Lab 4

Linear Regression Modeling

Lab
Due to Gradescope by Weds. June 10 11:59 PM

Introduction

In this lab, you’ll practice with simple and multiple regression modeling while continuing to reinforce data science / coding best practices mastered in the first half of the course.

Getting Started

By now you should be familiar with how to get started on a lab assignment by cloning the GitHub repo for the assignment.

Click to expand if you need a refresher on how to get started with a lab assignment.
  • Go to https://cmgr.oit.duke.edu/containers and login with your Duke NetID and Password.
  • Click STA199 under My reservations to log into your container. You should now see the RStudio environment.
  • Go to the course organization at github.com/sta199-su26 organization on GitHub. Click on the repo with the prefix lab-3. It contains the starter documents you need to complete the homework.
  • Click on the green CODE button, select Use SSH. Click on the clipboard icon to copy the repo URL.
  • In RStudio, go to File ➛ New Project ➛Version Control ➛ Git.
  • Copy and paste the URL of your assignment repo into the dialog box Repository URL. Again, please make sure to have SSH highlighted under Clone when you copy the address.
  • Click Create Project, and the files from your GitHub repo will be displayed in the Files pane in RStudio.

Open the lab-4.qmd template Quarto file and update the authors field to add your name (first and last). Render the document. Examine the rendered document and make sure your name is updated in the document. Commit your changes with a meaningful commit message and push to GitHub.

Click to expand if you need a refresher on assignment guidelines.

Code Guidelines:

As we’ve discussed in the lecture, your plots should include an informative title, axes and legends should have human-readable labels, and aesthetic choices should be carefully considered.

Additionally, code should follow the tidyverse style. In particular,

  • there should be spaces before and line breaks after each + when building a ggplot,

  • there should also be spaces before and line breaks after each |> in a data transformation pipeline,

  • code should be properly indented,

  • there should be spaces around = signs and spaces after commas.

Furthermore, all code should be visible in the PDF output, i.e., should not run off the page on the PDF. Long lines that run off the page should be split across multiple lines with line breaks.

As you complete the lab and other assignments in this course, remember to develop a sound workflow for reproducible data analysis. This assignment will periodically remind you to render, commit, and push your changes to GitHub. You should have at least 3 commits with meaningful commit messages by the end of the assignment.

Packages

In this part you will work with the tidyverse and tidymodels packages.

Part 1: Parasites

Parasites can cause infectious disease – but not all animals are affected by the same parasites. Some parasites are present in a multitude of species and others are confined to a single host. It is hypothesized that closely related hosts are more likely to share the same parasites. More specifically, it is thought that closely related hosts will live in similar environments and have similar genetic makeup that coincides with optimal conditions for the same parasite to flourish.

In this part of the lab, you will explore how much evolutionary history predicts parasite similarity.

The dataset comes from an Ecology Letters paper by Cooper at al. (2012) entitled “Phylogenetic host specificity and understanding parasite sharing in primates” located here. The goal of the paper was to identify the ability of evolutionary history and ecological traits to characterize parasite host specificity.

Each row of the data contains two species, species1 and species2.

Subsequent columns describe metrics that compare the species:

  • divergence_time: how many (millions) of years ago the two species diverged. i.e., how many million years ago they were the same species.

  • distance: geodesic distance between species geographic range centroids (in kilometers)

  • BMdiff: difference in body mass between the two species (in grams)

  • precdiff: difference in mean annual precipitation across the two species geographic ranges (mm)

  • parsim: a measure of parasite similarity (proportion of parasites shared between species, ranges between 0 to 1.)

The data are available in parasites.csv in your data folder.

Question 1

Let’s start by reading in the parasites data and examining the relationship between divergence_time and parsim.

  1. Load the data and save the data frame as parasites.

  2. Based on the goals of the analysis, which variable should we use as the outcome variable?

  3. Visualize the relationship between the two variables, making sure to place the outcome (y) and predictor (x) variables on the appropriate axes.

  4. Use the visualization to describe the relationship between the two variables.

Question 2

Next, model this relationship.

  1. Fit the model and write the fitted equation using proper notation (i.e., using LaTeX).

  2. Interpret the slope and the intercept in the context of the data.

  3. Recreate the visualization from the previous question, this time adding a regression line to the visualization.

  4. What do you notice about the estimated regression line that may be strange, particularly with respect to predictions for very large divergence times?

Question 3

Since parsim takes values between 0 and 1, but predicted values from a linear regression model can range between (−∞,+∞), we will transform parsim using a logit transformation. This transformation is commonly used for proportion data because it maps values between (0,1) to values between (−∞,+∞). We will learn more about this transformation in our forthcoming study of logistic regression.

  1. Using mutate, create a new variable in the parasites data frame called transformed_parsim that is calculated as log(parsim/(1-parsim)). Add this variable to your data frame.

