2  Linear (Gaussian) Models

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Configuring R

Functions from these packages will be used throughout this document:

library(conflicted) # check for conflicting function definitions
# library(printr) # inserts help-file output into markdown output
library(rmarkdown) # Convert R Markdown documents into a variety of formats.
library(pander) # format tables for markdown
library(ggplot2) # graphics
library(ggeasy) # help with graphics
library(ggfortify) # help with graphics
library(dplyr) # manipulate data
library(tibble) # `tibble`s extend `data.frame`s
library(magrittr) # `%>%` and other additional piping tools
library(haven) # import Stata files
library(knitr) # format R output for markdown
library(tidyr) # Tools to help to create tidy data
library(plotly) # interactive graphics
library(dobson) # datasets from Dobson and Barnett 2018
library(parameters) # format model output tables for markdown
library(haven) # import Stata files
library(latex2exp) # use LaTeX in R code (for figures and tables)
library(fs) # filesystem path manipulations
library(survival) # survival analysis
library(survminer) # survival analysis graphics
library(KMsurv) # datasets from Klein and Moeschberger
library(parameters) # format model output tables for
library(webshot2) # convert interactive content to static for pdf
library(forcats) # functions for categorical variables ("factors")
library(stringr) # functions for dealing with strings
library(lubridate) # functions for dealing with dates and times

Here are some R settings I use in this document:

rm(list = ls()) # delete any data that's already loaded into R

conflicts_prefer(dplyr::filter)
ggplot2::theme_set(
  ggplot2::theme_bw() + 
        # ggplot2::labs(col = "") +
    ggplot2::theme(
      legend.position = "bottom",
      text = ggplot2::element_text(size = 12, family = "serif")))

knitr::opts_chunk$set(message = FALSE)
options('digits' = 4)

panderOptions("big.mark", ",")
pander::panderOptions("table.emphasize.rownames", FALSE)
pander::panderOptions("table.split.table", Inf)
conflicts_prefer(dplyr::filter) # use the `filter()` function from dplyr() by default
legend_text_size = 9

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Note

This content is adapted from:

• Dobson and Barnett (2018), Chapters 2-6
• Dunn and Smyth (2018), Chapters 2-3
• Vittinghoff et al. (2012), Chapter 4

There are numerous textbooks specifically for linear regression, including:

• Kutner et al. (2005): used for UCLA Biostatistics MS level linear models class
• Chatterjee and Hadi (2015): used for Stanford MS-level linear models class
• Seber and Lee (2012): used for UCLA Biostatistics PhD level linear models class and UC Davis STA 108.
• Kleinbaum et al. (2014): same first author as Kleinbaum and Klein (2010) and Kleinbaum and Klein (2012)
• Weisberg (2005)

2.1 Overview

2.1.1 Why this course includes linear regression

• This course is about generalized linear models (for non-Gaussian outcomes)

• UC Davis STA 108 (“Applied Statistical Methods: Regression Analysis”) is a prerequisite for this course, so everyone here should have some understanding of linear regression already.

• We will review linear regression to:
  • make sure everyone is caught up
  • to provide an epidemiological perspective on model interpretation.

2.1.2 Chapter overview

• Section 2.2: how to interpret linear regression models

• Section 2.3: how to estimate linear regression models

• Section 2.4: how to quantify uncertainty about our estimates

• Section 2.8: how to tell if your model is insufficiently complex

2.2 Understanding Gaussian Linear Regression Models

2.2.1 Motivating example: birthweights and gestational age

Suppose we want to learn about the distributions of birthweights (outcome \(Y\)) for (human) babies born at different gestational ages (covariate \(A\)) and with different chromosomal sexes (covariate \(S\)) (Dobson and Barnett (2018) Example 2.2.2).

Data as table

library(dobson)
data("birthweight", package = "dobson")
birthweight |> knitr::kable()

Table 2.1: birthweight data (Dobson and Barnett (2018) Example 2.2.2)

boys gestational age	boys weight	girls gestational age	girls weight
40	2968	40	3317
38	2795	36	2729
40	3163	40	2935
35	2925	38	2754
36	2625	42	3210
37	2847	39	2817
41	3292	40	3126
40	3473	37	2539
37	2628	36	2412
38	3176	38	2991
40	3421	39	2875
38	2975	40	3231

Reshape data for graphing

bw = 
  birthweight |> 
  pivot_longer(
    cols = everything(),
    names_to = c("sex", ".value"),
    names_sep = "s "
  ) |> 
  rename(age = `gestational age`) |> 
  mutate(
    sex = sex |> 
      case_match(
        "boy" ~ "male",
        "girl" ~ "female") |> 
      factor(levels = c("female", "male")))

bw

Table 2.2: birthweight data reshaped

Data as graph

plot1 = bw |> 
  ggplot(aes(
    x = age, 
    y = weight,
    linetype = sex,
    shape = sex,
    col = sex))  +
  theme_bw() +
  xlab("Gestational age (weeks)") +
  ylab("Birthweight (grams)") +
  theme(legend.position = "bottom") +
  # expand_limits(y = 0, x = 0) +
  geom_point(alpha = .7)
print(plot1 + facet_wrap(~ sex))

Figure 2.1: birthweight data (Dobson and Barnett (2018) Example 2.2.2)

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Data notation

Let’s define some notation to represent this data.

• \(Y\): birthweight (measured in grams)

• \(S\): chromosomal sex: “male” (XY) or “female” (XX)

• \(M\): indicator variable for \(S\) = “male”¹

• \(M = 0\) if female (XX)

• \(M = 1\) if male (XY)

• \(F\): indicator variable for \(S\) = “female”²

• \(F = 1\) if female (XX)

• \(F = 0\) if male (XY)

• \(A\): estimated gestational age at birth (measured in weeks).

Note

Female is the reference level for the categorical variable \(S\) (chromosomal sex) and corresponding indicator variable \(M\) . The choice of a reference level is arbitrary and does not limit what we can do with the resulting model; it only makes it more computationally convenient to make inferences about comparisons involving that reference group.

2.2.2 Parallel lines regression

We don’t have enough data to model the distribution of birth weight separately for each combination of gestational age and sex, so let’s instead consider a (relatively) simple model for how that distribution varies with gestational age and sex:

\[p(Y=y|A=a,S=s) \ \sim_{\text{iid}}\ N(\mu(a,s), \sigma^2)\]

\[ \begin{aligned} \mu(a,s) &\stackrel{\text{def}}{=}\mathbb{E}\left[Y|A=a, S=s\right] \\ &= \beta_0 + \beta_A a+ \beta_M m \end{aligned} \tag{2.1}\]

Table 2.3 shows the parameter estimates from R. Figure 2.2 shows the estimated model, superimposed on the data.

bw_lm1 = lm(
  formula = weight ~ sex + age, 
  data = bw)

bw_lm1 |> 
  parameters() |>
  print_md(
    include_reference = TRUE,
    # show_sigma = TRUE,
    select = "{estimate}")

Table 2.3: Estimate of Model 2.1 for birthweight data

Parameter	Estimate
(Intercept)	-1773.32
sex (female)	0.00
sex (male)	163.04
age	120.89

bw = 
  bw |> 
  mutate(`E[Y|X=x]` = fitted(bw_lm1)) |> 
  arrange(sex, age)

plot2 = 
  plot1 %+% bw +
  geom_line(aes(y = `E[Y|X=x]`))

print(plot2)

Figure 2.2: Parallel-slopes model of birthweight

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Model assumptions and predictions

To learn what this model is assuming, let’s plug in a few values.

Exercise 2.1 According to this model, what’s the mean birthweight for a female born at 36 weeks?

Table 2.4: Estimated coefficients for model 2.1

coef(bw_lm1)
#> (Intercept)     sexmale         age 
#>     -1773.3       163.0       120.9

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Solution.  

pred_female = coef(bw_lm1)["(Intercept)"] + coef(bw_lm1)["age"]*36
coef(bw_lm1)
#> (Intercept)     sexmale         age 
#>     -1773.3       163.0       120.9
# print(pred_female)
### built-in prediction: 
# predict(bw_lm1, newdata = tibble(sex = "female", age = 36))

\[ \begin{aligned} E[Y|A = 0, A = 36] &= \beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36 \\ &= 2578.8739 \end{aligned} \]

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Exercise 2.2 What’s the mean birthweight for a male born at 36 weeks?

coef(bw_lm1)
#> (Intercept)     sexmale         age 
#>     -1773.3       163.0       120.9

---

Solution.  

pred_male = 
  coef(bw_lm1)["(Intercept)"] + 
  coef(bw_lm1)["sexmale"] + 
  coef(bw_lm1)["age"]*36
coef(bw_lm1)
#> (Intercept)     sexmale         age 
#>     -1773.3       163.0       120.9

\[ \begin{aligned} E[Y|M = 1, A = 36] &= \beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36 \\ &= 2741.9132 \end{aligned} \]

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Exercise 2.3 What’s the difference in mean birthweights between males born at 36 weeks and females born at 36 weeks?

coef(bw_lm1)
#> (Intercept)     sexmale         age 
#>     -1773.3       163.0       120.9

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Solution. \[ \begin{aligned} & E[Y|M = 1, A = 36] - E[Y|M = 0, A = 36]\\ &= 2741.9132 - 2578.8739\\ &= 163.0393 \end{aligned} \]

Shortcut:

\[ \begin{aligned} & E[Y|M = 1, A = 36] - E[Y|M = 0, A = 36]\\ &= (\beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36) - (\beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36) \\ &= \beta_M \\ &= 163.0393 \end{aligned} \]

Note that age doesn’t show up in this difference: in other words, according to this model, the difference between females and males with the same gestational age is the same for every age.

