From 1b65542fa228cf636d4ea510f6e16f9640a57f1c Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Sat, 18 Apr 2026 00:48:08 +0000
Subject: [PATCH 01/14] Initial plan


From d3b27d3d441ed9384e75bce61fbf602afab09e84 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Sat, 18 Apr 2026 00:57:44 +0000
Subject: [PATCH 02/14] docs: add independence diagnostics section for linear
 regression

Agent-Logs-Url: https://github.com/d-morrison/rme/sessions/77f22306-4128-4852-8346-93455180c8bb

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>
---
 .../_sec_linreg_diagnostics.qmd               | 37 +++++++++++++++++++
 references.bib                                | 32 ++++++++++++++++
 2 files changed, 69 insertions(+)

diff --git a/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd b/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
index 3543712f0..26672f0b0 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
@@ -8,6 +8,43 @@ This section is adapted from @dobson4e [§6.2-6.3] and
 
 {{< include _subfiles/Linear-models-overview/_sec-linreg-assumptions.qmd >}}
 
+### Diagnostics for the independence assumption
+
+The independence assumption means that residual noise terms are not correlated across observations,
+after conditioning on the predictors in the model.
+
+For data with a natural ordering
+(for example, time, spatial location, or clinic visit sequence),
+we usually assess independence with both plots and formal tests
+[@kutner2005applied, Chapter 12; @chatterjee2015regression, Chapter 6; @wp:regvalidation].
+
+Common diagnostics include:
+
+1. Residuals versus observation order (or time) plots:
+patterns, runs, or drifts can indicate dependence
+[@kutner2005applied, Chapter 12].
+
+2. Correlograms:
+plot the sample autocorrelation function (ACF),
+and often the partial autocorrelation function (PACF),
+to look for serial structure
+[@wp:correlogram; @chatterjee2015regression, Chapter 6].
+
+3. Durbin-Watson test:
+tests for first-order serial correlation in regression residuals
+[@wp:durbinwatson].
+
+4. Breusch-Godfrey test:
+tests for higher-order serial correlation,
+and is more flexible than Durbin-Watson in many regression settings
+[@wp:breuschgodfrey].
+
+No single diagnostic is definitive.
+In practice,
+we combine visual and formal diagnostics,
+and interpret them in the context of study design
+[@kutner2005applied, Chapter 12; @draper2014applied, Chapter 11].
+
 ### Direct visualization
 
 ::: notes
diff --git a/references.bib b/references.bib
index 3bb731149..bf67fa998 100644
--- a/references.bib
+++ b/references.bib
@@ -1365,6 +1365,38 @@ @misc{wp:statinference
     note = "[Online; accessed 7-September-2025]"
   }
 
+  @misc{wp:regvalidation,
+    author = "{Wikipedia contributors}",
+    title = "Regression validation --- {Wikipedia}{,} The Free Encyclopedia",
+    year = "2026",
+    url = "https://en.wikipedia.org/wiki/Regression_validation",
+    note = "[Online; accessed 18-April-2026]"
+  }
+
+  @misc{wp:durbinwatson,
+    author = "{Wikipedia contributors}",
+    title = "Durbin--Watson statistic --- {Wikipedia}{,} The Free Encyclopedia",
+    year = "2026",
+    url = "https://en.wikipedia.org/wiki/Durbin%E2%80%93Watson_statistic",
+    note = "[Online; accessed 18-April-2026]"
+  }
+
+  @misc{wp:correlogram,
+    author = "{Wikipedia contributors}",
+    title = "Correlogram --- {Wikipedia}{,} The Free Encyclopedia",
+    year = "2026",
+    url = "https://en.wikipedia.org/wiki/Correlogram",
+    note = "[Online; accessed 18-April-2026]"
+  }
+
+  @misc{wp:breuschgodfrey,
+    author = "{Wikipedia contributors}",
+    title = "Breusch--Godfrey test --- {Wikipedia}{,} The Free Encyclopedia",
+    year = "2026",
+    url = "https://en.wikipedia.org/wiki/Breusch%E2%80%93Godfrey_test",
+    note = "[Online; accessed 18-April-2026]"
+  }
+
 @article{heinze2018variable,
   title={Variable selection -- {A} review and recommendations for the practicing statistician},
   author={Heinze, Georg and Wallisch, Christine and Dunkler, Daniela},

