Simple Linear Regression

Prof. Maria Tackett

January 15, 2026

Announcements

• No lab Monday, January 20 - Martin Luther King Jr. Holiday
• No final exam in this class
  • No in-person activities after LDOC

Topics

• Use simple linear regression to describe the relationship between a quantitative predictor and quantitative response variable.

• Estimate the slope and intercept of the regression line using the least squares method.

• Interpret the slope and intercept of the regression line.

• Predict the response given a value of the predictor variable.

• Fit linear regression models in R

Computation set up

# load packages
library(tidyverse)       # for data wrangling
library(tidymodels)      # for modeling
library(knitr)          # for formatting tables

# set default theme and larger font size for ggplot2
ggplot2::theme_set(ggplot2::theme_bw(base_size = 16))

# set default figure parameters for knitr
knitr::opts_chunk$set(
  fig.width = 8,
  fig.asp = 0.618,
  fig.retina = 3,
  dpi = 300,
  out.width = "80%"
)

Data

Movie scores

• Data behind the FiveThirtyEight story Be Suspicious Of Online Movie Ratings, Especially Fandango’s
• Data originally from the fandango data frame in fivethirtyeight package. Available in movie-scores.csv
• Contains every film released in 2014 and 2015 that has at least 30 fan reviews on Fandango, an IMDb score, Rotten Tomatoes critic and user ratings, and Metacritic critic and user scores

Data overview

movie_scores <- read_csv("https://intro-regression.github.io/data/movie-scores.csv")

glimpse(movie_scores)

Rows: 146
Columns: 4
$ film           <chr> "Avengers: Age of Ultron", "Cinderella", "Ant-Man", "Do…
$ year           <dbl> 2015, 2015, 2015, 2015, 2015, 2015, 2015, 2014, 2015, 2…
$ critics_score  <dbl> 74, 85, 80, 18, 14, 63, 42, 86, 99, 89, 84, 82, 99, 51,…
$ audience_score <dbl> 86, 80, 90, 84, 28, 62, 53, 64, 82, 87, 77, 60, 79, 70,…

Movie scores data

The data set contains the “Tomatometer” score (critics_score) and audience score (audience_score) for 146 movies rated on rottentomatoes.com.

Movie ratings data

Goal: Fit a line to describe the relationship between the critics score and audience score.

Why fit a line?

We fit a line to accomplish one or both of the following:

Prediction

What is the audience score expected to be for an upcoming movie that received 35% from the critics?

Inference

Is the critics score a useful predictor of the audience score? By how much is the audience score expected to change for each additional point in the critics score?

Terminology

• Response, Y: variable describing the outcome of interest

• Predictor, X: variable we use to help understand the variability in the response

Regression model

A regression model is a function that describes the relationship between the response, \(Y\), and the predictor, \(X\).

\[\begin{aligned} Y &= \color{black}{\textbf{Model}} + \text{Error} \\[8pt] &= \color{black}{\mathbf{f(X)}} + \epsilon \\[8pt] &= \color{black}{\mathbf{E(Y|X)}} + \epsilon\end{aligned}\]

Regression model

\[\begin{aligned} Y &= {\color{purple} \textbf{Model}} + \textbf{Error} \\[8pt] &= {\color{purple} \mathbf{f(X)}} + \epsilon \\[8pt] &= {\color{purple} \mathbf{E(Y|X)}} + \epsilon \\ \end{aligned}\]

\(E(Y|X) = \mu_{Y|X}\) is the mean value of \(Y\) given a particular value of \(X\).

Regression model

\[ \begin{aligned} Y &= \color{purple}{\textbf{Model}} + \color{blue}{\textbf{Error}} \\[8pt] &= \color{purple}{\mathbf{f(X)}} + \color{blue}{\boldsymbol{\epsilon}} \\[8pt] &= \color{purple}{\mathbf{E(Y|X)}} + \color{blue}{\boldsymbol{\epsilon}} \\ \end{aligned} \]

Simple linear regression (SLR)

SLR: Statistical model (Theoretical)

When we have a quantitative response, \(Y\), and a single quantitative predictor, \(X\), we can use a simple linear regression model to describe the relationship between \(Y\) and \(X\).

\[y_i = \beta_0 + \beta_1 x_i + \epsilon_i\]

• \(\beta_1\): True slope of the relationship between \(X\) and \(Y\)
• \(\beta_0\): True intercept of the relationship between \(X\) and \(Y\)
• \(\epsilon_i\): Error for the \(i^{th}\) observation

SLR: Regression equation (Fitted)

\[ \hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i \]

• \(\hat{\beta}_1\): Estimated slope of the relationship between \(X\) and \(Y\)
• \(\hat{\beta}_0\): Estimated intercept of the relationship between \(X\) and \(Y\)
• No error term!

Why is there no error term in the estimated regression equation?

