• Use lm() with two-group and three-group categorical predictors.
• Interpret the reference group and dummy-coded coefficients.
• Count dummy coefficients, beta coefficients, and total model parameters.
• Calculate group means from coefficients, then use predict() as a check.
• Read a coefficient t-test and an overall model F-test.

Complete the online lecture before the workshop and the Week 7 problem set afterwards.

In Weeks 5–6, each continuous predictor entered as a single linear term added one slope coefficient. A categorical predictor can add more than one coefficient:

One categorical predictor with G levels adds G − 1 coefficients. We still fit one intercept as well.

These are general linear models, whether predictors are continuous, categorical, or both. Generalised linear models extend this framework; that is a different term.

The lecture represents two population means using one regression equation:

\[ \mu_i = \beta_0 + \beta_1 X_i. \]

• For the reference group, \(X=0\): its mean is \(\beta_0\).
• For the other group, \(X=1\): its mean is \(\beta_0+\beta_1\).
• Therefore, \(\beta_1\) is the other group’s mean minus the reference mean.

lm() estimates these population parameters from our sample. The model assumes a common residual SD, sigma, around the group means.

Open your module’s RStudio Project and check that workshop_data is inside it.

1. Download week_07_workshop.R from NOW.
2. Move it into your project folder using your file explorer.
3. Open it in RStudio.

Wait for each exercise checkpoint. Run one line at a time and keep your coefficient output visible for the worked examples.

blomkvist <- read_csv("workshop_data/analysis_blomkvist.csv")
blomkvist <- drop_na(blomkvist, rt, meds_group)
count(blomkvist, meds_group)

meds_group already contains “medicated” and “not medicated”. We do not need to create a new grouping variable.

Do the medication groups differ in mean hand reaction time?

The prepared data are adapted from Blomkvist et al. (2017).

m_lm <- lm(rt ~ meds_group, data = blomkvist)
m_lm
m_lm_summary <- summary(m_lm)
m_lm_summary
coef(m_lm)

The formula has the same structure as Week 5. The predictor now contains group labels rather than numeric values.

Complete the data-loading steps and Part A1 with me.

Identify which coefficient is the reference mean and which is a difference.

With these text labels, R orders the levels alphabetically. Under default treatment coding, the first level is the reference and the remaining level gets a binary indicator.

“medicated” comes before “not medicated”, so it is coded 0; “not medicated” is coded 1.

Two levels give 2 - 1 = 1 dummy variable. In general, G levels give G - 1 dummy variables. R creates these codes inside lm(); we will read them, not change them.

coef(m_lm) prints the values we need:

            (Intercept) meds_groupnot medicated 
               738.8143               -157.4875 

Read the coefficient names to identify the reference mean and the other group’s difference.

738.814286
# [1] 738.8143
738.814286 + (-157.487488)
# [1] 581.3268

• Medicated mean: the intercept alone.
• Not-medicated mean: intercept + not-medicated difference.

Numerically: 738.8143 + (-157.4875) ≈ 581.3268 ms.

new_meds <- tibble(
  meds_group = c("medicated", "not medicated")
)
new_meds
predict(m_lm, newdata = new_meds)

Now that we have calculated the means by hand, use predict() to check them. tibble() supplies one row per group.

predict() uses the fitted model and returns one predicted mean per row. The column name and group labels must match the model.

predict() uses the fitted model directly. For the manual calculation, read and add the printed coefficient values.

Use the coefficient row for the group difference, not the intercept row.

m_lm_summary <- summary(m_lm)
m_lm_summary

Find the meds_groupnot medicated row. Read its t value and Pr(>|t|) columns, as in Week 6.

The null hypothesis is that the population group difference is 0.

This is a short connection to the lecture; lm() remains our main workflow.

m_ttest <- t.test(rt ~ meds_group, data = blomkvist,
                  var.equal = TRUE, mu = 0)
m_ttest$statistic
m_ttest$p.value

With the same rows and group ordering, t.test() calculates the reference mean minus the other mean; the model coefficient calculates the other mean minus the reference.

The t-values have opposite signs but the same magnitude. Their two-sided p-values agree.

In the regression equation, \(\mu_i\) is a population mean described by the intercept and group coefficient.

In t.test(), the argument mu specifies the hypothesised value:

• One-sample test: a hypothesised population mean.
• Two-sample test: a hypothesised difference between population means.