From 67607dd12a431413e564c8eecb1d508feff4fe44 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Sat, 18 Apr 2026 02:58:41 +0000
Subject: [PATCH 03/14] docs: add independence diagnostics math and worked
 examples

Agent-Logs-Url: https://github.com/d-morrison/rme/sessions/6bcdf40e-f97e-4fde-a0d3-49f15177becd

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>
---
 .../_sec_linreg_diagnostics.qmd               |  4 +
 .../_sec_linreg_independence_examples.qmd     | 76 +++++++++++++++++++
 .../_sec_linreg_independence_math.qmd         | 47 ++++++++++++
 3 files changed, 127 insertions(+)
 create mode 100644 _subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
 create mode 100644 _subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd

diff --git a/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd b/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
index 26672f0b0..fafbc5b22 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
@@ -39,6 +39,10 @@ tests for higher-order serial correlation,
 and is more flexible than Durbin-Watson in many regression settings
 [@wp:breuschgodfrey].
 
+{{< include _subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd >}}
+
+{{< include _subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd >}}
+
 No single diagnostic is definitive.
 In practice,
 we combine visual and formal diagnostics,
diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
new file mode 100644
index 000000000..285705754
--- /dev/null
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
@@ -0,0 +1,76 @@
+#### Example diagnostics with code and output
+
+The example below simulates ordered data with positively autocorrelated errors.
+That creates a setting where independence should fail.
+
+```{r}
+#| label: exm-independence-sim-data
+#| code-fold: false
+set.seed(204)
+
+n_obs <- 120
+time_index <- seq_len(n_obs)
+x <- seq(-1, 1, length.out = n_obs)
+err <- as.numeric(arima.sim(model = list(ar = 0.7), n = n_obs, sd = 1))
+y <- 2 + 1.5 * x + err
+
+indep_exm <- tibble::tibble(
+  time_index = time_index,
+  x = x,
+  y = y
+)
+
+indep_exm_lm <- lm(y ~ x, data = indep_exm)
+
+summary(indep_exm_lm)
+```
+
+Residuals versus order:
+
+```{r}
+#| label: fig-independence-resid-order
+#| fig-cap: "Residuals versus observation order for simulated data"
+indep_exm |>
+  dplyr::mutate(resid = resid(indep_exm_lm)) |>
+  ggplot(aes(x = time_index, y = resid)) +
+  geom_hline(yintercept = 0, linetype = "dashed", color = "gray50") +
+  geom_line(color = "steelblue") +
+  theme_classic() +
+  labs(
+    x = "Observation order",
+    y = "Residual"
+  )
+```
+
+Correlogram diagnostics:
+
+```{r}
+#| label: fig-independence-correlogram
+#| fig-cap: "Residual ACF and PACF for simulated data"
+op <- par(mfrow = c(1, 2))
+acf(
+  resid(indep_exm_lm),
+  main = "Residual ACF"
+)
+pacf(
+  resid(indep_exm_lm),
+  main = "Residual PACF"
+)
+par(op)
+```
+
+Durbin-Watson and Breusch-Godfrey tests:
+
+```{r}
+#| label: exm-independence-formal-tests
+#| code-fold: false
+library(lmtest)
+
+dwtest(indep_exm_lm, alternative = "two.sided")
+bgtest(indep_exm_lm, order = 4)
+```
+
+In this example,
+the residual-order plot and correlogram suggest serial dependence,
+and both formal tests typically reject independence.
+
diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd
new file mode 100644
index 000000000..ebb2c0cd7
--- /dev/null
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd
@@ -0,0 +1,47 @@
+#### Mathematical details for Durbin-Watson and Breusch-Godfrey
+
+Suppose the observations have a meaningful order,
+indexed by $t = 1, \ldots, n$.
+Let $e_t$ denote the OLS residual from a fitted regression model.
+
+For the Durbin-Watson diagnostic,
+the test statistic is
+$$
+\ba
+d
+&= \frac{\sum_{t = 2}^{n} (e_t - e_{t-1})^2}
+{\sum_{t = 1}^{n} e_t^2}
+\ea
+$$
+which is approximately
+$2(1 - \hat r_1)$,
+where $\hat r_1$ is the sample lag-1 residual autocorrelation
+[@kutner2005applied, Chapter 12; @chatterjee2015regression, Chapter 6].
+
+The Durbin-Watson null and alternatives are typically framed through
+an AR(1) error model:
+$$
+\err_t = \rho \err_{t-1} + u_t.
+$$
+The null is $H_0: \rho = 0$,
+and alternatives can be one-sided or two-sided,
+depending on whether we suspect positive,
+negative,
+or any serial correlation
+[@kutner2005applied, Chapter 12].
+
+For the Breusch-Godfrey test of order $p$,
+we run an auxiliary regression of residuals on:
+the original regressors,
+and lagged residuals $e_{t-1}, \ldots, e_{t-p}$.
+If $R^2_{\text{aux}}$ is from that auxiliary model,
+the LM statistic is
+$$
+\text{LM} = (n - p)R^2_{\text{aux}},
+$$
+and under
+$H_0: \rho_1 = \cdots = \rho_p = 0$,
+it is asymptotically
+$\chi^2_p$
+[@chatterjee2015regression, Chapter 6; @draper2014applied, Chapter 11].
+

