RECAP: Linear regression with a single predictor

Lecture 15

2024-06-06

Goals

Review linear regression with a single predictor

Setup

library(tidyverse)
library(tidymodels)

Linear regression with a single predictor

Data overview

From the Fandango data set, we want to model audience movie ratings based on critics ratings.

movie_scores |>
  select(critics, audience)

# A tibble: 146 × 2
   critics audience
     <int>    <int>
 1      74       86
 2      85       80
 3      80       90
 4      18       84
 5      14       28
 6      63       62
 7      42       53
 8      86       64
 9      99       82
10      89       87
# ℹ 136 more rows

Regression model

A simple linear regression model is used to model the relationship between a quantitative outcome (\(Y\)) and a single quantitative predictor (\(X\)): \[\Large{Y = \beta_0 + \beta_1 X +\epsilon}\]

\(\beta_1\): True slope of the relationship between \(X\) and \(Y\)
\(\beta_0\): True intercept of the relationship between \(X\) and \(Y\)
\(\epsilon\): Error term

Fandango data: least squares line

Where the solid line minimizes the sum of the squared residuals.

Linear regression model output

movies_fit <- linear_reg() |>
  fit(audience ~ critics, data = movie_scores)

tidy(movies_fit)

# A tibble: 2 × 5
  term        estimate std.error statistic  p.value
  <chr>          <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)   32.3      2.34        13.8 4.03e-28
2 critics        0.519    0.0345      15.0 2.70e-31

Slope and intercept

Interpreting slope & intercept

\[\widehat{\text{audience}} = 32.3 + 0.519 \times \text{critics}\]

Slope: For every one point increase in the critics score, we expect the audience score to be higher by 0.519 points, on average.
Intercept: If the critics score is 0 points, we expect the audience score to be 32.3 points.

Is the intercept meaningful?

✅ The intercept is meaningful in context of the data if

the predictor can feasibly take values equal to or near zero or
the predictor has values near zero in the observed data

🛑 Otherwise, it might not be meaningful!