QML - Week 5

Regression models: the basics

Stefano Coretta

Word frequency and reaction times

What is the relationship between a word’s lexical frequency and reaction times in a lexical decision task in Croatian?

Which relationship?

Reaction times

# A tibble: 2,612 × 3
   word_string  rt_milliseconds_mean word_frequency
   <chr>                       <dbl>          <dbl>
 1 ribnjak                      575.           5755
 2 dostupnost                   624.          15564
 3 deformacija                  706.           6016
 4 antioksidans                 914.           7646
 5 zapis                        623.          44327
 6 kvadrat                      549.          36394
 7 presedan                     987.           7786
 8 zapremnina                  1035.           1434
 9 prerada                      750.          18759
10 general                      625.          96640
# ℹ 2,602 more rows

Word frequency

Word frequency: logged

Word frequency and RTs

Gaussian model of RT

\[ RT \sim Gaussian(\mu, \sigma) \]

But we want to know what happens to RTs depending on the value of lexical frequency…

Then we let the mean \(\mu\) vary by lexical frequency!

\[ \begin{align} RT & \sim Gaussian(\mu, \sigma)\\ \mu & = \beta_0 + \beta_1 \cdot logf \end{align} \]

But what are those \(\beta_0\) and \(\beta_1\)?

The equation of a line

\[ y = \beta_0 + \beta_1 \cdot x \]

  • \(\beta_0\) is the line intercept: the \(y\) value when \(x\) is 0 zero.

  • \(\beta_1\) is the line slope: the change in \(y\) for each unit-increase of \(x\).

  • Go to Linear models illustrated.

Regression model

\[ \begin{align} RT & \sim Gaussian(\mu, \sigma)\\ \mu & = \beta_0 + \beta_1 \cdot logf & \text{[Regression equation]} \end{align} \]

  • A regression model is a model based on the equation of a line.

  • The model estimates \(\beta_0\) (the intercept) and \(\beta_1\) (the slope) from the data (i.e. the observed \(RT\) and \(logf\) values).

  • \(\beta_0\), intercept

    • Mean RT value when logged frequency is 0 zero (i.e. when word frequency is 1; exp(0) = 1).

  • \(\beta_1\), slope

    • Change in mean RT for each unit increase of log-frequency (when log-frequency goes from \(x\) to \(x + 1\)).

Word frequency and reaction times (bis)

What is the relationship between a word’s lexical frequency and reaction times in a lexical decision task in Croatian?

  • When log-frequency is 0, the mean RTs are between 1084 and 1129 ms at 95% confidence.

  • For each unit increase of log-frequency, the mean RTs decrease by 43-48 ms, at 95% confidence.

Correlation in NOT causation

Be careful!

  • Correlation between two variables: they co-vary, i.e. they show a systematic association (their values tend to vary together in a consistent pattern).

  • Spurious correlations: two variables can look correlated because of bias from another variable.

  1. Number of plant names in a language vs. biodiversity of the region
    • Languages in biodiverse regions have more words for plants.
    • Mediator: cultural reliance on plants.
  2. Language endangerment vs. economic development
    • Higher economic development is associated with greater language endangerment.
    • Confounder: colonial history.
  3. Language prestige vs government policy
    • Collider: High prestige languages and officially supported languages each attract learners.
    • If you only look at languages with many learners, prestige and policy might appear related even if they’re not causally connected.

But it is if you use causal inference…

Causal inference

  • Correlation can be interpreted causally if you adopt a causal inference approach.

  • We won’t treat causal inference in this course due to time, but you can learn about it in McElreath’s textbook Statistical Rethinking. I will also run a Causal Inference introduction workshop in Semester 2.