From 5969608cbe4c12913632a8c190fe81e82d2a8c13 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Sat, 18 Apr 2026 03:00:38 +0000
Subject: [PATCH 04/14] docs: use subfiles for independence test math and
 examples

Agent-Logs-Url: https://github.com/d-morrison/rme/sessions/6bcdf40e-f97e-4fde-a0d3-49f15177becd

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>
---
 .../_sec_linreg_independence_examples.qmd                  | 7 ++-----
 1 file changed, 2 insertions(+), 5 deletions(-)

diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
index 285705754..a72879784 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
@@ -64,13 +64,10 @@ Durbin-Watson and Breusch-Godfrey tests:
 ```{r}
 #| label: exm-independence-formal-tests
 #| code-fold: false
-library(lmtest)
-
-dwtest(indep_exm_lm, alternative = "two.sided")
-bgtest(indep_exm_lm, order = 4)
+lmtest::dwtest(indep_exm_lm, alternative = "two.sided")
+lmtest::bgtest(indep_exm_lm, order = 4)
 ```
 
 In this example,
 the residual-order plot and correlogram suggest serial dependence,
 and both formal tests typically reject independence.
-

From 360d4eb86208e1f00329c2933499596497087b25 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Sat, 18 Apr 2026 03:02:42 +0000
Subject: [PATCH 05/14] docs: clarify AR error notation and dependent-example
 naming

Agent-Logs-Url: https://github.com/d-morrison/rme/sessions/6bcdf40e-f97e-4fde-a0d3-49f15177becd

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>
---
 .../_sec_linreg_independence_examples.qmd      | 18 +++++++++---------
 .../_sec_linreg_independence_math.qmd          |  6 ++++--
 2 files changed, 13 insertions(+), 11 deletions(-)

diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
index a72879784..1bcef6e9c 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
@@ -14,15 +14,15 @@ x <- seq(-1, 1, length.out = n_obs)
 err <- as.numeric(arima.sim(model = list(ar = 0.7), n = n_obs, sd = 1))
 y <- 2 + 1.5 * x + err
 
-indep_exm <- tibble::tibble(
+serial_dep_exm <- tibble::tibble(
   time_index = time_index,
   x = x,
   y = y
 )
 
-indep_exm_lm <- lm(y ~ x, data = indep_exm)
+serial_dep_exm_lm <- lm(y ~ x, data = serial_dep_exm)
 