For our two-sample comparison, mu = 0 tests equal group means. It does not say that either group mean is 0.

Equal group means correspond to a group coefficient of 0, not an intercept of 0.

Complete Part A2 with me.

Stop after Part A2.

After the break, we will extend the same logic to three groups.

Alphabetical order of the text levels: former smoker → non-smoker → smoker.

The first level, “former smoker”, is the reference: both its codes are 0. Three levels give 3 - 1 = 2 dummy variables. Each other level has a 1 in its own column and a 0 in the other.

R creates these codes automatically. Keep the text labels; one numeric 0, 1, 2 column would impose equally spaced predicted means.

blomkvist_smoker <- read_csv(
  "workshop_data/analysis_blomkvist.csv")
blomkvist_smoker <- drop_na(blomkvist_smoker, rt, smoker)
m_smoker <- lm(rt ~ smoker, data = blomkvist_smoker)
coef(m_smoker)

Start Part A3 with me. Use the intercept and the two differences to calculate the group means by hand.

coef(m_smoker) prints the values we need:

     (Intercept) smokernon-smoker     smokersmoker 
       652.51316        -17.95079        -19.06444 

Choose the difference for the named group. Do not add both differences to the intercept.

652.513158
# [1] 652.5132
652.513158 + (-17.950786)
# [1] 634.5624
652.513158 + (-19.064440)
# [1] 633.4487

These give the former-smoker, non-smoker, and smoker means, respectively.

• Non-smoker: 652.5132 + (-17.9508) ≈ 634.5624 ms.
• Smoker: 652.5132 + (-19.0644) ≈ 633.4487 ms.

In Part A3, choose the printed coefficient values for each named group, then continue to the overall test at Checkpoint 3.

For rt ~ smoker, rt is the outcome and smoker is one categorical predictor with three levels.

Here K = 2 counts the predictor coefficients, excluding the intercept. It does not count the original predictor variables. Sigma is a parameter, but not a beta coefficient or a predictor.

For a group-only model with G groups, distinguish:

The residual df is n − K − 1 = n − G. Count beta coefficients when working out residual df; do not subtract sigma as well.

Part A3: count predictors, coefficients, and parameters for numeric, categorical, and mixed examples. All examples have an intercept and one common sigma.

m_smoker_summary <- summary(m_smoker)
m_smoker_summary$fstatistic
smoker_anova <- anova(m_smoker)
smoker_anova

With one fitted model, anova() shows its analysis-of-variance table. F value and Pr(>F) are the overall group-test statistic and p-value.

Read those values from the smoker row. Its Sum Sq is ESS; the Residuals Sum Sq is RSS. Together they make TSS.

The null hypothesis is that all three population means are equal. A coefficient t-test instead compares one group with the reference.

The two group coefficients are differences from the reference; they are not a test of every possible pair of groups.

Locate the right row or overall test before reading a number. An overall F-test does not tell you which particular groups differ.

Finish Part A3 with me.

The two coefficient tests do not replace the overall F-test.

Complete Part B with the Chinese lexical-decision data.

• Fit rt ~ accuracy_group and calculate both means from its coefficients.
• Read the group coefficient’s t-statistic and p-value.
• Then fit rt ~ list_group using all four prepared list groups.
• Calculate all four means from the intercept and group differences, then read the overall F-test.

Begin Part C if you finish Part B; otherwise complete it after the workshop. It consolidates core skills and is separate from the optional stretch script.

• How do coefficient names reveal the reference group?
• Why are there three group coefficients for four list groups?
• How does an overall F-test differ from one coefficient’s t-test?

The Week 7 problem set asks for two-group predictions and a three-group overall test. Use the same logic even when the data set and labels change.

Recommended reading: Andrews (2021), Chapter 9, Normal Linear Models.

Complete the Week 7 problem set after this workshop. Calculate group means from the intercept and group differences; use predict() when a check is helpful. No downloaded dummy-coding function or extra prediction package is needed.

The first three questions use a two-group model. The remaining questions use three groups and ask for an overall test. Apply the same workflow to the new data and group labels.

• The intercept is the reference-group mean in a group-only model.
• Each dummy coefficient is a difference from that mean.
• A predictor with G groups needs G - 1 dummy coefficients.
• A group-only normal model has G beta coefficients and one shared sigma.
• Group predictions, coefficient tests, and overall tests answer different questions.