Discuss deviance calculation with replicates; clarify saturation vs. full parametrization

[Copilot]

• Fix the Gaussian deviance section to say "one free parameter per distinct covariate pattern"
• Add new subsection "Saturated vs. fully parametrized models with replicates"
• Add note about $0\log(0) = 0$ convention for boundary cases in Binomial deviance formula
• Clarify why $\ell_{\text{fp}} = 0$ for ungrouped Bernoulli data: R uses the fully parametrized model as reference (one param per observation), not the saturated model (one per pattern); they coincide only when $q = n$
• Add callout noting that $\ell_{\text{sat}} < 0$ when covariate patterns repeat ($q < n$) — the saturated model sets $\hat\pi_k = y_k/n_k$ per pattern; R's ungrouped deviance uses the fully parametrized reference in all cases
• Add R code chunk verifying deviance() == -2*logLik() for ungrouped data with repeated covariate patterns, demonstrating that R uses $\ell_{\text{fp}} = 0$ as its reference
• Correct GOF test explanation: the fundamental issue is not that $n_k$ never grows large (it can when patterns repeat), but that R's ungrouped deviance() returns $-2\ell(\hat\beta)$ (using $\ell_{\text{fp}} = 0$) rather than $2(\ell_{\text{sat}} - \ell(\hat\beta))$; these differ by $-2\ell_{\text{sat}} > 0$ whenever $q < n$, making R's value the wrong statistic to compare against $\chi^2(q - p)$ regardless of $n_k$; add note to use grouped data for a valid GOF test
• Add references to Dunn & Smyth (2018) "Generalized Linear Models with Examples in R" (Chapter 9) and the Stats Stack Exchange thread on Bernoulli deviance in R for further reading
• Update Summary table with corrected deviance formula
• Add ungrouped, leq, vx, fp, infty, GOF to inst/WORDLIST
• Lint: no issues; Spellcheck: no errors

[d-morrison]

@copilot let's refer readers to https://stats.stackexchange.com/questions/626597/is-there-a-justification-for-the-bernoulli-deviance-in-the-r-stats-package and https://link.springer.com/chapter/10.1007/978-1-4419-0118-7_9#Sec10 for more information

[Copilot]

@copilot let's refer readers to https://stats.stackexchange.com/questions/626597/is-there-a-justification-for-the-bernoulli-deviance-in-the-r-stats-package and https://link.springer.com/chapter/10.100...

Added in commit 86de9cb. The callout now ends with:

For further discussion, see @dunn2018generalized [Chapter 9] and this Stats Stack Exchange thread.

The Springer link (978-1-4419-0118-7) corresponds to Dunn & Smyth (2018) "Generalized Linear Models with Examples in R", which was already in references.bib as dunn2018generalized.

[github-actions]

PR Preview Action v1.8.1
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_{Built to branch gh-pages at 2026-04-10 19:21 UTC.}

[Copilot]

Pull request overview

Clarifies how deviance is computed/interpreted when covariate patterns repeat (replicates), with emphasis on the distinction between saturated vs. fully parametrized reference models and how R’s deviance() behaves for grouped vs. ungrouped Binomial/Bernoulli data.

Changes:

• Refines the Gaussian deviance discussion to be in terms of distinct covariate patterns and adds a new subsection contrasting saturated vs. fully parametrized models under replicates.
• Adds narrative + R examples demonstrating deviance(fit) == -2*logLik(fit) for ungrouped Bernoulli data with repeated patterns, and explains implications for GOF testing.
• Extends inst/WORDLIST with additional technical terms/acronyms.

Reviewed changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 4 comments.

File	Description
_sec_compare_lm_glm_deviance.qmd	Adds the replicates-focused explanation and examples; updates the summary table to reflect the clarified deviance reference.
inst/WORDLIST	Adds new allowlisted technical tokens used by the updated prose/math.

[Copilot]

inst/WORDLIST already contains GitHub, but this change adds github (lowercase) and also introduces duplicate entries for frac/github later in the file. Duplicates/inconsistent casing can make spellcheck behavior harder to reason about and may allow incorrect capitalization in prose; consider removing the duplicates and keeping only the preferred casing (e.g., GitHub).