-summary(indep_exm_lm)
+summary(serial_dep_exm_lm)
 ```
 
 Residuals versus order:
@@ -30,8 +30,8 @@ Residuals versus order:
 ```{r}
 #| label: fig-independence-resid-order
 #| fig-cap: "Residuals versus observation order for simulated data"
-indep_exm |>
-  dplyr::mutate(resid = resid(indep_exm_lm)) |>
+serial_dep_exm |>
+  dplyr::mutate(resid = resid(serial_dep_exm_lm)) |>
   ggplot(aes(x = time_index, y = resid)) +
   geom_hline(yintercept = 0, linetype = "dashed", color = "gray50") +
   geom_line(color = "steelblue") +
@@ -49,11 +49,11 @@ Correlogram diagnostics:
 #| fig-cap: "Residual ACF and PACF for simulated data"
 op <- par(mfrow = c(1, 2))
 acf(
-  resid(indep_exm_lm),
+  resid(serial_dep_exm_lm),
   main = "Residual ACF"
 )
 pacf(
-  resid(indep_exm_lm),
+  resid(serial_dep_exm_lm),
   main = "Residual PACF"
 )
 par(op)
@@ -64,8 +64,8 @@ Durbin-Watson and Breusch-Godfrey tests:
 ```{r}
 #| label: exm-independence-formal-tests
 #| code-fold: false
-lmtest::dwtest(indep_exm_lm, alternative = "two.sided")
-lmtest::bgtest(indep_exm_lm, order = 4)
+lmtest::dwtest(serial_dep_exm_lm, alternative = "two.sided")
+lmtest::bgtest(serial_dep_exm_lm, order = 4)
 ```
 
 In this example,
diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd
index ebb2c0cd7..bb771839b 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd
@@ -21,8 +21,11 @@ where $\hat r_1$ is the sample lag-1 residual autocorrelation
 The Durbin-Watson null and alternatives are typically framed through
 an AR(1) error model:
 $$
-\err_t = \rho \err_{t-1} + u_t.
+\eps_t = \rho \eps_{t-1} + u_t.
 $$
+Here,
+$\eps_t$ is the autocorrelated regression error process,
+and $u_t$ is a white-noise innovation term.
 The null is $H_0: \rho = 0$,
 and alternatives can be one-sided or two-sided,
 depending on whether we suspect positive,
@@ -44,4 +47,3 @@ $H_0: \rho_1 = \cdots = \rho_p = 0$,
 it is asymptotically
 $\chi^2_p$
 [@chatterjee2015regression, Chapter 6; @draper2014applied, Chapter 11].
-

From 017b84acceedd98bf758e903d6fe71c7ee47a4b3 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Sat, 18 Apr 2026 03:04:37 +0000
Subject: [PATCH 06/14] docs: tighten independence diagnostics explanations

Agent-Logs-Url: https://github.com/d-morrison/rme/sessions/6bcdf40e-f97e-4fde-a0d3-49f15177becd

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>
---
 .../_sec_linreg_independence_examples.qmd                     | 4 +++-
 .../Linear-models-overview/_sec_linreg_independence_math.qmd  | 3 +++
 2 files changed, 6 insertions(+), 1 deletion(-)

diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
index 1bcef6e9c..0bf0d4aff 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
@@ -2,6 +2,8 @@
 
 The example below simulates ordered data with positively autocorrelated errors.
 That creates a setting where independence should fail.
+We generate the error term from an AR(1) process with coefficient 0.7,
+matching the notation in the mathematical details section.
 
 ```{r}
 #| label: exm-independence-sim-data
@@ -70,4 +72,4 @@ lmtest::bgtest(serial_dep_exm_lm, order = 4)
 
 In this example,
 the residual-order plot and correlogram suggest serial dependence,
-and both formal tests typically reject independence.
+and both formal tests reject independence.
diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd
index bb771839b..09e605c67 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd
@@ -17,6 +17,9 @@ which is approximately
 $2(1 - \hat r_1)$,
 where $\hat r_1$ is the sample lag-1 residual autocorrelation
 [@kutner2005applied, Chapter 12; @chatterjee2015regression, Chapter 6].
+This approximation is most useful in standard large-sample linear-model settings,
+and is less straightforward in models with lagged outcomes
+[@kutner2005applied, Chapter 12].
 
 The Durbin-Watson null and alternatives are typically framed through
 an AR(1) error model:

From 3661078c288ecf2f8c1dd1a04267be7626a3e433 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Sat, 18 Apr 2026 03:06:27 +0000
Subject: [PATCH 07/14] docs: make correlogram par reset robust

Agent-Logs-Url: https://github.com/d-morrison/rme/sessions/6bcdf40e-f97e-4fde-a0d3-49f15177becd

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>
---
 .../_sec_linreg_independence_examples.qmd                       | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
index 0bf0d4aff..ea61832a4 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
@@ -50,6 +50,7 @@ Correlogram diagnostics:
 #| label: fig-independence-correlogram
 #| fig-cap: "Residual ACF and PACF for simulated data"
 op <- par(mfrow = c(1, 2))
+on.exit(par(op), add = TRUE)
 acf(
   resid(serial_dep_exm_lm),
   main = "Residual ACF"
@@ -58,7 +59,6 @@ pacf(
   resid(serial_dep_exm_lm),
   main = "Residual PACF"
 )
-par(op)
 ```
 
