HW 02: Palmer penguins

due Monday, October 2 at 11:59pm

Introduction

In this analysis you will use simple and multiple linear regression to analyze relationships between variables in three different scenarios.

Learning goals

In this assignment, you will…

  • Fit and interpret simple and multiple linear regression models
  • Evaluate and compare simple and multiple linear regression models
  • Continue developing a workflow for reproducible data analysis.

Getting started

The repo for this assignment is available on GitHub at github.com/sta210-fa23 and starts with the prefix hw-02. See Lab 01 for more detailed instructions on getting started.

Packages

The following packages will be used in this assignment:

library(tidyverse)
library(tidymodels) 
library(knitr) 
library(palmerpenguins)
Important

All narrative should be written in complete sentences, and all visualizations should have informative titles and axis labels.

Data: Palmer Penguins

Data were collected and made available by Dr. Kristen Gorman and the Palmer Station, Antarctica LTER, a member of the Long Term Ecological Research Network. (Gorman, Williams, and Fraser 2014)


These data can be found in the palmerpenguins package. We’re going to be working with the penguins dataset from this package. The dataset contains data for 344 penguins. There are 3 different species of penguins in this dataset, collected from 3 islands in the Palmer Archipelago, Antarctica. Click here to see the codebook.

Exercises

Exercise 1

Our eventual goal is to fit a model predicting body mass (which is more difficult to measure) from island, bill length, bill depth, flipper length, species, and sex.

We will start by preparing the data.

  • Use the drop_na() function to remove any observations from the penguins data frame that has missing values. Your resulting data frame should have 333 observations.
Note

For the purposes of this assignment, we will treat the missingness as random, i.e., this is still a representative sample even though we removed observations with missing values.

Exercise 2

  • Use the data to fit a model predicting body mass (which can be difficult to measure) from bill length.

  • Neatly display the model using 3 digits.

  • Write the estimated regression equation. Use the variable names in your equation.

  • Interpret the slope in the context of the data.

  • Calculate the \(R^2\) value.

Exercise 3

  • Plot the fitted values vs. the residual values for the model in the previous exercise.

  • Make another plot of the histogram of the residuals.

  • Is inference for linear regression appropriate? Why or why not?

Exercise 4

  • The p-value in Exercise 2 indicates statistical significance. Does this mean the model is a good fit for the data?

Exercise 5

  • Now, use the data to fit a model predicting body mass (which can be difficult to measure) from bill length, bill depth, flipper length, island, species, and sex. Only include main effects, i.e., no interaction terms, in this model.

  • Neatly display the model using 3 digits.

  • Write the estimated regression equation. Use the variable names in your equation.

  • Calculate the \(R^2\) value. Why is it larger than the previous one? Interpret this value in context of the data and the model.

Tip

The code for fitting multiple linear regression models is the same as simple linear regression. Use + to add multiple predictor variables to the model.

Exercise 6

  • Plot the fitted values vs. the residual values for the model in the previous exercise.

  • Make another plot of the histogram of the residuals.

  • Is inference for linear regression appropriate? Why or why not?

Exercise 7

  • Interpret the intercept in the context of the data.

  • Interpret the coefficient of sex in the context of the data.

  • Interpret coefficient (slope) of bill length in the context of the data.

  • Is the coefficient of bill length the same as in Exercise 2. Briefly explain why or why not?

Exercise 8

  • Calculate the residual for a female Gentoo penguin on the Biscoe island that weighs 4450 grams with the following body measurements: bill_length_mm = 43.2, bill_depth_mm = 14.5, flipper_length_mm = 208.

  • Does the model overpredict or underpredict this penguin’s mass?

Exercise 9

Fill in the starter code below to write the function predict_boots which takes a bootstrap sample, calculates \(R^2\) terms for the two models listed below. Set the seed to 29.

  • Model 1: The model from Exercise 5. Model 2: The model from Exercise 5, without the island term.

  • The last line of the code runs the function predict_boots for 1000 iterations and saves the output as the object r_squared_diffs. Create a new variable that calculates the difference between the \(R^2\) values for the two models.

  • Make a histogram of the differences.

  • What do you notice about the histogram?

  • Which model is best for predicting body mass of a penguin? Briefly explain.

set.seed(29)

predict_boots <- function(i) {
  # take a bootstrap sample
  boot <- penguins |>
    slice_sample(n = nrow(penguins), replace = TRUE) |>
    mutate(boot_samp = i)
  
  #fit the model with island to the bootstrap sample
  fit_island <- linear_reg() |>
    fit(_____, data = boot)
  
  # fit the model without island to the bootstrap sample
  fit_no_island <- linear_reg() |>
    fit(_______, data = boot)
  
  #save the rsq values for each model
  tibble(
    island_r2 = glance(fit_island)$r.squared,
    no_island_r2 = glance(fit_no_island)$r.squared
  )
}

# run the function 1000 times and save the differences between
r_squared_diffs <- map_df(1:1000, predict_boots)

Exercise 10

  • Use the model from the previous exercise without island, and set the seed to 210.

  • Fit the model to 1000 bootstrap samples.

  • Make a scatterplot of the estimated coefficients for flipper length (y-axis) vs. bill length (x-axis).

  • Calculate a 95% confidence interval for the difference between these coefficient values. Is there evidence that either bill length or flipper length has a larger influence on the model? Briefly explain.

Tip

The code for bootstrapping in multiple linear regression is the same as the code for simple linear regression.

Submission

Warning

Before you wrap up the assignment, make sure all documents are updated on your GitHub repo. We will be checking these to make sure you have been practicing how to commit and push changes.

Remember – you must turn in a PDF file to the Gradescope page before the submission deadline for full credit.

To submit your assignment:

  • Go to http://www.gradescope.com and click Log in in the top right corner.
  • Click School Credentials ➡️ Duke NetID and log in using your NetID credentials.
  • Click on your STA 210 course.
  • Click on the assignment, and you’ll be prompted to submit it.
  • Mark the pages associated with each exercise. All of the pages of your lab should be associated with at least one question (i.e., should be “checked”).
  • Select the first page of your PDF submission to be associated with the “Workflow & formatting” section.

Grading (50 points)

Component Points
Ex 1 - 10 47
Workflow & formatting 31

References

Gorman, Kristen B., Tony D. Williams, and William R. Fraser. 2014. “Ecological Sexual Dimorphism and Environmental Variability Within a Community of Antarctic Penguins (Genus Pygoscelis).” Edited by André Chiaradia. PLoS ONE 9 (3): e90081. https://doi.org/10.1371/journal.pone.0090081.

Footnotes

  1. The “Workflow & formatting” grade is to assess the reproducible workflow and document format. This includes having at least 3 informative commit messages, a neatly organized document with readable code and your name and the date in the YAML.↩︎