Welcome to the Course

• For today:
• Syllabus review, expectations and questions
• Getting started with R and Github
• Overview of the linear model

GitHub Overview

GitHub is a web based platform to host data, source code, projects. It’s useful for a variety of reasons, some perhaps more important than others in your own research.

• Version Control: Track changes to your code and documents
• Collaboration: Work with others on the same project
• Project Management: Organize tasks and track progress
• Backup: Store your work in the cloud
• Questions, Instructions

Why Use GitHub?

For Your Research

• GitHub makes computer code transparent, reproducible, and collaborative. It’s transparent because it’s open to the public, it’s reproducible because it allows others to recreate and expand upon your work, and it’s collaborative because it creates a platform for users to jointly contribute to a project.

• Version Control. It allows you to track every change you make. It’s trivial to revert back to previous code.

Repository (Repo)

A repository is a folder that contains: - Your code files - Data files - Documentation (README.md) - Configuration files - Complete history of changes - Let’s have a look at the structure of a repo

Example Structure

my_analysis/
├── README.md
├── data/
│   ├── raw_data.csv
│   └── processed_data.csv
├── code/
│   ├── 01_load_data.R
│   ├── 02_analysis.R
│   └── 03_visualize.R
└── output/
    ├── figures/
    └── tables/

Setup

• Create an account at github.com
• Log into your account
• Search for this repository, fork it to your account, and clone it to your computer
• And, createa sample repository to practice with.

RStudio

• Generally you should work from the terminal to use Git commands.
• You can access the terminal from within RStudio: Tools \(\rightarrow\) Terminal \(\rightarrow\) New Terminal
• Alternatively, when using RStudio, you can use the Git tab to perform Git operations via a graphical interface, the Git tab.
• We’ll use Git a lot in this course, but instead of an exhaustive presentation at the beginning, let’s take things in steps, starting with accessing the files for this course.

Understanding Git Workflows

Three Key Concepts

• Forking: Creating your own copy of someone else’s repository
• Committing: Saving changes to your local repository
• Push/Pull: Synchronizing between local and remote repositories

Forking vs. Cloning

Fork

• Creates a copy of a repository under your GitHub account
• A forked repository is useful for collaboration
• Use when you want to contribute to someone else’s project
• Common in open-source collaboration

Clone

• Downloads a repository to your local computer
• You can clone your own repos or others’ repos
• Changes are made locally until you push them
• Clones are “read-only” unless it is a repo in your account.
• Fork + Clone = Collaboration
• A Fork + Clone allows you to generate a “pull request”

The Commit Workflow

What is a Commit?

• A snapshot of your changes at a point in time
• Each commit has a unique ID and message
• Creates a history of your work

Basic Terminal Commands

# Check status of your files
git status

# Add files to staging area
git add data_683.R
git add .  # adds all changed files

# Commit changes with a message
git commit -m "My new 683 repo"

Pull, Push, and Staying in Sync

The Basic Workflow

# 1. Pull latest changes from remote
git pull origin main

# 2. Make your changes to files
# (edit your code, data, etc.)

# 3. Stage and commit changes
git add .
git commit -m "Update analysis"

# 4. Push changes to remote
git push origin main

Some Practical Advice:: Always pull before you start working to avoid conflicts!

When to Use What?

The Linear Regression Model

Linear regression is one of the most widely used statistical methods for several reasons:

• Straightforward and foundational - ideal starting point for advanced techniques
• Data Exploration and Inference - descriptive analysis, exploratory data analysis, inference
• Practical utility - hypothesis testing and predictive modeling

The Logic Underlying the Regression Model

A Brief (Troubled) History

• Francis Galton (1886) used regression and coined the term to denote a “regression to the mean” often in the heritability of physical charateristics.
• For instance, he compared characteristics using inter-generational data (e.g., average height in father-son pairs; Pearl & Mackenzie 2018)
• Observed “regression to the mean”: If a father is 5’2’’ (unusually short), his son tends to be short but closer to the population average (taller than the father, closer to the mean; problematically referred to as “regression to mediocrity” by Galton)
• Galton was a founder of the eugenics movement
• The broader method – a statistical model that can be used to quantify the relationship between two variables – is value neutral
• But a word of caution: Statistical methods are routinely abused and misinterpreted, sometimes in tragic and unexpected ways; all the more reason to thoroughly understand how they work

A Linear Model

• Linear regression is a common technique in the social sciences.
• Distills complex relationships to a simple linear relationship.

\[y_i = \alpha + \beta x_i\]

• Intercept (\(\alpha\)): The point at which the regression line crosses the \(y\)-axis, or the value of \(y\) when \(x=0\).

• Slope (\(\beta\)): The change in \(y\) for every unit change in \(x\). Formally: \(\frac{\partial y}{\partial x} = \beta\)

• Examples…..

Some Common Misconceptions

Reverse Causality

\(x\) may be caused by \(y\) rather than the reverse.

Confounding

\(x\) may be related to \(y\), but there is a common variable affecting both.

Mediation

A relationship may be indirect through another variable.

The Challenge

It is difficult to sort these out using a regression model with observed cross-sectional data.

In most applications, we observe the outcome of a process. It is often difficult to make claims about the process itself.

Observation versus Manipulation

Experimental versus Observational

Is \(x\) observed or under the control of the researcher?

• If observational: We can examine whether a relationship exists, but that relationship does not imply causation.

Correlation versus Regression

The two are intimately related, but we make several key assumptions in the regression equation.

• Dependent variable (regressand): The outcome
• Explanatory variable (regressor): The predictor
• The explanatory variable is fixed and exogenous; the dependent variable is random and endogenous *

Data Types

Random/Stochastic Variables

Variables that take on a set of values drawn from a probability distribution.

Cross-Sectional Data

Units observed once. Indexed with \(i\): \(x_i\)

Time Series Data

A single unit observed multiple times. Indexed with \(t\): \(x_t\) - May trend (systematic change) or be stationary (randomly varying around a mean)

Panel Data

Multiple units observed over multiple time periods. Indexed with \(i\) and \(t\): \(x_{it}\)

Variable Types

Quantitative variables: Ratio and Interval
Qualitative variables: Ordinal and Nominal

The Linear Model

The basic functional form:

\[y_i = \alpha + \beta x_i\]

Where:

• \(\alpha\) = intercept (value of \(y\) when \(x=0\))
• \(\beta\) = slope (change in \(y\) for every unit change in \(x\))

More formally: \(\partial y / \partial x = \beta\)

The Stochastic Model

Real-world relationships are stochastic rather than deterministic

\[y_i = \alpha + \beta x_i + \epsilon_i\]

• \(\epsilon_i\) = error term
• Often assume: \(\epsilon_i \sim N(0, \sigma^2)\)
  • Normally distributed
  • Mean zero
  • Constant variance \(\sigma^2\)

Multiple Predictors

With multiple predictors:

\[y_i = \alpha + \beta_1 x_{1i} + \beta_2 x_{2i} + ... + \beta_k x_{ki} + \epsilon_i\]

Or more compactly:

\[y_i = \alpha + \sum_{j=1}^{k} \beta_j x_{ji} + \epsilon_i\]

Conditional Distribution

We model the conditional distribution of \(y\) given \(x\):

\[p(y | x_1, x_2, ..., x_k) = f(x_1, x_2, ..., x_k; \alpha, \beta_1, \beta_2, ..., \beta_k)\]

Key Question: What is the expected value of \(y\) given \(x_1, x_2, ..., x_k\)?

\[E(y | x_1, x_2, ..., x_k) = \alpha + \sum_{j=1}^{k} \beta_j x_{j}\]