That’s an assumption of the model; it’s built-in to the parametric structure, even before we plug in the estimated values of those parameters.

That’s why the lines are parallel.

2.2.3 Interactions

What if we don’t like that parallel lines assumption?

Then we need to allow an “interaction” between age \(A\) and sex \(S\):

\[ E[Y|A=a, S=s] = \beta_0 + \beta_A a+ \beta_M m + \beta_{AM} (a \cdot m) \tag{2.2}\]

Now, the slope of mean birthweight \(E[Y|A,S]\) with respect to gestational age \(A\) depends on the value of sex \(S\).

bw_lm2 = lm(weight ~ sex + age + sex:age, data = bw)
bw_lm2 |> 
  parameters() |>
  print_md(
    include_reference = TRUE,
    # show_sigma = TRUE,
    select = "{estimate}")

Table 2.5: Birthweight model with interaction term

Parameter	Estimate
(Intercept)	-2141.67
sex (female)	0.00
sex (male)	872.99
age	130.40
sex (male) × age	-18.42

bw = 
  bw |> 
  mutate(
    predlm2 = predict(bw_lm2)
  ) |> 
  arrange(sex, age)

plot1_interact = 
  plot1 %+% bw +
  geom_line(aes(y = predlm2))

print(plot1_interact)

Figure 2.3: Birthweight model with interaction term

Now we can see that the lines aren’t parallel.

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Here’s another way we could rewrite this model (by collecting terms involving \(S\)):

\[ E[Y|A, M] = \beta_0 + \beta_M M+ (\beta_A + \beta_{AM} M) A \]

Note

If you want to understand a coefficient in a model with interactions, collect terms for the corresponding variable, and you will see what other variables are interacting with the variable you are interested in.

In this case, the coefficient \(S\) is interacting with \(A\). So the slope of \(Y\) with respect to \(A\) depends on the value of \(M\).

According to this model, there is no such thing as “the slope of birthweight with respect to age”. There are two slopes, one for each sex.³ We can only talk about “the slope of birthweight with respect to age among males” and “the slope of birthweight with respect to age among females”.

Then: that coefficient is the difference in means per unit change in its corresponding coefficient, when the other collected variables are set to 0.

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To learn what this model is assuming, let’s plug in a few values.

Exercise 2.4 According to this model, what’s the mean birthweight for a female born at 36 weeks?

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Solution.  

pred_female = coef(bw_lm2)["(Intercept)"] + coef(bw_lm2)["age"]*36

\[ E[Y|A = 0, X_2 = 36] = \beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36 + \beta_{AM} \cdot (0 * 36) = 2552.7333 \]

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Exercise 2.5 What’s the mean birthweight for a male born at 36 weeks?

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Solution.  

pred_male = 
  coef(bw_lm2)["(Intercept)"] + 
  coef(bw_lm2)["sexmale"] + 
  coef(bw_lm2)["age"]*36 + 
  coef(bw_lm2)["sexmale:age"] * 36

\[ \begin{aligned} E[Y|A = 0, X_2 = 36] &= \beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36 + \beta_{AM} \cdot 1 \cdot 36\\ &= 2762.7069 \end{aligned} \]

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Exercise 2.6 What’s the difference in mean birthweights between males born at 36 weeks and females born at 36 weeks?

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Solution. \[ \begin{aligned} & E[Y|M = 1, A = 36] - E[Y|M = 0, A = 36]\\ &= (\beta_0 + \beta_M \cdot 1+ \beta_A \cdot 36 + \beta_{AM} \cdot 1 \cdot 36)\\ &\ \ \ \ \ -(\beta_0 + \beta_M \cdot 0+ \beta_A \cdot 36 + \beta_{AM} \cdot 0 \cdot 36) \\ &= \beta_{S} + \beta_{AM}\cdot 36\\ &= 209.9736 \end{aligned} \]

Note that age now does show up in the difference: in other words, according to this model, the difference in mean birthweights between females and males with the same gestational age can vary by gestational age.

That’s how the lines in the graph ended up non-parallel.

2.2.4 Stratified regression

We could re-write the interaction model as a stratified model, with a slope and intercept for each sex:

\[ \mathbb{E}\left[Y|A=a, S=s\right] = \beta_M m + \beta_{AM} (a \cdot m) + \beta_F f + \beta_{AF} (a \cdot f) \tag{2.3}\]

Compare this stratified model with our interaction model, Equation 2.2:

\[ \mathbb{E}\left[Y|A=a, S=s\right] = \beta_0 + \beta_A a + \beta_M m + \beta_{AM} (a \cdot m) \]

In the stratified model, the intercept term \(\beta_0\) has been relabeled as \(\beta_F\).

bw_lm2 = lm(weight ~ sex + age + sex:age, data = bw)
bw_lm2 |> 
  parameters() |>
  print_md(
    include_reference = TRUE,
    # show_sigma = TRUE,
    select = "{estimate}")

Table 2.6: Birthweight model with interaction term

Parameter	Estimate
(Intercept)	-2141.67
sex (female)	0.00
sex (male)	872.99
age	130.40
sex (male) × age	-18.42

bw_lm_strat = 
  bw |> 
  lm(
    formula = weight ~ sex + sex:age - 1, 
    data = _)

bw_lm_strat |> 
  parameters() |>
  print_md(
    # show_sigma = TRUE,
    select = "{estimate}")

Table 2.7: Birthweight model - stratified betas

Parameter	Estimate
sex (female)	-2141.67
sex (male)	-1268.67
sex (female) × age	130.40
sex (male) × age	111.98

2.2.5 Curved-line regression

If we transform some of our covariates (\(X\)s) and plot the resulting model on the original covariate scale, we end up with curved regression lines:

bw_lm3 = lm(weight ~ sex:log(age) - 1, data = bw)
library(palmerpenguins)

ggpenguins <- 
  palmerpenguins::penguins |> 
  dplyr::filter(species == "Adelie") |> 
  ggplot(
    aes(x = bill_length_mm , y = body_mass_g)) +
  geom_point() + 
  xlab("Bill length (mm)") + 
  ylab("Body mass (g)")

ggpenguins2 = ggpenguins +
  stat_smooth(
    method = "lm",
    formula = y ~ log(x),
    geom = "smooth") +
  xlab("Bill length (mm)") + 
  ylab("Body mass (g)")


ggpenguins2 |> print()

Figure 2.4: palmerpenguins model with bill_length entering on log scale

2.3 Estimating Linear Models via Maximum Likelihood

2.3.1 Likelihood, log-likelihood, and score functions for linear regression

In EPI 203 and Appendix F, we learned how to fit outcome-only models of the form \(p(X=x|\theta)\) to iid data \(\tilde{x}= (x_1,…,x_n)\) using maximum likelihood estimation.

Now, we apply the same procedure to linear regression models:

\[ \mathcal L(\tilde{y}|\mathbf{x},\beta, \sigma^2) = \prod_{i=1}^n (2\pi\sigma^2)^{-1/2} \text{exp}\left\{-\frac{1}{2\sigma^2}(y_i - \tilde{x_i}'\beta)^2\right\} \tag{2.4}\]

\[ \ell(\tilde{y}|\mathbf{x},\beta, \sigma^2) = -\frac{n}{2}\text{log}\left\{\sigma^2\right\} - \frac{1}{2\sigma^2}\sum_{i=1}^n (y_i - \tilde{x_i}' \beta)^2 \tag{2.5}\]

\[ \ell'_{\beta}(\tilde{y}|\mathbf{x},\beta, \sigma^2) = - \frac{1}{2\sigma^2}\frac{\partial}{\partial \beta} \left(\sum_{i=1}^n (y_i - \tilde{x_i}^{\top} \beta)^2\right) \tag{2.6}\]

---

Let’s switch to matrix-vector notation:

\[ \sum_{i=1}^n (y_i - \tilde{x}_i^{\top} \tilde{\beta})^2 = (\tilde{y}- \mathbf{X}\tilde{\beta})'(\tilde{y}- \mathbf{X}\tilde{\beta}) \]

---

So

\[ \begin{aligned} (\tilde{y}- \mathbf{X}\tilde{\beta})'(\tilde{y}- \mathbf{X}\tilde{\beta}) &= (\tilde{y}' - \tilde{\beta}'X')(\tilde{y}- \mathbf{X}\tilde{\beta}) \\ &= y'y - \tilde{\beta}'X'y - y'\mathbf{X}\tilde{\beta}+\tilde{\beta}'\mathbf{X}'\mathbf{X}\beta \\ &= y'y - 2y'\mathbf{X}\beta +\beta'\mathbf{X}'\mathbf{X}\beta \end{aligned} \]

2.3.2 Deriving the linear regression score function

We will use some results from vector calculus:

\[ \begin{aligned} \frac{\partial}{\partial \beta}\left(\sum_{i=1}^n (y_i - x_i' \beta)^2\right) &= \frac{\partial}{\partial \beta}(\tilde{y}- X\beta)'(\tilde{y}- X\beta) \\ &= \frac{\partial}{\partial \beta} (y'y - 2y'X\beta +\beta'X'X\beta) \\ &= (- 2X'y +2X'X\beta) \\ &= - 2X'(y - X\beta) \\ &= - 2X'(y - \mathbb{E}[y]) \\ &= - 2X' \varepsilon(y) \end{aligned} \tag{2.7}\]

---

So if \(\ell(\beta,\sigma^2) =0\), then

\[ \begin{aligned} 0 &= (- 2X'y +2X'X\beta)\\ 2X'y &= 2X'X\beta\\ X'y &= X'X\beta\\ (X'X)^{-1}X'y &= \beta \end{aligned} \]

---

The second derivative matrix \(\ell_{\beta, \beta'} ''(\beta, \sigma^2;\mathbf X,\tilde{y})\) is negative definite at \(\beta = (X'X)^{-1}X'y\), so \(\hat \beta_{ML} = (X'X)^{-1}X'y\) is the MLE for \(\beta\).

