Announcements

Project due on 2 Nov (Fri) 23:59:59

Remaining office hours:

• 26 Oct (tomorrow)
• 2 Nov (Fri)

10am-12pm, Sequoia Hall Rm 207

Recap of session 7

• Making maps in R
  • geom_polygon
  • aes(x = long, y = lat, group = group)
• Joining datasets with left_join
  • df1 %>% left_join(df2, by = "key")

Agenda for today

• A crash course in statistics and modeling
  • Hypothesis testing
  • Linear regression

A very high level picture: for technical details, take STATS 60/STATS 101

Today’s dataset: Top 100 songs on Spotify

(Source: Spotify)

Tempo by mode: Is there a difference?

Hard to tell from the histograms:

Look at mean tempo for each mode

• Major: \(\approx 122\) bpm
• Minor: \(\approx 116\) bpm

Is this difference significant? What do we mean by significance anyway?

Structure of a hypothesis test

1. Start with a null hypothesis: An assumption on how the data is generated

Structure of a hypothesis test

1. Start with a null hypothesis: An assumption on how the data is generated
   
2. Based on this assumption, how likely were we to collect data as extreme as what we have?
   • p-value: probability of collecting data as extreme as ours (if null hypothesis is true)

Structure of a hypothesis test

1. Start with a null hypothesis: An assumption on how the data is generated
   
2. Based on this assumption, how likely were we to collect data as extreme as what we have?
   • p-value: probability of collecting data as extreme as ours (if null hypothesis is true)
3. Is the p-value considered low or not?
   • Threshold should depend on the context
   • Typical thresholds, 0.1, 0.05, 0.01

Structure of a hypothesis test

1. Start with a null hypothesis: An assumption on how the data is generated
   
2. Based on this assumption, how likely were we to collect data as extreme as what we have?
   • p-value: probability of collecting data as extreme as ours (if null hypothesis is true)
3. Is the p-value considered low or not?
   • Threshold should depend on the context
   • Typical thresholds, 0.1, 0.05, 0.01
4. If p-value is below threshold, 2 possible conclusions:
   • A rare event just happened, or
   • Our assumption in Step 1 was false

Tempo by mode: Is there a difference?

Two options:

• \(t\)-test
  • Null hypothesis: Mean tempo for songs in minor key is the same as that for songs in major key
  • Makes more assumptions on the data generation process (“parametric test”)
• Kolmogorov-Smirnov test
  • Null hypothesis: The distribution of tempo for songs in minor key is the same as that for songs in major key
  • Less assumptions on data generation process (“non-parametric test”), but rejecting the null gives less information

What is a model?

• A model is a simplified and idealized way to understand a system.
• R4DS: “The goal of a model is to provide a simple low-dimensional summary of a dataset. Ideally, the model will capture true “signals” (i.e. patterns generated by the phenomenon of interest), and ignore “noise” (i.e. random variation that you’re not interested in).”

Two steps to modeling

Step 1: Identify a family of models which express a generic pattern between your variables of interest.

Possible model family: Linear model, i.e. \(child = a_1 + a_2 \times parent\).

• Variables: child and parent
• Model parameters: \(a_1\) and \(a_2\)

Many other possible models: linear without intercept, quadratic, exponential, …

Different models within the linear model family

Each line corresponds to a choice of \(a_1\) and \(a_2\).

Two steps to modeling

Step 2: Find the model in this family that most closely matches your data.

That is, find specific values of \(a_1\) and \(a_2\) which make the model match the data most closely.

What do we mean by “closely matching the data”?

We choose \(a_1\) and \(a_2\) such that some objective function (loss function) is minimized.

Most common objective: Minimize the sum of squares of the black lines below.

(Source: uc-r.github.io)

Linear models in R

• Linear regression can be done with the lm function
• Syntax: lm(formula, data = df)
• Formulas look like y ~ x, which lm will translate to a function like \(y = a_1 + a_2 \cdot x\)

Models with categorical variables

Consider modeling valence ~ mode.

• Does the model \(valence = a_1 + a_2 \cdot mode\) make sense?
  • 3 + 4 \(\cdot\) “Major”??
• What R does:
  • Choose a baseline category (say, “Minor”)
  • Model \(valence = a_1 + a_2 \cdot modeMajor\), where \(modeMajor = \begin{cases} 1 &\text{if "Major"}, \\ 0 &\text{if "Minor"}. \end{cases}\)
• \(valence = a_1\) if Minor, \(valence = a_1 + a_2\) if Major

Additive models

Formula valence ~ loudness + mode translates to

• \(valence = a_1 + a_2 \cdot loudness + a_3 \cdot modeMajor\), where \(modeMajor = \begin{cases} 1 &\text{if "Major"}, \\ 0 &\text{if "Minor"}. \end{cases}\)
• \(valence = a_1 + a_2 \cdot loudness\) if Minor
• \(valence = (a_1 + a_3) + a_2 \cdot loudness\) if Major
• Same gradient, different intercept

Models with interaction

Formula valence ~ loudness * mode translates to

• \(valence = a_1 + a_2 \cdot loudness + a_3 \cdot modeMajor + \color{blue}{a_4 \cdot loudness \cdot modeMajor}\), where \(modeMajor = \begin{cases} 1 &\text{if "Major"}, \\ 0 &\text{if "Minor"}. \end{cases}\)
• \(valence = a_1 + a_2 \cdot loudness\) if Minor
• \(valence = (a_1 + a_3) + (a_2 + a_4) \cdot loudness\) if Major
• Different gradient, different intercept

Summary of the course

• Variable types
• Basic objects in R (vectors, lists, data frames)
• Plotting data with ggplot2
• Transforming and joining data with dplyr
• Importing and exporting data
• R projects, R scripts and R markdown
• Making maps
• Basic statistical testing and modeling

Where do we go from here?

• Read R for Data Science from cover to cover!
• Take short courses on DataCamp
• Writing your own functions and running simulations
• Advanced mapping with ggmap
• Advanced regression models with glmnet
• Interactive web apps with shiny
• Text analysis with tidytext
  • Recommmended text: Text Mining with R by Julia Silge and David Robinson (avaible online for free at tidytextmining.com)

Other Stanford courses

• Programming: CS 106A
• Statistical methods: STATS 60, STATS 101

Thank you! :)