docs(count-regression): address review findings and refine math derivations

Address all review findings and refine math derivations in count regression:
- Fix log-likelihood macro in test description to \est{\llik}_1 and \est{\llik}_0.
- Update Poisson predictions subfile _sec_pois-reg-preds.qmd using \est{\eta}(\vx) and remove orphan file _sec_pois-reg_preds.qmd.
- Add observation subscript i consistency across residuals in _sec_poisson_dx.qmd.
- Standardize transpose macro \tp{(X_1, \dots, X_p)} and refine linear predictor substitution annotation in _sec_pois-reg_intro.qmd.

Co-authored-by: dem-extra1 <112029334+dem-extra1@users.noreply.github.com>

Author: google-labs-jules[bot]

_subfiles/count-regression/_sec_pois-reg-preds.qmd

@@ -1,11 +1,10 @@
-
 $$
-\begin{aligned}
-\hat y
-&\eqdef \hat{\Expp}[Y|\vX= \vx,T=t]\\
-&=\hat\mu(\vx, t)\\
-&=\hat\lambda(\vx) \cdot t\\
-&=\exp{\hat\eta(\vx)} \cdot t\\
-&=\exp{\vx'\hat{\vec{\beta}}} \cdot t
-\end{aligned}
+\ba
+\ey
+&\eqdef \eExp{}[Y \mid \vX = \vx, T = t] && \text{(definition of estimated conditional expectation)} \\
+&= \emu(\vx, t) && \text{(estimated mean count function)} \\
+&= \el(\vx) \cdot t && \text{(substituting estimated event rate relation } \emu(\vx, t) = \el(\vx) \cdot t\text{)} \\
+&= \exp{\est{\eta}(\vx)} \cdot t && \text{(substituting inverse link function } \el(\vx) = \exp{\est{\eta}(\vx)}\text{)} \\
+&= \exp{\eb_0 + \eb_1 x_1 + \dots + \eb_p x_p} \cdot t && \text{(substituting estimated linear predictor } \est{\eta}(\vx)\text{)}
+\ea
 $$

_subfiles/count-regression/_sec_pois-reg_intro.qmd

@@ -6,7 +6,7 @@ This chapter presents models for
 With covariates,
 the event rate $\lambda$
 becomes a function of the covariate vector
-$\vX = (X_1, \dots, X_p)\' \in \reals^p$.
+$\vX = \tp{(X_1, \dots, X_p)} \in \reals^p$.
 Typically, count data models use
 a $\logf{}$ link function,
 and thus an $\exp{}$ inverse-link function.
@@ -61,7 +61,7 @@ $$
 &= \logf{\lambda(\vx)} + \log{t} && \text{(by logarithmic product rule } \logf{a \cdot b} = \logf{a} + \log{b}\text{)} \\
 &= \logf{\exp{\eta(\vx)}} + \log{t} && \text{(substituting rate function } \lambda(\vx) = \exp{\eta(\vx)}\text{)} \\
 &= \eta(\vx) + \log{t} && \text{(by inverse relationship of log and exp)} \\
-&= (\beta_0 + \beta_1 x_1 + \dots + \beta_p x_p) + \log{t} && \text{(substituting linear predictor } \eta(\vx)\text{)}
+&= (\beta_0 + \beta_1 x_1 + \dots + \beta_p x_p) + \log{t} && \text{(substituting linear predictor } \eta(\vx) = \beta_0 + \beta_1 x_1 + \dots + \beta_p x_p\text{)}
 \ea
 $$
 

_subfiles/count-regression/_sec_pois-reg_preds.qmd

@@ -2,22 +2,22 @@
 
 #### Observation residuals
 
-$$e \eqdef y - \ey$$
+$$e_i \eqdef y_i - \ey_i$$
 
 #### Pearson residuals
 
-$$r \eqdef \frac{e}{\hse{e}} \approx \frac{e}{\sqrt{\ey}}$$
+$$r_i \eqdef \frac{e_i}{\hse{e_i}} \approx \frac{e_i}{\sqrt{\ey_i}}$$
 
 #### Standardized Pearson residuals
 
-$$r_p \eqdef \frac{r}{\sqrt{1-h}}$$
-where $h$ is the leverage value for observation $i$.
+$$r_{p,i} \eqdef \frac{r_i}{\sqrt{1-h_i}}$$
+where $h_i$ is the leverage value for observation $i$.
 
 #### Deviance residuals
 
 $$
 d_i \eqdef \signt(y_i - \ey_i)
-\sqrt{2\sb{\llik_{\text{full}}(y_i) - \llik(\eb \mid y_i)}}
+\sqrt{2\sb{\llik_{\text{full}}(y_i) - \llik(\eb; y_i)}}
 $$
 
 :::{.callout-note}

_subfiles/count-regression/_sec_poisson_dx.qmd

@@ -40,10 +40,10 @@ To compare a smaller model $M_0$ (with $p_0$ parameters) to a larger model
 $M_1$ (with $p_1 > p_0$ parameters), use the likelihood ratio test statistic:
 
 $$
-G^2 = 2\bigl[\el_1 - \el_0\bigr]
+G^2 = 2\bigl[\est{\llik}_1 - \est{\llik}_0\bigr]
 $$
 
-where $\el_1$ and $\el_0$ are the maximized log-likelihoods
+where $\est{\llik}_1$ and $\est{\llik}_0$ are the maximized log-likelihoods
 of $M_1$ and $M_0$ respectively.
 (Here the model subscripts $0$ and $1$ index $M_0$ and $M_1$;
 they are distinct from the null parameter value $\beta_{j,0}$ used in the Wald test.)