---

Similarly (not shown):

\[ \hat\sigma^2_{ML} = \frac{1}{n} (Y-X\hat\beta)'(Y-X\hat\beta) \]

And

\[ \begin{aligned} \mathcal I_{\beta} &= E[-\ell_{\beta, \beta'} ''(Y|X,\beta, \sigma^2)]\\ &= \frac{1}{\sigma^2}X'X \end{aligned} \]

---

So:

\[ Var(\hat \beta) \approx (\mathcal I_{\beta})^{-1} = \sigma^2 (X'X)^{-1} \]

and

\[ \hat\beta \dot \sim N(\beta, \mathcal I_{\beta}^{-1}) \]

These are all results you have hopefully seen before.

---

In the Gaussian linear regression case, we also have exact results:

\[ \frac{\hat\beta_j}{\hat{\text{se}}\left(\hat\beta_j\right)} \ \sim \ t_{n-p} \]

---

In our model 2 above, \(\hat{\mathcal{I}}(\beta)\) is:

bw_lm2 |> vcov()
#>             (Intercept)  sexmale      age sexmale:age
#> (Intercept)     1353968 -1353968 -34871.0     34871.0
#> sexmale        -1353968  2596387  34871.0    -67211.0
#> age              -34871    34871    899.9      -899.9
#> sexmale:age       34871   -67211   -899.9      1743.5

If we take the square roots of the diagonals, we get the standard errors listed in the model output:

bw_lm2 |> vcov() |> diag() |> sqrt()
#> (Intercept)     sexmale         age sexmale:age 
#>     1163.60     1611.33       30.00       41.76

bw_lm2 |> parameters() |> print_md()

Table 2.8: Estimated model for birthweight data with interaction term

Parameter	Coefficient	SE	95% CI	t(20)	p
(Intercept)	-2141.67	1163.60	(-4568.90, 285.56)	-1.84	0.081
sex (male)	872.99	1611.33	(-2488.18, 4234.17)	0.54	0.594
age	130.40	30.00	(67.82, 192.98)	4.35	< .001
sex (male) × age	-18.42	41.76	(-105.52, 68.68)	-0.44	0.664

So we can do confidence intervals, hypothesis tests, and p-values exactly as in the one-variable case we looked at previously.

2.3.3 Residual Standard Deviation

\(\hat \sigma\) represents an estimate of the Residual Standard Deviation parameter, \(\sigma\). We can extract \(\hat \sigma\) from the fitted model, using the sigma() function:

sigma(bw_lm2)
#> [1] 180.6

---

\(\sigma\) is NOT “Residual standard error”

In the summary.lm() output, this estimate is labeled as "Residual standard error":

summary(bw_lm2)
#> 
#> Call:
#> lm(formula = weight ~ sex + age + sex:age, data = bw)
#> 
#> Residuals:
#>    Min     1Q Median     3Q    Max 
#> -246.7 -138.1  -39.1  176.6  274.3 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  -2141.7     1163.6   -1.84  0.08057 .  
#> sexmale        873.0     1611.3    0.54  0.59395    
#> age            130.4       30.0    4.35  0.00031 ***
#> sexmale:age    -18.4       41.8   -0.44  0.66389    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 181 on 20 degrees of freedom
#> Multiple R-squared:  0.643,  Adjusted R-squared:  0.59 
#> F-statistic:   12 on 3 and 20 DF,  p-value: 0.000101

---

However, this is a misnomer:

library(printr) # captures ? documentation
?stats::sigma

sigma

R Documentation

Extract Residual Standard Deviation 'Sigma'

Description

Extract the estimated standard deviation of the errors, the “residual standard deviation” (misnamed also “residual standard error”, e.g., in summary.lm()'s output, from a fitted model).

Many classical statistical models have a scale parameter, typically the standard deviation of a zero-mean normal (or Gaussian) random variable which is denoted as \sigma. sigma(.) extracts the estimated parameter from a fitted model, i.e., \hat\sigma.

Note

The misnomer “Residual standard error” has been part of too many R (and S) outputs to be easily changed there.

2.4 Inference about Gaussian Linear Regression Models

2.4.1 Motivating example: birthweight data

Research question: is there really an interaction between sex and age?

\(H_0: \beta_{AM} = 0\)

\(H_A: \beta_{AM} \neq 0\)

\(P(|\hat\beta_{AM}| > |-18.4172| \mid H_0)\) = ?

2.4.2 Wald tests and CIs

R can give you Wald tests for single coefficients and corresponding CIs:

bw_lm2 |> 
  parameters() |>
  print_md(
    include_reference = TRUE)

Parameter	Coefficient	SE	95% CI	t(20)	p
(Intercept)	-2141.67	1163.60	(-4568.90, 285.56)	-1.84	0.081
sex (female)	0.00
sex (male)	872.99	1611.33	(-2488.18, 4234.17)	0.54	0.594
age	130.40	30.00	(67.82, 192.98)	4.35	< .001
sex (male) × age	-18.42	41.76	(-105.52, 68.68)	-0.44	0.664

To understand what’s happening, let’s replicate these results by hand for the interaction term.

2.4.3 P-values

bw_lm2 |> 
  parameters(keep = "sexmale:age") |>
  print_md(
    include_reference = TRUE)

Parameter	Coefficient	SE	95% CI	t(20)	p
sex (male) × age	-18.42	41.76	(-105.52, 68.68)	-0.44	0.664

beta_hat = coef(summary(bw_lm2))["sexmale:age", "Estimate"]
se_hat = coef(summary(bw_lm2))["sexmale:age", "Std. Error"]
dfresid = bw_lm2$df.residual
t_stat = abs(beta_hat)/se_hat
pval_t = 
  pt(-t_stat, df = dfresid, lower.tail = TRUE) +
  pt(t_stat, df = dfresid, lower.tail = FALSE)

\[ \begin{aligned} &P\left( | \hat \beta_{AM} | > | -18.4172| \middle| H_0 \right) \\ &= \Pr \left( \left| \frac{\hat\beta_{AM}}{\hat{SE}(\hat\beta_{AM})} \right| > \left| \frac{-18.4172}{41.7558} \right| \middle| H_0 \right)\\ &= \Pr \left( \left| T_{20} \right| > 0.4411 | H_0 \right)\\ &= 0.6639 \end{aligned} \]

This matches the result in the table above.

2.4.4 Confidence intervals

bw_lm2 |> 
  parameters(keep = "sexmale:age") |>
  print_md(
    include_reference = TRUE)

Parameter	Coefficient	SE	95% CI	t(20)	p
sex (male) × age	-18.42	41.76	(-105.52, 68.68)	-0.44	0.664

q_t = qt(
  p = 0.975, 
  df = dfresid, 
  lower.tail = TRUE)

q_t = qt(
  p = 0.025, 
  df = dfresid, 
  lower.tail = TRUE)


confint_radius_t = 
  se_hat * q_t

confint_t = beta_hat + c(-1,1) * confint_radius_t

print(confint_t)
#> [1]   68.68 -105.52

This also matches.

2.4.5 Gaussian approximations

Here are the asymptotic (Gaussian approximation) equivalents:

2.4.6 P-values

bw_lm2 |> 
  parameters(keep = "sexmale:age") |>
  print_md(
    include_reference = TRUE)

Parameter	Coefficient	SE	95% CI	t(20)	p
sex (male) × age	-18.42	41.76	(-105.52, 68.68)	-0.44	0.664

pval_z = pnorm(abs(t_stat), lower = FALSE) * 2

print(pval_z)
#> [1] 0.6592

2.4.7 Confidence intervals

bw_lm2 |> 
  parameters(keep = "sexmale:age") |>
  print_md(
    include_reference = TRUE)

Parameter	Coefficient	SE	95% CI	t(20)	p
sex (male) × age	-18.42	41.76	(-105.52, 68.68)	-0.44	0.664

confint_radius_z = se_hat * qnorm(0.975, lower = TRUE)
confint_z = 
  beta_hat + c(-1,1) * confint_radius_z
print(confint_z)
#> [1] -100.26   63.42

2.4.8 Likelihood ratio statistics

logLik(bw_lm2)
#> 'log Lik.' -156.6 (df=5)
logLik(bw_lm1)
#> 'log Lik.' -156.7 (df=4)

lLR = (logLik(bw_lm2) - logLik(bw_lm1)) |> as.numeric()
delta_df = (bw_lm1$df.residual - df.residual(bw_lm2))


x_max = 1

---

d_lLR = function(x, df = delta_df) dchisq(x, df = df)

chisq_plot = 
  ggplot() + 
  geom_function(fun = d_lLR) +
  stat_function( fun = d_lLR, xlim = c(lLR, x_max), geom = "area", fill = "gray") +
  geom_segment(aes(x = lLR, xend = lLR, y = 0, yend = d_lLR(lLR)), col = "red") + 
  xlim(0.0001,x_max) + 
  ylim(0,4) + 
  ylab("p(X=x)") + 
  xlab("log(likelihood ratio) statistic [x]") +
  theme_classic()
chisq_plot |> print()

Figure 2.5: Chi-square distribution

---

Now we can get the p-value:

pchisq(
  q = 2*lLR, 
  df = delta_df, 
  lower = FALSE) |> 
  print()
#> [1] 0.6298

---

In practice you don’t have to do this by hand; there are functions to do it for you:

# built in
library(lmtest)
lrtest(bw_lm2, bw_lm1)

#Df	LogLik	Df	Chisq	Pr(>Chisq)
5	-156.6	NA	NA	NA
4	-156.7	-1	0.2323	0.6298

2.5 Goodness of fit

2.5.1 AIC and BIC

When we use likelihood ratio tests, we are comparing how well different models fit the data.