 Durbin-Watson and Breusch-Godfrey tests:

From e5083cdce65d4cbbbe9466f77d74c37864dd1aa9 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Sat, 18 Apr 2026 03:08:20 +0000
Subject: [PATCH 08/14] docs: rename simulated AR error variable for clarity

Agent-Logs-Url: https://github.com/d-morrison/rme/sessions/6bcdf40e-f97e-4fde-a0d3-49f15177becd

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>
---
 .../_sec_linreg_independence_examples.qmd                     | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
index ea61832a4..935b2ec6b 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
@@ -13,8 +13,8 @@ set.seed(204)
 n_obs <- 120
 time_index <- seq_len(n_obs)
 x <- seq(-1, 1, length.out = n_obs)
-err <- as.numeric(arima.sim(model = list(ar = 0.7), n = n_obs, sd = 1))
-y <- 2 + 1.5 * x + err
+ar_errors <- as.numeric(arima.sim(model = list(ar = 0.7), n = n_obs, sd = 1))
+y <- 2 + 1.5 * x + ar_errors
 
 serial_dep_exm <- tibble::tibble(
   time_index = time_index,

From b00d5d37d99ee71609602170d392c897d3d773ba Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Sat, 18 Apr 2026 03:10:06 +0000
Subject: [PATCH 09/14] docs: preserve full graphics state in correlogram
 example

Agent-Logs-Url: https://github.com/d-morrison/rme/sessions/6bcdf40e-f97e-4fde-a0d3-49f15177becd

