Regression models: the basics
What is the relationship between a word’s lexical frequency and reaction times in a lexical decision task in Croatian?
Data from Lexical decision times for nouns from the Croatian Psycholinguistic Database.
Lexical decision task (is it a real Croatian word?)
Reaction times.
Word frequency: counts from the Croatian web Corpus hrWaC.
\[ RT \sim Gaussian(\mu, \sigma) \]
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 \(\beta_1\)?
\[ y = \beta_0 + \beta_1 \cdot x \]
Go to Linear models illustrated.
\(\beta_0\) 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 \(x\).
\[ \begin{align} RT & \sim Gaussian(\mu, \sigma)\\ \mu & = \beta_0 + \beta_1 \cdot logf & \text{[Regression equation]} \end{align} \]
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
0 zero (i.e. when word frequency is 1; exp(0) = 1).\(\beta_1\), slope
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.
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.
Causal inference
Correlation can be interpreted causally if you adopt a causal inference approach.
Learn about it in McElreath’s textbook Statistical Rethinking. Also check STeW.