AE 08: Inference for Multiple Linear Regression

Rail Trails

Important

  • Open RStudio and create a subfolder in your AE folder called “AE-08”.

  • Go to the Canvas and locate your AE-08 assignment to get started.

  • Upload the ae-08.qmd and rail-trail.csv files into the folder you just created.

Packages + data

library(tidyverse)
library(ggformula)
library(broom)
library(knitr)
library(mosaic)
library(mosaicData)

rail_trail <- read_csv("../data/rail_trail.csv")

The data for this AE is based on data from the Pioneer Valley Planning Commission (PVPC) and is included in the mosaicData package. The PVPC collected data for ninety days from April 5, 2005 to November 15, 2005. Data collectors set up a laser sensor, with breaks in the laser beam recording when a rail-trail user passed the data collection station. More information can be found here, here, and here.

Response

  • volume: Number bikes that broke the laser beam

Analysis goal

The goals of this activity are to:

  • Perform inference for multiple linear regression
  • Conduct/interpret hypothesis tests
  • Construct/interpret confidence intervals
  • Determine whether the conditions for inference are satisfied in this multi-predictor setting.

Exercise 0

Import GGally and use ggpairs to plot all combinations of variables.

Exercise 1

Choose two explanatory variables which you think will best predict Volume based on the plot above. Write your two variables on the board. You may not use the same combination as another group. If you choose a temperature, only select one.

Exercise 2

Fit and display two linear regression models predicting volume from the two predictors you chose. In one, include an interaction term between the two variables and in the other, do not.

Exercise 3

Consider the model without an interaction term. Perform a hypothesis test on one of your explanatory variables (fill in the blanks where appropriate:

  1. Set hypothesis: \(H_0: \beta_{fill in} [fill in]\) vs. \(\beta_{fill in} [fill in]\). Restate these hypothesis in words: [fill in]
  2. Calculate test statistics and p-value: The test statistic is \([fill in]\). The p-value is \([fill in]\).
  3. State the conclusion: [fill in]

Exercise 4

Consider the model with an interaction term. Interpret the the p-value associated with the interaction term.

Exercise 5

Generate 95% confidence intervals for the model without an interaction term. Hint: use the tidy function with the argument conf.int = TRUE. Interpret the confidence interval for the same explanatory variable as Exercise 3 and explain why the combination of p-value and confidence interval makes sense.

Exercise 6

What does it mean for two things to be independent in statistics (feel free to use Google)? Do we think our p-values/confidence intervals are independent across variables?

Exercise 7

Important

If the intercept term was significant, use that model for the rest of this activity, otherwise use the model without the interaction term.

Generate a scatter plot of the residuals vs. the fitted values for this model.

Exercise 8

Plot the residuals vs. each of your predictors (two plots total).

Exercise 9

Based on the three plots you’ve made, do you think the linearity condition is satisfied?

Exerise 10

We check the constant variance assumption in the same way we do with SLR. To what extent does the constant variance condition seem to be satisfied?

Exercise 11

Generate a histogram and QQ-plot of the residuals. To what extent do you believe the normality condition is satisfied?

Exercise 12

How do you think would you go about checking the independence condition?

To submit the AE

Important
  • Render the document to produce the HTML file with all of your work from today’s class.
  • Upload your QMD and HTML files to the Canvas assignment.