    Note

    log() in R represents the nautral log.

  2. Then, visualize the relationship between divergence_time and transformed_parsim, making sure to place the outcome (y) and predictor (x) variables on the appropriate axes. Add a regression line to your visualization.

  3. Write a 1-2 sentence description of what you observe in the visualization.

Question 4

Which variable is the strongest individual predictor of parasite similarity between species?

To answer this question, begin by fitting a linear regression model to each of the following pairs of variables, where transformed_parsim is the outcome in each model. Do not report the model outputs in a tidy format but save each one as dt_model, dist_model, BM_model, and prec_model, respectively.

  • divergence_time and transformed_parsim

  • distance and transformed_parsim

  • BMdiff and transformed_parsim

  • precdiff and transformed_parsim

  1. Report the slopes for each of these models, clearly identifying which reported slope belongs to which model.

  2. To answer our question of interest, would it be useful to compare the slopes in each model to choose the variable that is the strongest predictor of parasite similarity? Why or why not?

Question 5

Now, what if we calculated \(R^2\) to help answer our question? To compare the explanatory power of each individual predictor, we will look at \(R^2\) between the models. \(R^2\) is a measure of how much of the variability in the response variable is explained by the model.

As you may have guessed from the name, \(R^2\) can be calculated by squaring the correlation when we have a simple linear regression model. The correlation r takes values -1 to 1, therefore, \(R^2\) takes values 0 to 1. Intuitively, if r=1 or −1, then \(R^2\)=1, indicating the model is a perfect fit for the data. If r≈0 then \(R^2\)≈0, indicating the model is a very bad fit for the data.

You can calculate \(R^2\) using the glance function. For example, you can calculate \(R^2\) for dt_model using the code glance(dt_model)$r.squared.

  1. Calculate and report \(R^2\) for each model fit in the previous exercise.

  2. To answer our question of interest, would it be useful to compare the \(R^2\) in each model to choose the variable that is the strongest predictor of parasite similarity? Why or why not? And if so, which variable is the strongest individual predictor of parasite similarity between species?

Part 2: Card & Krueger

Should the minimum wage be increased? I’m sure many of you have an opinion about this. When debating this policy issue, one thing we need to understand is how changes in the minimum wage affect employment. When the minimum wage is raised, does it create jobs? Does it put people out of work? Does it have no affect at all? An ECON 101 model of the labor market implies that minimum wage increases could put people out of work because they make it more expensive for firms to employ workers, and we tend to do less of something if it becomes more expensive (demand curves slope downwards). That’s a theoretical argument anyway, but this is ultimately an empirical question – one that has a long history in applied economics.

Indeed, the 2021 Nobel Prize in Economic Sciences was shared by economist David Card, in part for a famous paper he wrote with Alan Krueger that initiated the modern empirical literature on the minimum wage:

Card, David and Alan Krueger (1994): “Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania,” American Economic Review, Vol. 84, No. 4., pp. 772-793.

On April 1, 1992, New Jersey’s minimum wage rose from $4.25 to $5.05 per hour. Pennsylvania’s did not. In order to assess the impact of this change, Card and Krueger collected data from fast-food restaurants along the PA/NJ border, both before the policy change went into affect (February 1992) and after (November/December 1992). They figured that restaurants close to the border were probably indistinguishable in terms of their features, practices, clientele, work force, etc, and so if there were any noticeable differences in employment after the wage hike, they must be due to the wage hike itself and not some other confounding factor. This is what we call a natural experiment. The data in Card and Krueger are purely observational, but the main idea is that the arbitrary placement of otherwise similar restaurants on either side of the PA/NJ border acts as if a controlled, randomized experiment were performed, and so we can use these data to draw causal conclusions about the impact of minimum wage policy on employment. As you can imagine, a massive literature has emerged where statisticians and economists argue about when this is appropriate and how it should be done, but nevertheless, researchers nowadays love to sniff around like truffle pigs for cute natural experiments hidden in otherwise messy observational data sets.

For this part of your lab, you will play around with the original Card and Krueger data:

card_krueger <- read_csv("data/card-krueger.csv")
glimpse(card_krueger)
Rows: 820
Columns: 5
$ id    <dbl> 46, 46, 49, 49, 506, 506, 56, 56, 61, 61, 62, 62, 445, 445, 451,…
$ state <chr> "PA", "PA", "PA", "PA", "PA", "PA", "PA", "PA", "PA", "PA", "PA"…
$ time  <chr> "before", "after", "before", "after", "before", "after", "before…
$ wage  <dbl> NA, 4.30, NA, 4.45, NA, 5.00, 5.00, 5.25, 5.50, 4.75, 5.00, NA, …
$ fte   <dbl> 40.50, 24.00, 13.75, 11.50, 8.50, 10.50, 34.00, 20.00, 24.00, 35…

There are five columns:

  • id: a unique identifier for each restaurant;
  • state: which state is the restaurant in?
  • time: measurements collected before or after the NJ minimum wage increase?
  • wage: the starting wage in US dollars;
  • fte: full-time-equivalent employment, calculated as the number of full-time workers (including managers) plus 0.5 times the number of part-time workers.

