• What a normal linear model estimates
• Greek symbols and model notation
• Writing model formulas in RMarkdown
• Formula syntax in lm()
• Intercepts and slopes
• Continuous and categorical predictors
• tidy() for coefficient tables
• Estimate, uncertainty, and evidence
• Short model interpretation

A normal linear model can be written as:

\[ \begin{align} y_i &\sim \mathcal{N}(\mu_i, \sigma) \\ \mu_i &= \beta_0 + \beta_1 x_i \end{align} \]

This says:

• each observed outcome \(y_i\) comes from a normal distribution
• \(\mu_i\) is the model-predicted mean for observation \(i\)
• \(\sigma\) is the residual spread around the predicted mean
• \(\beta_0\) is the intercept
• \(\beta_1\) is the slope for predictor \(x_i\)

Hypothesis:

Reaction times vary by participant age.

We adapt the generic model by replacing \(y_i\) with reaction time and \(x_i\) with age.

\[ \text{rt}_i \sim \mathcal{N}(\mu_i, \sigma) \]

\[ \mu_i = \beta_0 + \beta_\text{age} \cdot \text{age}_i \]

This says that each participant’s reaction time varies around a predicted mean, and that predicted mean depends on age.

model_0 <- lm(rt ~ 1, data = blomkvist)
coef(model_0)

(Intercept) 
        640 

The intercept is the model’s estimate of the mean outcome.

model_age <- lm(rt ~ age, data = blomkvist)
coef(model_age)

(Intercept)         age 
     314.74        5.77 

The age coefficient estimates how much rt changes for a one-year increase in age.

The slope is a change per one-unit increase in continuous predictor.

Examples:

Always ask: one unit of what predictor, and what outcome scale?

For a categorical predictor, the slope is a difference from a reference group.

For example, if sex is coded with female as the reference group (intercept):

A positive categorical coefficient means the comparison group has a higher predicted outcome than the reference group. A negative coefficient means the comparison group has a lower predicted outcome than the reference group.

model_age_sex <- lm(rt ~ age + sex, data = blomkvist)
coef(model_age_sex)

(Intercept)         age     sexmale 
     339.62        5.74      -59.29 

Turn to the person next to you.

What do these results mean?

• What does the age coefficient tell us?
• What does the sex coefficient tell us?
• What would you need to know before interpreting sex?

For rt ~ age + sex:

• (Intercept) is the predicted outcome when numeric predictors are zero and categorical predictors are at their reference level
• age is the expected change in rt for a one-year increase in age, holding sex constant
• sexmale is the expected difference between male and female participants, holding age constant

Do not interpret a coefficient without asking: compared with what, and holding what constant?

For each row, what would you say in words?

Useful language: evidence for a positive effect, evidence for a negative effect, or uncertain because the interval includes zero.

If there is time, start part-1-transfer-chinese-ldt.Rmd as individual work. Otherwise, complete it as homework before Session 2.

Use part-1-transfer-own-data-template.Rmd after that if you already have a formative dataset.

The transfer work extends today’s core sequence with one interaction model:

1. read or write a model description
2. implement the model in lm()
3. extract estimates and confidence intervals with tidy()
4. interpret the interaction as asking whether one effect differs across groups

Reproduce the main workflow from today with your own formative dataset if you have one.

Try to:

• identify one continuous outcome
• identify one or two predictors
• write a possible lm() formula
• fit a first normal linear model
• create a tidy() coefficient table
• write two or three sentences interpreting the selected model

Use part-1-transfer-own-data-template.Rmd for guidance.

For a basic linear model report:

• research question
• formal model description
• final model formula
• brief explanation of why this model was used
• coefficient estimates and confidence intervals for the selected model
• interpretation of the selected model in context
• one useful coefficient table or plot that supports the selected model

Read after the workshop:

• Required: Mark Andrews, Doing Data Science in R, Chapter 9: Normal Linear Models (Andrews, 2021)
• Optional/advanced: Gelman, Hill, and Vehtari, Regression and Other Stories, Chapters 7 and 10 on linear regression (Gelman et al., 2020)

Optional refresher:

• Optional/foundation: Danielle Navarro, Learning Statistics with R, Chapter 15: Linear Regression https://learningstatisticswithr.com/book/regression.html
• Optional/foundation: Field, Miles, and Field, Discovering Statistics Using R, chapters on correlation and regression (Field et al., 2012)

Use these to revise what lm() estimates, how coefficients are interpreted, and how coefficient estimates support model interpretation.