Likelihood ratio tests require “nested” models: one must be a special case of the other.

If we have non-nested models, we can instead use the Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC):

• AIC = \(-2 * \ell(\hat\theta) + 2 * p\)

• BIC = \(-2 * \ell(\hat\theta) + p * \text{log}(n)\)

where \(\ell\) is the log-likelihood of the data evaluated using the parameter estimates \(\hat\theta\), \(p\) is the number of estimated parameters in the model (including \(\hat\sigma^2\)), and \(n\) is the number of observations.

You can calculate these criteria using the logLik() function, or use the built-in R functions:

AIC in R

-2 * logLik(bw_lm2) |> as.numeric() + 
  2*(length(coef(bw_lm2))+1) # sigma counts as a parameter here
#> [1] 323.2

AIC(bw_lm2)
#> [1] 323.2

BIC in R

-2 * logLik(bw_lm2) |> as.numeric() + 
  (length(coef(bw_lm2))+1) * log(nobs(bw_lm2))
#> [1] 329

BIC(bw_lm2)
#> [1] 329

Large values of AIC and BIC are worse than small values. There are no hypothesis tests or p-values associated with these criteria.

2.5.2 (Residual) Deviance

Let \(q\) be the number of distinct covariate combinations in a data set.

bw.X.unique = 
  bw |> 
  count(sex, age)

n_unique.bw  = nrow(bw.X.unique)

For example, in the birthweight data, there are \(q = 12\) unique patterns (Table 2.9).

bw.X.unique

Table 2.9: Unique covariate combinations in the birthweight data, with replicate counts

sex	age	n
female	36	2
female	37	1
female	38	2
female	39	2
female	40	4
female	42	1
male	35	1
male	36	1
male	37	2
male	38	3
male	40	4
male	41	1

---

Definition 2.1 (Replicates) If a given covariate pattern has more than one observation in a dataset, those observations are called replicates.

---

Example 2.1 (Replicates in the birthweight data) In the birthweight dataset, there are 2 replicates of the combination “female, age 36” (Table 2.9).

---

Exercise 2.7 (Replicates in the birthweight data) Which covariate pattern(s) in the birthweight data has the most replicates?

---

Solution 2.1 (Replicates in the birthweight data). Two covariate patterns are tied for most replicates: males at age 40 weeks and females at age 40 weeks. 40 weeks is the usual length for human pregnancy (Polin, Fox, and Abman (2011)), so this result makes sense.

bw.X.unique |> dplyr::filter(n == max(n))

sex	age	n
female	40	4
male	40	4

---

Saturated models

The most complicated model we could fit would have one parameter (a mean) for each covariate pattern, plus a variance parameter:

lm_max = 
  bw |> 
  mutate(age = factor(age)) |> 
  lm(
    formula = weight ~ sex:age - 1, 
    data = _)

lm_max |> 
  parameters() |> 
  print_md()

Table 2.10: Saturated model for the birthweight data

Parameter	Coefficient	SE	95% CI	t(12)	p
sex (male) × age35	2925.00	187.92	(2515.55, 3334.45)	15.56	< .001
sex (female) × age36	2570.50	132.88	(2280.98, 2860.02)	19.34	< .001
sex (male) × age36	2625.00	187.92	(2215.55, 3034.45)	13.97	< .001
sex (female) × age37	2539.00	187.92	(2129.55, 2948.45)	13.51	< .001
sex (male) × age37	2737.50	132.88	(2447.98, 3027.02)	20.60	< .001
sex (female) × age38	2872.50	132.88	(2582.98, 3162.02)	21.62	< .001
sex (male) × age38	2982.00	108.50	(2745.60, 3218.40)	27.48	< .001
sex (female) × age39	2846.00	132.88	(2556.48, 3135.52)	21.42	< .001
sex (female) × age40	3152.25	93.96	(2947.52, 3356.98)	33.55	< .001
sex (male) × age40	3256.25	93.96	(3051.52, 3460.98)	34.66	< .001
sex (male) × age41	3292.00	187.92	(2882.55, 3701.45)	17.52	< .001
sex (female) × age42	3210.00	187.92	(2800.55, 3619.45)	17.08	< .001

We call this model the full, maximal, or saturated model for this dataset.

We can calculate the log-likelihood of this model as usual:

logLik(lm_max)
#> 'log Lik.' -151.4 (df=13)

We can compare this model to our other models using chi-square tests, as usual:

lrtest(lm_max, bw_lm2)

#Df	LogLik	Df	Chisq	Pr(>Chisq)
13	-151.4	NA	NA	NA
5	-156.6	-8	10.36	0.241

The likelihood ratio statistic for this test is \[\lambda = 2 * (\ell_{\text{full}} - \ell) = 10.3554\] where:

• \(\ell_{\text{max}}\) is the log-likelihood of the full model: -151.4016
• \(\ell\) is the log-likelihood of our comparison model (two slopes, two intercepts): -156.5793

This statistic is called the deviance or residual deviance for our two-slopes and two-intercepts model; it tells us how much the likelihood of that model deviates from the likelihood of the maximal model.

The corresponding p-value tells us whether there we have enough evidence to detect that our two-slopes, two-intercepts model is a worse fit for the data than the maximal model; in other words, it tells us if there’s evidence that we missed any important patterns. (Remember, a nonsignificant p-value could mean that we didn’t miss anything and a more complicated model is unnecessary, or it could mean we just don’t have enough data to tell the difference between these models.)

2.5.3 Null Deviance

Similarly, the least complicated model we could fit would have only one mean parameter, an intercept:

\[\text E[Y|X=x] = \beta_0\] We can fit this model in R like so:

lm0 = lm(weight ~ 1, data = bw)

lm0 |> parameters() |> print_md()

Parameter	Coefficient	SE	95% CI	t(23)	p
(Intercept)	2967.67	57.58	(2848.56, 3086.77)	51.54	< .001

This model also has a likelihood:

logLik(lm0)
#> 'log Lik.' -169 (df=2)

And we can compare it to more complicated models using a likelihood ratio test:

lrtest(bw_lm2, lm0)

#Df	LogLik	Df	Chisq	Pr(>Chisq)
5	-156.6	NA	NA	NA
2	-169.0	-3	24.75	0

The likelihood ratio statistic for the test comparing the null model to the maximal model is \[\lambda = 2 * (\ell_{\text{full}} - \ell_{0}) = 35.1067\] where:

• \(\ell_{\text{0}}\) is the log-likelihood of the null model: -168.955
• \(\ell_{\text{full}}\) is the log-likelihood of the maximal model: -151.4016

In R, this test is:

lrtest(lm_max, lm0)

#Df	LogLik	Df	Chisq	Pr(>Chisq)
13	-151.4	NA	NA	NA
2	-169.0	-11	35.11	2e-04

This log-likelihood ratio statistic is called the null deviance. It tells us whether we have enough data to detect a difference between the null and full models.

2.6 Rescaling

2.6.1 Rescale age

bw = 
  bw |>
  mutate(
    `age - mean` = age - mean(age),
    `age - 36wks` = age - 36
  )

lm1c = lm(weight ~ sex + `age - 36wks`, data = bw)

lm2c = lm(weight ~ sex + `age - 36wks` + sex:`age - 36wks`, data = bw)

parameters(lm2c, ci_method = "wald") |> print_md()

Parameter	Coefficient	SE	95% CI	t(20)	p
(Intercept)	2552.73	97.59	(2349.16, 2756.30)	26.16	< .001
sex (male)	209.97	129.75	(-60.68, 480.63)	1.62	0.121
age - 36wks	130.40	30.00	(67.82, 192.98)	4.35	< .001
sex (male) × age - 36wks	-18.42	41.76	(-105.52, 68.68)	-0.44	0.664

Compare with what we got without rescaling:

parameters(bw_lm2, ci_method = "wald") |> print_md()

Parameter	Coefficient	SE	95% CI	t(20)	p
(Intercept)	-2141.67	1163.60	(-4568.90, 285.56)	-1.84	0.081
sex (male)	872.99	1611.33	(-2488.18, 4234.17)	0.54	0.594
age	130.40	30.00	(67.82, 192.98)	4.35	< .001
sex (male) × age	-18.42	41.76	(-105.52, 68.68)	-0.44	0.664

2.7 Prediction

2.7.1 Prediction for linear models

Definition 2.2 (Predicted value) In a regression model \(\text{p}(y|x)\), the predicted value of \(y\) given \(x\) is the estimated mean of \(Y\) given \(X\):

\[\hat y \stackrel{\text{def}}{=}\hat{\text{E}}\left[Y|X=x\right]\]

