Lecture 02 AE: Review of multiple linear regression

Introduction

Today’s data is from the Kentucky Derby, an annual 1.25-mile horse race held at the Churchill Downs race track in Louisville, KY. The data is in the file derbyplus.csv in the data folder. It contains information for races 1896 - 2017.

Response variable

• speed: Average speed of the winner in feet per second (ft/s)

Additional variable

• winner: Winning horse

Predictor variables

• year: Year of the race
• condition: Condition of the track (good, fast, slow)
• starters: Number of horses who raced

Goal: Understand variability in average winner speed based on characteristics of the race.

library(tidyverse)
library(tidymodels)
library(knitr)

derby <- read_csv("data/derbyplus.csv")

Part 1

Model 1: Main effects model

model1 <- lm(speed ~ starters + year + condition, data = derby)

tidy(model1) |>
  kable(digits = 3)

term	estimate	std.error	statistic	p.value
(Intercept)	8.197	4.508	1.818	0.072
starters	-0.005	0.017	-0.299	0.766
year	0.023	0.002	9.766	0.000
conditiongood	-0.443	0.231	-1.921	0.057
conditionslow	-1.543	0.161	-9.616	0.000

Ex 1

Write the equation of the fitted model.

[add response here]

Ex 2

Interpret the coefficient of conditionslow in the context of the data.

[add response here]

Ex 3

Does the intercept have a meaningful interpretation? If so, interpret it. If not…

• Refit the model so that the intercept has a meaningful interpretation.

• Interpret the intercept for the new model.

[add response here]

Model 2: Main effects + quadratic effect for year

Ex 4

Fit a model that includes all main effects and a quadratic term for year. Display the model.

[add response here]

Interpreting quadratic effects

Suppose you have the following model:

$$\hat{y}={\hat{\beta }}_0+{\hat{\beta }}_1{x}_1+{\hat{\beta }}_2{x}_2+{\hat{\beta }}_3{x}_2^2$$

The interpretation of a variable’s effect when there is a quadratic term in the model is

“When $x_2$ increases from a to b, $y$ is expected to change by ${\hat{\beta }}_2(b-a)+{\hat{\beta }}_3(b^2-a^2)$, holding $x_1$ constant.”

Ex 5

Interpret the effect of year for the 5 most recent years (2013 - 2017).

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Model 3: Main effects + interaction between year and condition

Ex 6

Fit a model that includes all main effects and an interaction between year and condition. Do not include a quadratic term for year. Display the model.

[add response here]

Ex 7

Interpret yearnew:conditiongood.

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Ex 8

Interpret the effect of year for slow track conditions.

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Part 2

Ex 9

Display Model 3 with the 90% confidence intervals for the coefficients.

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Ex 10

Complete the following for your assigned variable:

1. State the null and alternative hypotheses in words.
2. What is the test statistic? What does this value tell you?
3. What is the conclusion of the hypothesis test? Use the decision-making threshold $\alpha$ that corresponds to the confidence interval.
4. Interpret the confidence interval.

Choose one person to put your group’s response in Slack.

Choose one person to share your group’s response with the class.

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