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>
---
 .../_sec_linreg_independence_examples.qmd                      | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
index 935b2ec6b..0ed3cd7a1 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
@@ -49,7 +49,8 @@ Correlogram diagnostics:
 ```{r}
 #| label: fig-independence-correlogram
 #| fig-cap: "Residual ACF and PACF for simulated data"
-op <- par(mfrow = c(1, 2))
+op <- par(no.readonly = TRUE)
+par(mfrow = c(1, 2))
 on.exit(par(op), add = TRUE)
 acf(
   resid(serial_dep_exm_lm),

From a6673aea0e8eef026ae99ff4a1e3534a16d5a9a3 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Sat, 18 Apr 2026 03:11:58 +0000
Subject: [PATCH 10/14] docs: simplify time index setup in independence example

Agent-Logs-Url: https://github.com/d-morrison/rme/sessions/6bcdf40e-f97e-4fde-a0d3-49f15177becd

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>
---
 .../_sec_linreg_independence_examples.qmd                    | 5 ++---
 1 file changed, 2 insertions(+), 3 deletions(-)

diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
index 0ed3cd7a1..182150a66 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
@@ -11,13 +11,12 @@ matching the notation in the mathematical details section.
 set.seed(204)
 
 n_obs <- 120
-time_index <- seq_len(n_obs)
 x <- seq(-1, 1, length.out = n_obs)
 ar_errors <- as.numeric(arima.sim(model = list(ar = 0.7), n = n_obs, sd = 1))
 y <- 2 + 1.5 * x + ar_errors
 
 serial_dep_exm <- tibble::tibble(
-  time_index = time_index,
+  time_index = seq_len(n_obs),
   x = x,
   y = y
 )
@@ -50,8 +49,8 @@ Correlogram diagnostics:
 #| label: fig-independence-correlogram
 #| fig-cap: "Residual ACF and PACF for simulated data"
 op <- par(no.readonly = TRUE)
-par(mfrow = c(1, 2))
 on.exit(par(op), add = TRUE)
+par(mfrow = c(1, 2))
 acf(
   resid(serial_dep_exm_lm),
   main = "Residual ACF"

From 8143f6e37df99e91734e709baf8d764e70b16037 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Sat, 18 Apr 2026 23:36:24 +0000
Subject: [PATCH 11/14] docs: add real-data independence diagnostics examples

Agent-Logs-Url: https://github.com/d-morrison/rme/sessions/66346382-6b8d-4b3b-8ec2-71de2d6d2c90

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>
---
 .../_sec_linreg_independence_examples.qmd     | 57 +++++++++++++++++++
 1 file changed, 57 insertions(+)

diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
index 182150a66..70342423f 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
@@ -73,3 +73,60 @@ lmtest::bgtest(serial_dep_exm_lm, order = 4)
 In this example,
 the residual-order plot and correlogram suggest serial dependence,
 and both formal tests reject independence.
+
+#### Real-data examples
+
+We can also run the same diagnostics on real ordered data sets.
+
+##### Nile annual river flow
+
+```{r}
+#| label: exm-independence-real-nile
+#| code-fold: false
+nile_df <- tibble::tibble(
+  year = as.numeric(time(datasets::Nile)),
+  flow = as.numeric(datasets::Nile)
+)
+
+nile_lm <- lm(flow ~ year, data = nile_df)
+
+summary(nile_lm)
+lmtest::dwtest(nile_lm, alternative = "two.sided")
+lmtest::bgtest(nile_lm, order = 4)
+```
+
+```{r}
+#| label: fig-independence-real-nile-resid-order
+#| fig-cap: "Residuals versus year for Nile flow model"
+nile_df |>
+  dplyr::mutate(resid = resid(nile_lm)) |>
+  ggplot(aes(x = year, y = resid)) +
+  geom_hline(yintercept = 0, linetype = "dashed", color = "gray50") +
+  geom_line(color = "steelblue") +
+  theme_classic() +
+  labs(
+    x = "Year",
+    y = "Residual"
+  )
+```
+
+##### Mauna Loa atmospheric carbon dioxide (`co2`)
+
+```{r}
+#| label: exm-independence-real-co2
+#| code-fold: false
+co2_df <- tibble::tibble(
+  decimal_year = as.numeric(time(datasets::co2)),
+  co2_ppm = as.numeric(datasets::co2)
+)
+
+co2_lm <- lm(co2_ppm ~ decimal_year, data = co2_df)
+
+summary(co2_lm)
+lmtest::dwtest(co2_lm, alternative = "two.sided")
+lmtest::bgtest(co2_lm, order = 12)
+```
+
+For both real-data examples,
+the diagnostics indicate residual dependence,
+consistent with their time-ordered structure.

From f6b5e5fa28bd99810c23813d5ef4789c9d232cc7 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Sat, 18 Apr 2026 23:55:13 +0000
Subject: [PATCH 12/14] docs: replace wikipedia primary citations with textbook
 sources

Agent-Logs-Url: https://github.com/d-morrison/rme/sessions/e54eadb3-a6c1-44ac-a1b5-e645fdfc17ed

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>
---
 .../_sec_linreg_diagnostics.qmd               |  8 ++---
 references.bib                                | 32 -------------------
 2 files changed, 4 insertions(+), 36 deletions(-)

diff --git a/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd b/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