---

For linear models, the predicted value can be straightforwardly calculated by multiplying each predictor value \(x_j\) by its corresponding coefficient \(\beta_j\) and adding up the results:

\[ \begin{aligned} \hat Y &= \hat E[Y|X=x] \\ &= x'\hat\beta \\ &= \hat\beta_0\cdot 1 + \hat\beta_1 x_1 + ... + \hat\beta_p x_p \end{aligned} \]

---

2.7.2 Example: prediction for the birthweight data

X = c(1,1,40)
sum(X * coef(bw_lm1))
#> [1] 3225

R has built-in functions for prediction:

x = tibble(age = 40, sex = "male")
bw_lm1 |> predict(newdata = x)
#>    1 
#> 3225

If you don’t provide newdata, R will use the covariate values from the original dataset:

predict(bw_lm1)
#>    1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16 
#> 3225 3062 2984 2579 3225 3062 2621 2821 2742 3304 2863 2942 3346 3062 3225 2700 
#>   17   18   19   20   21   22   23   24 
#> 2863 2579 2984 2821 3225 2942 2984 3062

These special predictions are called the fitted values of the dataset:

Definition 2.3 For a given dataset \((\tilde{Y}, \mathbf{X})\) and corresponding fitted model \(\text{p}_{\hat \beta}(\tilde{y}|\mathbf{x})\), the fitted value of \(y_i\) is the predicted value of \(y\) when \(\tilde{X}=\tilde{x}_i\) using the estimate parameters \(\hat \beta\).

R has an extra function to get these values:

fitted(bw_lm1)
#>    1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16 
#> 3225 3062 2984 2579 3225 3062 2621 2821 2742 3304 2863 2942 3346 3062 3225 2700 
#>   17   18   19   20   21   22   23   24 
#> 2863 2579 2984 2821 3225 2942 2984 3062

2.7.3 Quantifying uncertainty in predictions

bw_lm1 |> 
  predict(
    newdata = x,
    se.fit = TRUE)
#> $fit
#>    1 
#> 3225 
#> 
#> $se.fit
#> [1] 61.46
#> 
#> $df
#> [1] 21
#> 
#> $residual.scale
#> [1] 177.1

This is a list(); you can extract the elements with $ or magrittr::use_series():

bw_lm1 |> 
  predict(
    newdata = x,
    se.fit = TRUE) |> 
  use_series(se.fit)
#> [1] 61.46

You can get confidence intervals for \(\mathbb{E}\left[Y|X=x\right]\):

bw_lm1 |> predict(
  newdata = x,
  interval = "confidence")

fit	lwr	upr
3225	3098	3353

You can also get prediction intervals for the value of an individual outcome \(Y\):

bw_lm1 |> 
  predict(newdata = x, interval = "predict")

fit	lwr	upr
3225	2836	3615

The warning from the last command is: “predictions on current data refer to future responses” (since you already know what happened to the current data, and thus don’t need to predict it).

See ?predict.lm for more.

2.8 Diagnostics

Tip

This section is adapted from Dobson and Barnett (2018, secs. 6.2–6.3) and Dunn and Smyth (2018) Chapter 3.

2.8.1 Assumptions in linear regression models

\[Y|\tilde{X}\ \sim_{⫫}\ N(\tilde{X}'\beta,\sigma^2)\]

1. Normality: The distribution conditional on a given \(X\) value is normal

2. Correct Functional Form: The conditional means have the structure

\[E[Y|\tilde{X} = \tilde{x}] = \tilde{x}'\beta\] 3. Homoskedasticity: The variance \(\sigma^2\) is constant (with respect to \(\tilde{x}\))

4. Independence: The observations are statistically independent

2.8.2 Direct visualization

The most direct way to examine the fit of a model is to compare it to the raw observed data.

bw = 
  bw |> 
  mutate(
    predlm2 = predict(bw_lm2)
  ) |> 
  arrange(sex, age)

plot1_interact = 
  plot1 %+% bw +
  geom_line(aes(y = predlm2))

print(plot1_interact)

Figure 2.6: Birthweight model with interaction term

It’s not easy to assess these assumptions from this model. If there are multiple continuous covariates, it becomes even harder to visualize the raw data.

2.8.3 Residuals

Maybe we can transform the data and model in some way to make it easier to inspect.

Definition 2.4 (Residual noise) The residual noise in a probabilistic model \(p(Y)\) is the difference between an observed value \(y\) and its distributional mean:

\[\varepsilon(y) \stackrel{\text{def}}{=}y - \mathbb{E}\left[Y\right] \tag{2.8}\]

We use the same notation for residual noise that we used for errors. \(\mathbb{E}\left[Y\right]\) can be viewed as an estimate of \(Y\), before \(y\) is observed. Conversely, each observation \(y\) can be viewed as an estimate of \(\mathbb{E}\left[Y\right]\) (albeit an imprecise one, individually, since \(n=1\)).

We can rearrange Equation 2.8 to view \(y\) as the sum of its mean plus the residual noise:

\[y = \mathbb{E}\left[Y\right] + \varepsilon{y}\]

---

Theorem 2.1 (Residuals in Gaussian models) If \(Y\) has a Gaussian distribution, then \(\varepsilon(Y)\) also has a Gaussian distribution, and vice versa.

Proof. Left to the reader.

---

Definition 2.5 (Residual errors of a fitted model value) The residual of a fitted value \(\hat y\) (shorthand: “residual”) is its error: \[ \begin{aligned} e(\hat y) &\stackrel{\text{def}}{=}\varepsilon\left(\hat y\right) \\&= y - \hat y \end{aligned} \]

\(e(\hat y)\) can be seen as the maximum likelihood estimate of the residual noise:

\[ \begin{aligned} e(\hat y) &= y - \hat y \\ &= \hat\varepsilon_{ML} \end{aligned} \]

---

General characteristics of residuals

Theorem 2.2 For unbiased estimators \(\hat\theta\):

\[\mathbb{E}\left[e(y)\right] = 0 \tag{2.9}\] \[\text{Var}\left(e(y)\right) \approx \sigma^2 \tag{2.10}\]

Proof.  

Equation 2.9:

\[ \begin{aligned} \mathbb{E}\left[e(y)\right] &= \mathbb{E}\left[y - \hat y\right] \\ &= \mathbb{E}\left[y\right] - \mathbb{E}\left[\hat y\right] \\ &= \mathbb{E}\left[y\right] - \mathbb{E}\left[y\right] \\ &= 0 \end{aligned} \]

Equation 2.10:

\[ \begin{aligned} \text{Var}\left(e(y)\right) &= \text{Var}\left(y - \hat y\right) \\ &= \text{Var}\left(y\right) + \text{Var}\left(\hat y\right) - 2 \text{Cov}\left(y, \hat y\right) \\ &{\dot{\approx}} \text{Var}\left(y\right) + 0 - 2 \cdot 0 \\ &= \text{Var}\left(y\right) \\ &= \sigma^2 \end{aligned} \]

---

Characteristics of residuals in Gaussian models

With enough data and a correct model, the residuals will be approximately Guassian distributed, with variance \(\sigma^2\), which we can estimate using \(\hat\sigma^2\): that is:

\[ e_i \ \sim_{\text{iid}}\ N(0, \hat\sigma^2) \]

---

Example 2.2 (residuals in birthweight data) R provides a function for residuals:

resid(bw_lm2)
#>       1       2       3       4       5       6       7       8       9      10 
#>  176.27 -140.73 -144.13  -59.53  177.47 -126.93  -68.93  242.67 -139.33   51.67 
#>      11      12      13      14      15      16      17      18      19      20 
#>  156.67 -125.13  274.28 -137.71  -27.69 -246.69 -191.67  189.33  -11.67 -242.64 
#>      21      22      23      24 
#>  -47.64  262.36  210.36  -30.62

Exercise 2.8 Check R’s output by computing the residuals directly.

Solution.  

bw$weight - fitted(bw_lm2)
#>       1       2       3       4       5       6       7       8       9      10 
#>  176.27 -140.73 -144.13  -59.53  177.47 -126.93  -68.93  242.67 -139.33   51.67 
#>      11      12      13      14      15      16      17      18      19      20 
#>  156.67 -125.13  274.28 -137.71  -27.69 -246.69 -191.67  189.33  -11.67 -242.64 
#>      21      22      23      24 
#>  -47.64  262.36  210.36  -30.62

This matches R’s output!

---

Graph the residuals

bw = bw |> 
  mutate(resids_intxn = 
           weight - fitted(bw_lm2))

plot_bw_resid =
  bw |> 
  ggplot(aes(
    x = age, 
    y = resids_intxn,
    linetype = sex,
    shape = sex,
    col = sex))  +
  theme_bw() +
  xlab("Gestational age (weeks)") +
  ylab("residuals (grams)") +
  theme(legend.position = "bottom") +
  # expand_limits(y = 0, x = 0) +
  geom_point(alpha = .7)
print(plot_bw_resid + facet_wrap(~ sex))

Figure 2.7: Residuals of interaction model for birthweight data

---

Definition 2.6 (Standardized residuals) \[r_i = \frac{e_i}{\widehat{SD}(e_i)}\]

Hence, with enough data and a correct model, the standardized residuals will be approximately standard Gaussian; that is,

\[ r_i \ \sim_{\text{iid}}\ N(0,1) \]

2.8.4 Marginal distributions of residuals

To look for problems with our model, we can check whether the residuals \(e_i\) and standardized residuals \(r_i\) look like they have the distributions that they are supposed to have, according to the model.

