Today

• What an interaction is, and why lines stop being parallel
• Categorical × continuous interactions
• Categorical × categorical interactions
• Continuous × continuous interactions and curved regression planes
• Why centering and sum coding matter so much here
• A word on higher-order interactions

So far

Lectures 6 and 7: several predictors, each with one effect.

Interactions in R

* adds both predictors and their interaction:

lm(y ~ x * group)            # is the same as
lm(y ~ x + group + x:group)

Discussion

Interactions are common in linguistics: a frequency effect can be larger for L2 speakers, or a vowel can lengthen more before voiced consonants in one dialect than in another.

• Think of a study in your own field where the effect of one variable could depend on another.
• Sketch the plot you would expect with an interaction, and without one.

The data

A simulated lexical decision task with first-language (L1) and second-language (L2) speakers:

set.seed(10)
lex <- tibble(group    = rep(c("L1", "L2"), each = 200),
              log_freq = runif(400, 1, 6)) |>
  mutate(rt = if_else(group == "L1", 700 - 25 * log_freq,
                                     900 - 55 * log_freq) + rnorm(400, 0, 40))

The data

Two different slopes. Can last week’s model capture that?

Without an interaction

m_add <- lm(rt ~ log_freq + group, data = lex)
coef(m_add)

(Intercept)    log_freq     groupL2 
  768.58134   -44.00400    92.00191 

One slope for both groups: too shallow for L2, too steep for L1.

With an interaction

m_int <- lm(rt ~ log_freq * group, data = lex)
coef(m_int)

     (Intercept)         log_freq          groupL2 log_freq:groupL2 
       711.46630        -27.05222        201.53411        -31.43419 

The “main effect” trap

groupL2 = 202 ms: is that the L1–L2 difference?

Centering rescues the coefficients

From lecture 5: centre the continuous predictor, so that 0 means “average frequency”.

lex <- lex |> mutate(log_freq_c = log_freq - mean(log_freq))
coef(lm(rt ~ log_freq_c * group, data = lex))

       (Intercept)         log_freq_c            groupL2 
         616.93726          -27.05222           91.69311 
log_freq_c:groupL2 
         -31.43419 

Did the interaction help?

c(additive    = glance(m_add)$adj.r.squared,
  interaction = glance(m_int)$adj.r.squared)

   additive interaction 
  0.7078173   0.7709118 

The data

Vowel duration by voicing of the following consonant and speaking style, simulated:

set.seed(11)
cells <- tribble(
  ~style,   ~voicing,    ~mu,
  "casual", "voiceless", 110,
  "casual", "voiced",    145,
  "clear",  "voiceless", 125,
  "clear",  "voiced",    180
)
vs <- cells |>
  uncount(40) |>                                    # 40 tokens per cell
  mutate(duration = round(rnorm(n(), mu, 18)),
         voicing  = fct_relevel(voicing, "voiceless")) |>
  select(-mu)

Four cell means

vs |> summarise(mean = mean(duration), .by = c(style, voicing))

# A tibble: 4 × 3
  style  voicing    mean
  <chr>  <fct>     <dbl>
1 casual voiceless  104.
2 casual voiced     146.
3 clear  voiceless  125.
4 clear  voiced     182.

An interaction plot

Clear speech lengthens vowels, more so before voiced consonants: the lines are not parallel.

Rebuilding the cell means

b <- coef(m_vs)
c(voiceless_casual = b[[1]],
  voiced_casual    = b[[1]] + b[[2]],
  voiceless_clear  = b[[1]] + b[[3]],
  voiced_clear     = b[[1]] + b[[2]] + b[[3]] + b[[4]])

voiceless_casual    voiced_casual  voiceless_clear     voiced_clear 
         104.175          145.700          124.850          181.800 

Sum coding, as promised

vs_sum <- vs |> mutate(style = factor(style))
contrasts(vs_sum$voicing) <- contr.sum(2)
contrasts(vs_sum$style)   <- contr.sum(2)
coef(lm(duration ~ voicing * style, data = vs_sum))

    (Intercept)        voicing1          style1 voicing1:style1 
      139.13125       -24.61875       -14.19375         3.85625 

Which coding?

With treatment coding, each effect describes the reference level of the other predictor: the voicing effect in casual speech. With sum coding, each effect is averaged over the levels of the other predictor: the voicing effect across both styles.

The data

Back to word frequency and length, simulated, now with an interaction:

set.seed(12)
w <- tibble(log_freq = runif(400, 1, 6),
            length   = round(runif(400, 3, 12))) |>
  mutate(lf_c  = log_freq - mean(log_freq),
         len_c = length - mean(length),
         rt    = 760 - 30 * lf_c + 14 * len_c - 5 * lf_c * len_c + rnorm(400, 0, 40))

Uncentred: a surprise

coef(lm(rt ~ log_freq * length, data = w))

    (Intercept)        log_freq          length log_freq:length 
     620.941597        9.734164       31.430518       -5.103751 

coef(lm(rt ~ lf_c * len_c, data = w))

(Intercept)        lf_c       len_c  lf_c:len_c 
 756.910198  -28.288779   13.715687   -5.103751 

Simple slopes

The effect of length at three frequencies. Compare lecture 6: the lines now fan out.

A twisted plane

Without an interaction, the contour lines were straight and parallel (lecture 6). With one, they bend: the plane is twisted.

Three-way interactions

lm(y ~ a * b * c)

Reading interactions: a checklist

1. Plot the data: do the lines or groups look non-parallel?
2. Centre continuous predictors and choose your coding before fitting
3. Read the interaction term as a change in an effect
4. Read lower-order terms as simple effects, at 0 or the reference level
5. Plot the model predictions as well as the coefficients

Exercises

With palmerpenguins::penguins, after drop_na():

1. Model body_mass_g by flipper_length_mm * species. Plot the fitted lines.
2. What is the flipper slope for Gentoo penguins?
3. Centre flipper length and refit. Which coefficients changed?
4. Model body_mass_g by species * sex, first with treatment coding, then with sum coding. Interpret the intercept of each.

Summary

Slides and code

These slides

mcanzi.de/ws26_statistics/08_interactions.html

Code, data and all lectures

github.com/mcanzi/ws26_statistics