp (Y(0), Y(1)) \mid \vec{L}$$
 
 Intuitively, this means that among individuals with the same
 covariate values,
@@ -43,7 +43,7 @@ there are no unmeasured common causes of treatment and outcome.
 
 **Unconditional exchangeability** ($A \perp\!\!\!\perp (Y(0), Y(1))$)
 holds in perfectly randomized trials.
-**Conditional exchangeability** requires adjusting for $\vL$.
+**Conditional exchangeability** requires adjusting for $\vec{L}$.
 
 :::
 
@@ -57,8 +57,8 @@ The **positivity** assumption states that
 every individual has a positive probability of receiving
 each treatment level,
 conditional on their covariate values:
-$$0 < \Pf{A = 1 \mid \vL = \vl} < 1
-\quad \text{for all } \vl \text{ in the support of } \vL$$
+$$0 < \Pf{A = 1 \mid \vec{L} = \vec{l}} < 1
+\quad \text{for all } \vec{l} \text{ in the support of } \vec{L}$$
 
 Without positivity,
 some subgroups have no treated (or untreated) members,

_subfiles/causal-inference/_sec_propensity_scores.qmd

@@ -2,7 +2,7 @@
 
 **Propensity score methods** are an alternative approach
 to confounding adjustment in observational studies.
-They replace adjustment for a high-dimensional confounder set $\vL$
+They replace adjustment for a high-dimensional confounder set $\vec{L}$
 with adjustment for a single scalar summary:
 the propensity score.
 
@@ -13,13 +13,13 @@ the propensity score.
 
 The **propensity score** is the conditional probability of treatment
 given the observed covariates:
-$$e(\vL) \eqdef \Pf{A = 1 \mid \vL}$$
+$$e(\vec{L}) \eqdef \Pf{A = 1 \mid \vec{L}}$$
 
 @rosenbaum1983central showed that
-if $A \perp\!\!\!\perp (Y(0), Y(1)) \mid \vL$
-(conditional exchangeability given $\vL$),
+if $A \perp\!\!\!\perp (Y(0), Y(1)) \mid \vec{L}$
+(conditional exchangeability given $\vec{L}$),
 then also:
-$$A \perp\!\!\!\perp (Y(0), Y(1)) \mid e(\vL)$$
+$$A \perp\!\!\!\perp (Y(0), Y(1)) \mid e(\vec{L})$$
 
 In other words, the propensity score is a **balancing score**:
 conditioning on the propensity score is sufficient
@@ -33,10 +33,10 @@ The propensity score is estimated by fitting a model
 for the probability of treatment given covariates,
 typically using logistic regression:
 
-$$\text{logit}(\hat{e}(\vL_i)) =
+$$\text{logit}(\hat{e}(\vec{L}_i)) =
 \hat{\beta}_0 + \hat{\beta}_1 L_{i1} + \cdots + \hat{\beta}_p L_{ip}$$
 
-The estimated propensity score $\hat{e}(\vL_i)$
+The estimated propensity score $\hat{e}(\vec{L}_i)$
 is the predicted probability of treatment for each individual.
 
 ## Propensity score methods {#sec-ps-methods}
@@ -68,8 +68,8 @@ There are four main ways to use the propensity score:
 
 The **inverse probability weighted (IPW)** estimator
 of the ATE uses weights:
-$$w_i \eqdef \frac{A_i}{\hat{e}(\vL_i)} +
-\frac{1 - A_i}{1 - \hat{e}(\vL_i)}$$
+$$w_i \eqdef \frac{A_i}{\hat{e}(\vec{L}_i)} +
+\frac{1 - A_i}{1 - \hat{e}(\vec{L}_i)}$$
 
 The IPW estimator of $\E{Y(a)}$ is:
 $$\Est{\E{Y(a)}}_{\text{IPW}} \eqdef

_subfiles/causal-inference/_sec_regression_adjustment.qmd

@@ -3,7 +3,7 @@
 Under the causal assumptions
 (consistency, conditional exchangeability, and positivity),
 the causal effect can be estimated
-by adjusting for the confounders $\vL$
+by adjusting for the confounders $\vec{L}$
 in a regression model.
 
 ## Direct standardization (G-computation) {#sec-causal-g-computation}
@@ -13,11 +13,11 @@ The **G-computation** estimator
 estimates the ATE by:
 
 1. Fitting a regression model:
-   $\emu(a, \vl) \eqdef \Est{\E{Y \mid A = a, \vL = \vl}}$
+   $\emu(a, \vec{l}) \eqdef \Est{\E{Y \mid A = a, \vec{L} = \vec{l}}}$
 
 2. Predicting the potential outcome mean for each individual
    under both treatment levels $a = 1$ and $a = 0$:
-   $$\hat{Y}_i(a) \eqdef \emu(a, \vL_i)$$
+   $$\hat{Y}_i(a) \eqdef \emu(a, \vec{L}_i)$$
 
 3. Averaging over the study population:
    $$\Est{\E{Y(a)}} \eqdef \frac{1}{n} \sum_{i=1}^n \hat{Y}_i(a)$$
@@ -47,7 +47,7 @@ and the regression model is linear,
 G-computation simplifies to the
 **adjusted treatment coefficient** from a linear regression:
 
-$$Y = \beta_0 + \beta_A A + \tp{\vbeta_L} \vL + \varepsilon$$
+$$Y = \beta_0 + \beta_A A + \tp{\vbeta_L} \vec{L} + \varepsilon$$
 
 Under consistency, conditional exchangeability,
 and positivity,
@@ -59,10 +59,10 @@ $$
 \text{ATE}
 &\eqdef \E{Y(1) - Y(0)} && \text{(definition of ATE)}
 \\&= \E{Y(1)} - \E{Y(0)} && \text{(linearity of expectation)}
-\\&= \E{\E{Y(1) \mid \vL}} - \E{\E{Y(0) \mid \vL}} && \text{(law of iterated expectations)}
-\\&= \E{\E{Y \mid A = 1, \vL}} - \E{\E{Y \mid A = 0, \vL}} && \text{(consistency and conditional exchangeability)}
-\\&= \E{\beta_0 + \beta_A (1) + \tp{\vbeta_L} \vL} - \E{\beta_0 + \beta_A (0) + \tp{\vbeta_L} \vL} && \text{(linear outcome model specification)}
-\\&= \left(\beta_0 + \beta_A + \tp{\vbeta_L} \E{\vL}\right) - \left(\beta_0 + \tp{\vbeta_L} \E{\vL}\right) && \text{(linearity of expectation)}
+\\&= \E{\E{Y(1) \mid \vec{L}}} - \E{\E{Y(0) \mid \vec{L}}} && \text{(law of iterated expectations)}
+\\&= \E{\E{Y \mid A = 1, \vec{L}}} - \E{\E{Y \mid A = 0, \vec{L}}} && \text{(consistency and conditional exchangeability)}
+\\&= \E{\beta_0 + \beta_A (1) + \tp{\vbeta_L} \vec{L}} - \E{\beta_0 + \beta_A (0) + \tp{\vbeta_L} \vec{L}} && \text{(linear outcome model specification)}
+\\&= \left(\beta_0 + \beta_A + \tp{\vbeta_L} \E{\vec{L}}\right) - \left(\beta_0 + \tp{\vbeta_L} \E{\vec{L}}\right) && \text{(linearity of expectation)}
 \\&= \beta_A && \text{(algebraic simplification)}
 \ea
 $$