---

Standardized residuals in R

rstandard(bw_lm2)
#>        1        2        3        4        5        6        7        8 
#>  1.15982 -0.92601 -0.87479 -0.34723  1.03507 -0.73473 -0.39901  1.43752 
#>        9       10       11       12       13       14       15       16 
#> -0.82539  0.30606  0.92807 -0.87616  1.91428 -0.86559 -0.16430 -1.46376 
#>       17       18       19       20       21       22       23       24 
#> -1.11016  1.09658 -0.06761 -1.46159 -0.28696  1.58040  1.26717 -0.19805
resid(bw_lm2)/sigma(bw_lm2)
#>        1        2        3        4        5        6        7        8 
#>  0.97593 -0.77920 -0.79802 -0.32962  0.98258 -0.70279 -0.38166  1.34357 
#>        9       10       11       12       13       14       15       16 
#> -0.77144  0.28606  0.86741 -0.69282  1.51858 -0.76244 -0.15331 -1.36584 
#>       17       18       19       20       21       22       23       24 
#> -1.06123  1.04825 -0.06463 -1.34341 -0.26376  1.45262  1.16471 -0.16954

These are not quite the same, because R is doing something more complicated and precise to get the standard errors. Let’s not worry about those details for now; the difference is pretty small in this case:

---

rstandard_compare_plot = 
  tibble(
    x = resid(bw_lm2)/sigma(bw_lm2), 
    y = rstandard(bw_lm2)) |> 
  ggplot(aes(x = x, y = y)) +
  geom_point() + 
  theme_bw() +
  coord_equal() + 
  xlab("resid(bw_lm2)/sigma(bw_lm2)") +
  ylab("rstandard(bw_lm2)") +
  geom_abline(
    aes(
      intercept = 0,
      slope = 1, 
      col = "x=y")) +
  labs(colour="") +
  scale_colour_manual(values="red")

print(rstandard_compare_plot)

---

Let’s add these residuals to the tibble of our dataset:

bw = 
  bw |> 
  mutate(
    fitted_lm2 = fitted(bw_lm2),
    
    resid_lm2 = resid(bw_lm2),
    # resid_lm2 = weight - fitted_lm2,
    
    std_resid_lm2 = rstandard(bw_lm2),
    # std_resid_lm2 = resid_lm2 / sigma(bw_lm2)
  )

bw |> 
  select(
    sex,
    age,
    weight,
    fitted_lm2,
    resid_lm2,
    std_resid_lm2
  )

sex	age	weight	fitted_lm2	resid_lm2	std_resid_lm2
female	36	2729	2553	176.27	1.1598
female	36	2412	2553	-140.73	-0.9260
female	37	2539	2683	-144.13	-0.8748
female	38	2754	2814	-59.53	-0.3472
female	38	2991	2814	177.47	1.0351
female	39	2817	2944	-126.93	-0.7347
female	39	2875	2944	-68.93	-0.3990
female	40	3317	3074	242.67	1.4375
female	40	2935	3074	-139.33	-0.8254
female	40	3126	3074	51.67	0.3061
female	40	3231	3074	156.67	0.9281
female	42	3210	3335	-125.13	-0.8762
male	35	2925	2651	274.28	1.9143
male	36	2625	2763	-137.71	-0.8656
male	37	2847	2875	-27.69	-0.1643
male	37	2628	2875	-246.69	-1.4638
male	38	2795	2987	-191.67	-1.1102
male	38	3176	2987	189.33	1.0966
male	38	2975	2987	-11.67	-0.0676
male	40	2968	3211	-242.64	-1.4616
male	40	3163	3211	-47.64	-0.2870
male	40	3473	3211	262.36	1.5804
male	40	3421	3211	210.36	1.2672
male	41	3292	3323	-30.62	-0.1981

---

Now let’s build histograms:

resid_marginal_hist = 
  bw |> 
  ggplot(aes(x = resid_lm2)) +
  geom_histogram()

print(resid_marginal_hist)

Figure 2.8: Marginal distribution of (nonstandardized) residuals

Hard to tell with this small amount of data, but I’m a bit concerned that the histogram doesn’t show a bell-curve shape.

---

std_resid_marginal_hist = 
  bw |> 
  ggplot(aes(x = std_resid_lm2)) +
  geom_histogram()

print(std_resid_marginal_hist)

Figure 2.9: Marginal distribution of standardized residuals

This looks similar, although the scale of the x-axis got narrower, because we divided by \(\hat\sigma\) (roughly speaking).

Still hard to tell if the distribution is Gaussian.

---

2.8.5 QQ plot of standardized residuals

Another way to assess normality is the QQ plot of the standardized residuals versus normal quantiles:

library(ggfortify) 
# needed to make ggplot2::autoplot() work for `lm` objects

qqplot_lm2_auto = 
  bw_lm2 |> 
  autoplot(
    which = 2, # options are 1:6; can do multiple at once
    ncol = 1) +
  theme_classic()

print(qqplot_lm2_auto)

If the Gaussian model were correct, these points should follow the dotted line.

Fig 2.4 panel (c) in Dobson and Barnett (2018) is a little different; they didn’t specify how they produced it, but other statistical analysis systems do things differently from R.

See also Dunn and Smyth (2018) §3.5.4.

---

QQ plot - how it’s built

Let’s construct it by hand:

bw = bw |> 
  mutate(
    p = (rank(std_resid_lm2) - 1/2)/n(), # "Blom's method"
    expected_quantiles_lm2 = qnorm(p)
  )

qqplot_lm2 = 
  bw |> 
  ggplot(
    aes(
      x = expected_quantiles_lm2, 
      y = std_resid_lm2, 
      col = sex, 
      shape = sex)
  ) + 
  geom_point() +
  theme_classic() +
  theme(legend.position='none') + # removing the plot legend
  ggtitle("Normal Q-Q") +
  xlab("Theoretical Quantiles") + 
  ylab("Standardized residuals")

# find the expected line:

ps <- c(.25, .75)                  # reference probabilities
a <- quantile(rstandard(bw_lm2), ps)  # empirical quantiles
b <- qnorm(ps)                     # theoretical quantiles

qq_slope = diff(a)/diff(b)
qq_intcpt = a[1] - b[1] * qq_slope

qqplot_lm2 = 
  qqplot_lm2 +
  geom_abline(slope = qq_slope, intercept = qq_intcpt)

print(qqplot_lm2)

---

2.8.6 Conditional distributions of residuals

If our Gaussian linear regression model is correct, the residuals \(e_i\) and standardized residuals \(r_i\) should have:

• an approximately Gaussian distribution, with:
• a mean of 0
• a constant variance

This should be true for every value of \(x\).

---

If we didn’t correctly guess the functional form of the linear component of the mean, \[\text{E}[Y|X=x] = \beta_0 + \beta_1 X_1 + ... + \beta_p X_p\]

Then the the residuals might have nonzero mean.

Regardless of whether we guessed the mean function correctly, ther the variance of the residuals might differ between values of \(x\).

---

Residuals versus fitted values

To look for these issues, we can plot the residuals \(e_i\) against the fitted values \(\hat y_i\) (Figure 2.10).

autoplot(bw_lm2, which = 1, ncol = 1) |> print()

Figure 2.10: birthweight model (Equation 2.2): residuals versus fitted values

If the model is correct, the blue line should stay flat and close to 0, and the cloud of dots should have the same vertical spread regardless of the fitted value.

If not, we probably need to change the functional form of linear component of the mean, \[\text{E}[Y|X=x] = \beta_0 + \beta_1 X_1 + ... + \beta_p X_p\]

---

Example: PLOS Medicine title length data

(Adapted from Dobson and Barnett (2018), §6.7.1)

data(PLOS, package = "dobson")
library(ggplot2)
fig1 = 
  PLOS |> 
  ggplot(
    aes(x = authors,
        y = nchar)
  ) +
  geom_point() +
  theme(legend.position = "bottom") +
  labs(col = "") +
  guides(col=guide_legend(ncol=3))
fig1

Figure 2.11: Number of authors versus title length in PLOS Medicine articles

---

Linear fit

lm_PLOS_linear = lm(
  formula = nchar ~ authors, 
  data = PLOS)

fig2 = fig1 +
  geom_smooth(
    method = "lm", 
              fullrange = TRUE,
              aes(col = "lm(y ~ x)"))
fig2

library(ggfortify)
autoplot(lm_PLOS_linear, which = 1, ncol = 1)

Figure 2.12: Number of authors versus title length in PLOS Medicine, with linear model fit

(a) Data and fit

(b) Residuals vs fitted

---

Quadratic fit

lm_PLOS_quad = lm(
  formula = nchar ~ authors + I(authors^2), 
  data = PLOS)

fig3 = 
  fig2 + 
geom_smooth(
    method = "lm",
    fullrange = TRUE,
    formula = y ~ x + I(x ^ 2),
    aes(col = "lm(y ~ x + I(x^2))")
  )
fig3

autoplot(lm_PLOS_quad, which = 1, ncol = 1)

Figure 2.13: Number of authors versus title length in PLOS Medicine, with quadratic model fit

(a) Data and fit

(b) Residuals vs fitted

---

Linear versus quadratic fits

library(ggfortify)
autoplot(lm_PLOS_linear, which = 1, ncol = 1)

autoplot(lm_PLOS_quad, which = 1, ncol = 1)