The full dataset is available on David Card’s website, and it contains more information than this if you want to keep exploring.

Background (Optional)

Acquire some domain knowledge! As we know, it’s never a good idea to blunder into a data analysis without some subject-matter expertise, so here is some suggested reading:

  • In the WSJ this past spring semester: “The 30-Year Debate Over the Minimum Wage Is Still Not Settled;”
  • The original Card and Krueger paper is here. Their bottom line was “[w]e find no indication that the rise in the minimum wage reduced employment,” which runs counter to the usual ECON 101 story. People have been arguing about this ever since;
  • This recent survey article attempts to summarize the state of the literature on (dis?)employment effects of the minimum wage;
  • The award citation for Card’s Nobel has useful summaries: popular, advanced.

We are not grading this, and we’ll never know if you did it or not, but you should definitely go exploring if this area interests you. Furthermore, while you may not do the reading we are suggesting here, you should definitely do a little outside reading in the domain relevant to your final project.

Question 6

  1. Copy / paste the code provided above to load in the data and store the data frame as card_krueger. Then, update the data frame using appropriate methods to relevel each of the state and time variables so that "PA" and "before" are the baselines, respectively.

  2. How many restaurants were sampled in each state? Use code to support your response.

  3. Compute the median wage and the median employment in each state before and after the policy change.

  4. Create a faceted density plot displaying wage according to both state and time. We want two panels stacked vertically, one for each time period (before and after the policy change). Within each panel, we want two densities, one for each state. Comment on any patterns you notice.

  5. Create a faceted density plot displaying fte according to both state and time (similar to part d). Comment on any patterns you notice.

Question 7

  1. Use pivot_wider to create a new data frame card_krueger_wide that has six columns: id, state, wage_before, wage_after, fte_before, and fte_after.

  2. Modify and update card_krueger_wide by discarding any rows that have a missing value in any of the following four columns: wage_before, wage_after, fte_before, and fte_after.

  3. Add a new variable emp_diff to card_krueger_wide which measures the change in fte after the new law took effect (fte_after - fte_before)

  4. Add a new variable gap to card_krueger_wide which is constructed in the following way:

  • gap equals zero for stores in Pennsylvania;
  • gap equals zero for stores in New Jersey whose starting wage before the policy change was already higher than the new minimum;
  • gap equals \((5.05 - \text{wage}_{\text{before}})/\text{wage}_{\text{before}}\) for all other stores in New Jersey.

Card and Krueger introduced gap as an alternative measure of the impact of the minimum wage at each store. In their words:

\(\text{GAP}_i\) is the proportional increase in wages at store \(i\) necessary to meet the new minimum rate. Variation in GAP, reflects both the New Jersey-Pennsylvania contrast and differences within New Jersey based on reported starting wages in wave 1. Indeed, the value of GAP, is a strong predictor of the actual proportional wage change between waves 1 and 2 (\(R^2=0.75\)), and conditional on GAP, there is no difference in wage behavior between stores in New Jersey and Pennsylvania

Question 8

  1. Create side-by-side boxplots of emp_diff for each state.

  2. Fit a linear model that predicts emp_diff (y) from state (x) and save the model object. Then, provide the tidy summary output.

  3. Write the estimated least squares regression line below using proper notation.

  4. Interpret the intercept in the context of the data and the research question. Is the intercept meaningful in this context? Why or why not?

  5. Interpret the slope in the context of the data and the research question.

Question 9

  1. Create a scatter plot of gap (x) versus emp_diff (y).

  2. Fit a linear model that predicts emp_diff from gap and save the model object. Then, provide the tidy summary output.

  3. Write the estimated least squares regression line below using proper notation.

  4. Interpret the intercept in the context of the data and the research question. Is the intercept meaningful in this context? Why or why not?

  5. Interpret the slope in the context of the data and the research question.

Wrap-up

Warning

Before you wrap up the assignment, make sure that you render, commit, and push one final time so that the final versions of both your .qmd file and the rendered PDF are pushed to GitHub and your Git pane is empty. We will be checking these to make sure you have been practicing how to commit and push changes.

Grading and Feedback

Reminders:

  • Questions will be graded for accuracy and completeness

  • Partial credit will be given where appropriate

  • There are also workflow points for:

    • committing at least three times as you work through your lab

    • having your final version of .qmd and .pdf files in your GitHub repository

    • selecting pages corresponding to each question in Gradescope