[Copilot]

The note says patterns with y_k = 0 or y_k = n_k “contribute zero to the sum,” but in the Binomial deviance expression only the 0·log(0) component is taken as 0; the other term generally remains nonzero (e.g., if y_k=0, the (n_k-y_k) log((1-y_k/n_k)/(1-\hat\pi_k)) term is still present). Reword to avoid implying the entire bracketed contribution is 0.

Suggested change

	By convention, terms with $y_k = 0$ or $y_k = n_k$ contribute zero to the sum,
	since $0\log(0) = 0$ in the limit.
	By convention,
	when $y_k = 0$ or $y_k = n_k$,
	the corresponding component of the form $0\log(0)$ is taken to equal 0
	in the limiting sense.
	The other component in the bracket generally remains nonzero
	and still contributes to the sum.

[Copilot]

“When patterns do repeat ($q &lt; n$) … giving $\ell_{\text{sat}} &lt; 0$” isn’t always true: with replicates you can still have $\ell_{\text{sat}} = 0$ if every repeated pattern is all-0 or all-1 (i.e., each pattern has $y_k \in {0,n_k}$ so the Binomial pmf at $\hat\pi_k=y_k/n_k$ is 1). Consider qualifying this as “$\ell_{\text{sat}} < 0$ whenever at least one pattern has both outcomes (0 < y_k < n_k).”

Suggested change

	When patterns do repeat ($q < n$), the saturated model sets $\hat\pi_k = y_k/n_k$ per pattern,
	giving $\ell_{\text{sat}} < 0$.
	`deviance()` for ungrouped data **cannot** be used as a goodness-of-fit test
	against the $\chi^2$ distribution when $q < n$.
	The correct GOF statistic is $2(\ell_{\text{sat}} - \ell(\hat\beta))$,
	but R's `deviance()` for ungrouped data returns $-2\ell(\hat\beta)$ (using $\ell_{\text{fp}} = 0$).
	These two quantities differ by $-2\ell_{\text{sat}} > 0$ whenever $q < n$.
	When patterns do repeat ($q < n$), the saturated model sets $\hat\pi_k = y_k/n_k$ per pattern.
	If at least one pattern has both outcomes
	($0 < y_k < n_k$),
	then $\ell_{\text{sat}} < 0$.
	`deviance()` for ungrouped data **cannot** be used as a goodness-of-fit test
	against the $\chi^2$ distribution when $q < n$.
	The correct GOF statistic is $2(\ell_{\text{sat}} - \ell(\hat\beta))$,
	but R's `deviance()` for ungrouped data returns $-2\ell(\hat\beta)$ (using $\ell_{\text{fp}} = 0$).
	These two quantities differ by $-2\ell_{\text{sat}} > 0$
	whenever at least one repeated pattern has both outcomes
	($0 < y_k < n_k$).

[Copilot]

The claim that ungrouped deviance() “cannot be used as a goodness-of-fit test against the $\chi^2$ distribution when $q &lt; n$” is too broad. The issue you describe is that it’s the wrong statistic to compare to $\chi^2(q-p)$ (lack-of-fit by covariate pattern); ungrouped deviance is typically compared to $\chi^2(n-p)$ instead. Suggest rephrasing to explicitly mention the df/statistic mismatch rather than implying no $\chi^2$ GOF use at all.

Suggested change

	`deviance()` for ungrouped data **cannot** be used as a goodness-of-fit test
	against the $\chi^2$ distribution when $q < n$.
	The correct GOF statistic is $2(\ell_{\text{sat}} - \ell(\hat\beta))$,
	but R's `deviance()` for ungrouped data returns $-2\ell(\hat\beta)$ (using $\ell_{\text{fp}} = 0$).
	When $q < n$, `deviance()` for ungrouped data is **not** the correct statistic
	for the covariate-pattern goodness-of-fit test based on $\chi^2(q - p)$.
	For that lack-of-fit test, the correct statistic is
	$2(\ell_{\text{sat}} - \ell(\hat\beta))$,
	but R's `deviance()` for ungrouped data returns $-2\ell(\hat\beta)$
	(using $\ell_{\text{fp}} = 0$).