Figure 2.14: Residuals versus fitted plot for linear and quadratic fits to PLOS data

(a) Linear

(b) Quadratic

---

Cubic fit

lm_PLOS_cub = lm(
  formula = nchar ~ authors + I(authors^2) + I(authors^3), 
  data = PLOS)

fig4 = 
  fig3 + 
geom_smooth(
    method = "lm",
    fullrange = TRUE,
    formula = y ~ x + I(x ^ 2) + I(x ^ 3),
    aes(col = "lm(y ~ x + I(x^2) + I(x ^ 3))")
  )
fig4

autoplot(lm_PLOS_cub, which = 1, ncol = 1)

Figure 2.15: Number of authors versus title length in PLOS Medicine, with cubic model fit

(a) Data and fit

(b) Residuals vs fitted

---

Logarithmic fit

lm_PLOS_log = lm(nchar ~ log(authors), data = PLOS)

fig5 = fig4 + 
  geom_smooth(
    method = "lm",
    fullrange = TRUE,
    formula = y ~ log(x),
    aes(col = "lm(y ~ log(x))")
  )
fig5

autoplot(lm_PLOS_log, which = 1, ncol = 1)

Figure 2.16: logarithmic fit

(a) Data and fit

(b) Residuals vs fitted

---

Model selection

anova(lm_PLOS_linear, lm_PLOS_quad)

Table 2.11: linear vs quadratic

Res.Df	RSS	Df	Sum of Sq	F	Pr(>F)
876	947502	NA	NA	NA	NA
875	880950	1	66552	66.1	0

anova(lm_PLOS_quad, lm_PLOS_cub)

Table 2.12: quadratic vs cubic

Res.Df	RSS	Df	Sum of Sq	F	Pr(>F)
875	880950	NA	NA	NA	NA
874	865933	1	15018	15.16	1e-04

---

AIC/BIC

AIC(lm_PLOS_quad)
#> [1] 8568
AIC(lm_PLOS_cub)
#> [1] 8555

AIC(lm_PLOS_cub)
#> [1] 8555
AIC(lm_PLOS_log)
#> [1] 8544

BIC(lm_PLOS_cub)
#> [1] 8578
BIC(lm_PLOS_log)
#> [1] 8558

---

Extrapolation is dangerous

fig_all = fig5 +
  xlim(0, 60)
fig_all

Figure 2.17: Number of authors versus title length in PLOS Medicine

---

Scale-location plot

We can also plot the square roots of the absolute values of the standardized residuals against the fitted values (Figure 2.18).

autoplot(bw_lm2, which = 3, ncol = 1) |> print()

Figure 2.18: Scale-location plot of birthweight data

Here, the blue line doesn’t need to be near 0, but it should be flat. If not, the residual variance \(\sigma^2\) might not be constant, and we might need to transform our outcome \(Y\) (or use a model that allows non-constant variance).

---

Residuals versus leverage

We can also plot our standardized residuals against “leverage”, which roughly speaking is a measure of how unusual each \(x_i\) value is. Very unusual \(x_i\) values can have extreme effects on the model fit, so we might want to remove those observations as outliers, particularly if they have large residuals.

autoplot(bw_lm2, which = 5, ncol = 1) |> print()

birthweight model with interactions (Equation 2.2): residuals versus leverage

The blue line should be relatively flat and close to 0 here.

---

2.8.7 Diagnostics constructed by hand

bw = 
  bw |> 
  mutate(
    predlm2 = predict(bw_lm2),
    residlm2 = weight - predlm2,
    std_resid = residlm2 / sigma(bw_lm2),
    # std_resid_builtin = rstandard(bw_lm2), # uses leverage
    sqrt_abs_std_resid = std_resid |> abs() |> sqrt()
    
  )

Residuals vs fitted

resid_vs_fit = bw |> 
  ggplot(
    aes(x = predlm2, y = residlm2, col = sex, shape = sex)
  ) + 
  geom_point() +
  theme_classic() +
  geom_hline(yintercept = 0)

print(resid_vs_fit)

Standardized residuals vs fitted

bw |> 
  ggplot(
    aes(x = predlm2, y = std_resid, col = sex, shape = sex)
  ) + 
  geom_point() +
  theme_classic() +
  geom_hline(yintercept = 0)

Standardized residuals vs gestational age

bw |> 
  ggplot(
    aes(x = age, y = std_resid, col = sex, shape = sex)
  ) + 
  geom_point() +
  theme_classic() +
  geom_hline(yintercept = 0)

sqrt(abs(rstandard())) vs fitted

Compare with autoplot(bw_lm2, 3)

bw |> 
  ggplot(
    aes(x = predlm2, y = sqrt_abs_std_resid, col = sex, shape = sex)
  ) + 
  geom_point() +
  theme_classic() +
  geom_hline(yintercept = 0)

2.9 Model selection

(adapted from Dobson and Barnett (2018) §6.3.3; for more information on prediction, see James et al. (2013) and Harrell (2015)).

If we have a lot of covariates in our dataset, we might want to choose a small subset to use in our model.

There are a few possible metrics to consider for choosing a “best” model.

2.9.1 Mean squared error

We might want to minimize the mean squared error, \(\text E[(y-\hat y)^2]\), for new observations that weren’t in our data set when we fit the model.

Unfortunately, \[\frac{1}{n}\sum_{i=1}^n (y_i-\hat y_i)^2\] gives a biased estimate of \(\text E[(y-\hat y)^2]\) for new data. If we want an unbiased estimate, we will have to be clever.

---

Cross-validation

data("carbohydrate", package = "dobson")
library(cvTools)
full_model <- lm(carbohydrate ~ ., data = carbohydrate)
cv_full = 
  full_model |> cvFit(
    data = carbohydrate, K = 5, R = 10,
    y = carbohydrate$carbohydrate)

reduced_model = update(full_model, 
                       formula = ~ . - age)

cv_reduced = 
  reduced_model |> cvFit(
    data = carbohydrate, K = 5, R = 10,
    y = carbohydrate$carbohydrate)

---

results_reduced = 
  tibble(
      model = "wgt+protein",
      errs = cv_reduced$reps[])
results_full = 
  tibble(model = "wgt+age+protein",
           errs = cv_full$reps[])

cv_results = 
  bind_rows(results_reduced, results_full)

cv_results |> 
  ggplot(aes(y = model, x = errs)) +
  geom_boxplot()

---

comparing metrics

compare_results = tribble(
  ~ model, ~ cvRMSE, ~ r.squared, ~adj.r.squared, ~ trainRMSE, ~loglik,
  "full", cv_full$cv, summary(full_model)$r.squared,  summary(full_model)$adj.r.squared, sigma(full_model), logLik(full_model) |> as.numeric(),
  "reduced", cv_reduced$cv, summary(reduced_model)$r.squared, summary(reduced_model)$adj.r.squared, sigma(reduced_model), logLik(reduced_model) |> as.numeric())

compare_results

model	cvRMSE	r.squared	adj.r.squared	trainRMSE	loglik
full	6.808	0.4805	0.3831	5.956	-61.84
reduced	6.516	0.4454	0.3802	5.971	-62.49

---

anova(full_model, reduced_model)

Res.Df	RSS	Df	Sum of Sq	F	Pr(>F)
16	567.7	NA	NA	NA	NA
17	606.0	-1	-38.36	1.081	0.3139

---

stepwise regression

library(olsrr)
olsrr:::ols_step_both_aic(full_model)
#> 
#> 
#>                              Stepwise Summary                              
#> -------------------------------------------------------------------------
#> Step    Variable         AIC        SBC       SBIC       R2       Adj. R2 
#> -------------------------------------------------------------------------
#>  0      Base Model     140.773    142.764    83.068    0.00000    0.00000 
#>  1      protein (+)    137.950    140.937    80.438    0.21427    0.17061 
#>  2      weight (+)     132.981    136.964    77.191    0.44544    0.38020 
#> -------------------------------------------------------------------------
#> 
#> Final Model Output 
#> ------------------
#> 
#>                          Model Summary                          
#> ---------------------------------------------------------------
#> R                       0.667       RMSE                 5.505 
#> R-Squared               0.445       MSE                 35.648 
#> Adj. R-Squared          0.380       Coef. Var           15.879 
#> Pred R-Squared          0.236       AIC                132.981 
#> MAE                     4.593       SBC                136.964 
#> ---------------------------------------------------------------
#>  RMSE: Root Mean Square Error 
#>  MSE: Mean Square Error 
#>  MAE: Mean Absolute Error 
#>  AIC: Akaike Information Criteria 
#>  SBC: Schwarz Bayesian Criteria 
#> 
#>                                ANOVA                                
#> -------------------------------------------------------------------
#>                 Sum of                                             
#>                Squares        DF    Mean Square      F        Sig. 
#> -------------------------------------------------------------------
#> Regression     486.778         2        243.389    6.827    0.0067 
#> Residual       606.022        17         35.648                    
#> Total         1092.800        19                                   
#> -------------------------------------------------------------------
#> 
#>                                   Parameter Estimates                                    
#> ----------------------------------------------------------------------------------------
#>       model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
#> ----------------------------------------------------------------------------------------
#> (Intercept)    33.130        12.572                  2.635    0.017     6.607    59.654 
#>     protein     1.824         0.623        0.534     2.927    0.009     0.509     3.139 
#>      weight    -0.222         0.083       -0.486    -2.662    0.016    -0.397    -0.046 
#> ----------------------------------------------------------------------------------------

---

Lasso

\[\arg min_{\theta} \ell(\theta) + \lambda \sum_{j=1}^p|\beta_j|\]

library(glmnet)
y = carbohydrate$carbohydrate
x = carbohydrate |> 
  select(age, weight, protein) |> 
  as.matrix()
fit = glmnet(x,y)

---

autoplot(fit, xvar = 'lambda')

Figure 2.19: Lasso selection

---

cvfit = cv.glmnet(x,y)
plot(cvfit)

---

coef(cvfit, s = "lambda.1se")
#> 4 x 1 sparse Matrix of class "dgCMatrix"
#>                   s1
#> (Intercept) 34.30916
#> age          .      
#> weight      -0.08001
#> protein      0.76405

2.10 Categorical covariates with more than two levels

2.10.1 Example: birthweight

In the birthweight example, the variable sex had only two observed values:

unique(bw$sex)
#> [1] female male  
#> Levels: female male

If there are more than two observed values, we can’t just use a single variable with 0s and 1s.