Computing estimates \(\hat{\beta}_1\) and \(\hat{\beta}_0\)

Residuals

\[\text{residual} = \text{observed} - \text{predicted} = y_i - \hat{y}_i\]

Least squares line

• The residual for the \(i^{th}\) observation is

\[e_i = \text{observed} - \text{predicted} = y_i - \hat{y}_i\]

• The sum of squared residuals is

\[e^2_1 + e^2_2 + \dots + e^2_n\]

• The Ordinary Least Squares (OLS) line is the one that minimizes the sum of squared residuals

Least-squares estimator of \(\hat{\beta}_0\)

Click here for full details on finding \(\hat{\beta}_0\) and \(\hat{\beta}_1\) in simple linear regression.

Slope and intercept

Properties of least squares regression

• The regression line goes through the center of mass point, the coordinates corresponding to average \(X\) and average \(Y\): \(\hat{\beta}_0 = \bar{y} - \hat{\beta}_1\bar{x}\)

• The slope has the same sign as the correlation coefficient: \(\hat{\beta}_1 = r \frac{s_Y}{s_X}\)

• The sum of the residuals is approximately zero: \(\sum_{i = 1}^n e_i \approx 0\)

• The residuals and \(X\) values are uncorrelated

Estimating the slope

\[\large{\hat{\beta}_1 = r \frac{s_Y}{s_X}}\]

\[\begin{aligned} s_X &= 30.1688 \\ s_Y &= 20.0244 \\ r &= 0.7814 \end{aligned}\]

\[\begin{aligned} \hat{\beta}_1 &= 0.7814 \times \frac{20.0244}{30.1688} \\ &= 0.5187\end{aligned}\]

Estimating the intercept

\[\large{\hat{\beta}_0 = \bar{y} - \hat{\beta}_1\bar{x}}\]

\[\begin{aligned} &\bar{x} = 60.8493 \\ &\bar{y} = 63.8767 \\ &\hat{\beta}_1 = 0.5187 \end{aligned}\]

\[\begin{aligned}\hat{\beta}_0 &= 63.8767 - 0.5187 \times 60.8493 \\ &= 32.3142 \end{aligned}\]

Interpretation

\[ \widehat{\text{audience}} = 32.3142 + 0.5187 \times \text{critics} \]

• Question
• Submit

Submit your responses to the following:

• The slope of the model for predicting audience score from critics score is 0.5187 . Which of the following is the best interpretation of this value?

• 32.3142 is the predicted mean audience score for what type of movies?

🔗 https://forms.office.com/r/NYMZZguyqP

Does it make sense to interpret the intercept?

✅ The intercept is meaningful in the context of the data if

• the predictor can feasibly take values equal to or near zero, or

• there are values near zero in the observed data.

🛑 Otherwise, the intercept may not be meaningful!

Prediction

Making a prediction

Suppose that a movie has a critics score of 70. According to this model, what is the movie’s predicted audience score?

\[\begin{aligned} \widehat{\text{audience_score}} &= 32.3142 + 0.5187 \times \text{critics_score} \\ &= 32.3142 + 0.5187 \times 70 \\ &= 68.6232 \end{aligned}\]

Caution

Using the model to predict for values outside the range of the original data is extrapolation. Why do we want to avoid extrapolation?

Linear regression in R

Fit the model

Use the lm() function to fit a linear regression model

movie_fit <- lm(audience_score ~ critics_score, data = movie_scores)
movie_fit

Call:
lm(formula = audience_score ~ critics_score, data = movie_scores)

Coefficients:
  (Intercept)  critics_score  
      32.3155         0.5187  

Tidy results

Use the tidy() function from the broom R package to “tidy” the model output

movie_fit <- lm(audience_score ~ critics_score, data = movie_scores)
tidy(movie_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_score    0.519    0.0345      15.0 2.70e-31

Format results

Use the kable() function from the knitr package to neatly format the results

movie_fit <- lm(audience_score ~ critics_score, data = movie_scores)
tidy(movie_fit) |>
  kable(digits = 3)

term	estimate	std.error	statistic	p.value
(Intercept)	32.316	2.343	13.795	0
critics_score	0.519	0.035	15.028	0

Prediction

Use the predict() function to calculate predictions for new observations

Single observation

new_movie <- tibble(critics_score = 70)
predict(movie_fit, new_movie)

       1 
68.62297 

Prediction

Use the predict() function to calculate predictions for new observations

Multiple observations

more_new_movies <- tibble(critics_score = c(24,70, 85))
predict(movie_fit, more_new_movies)

       1        2        3 
44.76379 68.62297 76.40313 

Application exercise

📋 sta210-sp26.github.io/ae/ae-03-slr.html

• Find your ae-03 repo in the course GitHub organization.

• If you do not see an ae-03 repo, use the link to create one: https://classroom.github.com/a/JpQEbKBU

Recap

• Used simple linear regression to describe the relationship between a quantitative predictor and quantitative response variable.

• Used the least squares method to estimate the slope and intercept.

• Interpreted the slope and intercept.

• Predicted the response given a value of the predictor variable.

• Used R to fit the regression line and calculate predictions

Next Class

• Inference: Bootstrap confidence intervals

• Complete Lecture 04 Prepare