index fafbc5b22..31a85e859 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
@@ -16,7 +16,7 @@ after conditioning on the predictors in the model.
 For data with a natural ordering
 (for example, time, spatial location, or clinic visit sequence),
 we usually assess independence with both plots and formal tests
-[@kutner2005applied, Chapter 12; @chatterjee2015regression, Chapter 6; @wp:regvalidation].
+[@kutner2005applied, Chapter 12; @chatterjee2015regression, Chapter 6].
 
 Common diagnostics include:
 
@@ -28,16 +28,16 @@ patterns, runs, or drifts can indicate dependence
 plot the sample autocorrelation function (ACF),
 and often the partial autocorrelation function (PACF),
 to look for serial structure
-[@wp:correlogram; @chatterjee2015regression, Chapter 6].
+[@chatterjee2015regression, Chapter 6].
 
 3. Durbin-Watson test:
 tests for first-order serial correlation in regression residuals
-[@wp:durbinwatson].
+[@chatterjee2015regression, Chapter 9; @kutner2005applied, Chapter 12].
 
 4. Breusch-Godfrey test:
 tests for higher-order serial correlation,
 and is more flexible than Durbin-Watson in many regression settings
-[@wp:breuschgodfrey].
+[@chatterjee2015regression, Chapter 9; @kutner2005applied, Chapter 12].
 
 {{< include _subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd >}}
 
diff --git a/references.bib b/references.bib
index bf67fa998..3bb731149 100644
--- a/references.bib
+++ b/references.bib
@@ -1365,38 +1365,6 @@ @misc{wp:statinference
     note = "[Online; accessed 7-September-2025]"
   }
 
-  @misc{wp:regvalidation,
-    author = "{Wikipedia contributors}",
-    title = "Regression validation --- {Wikipedia}{,} The Free Encyclopedia",
-    year = "2026",
-    url = "https://en.wikipedia.org/wiki/Regression_validation",
-    note = "[Online; accessed 18-April-2026]"
-  }
-
-  @misc{wp:durbinwatson,
-    author = "{Wikipedia contributors}",
-    title = "Durbin--Watson statistic --- {Wikipedia}{,} The Free Encyclopedia",
-    year = "2026",
-    url = "https://en.wikipedia.org/wiki/Durbin%E2%80%93Watson_statistic",
-    note = "[Online; accessed 18-April-2026]"
-  }
-
-  @misc{wp:correlogram,
-    author = "{Wikipedia contributors}",
-    title = "Correlogram --- {Wikipedia}{,} The Free Encyclopedia",
-    year = "2026",
-    url = "https://en.wikipedia.org/wiki/Correlogram",
-    note = "[Online; accessed 18-April-2026]"
-  }
-
-  @misc{wp:breuschgodfrey,
-    author = "{Wikipedia contributors}",
-    title = "Breusch--Godfrey test --- {Wikipedia}{,} The Free Encyclopedia",
-    year = "2026",
-    url = "https://en.wikipedia.org/wiki/Breusch%E2%80%93Godfrey_test",
-    note = "[Online; accessed 18-April-2026]"
-  }
-
 @article{heinze2018variable,
   title={Variable selection -- {A} review and recommendations for the practicing statistician},
   author={Heinze, Georg and Wallisch, Christine and Dunkler, Daniela},

From d98ff694ed8007a752a7ce686877f85820af612c Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Sun, 19 Apr 2026 00:28:53 +0000
Subject: [PATCH 13/14] docs: apply remaining review-thread fixes for
 independence diagnostics

Agent-Logs-Url: https://github.com/d-morrison/rme/sessions/8530f689-5c1f-420b-805b-a3dfc5943256

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>
---
 .../_sec_linreg_diagnostics.qmd               |  7 ++++--
 .../_sec_linreg_independence_examples.qmd     | 24 ++++++++++---------
 2 files changed, 18 insertions(+), 13 deletions(-)

diff --git a/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd b/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
index 31a85e859..1c2e1012f 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
@@ -10,8 +10,11 @@ This section is adapted from @dobson4e [§6.2-6.3] and
 
 ### Diagnostics for the independence assumption
 
-The independence assumption means that residual noise terms are not correlated across observations,
-after conditioning on the predictors in the model.
+The independence assumption means that the model errors are independent,
+or at least uncorrelated,
+across observations after conditioning on the predictors in the model.
+Because those errors are unobserved,
+we usually assess this assumption using residual-based plots and formal tests.
 
 For data with a natural ordering
 (for example, time, spatial location, or clinic visit sequence),
diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
index 70342423f..fbe8574e7 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
@@ -48,17 +48,19 @@ Correlogram diagnostics:
 ```{r}
 #| label: fig-independence-correlogram
 #| fig-cap: "Residual ACF and PACF for simulated data"
-op <- par(no.readonly = TRUE)
