docs(count-regression): extract quasipoisson subfile, use \ba/\ea macros

Addresses two review findings on #1138:

- Extract the quasipoisson section's prose from chapters/count-regression.qmd
  into _subfiles/count-regression/_sec-quasipoisson.qmd, matching the pattern
  every other section in the chapter already follows. The parent keeps the
  heading and gains an include; the chapter drops from 180 to 134 lines.
  Verified content-neutral: a sorted comparison of the old chapter against the
  new chapter plus the new subfile is identical.

- Convert the one remaining raw \begin{aligned}/\end{aligned} block in
  _sec_pois-reg_intro.qmd to the \ba/\ea macros (latex-macros/macros.qmd:140-141),
  matching every sibling derivation this PR already converted. Confirmed in the
  rendered HTML that \ba expands to \begin{aligned} as expected.

The third finding (unverified @vittinghoff2e citations) is rebutted separately
on the PR: all three claims check out against the source text.

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

Author: d-morrison

_subfiles/count-regression/_sec-quasipoisson.qmd

@@ -0,0 +1,48 @@
+Another way to handle overdispersion ---
+rather than switching to the negative binomial distributional family ---
+is the quasipoisson approach.
+It assumes less *model structure* than the negative binomial does:
+rather than committing to a complete probability distribution
+and estimating by maximum likelihood,
+it assumes only the mean-variance relationship
+$\Var{Y \mid \vX=\vx, T=t} = \theta \mu(\vx, t)$,
+for a dispersion parameter $\theta$.
+It pairs that single assumption with a moment-based *inference method*,
+estimating $\theta$ from the Pearson residuals.
+
+While point estimates for regression coefficients $\evb$
+remain identical to standard Poisson regression,
+their estimated standard errors are scaled by $\sqrt{\eth}$.
+This approach provides valid standard errors and $p$-values
+when overdispersion is multiplicative.
+That validity comes from the assumed mean-variance relationship itself.
+A sandwich (robust) variance estimator also uses the residuals,
+but it does not require the model's variance function to be correct;
+it accumulates the squared residuals across observations
+instead of scaling the variance formula the model supplied.
+Its validity therefore does not depend on that variance function
+being the right one,
+only on the mean model being approximately correct
+[@vittinghoff2e, §4.7.3.6 for linear regression;
+@vittinghoff2e, §8.3.1 for the generalized linear model case].
+The quasipoisson scaling has no such guarantee:
+its single $\eth$ is estimated under the assumption of proportionality,
+so if the variance is not proportional to the mean,
+the scaled standard errors are simply wrong.
+
+Robust standard errors are not a free improvement, though.
+For linear regression, @vittinghoff2e [§4.7.3.6] reports simulations
+in which robust standard errors can be too small
+in samples as large as 250 observations,
+and recommends the more conservative HC3 estimator at those sizes.
+The specific remedy is a linear-model construction,
+but the caution about small samples is worth carrying over
+to the sample sizes many epidemiological studies actually have.
+
+The quasipoisson approach is simpler to implement
+than the negative binomial model,
+but provides less information than a full negative binomial likelihood:
+it does not specify a full parametric distribution
+for prediction intervals or model likelihood comparisons.
+
+See `?quasipoisson` in R for implementation details.

_subfiles/count-regression/_sec_pois-reg_intro.qmd

@@ -15,12 +15,12 @@ to the event rate and linear predictor as:
 :::
 
 $$
-\begin{aligned}
+\ba
 \Expp[Y \mid \vX = \vx, T = t] &= \mu(\vx,t) \\
 \mu(\vx,t) &= \lambda(\vx)\cdot t \\
 \lambda(\vx) &= \exp{\eta(\vx)} \\
 \eta(\vx) &= \beta_0 + \beta_1 x_1 + \dots + \beta_p x_p
-\end{aligned}
+\ea
 $${#eq-mean-poisson}
 
 ::: notes

chapters/count-regression.qmd

@@ -113,54 +113,7 @@ for the count component.
 
 ## Quasipoisson regression
 
-Another way to handle overdispersion ---
-rather than switching to the negative binomial distributional family ---
-is the quasipoisson approach.
-It assumes less *model structure* than the negative binomial does:
-rather than committing to a complete probability distribution
-and estimating by maximum likelihood,
-it assumes only the mean-variance relationship
-$\Var{Y \mid \vX=\vx, T=t} = \theta \mu(\vx, t)$,
-for a dispersion parameter $\theta$.
-It pairs that single assumption with a moment-based *inference method*,
-estimating $\theta$ from the Pearson residuals.
-
-While point estimates for regression coefficients $\evb$
-remain identical to standard Poisson regression,
-their estimated standard errors are scaled by $\sqrt{\eth}$.
-This approach provides valid standard errors and $p$-values
-when overdispersion is multiplicative.
-That validity comes from the assumed mean-variance relationship itself.
-A sandwich (robust) variance estimator also uses the residuals,
-but it does not require the model's variance function to be correct;
-it accumulates the squared residuals across observations
-instead of scaling the variance formula the model supplied.
-Its validity therefore does not depend on that variance function
-being the right one,
-only on the mean model being approximately correct
-[@vittinghoff2e, §4.7.3.6 for linear regression;
-@vittinghoff2e, §8.3.1 for the generalized linear model case].
-The quasipoisson scaling has no such guarantee:
-its single $\eth$ is estimated under the assumption of proportionality,
-so if the variance is not proportional to the mean,
-the scaled standard errors are simply wrong.
-
-Robust standard errors are not a free improvement, though.
-For linear regression, @vittinghoff2e [§4.7.3.6] reports simulations
-in which robust standard errors can be too small
-in samples as large as 250 observations,
-and recommends the more conservative HC3 estimator at those sizes.
-The specific remedy is a linear-model construction,
-but the caution about small samples is worth carrying over
-to the sample sizes many epidemiological studies actually have.
-
-The quasipoisson approach is simpler to implement
-than the negative binomial model,
-but provides less information than a full negative binomial likelihood:
-it does not specify a full parametric distribution
-for prediction intervals or model likelihood comparisons.
-
-See `?quasipoisson` in R for implementation details.
+{{< include _subfiles/count-regression/_sec-quasipoisson.qmd >}}
 
 # More on count regression