2.10.2

For example, Table 2.13 shows the (in)famous iris data (Anderson (1935)), and Table 2.14 provides summary statistics. The data include three species: “setosa”, “versicolor”, and “virginica”.

head(iris)

Table 2.13: The iris data

Sepal.Length	Sepal.Width	Petal.Length	Petal.Width	Species
5.1	3.5	1.4	0.2	setosa
4.9	3.0	1.4	0.2	setosa
4.7	3.2	1.3	0.2	setosa
4.6	3.1	1.5	0.2	setosa
5.0	3.6	1.4	0.2	setosa
5.4	3.9	1.7	0.4	setosa

library(table1)
table1(
  x = ~ . | Species,
  data = iris,
  overall = FALSE
)

Table 2.14: Summary statistics for the iris data

	setosa (N=50)	versicolor (N=50)	virginica (N=50)
Sepal.Length
Mean (SD)	5.01 (0.352)	5.94 (0.516)	6.59 (0.636)
Median [Min, Max]	5.00 [4.30, 5.80]	5.90 [4.90, 7.00]	6.50 [4.90, 7.90]
Sepal.Width
Mean (SD)	3.43 (0.379)	2.77 (0.314)	2.97 (0.322)
Median [Min, Max]	3.40 [2.30, 4.40]	2.80 [2.00, 3.40]	3.00 [2.20, 3.80]
Petal.Length
Mean (SD)	1.46 (0.174)	4.26 (0.470)	5.55 (0.552)
Median [Min, Max]	1.50 [1.00, 1.90]	4.35 [3.00, 5.10]	5.55 [4.50, 6.90]
Petal.Width
Mean (SD)	0.246 (0.105)	1.33 (0.198)	2.03 (0.275)
Median [Min, Max]	0.200 [0.100, 0.600]	1.30 [1.00, 1.80]	2.00 [1.40, 2.50]

---

If we want to model Sepal.Length by species, we could create a variable \(X\) that represents “setosa” as \(X=1\), “virginica” as \(X=2\), and “versicolor” as \(X=3\).

data(iris) # this step is not always necessary, but ensures you're starting  
# from the original version of a dataset stored in a loaded package

iris = 
  iris |> 
  tibble() |>
  mutate(
    X = case_when(
      Species == "setosa" ~ 1,
      Species == "virginica" ~ 2,
      Species == "versicolor" ~ 3
    )
  )

iris |> 
  distinct(Species, X)

Table 2.15: iris data with numeric coding of species

Species	X
setosa	1
versicolor	3
virginica	2

Then we could fit a model like:

iris_lm1 = lm(Sepal.Length ~ X, data = iris)
iris_lm1 |> parameters() |> print_md()

Table 2.16: Model of iris data with numeric coding of Species

Parameter	Coefficient	SE	95% CI	t(148)	p
(Intercept)	4.91	0.16	(4.60, 5.23)	30.83	< .001
X	0.47	0.07	(0.32, 0.61)	6.30	< .001

2.10.3 Let’s see how that model looks:

iris_plot1 = iris |> 
  ggplot(
    aes(
      x = X, 
      y = Sepal.Length)
  ) +
  geom_point(alpha = .1) +
  geom_abline(
    intercept = coef(iris_lm1)[1], 
    slope = coef(iris_lm1)[2]) +
  theme_bw(base_size = 18)
print(iris_plot1)

Figure 2.20: Model of iris data with numeric coding of Species

We have forced the model to use a straight line for the three estimated means. Maybe not a good idea?

2.10.4 Let’s see what R does with categorical variables by default:

iris_lm2 = lm(Sepal.Length ~ Species, data = iris)
iris_lm2 |> parameters() |> print_md()

Table 2.17: Model of iris data with Species as a categorical variable

Parameter	Coefficient	SE	95% CI	t(147)	p
(Intercept)	5.01	0.07	(4.86, 5.15)	68.76	< .001
Species (versicolor)	0.93	0.10	(0.73, 1.13)	9.03	< .001
Species (virginica)	1.58	0.10	(1.38, 1.79)	15.37	< .001

2.10.5 Re-parametrize with no intercept

If you don’t want the default and offset option, you can use “-1” like we’ve seen previously:

iris.lm2b = lm(Sepal.Length ~ Species - 1, data = iris)
iris.lm2b |> parameters() |> print_md()

Parameter	Coefficient	SE	95% CI	t(147)	p
Species (setosa)	5.01	0.07	(4.86, 5.15)	68.76	< .001
Species (versicolor)	5.94	0.07	(5.79, 6.08)	81.54	< .001
Species (virginica)	6.59	0.07	(6.44, 6.73)	90.49	< .001

2.10.6 Let’s see what these new models look like:

iris_plot2 = 
  iris |> 
  mutate(
    predlm2 = predict(iris_lm2)) |> 
  arrange(X) |> 
  ggplot(aes(x = X, y = Sepal.Length)) +
  geom_point(alpha = .1) +
  geom_line(aes(y = predlm2), col = "red") +
  geom_abline(
    intercept = coef(iris_lm1)[1], 
    slope = coef(iris_lm1)[2]) + 
  theme_bw(base_size = 18)

print(iris_plot2)

Figure 2.21

2.10.7 Let’s see how R did that:

formula(iris_lm2)
#> Sepal.Length ~ Species
model.matrix(iris_lm2) |> as_tibble() |> unique()

(Intercept)	Speciesversicolor	Speciesvirginica
1	0	0
1	1	0
1	0	1

This is called a “corner point parametrization”.

formula(iris.lm2b)
#> Sepal.Length ~ Species - 1
model.matrix(iris.lm2b) |> as_tibble() |> unique()

Speciessetosa	Speciesversicolor	Speciesvirginica
1	0	0
0	1	0
0	0	1

This can be called a “group point parametrization”.

There are more options; see Dobson and Barnett (2018) §6.4.1 and the codingMatrices package vignette (Venables (2023)).

2.11 Ordinal covariates

(c.f. Dobson and Barnett (2018) §2.4.4)

---

We can create ordinal variables in R using the ordered() function⁴.

Example 2.3  

url = paste0(
  "https://regression.ucsf.edu/sites/g/files/tkssra6706/",
  "f/wysiwyg/home/data/hersdata.dta")
library(haven)
hers = read_dta(url)

hers |> head()

Table 2.18: HERS dataset

HT	age	raceth	nonwhite	smoking	drinkany	exercise	physact	globrat	poorfair	medcond	htnmeds	statins	diabetes	dmpills	insulin	weight	BMI	waist	WHR	glucose	weight1	BMI1	waist1	WHR1	glucose1	tchol	LDL	HDL	TG	tchol1	LDL1	HDL1	TG1	SBP	DBP	age10
0	70	2	1	0	0	0	5	3	0	0	1	1	0	0	0	73.8	23.69	96.0	0.932	84	73.6	23.63	93.0	0.912	94	189	122.4	52	73	201	137.6	48	77	138	78	7.0
0	62	2	1	0	0	0	1	3	0	1	1	0	0	0	0	70.9	28.62	93.0	0.964	111	73.4	28.89	95.0	0.964	78	307	241.6	44	107	216	150.6	48	87	118	70	6.2
1	69	1	0	0	0	0	3	3	0	0	1	0	1	0	0	102.0	42.51	110.2	0.782	114	96.1	40.73	103.0	0.774	98	254	166.2	57	154	254	156.0	66	160	134	78	6.9
0	64	1	0	1	1	0	1	3	0	1	1	0	0	0	0	64.4	24.39	87.0	0.877	94	58.6	22.52	77.0	0.802	93	204	116.2	56	159	207	122.6	57	137	152	72	6.4
0	65	1	0	0	0	0	2	3	0	0	0	0	0	0	0	57.9	21.90	77.0	0.794	101	58.9	22.28	76.5	0.757	92	214	150.6	42	107	235	172.2	35	139	175	95	6.5
1	68	2	1	0	1	0	3	3	0	0	0	0	0	0	0	60.9	29.05	96.0	1.000	116	57.7	27.52	86.0	0.910	115	212	137.8	52	111	202	126.6	53	112	174	98	6.8

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# C(contr = codingMatrices::contr.diff)

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1. \(M\) is implicitly a deterministic function of \(S\)

2. \(F\) is implicitly a deterministic function of \(S\)

3. using the definite article “the” would mean there is only one slope.

4. or equivalently, factor(ordered = TRUE)