-on.exit(par(op), add = TRUE)
-par(mfrow = c(1, 2))
-acf(
-  resid(serial_dep_exm_lm),
-  main = "Residual ACF"
-)
-pacf(
-  resid(serial_dep_exm_lm),
-  main = "Residual PACF"
-)
+local({
+  op <- par(no.readonly = TRUE)
+  on.exit(par(op), add = TRUE)
+  par(mfrow = c(1, 2))
+  acf(
+    resid(serial_dep_exm_lm),
+    main = "Residual ACF"
+  )
+  pacf(
+    resid(serial_dep_exm_lm),
+    main = "Residual PACF"
+  )
+})
 ```
 
 Durbin-Watson and Breusch-Godfrey tests:

From 26c428fedd9f2deff35afe6e9aa265da58e47a2d Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Mon, 20 Apr 2026 07:54:50 +0000
Subject: [PATCH 14/14] docs: use theorem-variant divs in independence
 diagnostics section

Agent-Logs-Url: https://github.com/d-morrison/rme/sessions/af609840-ccc3-42e1-b230-46a09d70378b

Co-authored-by: d-morrison <2474437+d-morrison@users.noreply.github.com>
---
 .../_sec_linreg_diagnostics.qmd                   |  4 +++-
 .../_sec_linreg_independence_examples.qmd         |  8 ++++++--
 .../_sec_linreg_independence_math.qmd             | 15 +++++++++++----
 3 files changed, 20 insertions(+), 7 deletions(-)

diff --git a/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd b/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
index 1c2e1012f..a51beb2b9 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_diagnostics.qmd
@@ -21,7 +21,8 @@ For data with a natural ordering
 we usually assess independence with both plots and formal tests
 [@kutner2005applied, Chapter 12; @chatterjee2015regression, Chapter 6].
 
-Common diagnostics include:
+:::{#def-independence-diagnostics}
+#### Common diagnostics for independence
 
 1. Residuals versus observation order (or time) plots:
 patterns, runs, or drifts can indicate dependence
@@ -41,6 +42,7 @@ tests for first-order serial correlation in regression residuals
 tests for higher-order serial correlation,
 and is more flexible than Durbin-Watson in many regression settings
 [@chatterjee2015regression, Chapter 9; @kutner2005applied, Chapter 12].
+:::
 
 {{< include _subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd >}}
 
diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
index fbe8574e7..15e053d3f 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_examples.qmd
@@ -1,4 +1,5 @@
-#### Example diagnostics with code and output
+:::{#exm-independence-diagnostics-simulated}
+#### Simulated example for independence diagnostics
 
 The example below simulates ordered data with positively autocorrelated errors.
 That creates a setting where independence should fail.
@@ -75,8 +76,10 @@ lmtest::bgtest(serial_dep_exm_lm, order = 4)
 In this example,
 the residual-order plot and correlogram suggest serial dependence,
 and both formal tests reject independence.
+:::
 
-#### Real-data examples
+:::{#exm-independence-diagnostics-real-data}
+#### Real-data examples for independence diagnostics
 
 We can also run the same diagnostics on real ordered data sets.
 
@@ -132,3 +135,4 @@ lmtest::bgtest(co2_lm, order = 12)
 For both real-data examples,
 the diagnostics indicate residual dependence,
 consistent with their time-ordered structure.
+:::
diff --git a/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd b/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd
index 09e605c67..100b73e01 100644
--- a/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd
+++ b/_subfiles/Linear-models-overview/_sec_linreg_independence_math.qmd
@@ -4,8 +4,10 @@ Suppose the observations have a meaningful order,
 indexed by $t = 1, \ldots, n$.
 Let $e_t$ denote the OLS residual from a fitted regression model.
 
-For the Durbin-Watson diagnostic,
-the test statistic is
+:::{#def-durbin-watson-diagnostic}
+#### Durbin-Watson diagnostic
+
+The test statistic is
 $$
 \ba
 d
@@ -21,7 +23,7 @@ This approximation is most useful in standard large-sample linear-model settings
 and is less straightforward in models with lagged outcomes
 [@kutner2005applied, Chapter 12].
 
-The Durbin-Watson null and alternatives are typically framed through
+The null and alternatives are typically framed through
 an AR(1) error model:
 $$
 \eps_t = \rho \eps_{t-1} + u_t.
@@ -35,8 +37,12 @@ depending on whether we suspect positive,
 negative,
 or any serial correlation
 [@kutner2005applied, Chapter 12].
+:::
+
+:::{#def-breusch-godfrey-diagnostic}
+#### Breusch-Godfrey diagnostic
 
-For the Breusch-Godfrey test of order $p$,
+For a test of order $p$,
 we run an auxiliary regression of residuals on:
 the original regressors,
 and lagged residuals $e_{t-1}, \ldots, e_{t-p}$.
@@ -50,3 +56,4 @@ $H_0: \rho_1 = \cdots = \rho_p = 0$,
 it is asymptotically
 $\chi^2_p$
 [@chatterjee2015regression, Chapter 6; @draper2014applied, Chapter 11].
+:::