AE 03: Simulation-Based Inference

Bikeshare

Important

  • Open RStudio and create a subfolder in your AE folder called “AE-03”

  • Go to the Canvas and locate your AE 03 assignment to get started.

  • Upload the ae-03-sbi.qmd and dcbikeshare.csv files into the folder you just created.

library(tidyverse)
library(ggformula)
library(broom)
library(openintro)
library(kableExtra)

Data

Our dataset contains daily rentals from the Capital Bikeshare in Washington, DC in 2011 and 2012. It was obtained from the dcbikeshare data set in the dsbox R package.

We will focus on the following variables in the analysis:

  • count: total bike rentals
  • temp_orig: Temperature in degrees Celsius
bikeshare <- read_csv("../data/dcbikeshare.csv") |>
  mutate(season = case_when(
    season == 1 ~ "winter",
    season == 2 ~ "spring",
    season == 3 ~ "summer",
    season == 4 ~ "fall"
  ),
  season = factor(season))
Rows: 731 Columns: 17
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl  (16): instant, season, yr, mnth, holiday, weekday, workingday, weathers...
date  (1): dteday

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
glimpse(bikeshare)
Rows: 731
Columns: 17
$ instant    <dbl> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, …
$ dteday     <date> 2011-01-01, 2011-01-02, 2011-01-03, 2011-01-04, 2011-01-05…
$ season     <fct> winter, winter, winter, winter, winter, winter, winter, win…
$ yr         <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ mnth       <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,…
$ holiday    <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,…
$ weekday    <dbl> 6, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4,…
$ workingday <dbl> 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1,…
$ weathersit <dbl> 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2,…
$ temp       <dbl> 0.3441670, 0.3634780, 0.1963640, 0.2000000, 0.2269570, 0.20…
$ atemp      <dbl> 0.3636250, 0.3537390, 0.1894050, 0.2121220, 0.2292700, 0.23…
$ hum        <dbl> 0.805833, 0.696087, 0.437273, 0.590435, 0.436957, 0.518261,…
$ windspeed  <dbl> 0.1604460, 0.2485390, 0.2483090, 0.1602960, 0.1869000, 0.08…
$ casual     <dbl> 331, 131, 120, 108, 82, 88, 148, 68, 54, 41, 43, 25, 38, 54…
$ registered <dbl> 654, 670, 1229, 1454, 1518, 1518, 1362, 891, 768, 1280, 122…
$ count      <dbl> 985, 801, 1349, 1562, 1600, 1606, 1510, 959, 822, 1321, 126…
$ temp_orig  <dbl> 14.110847, 14.902598, 8.050924, 8.200000, 9.305237, 8.37826…

Exercises

In this activity, each group will be assigned a season to use:

  • Group 1: winter
  • Group 2: spring
  • Group 3: summer
  • Group 4: fall

Exercise 0

Filter the bikeshare data set so that it only contains observations from your assigned season. Make sure to give the new data set a different name.

Exercise 1

Conduct a little EDA. Generate a scatter plot of the number of bike rentals vs the temperature for your season by filling in the blanks in the code below. What do you think alpha does?

gf______(______, data = ______, alpha = 0.7) |> 
  gf_labs(
    x = "Temperature (C)",
    y = "Number of Bike Rentals",
  )

Exercise 2

Fit a simple linear regression model, display the results to two decimal places, and be prepared to discuss the interpretation of the resulting slope and intercept.

model_fit <- lm(_____ ~ ______, data = _____)

tidy(_____) |>
  kable(digits = _____)

Exercise 3

Load the infer package and calculate the observed fit (slope)

observed_fit <- ____ |>
  specify(count ~ temp_orig) |>
  fit()

observed_fit

Exercise 4

Take n bootstrap samples and fit models to each one.

Fill in the code, then set eval: true .

n <- 100
set.seed(212)

boot_fits <- ______ |>
  specify(______) |>
  generate(reps = ____, type = "bootstrap") |>
  fit()

boot_fits
  • Why do we set a seed before taking the bootstrap samples?

Exercise 5 (Challenging)

Make a histogram of the bootstrap samples to visualize the bootstrap distribution.

# Code for histogram

Exercise 6

Compute and interpret the 95% confidence interval as the middle 95% of the bootstrap distribution.

Fill in the code, then set eval: true .

get_confidence_interval(
  boot_fits,
  point_estimate = _____,
  level = ____,
  type = "percentile"
)

Exercise 7

Modify the code from Exercise 6 to create a 90% confidence interval.

# Paste code for 90% confidence interval

Exercise 8

Modify the code from Exercise 6 to create a 90% confidence interval.

# Paste code for 90% confidence interval

Exercise 9

  • Which confidence level produces the most accurate confidence interval (90%, 95%, 99%)? Explain

  • Which confidence level produces the most precise confidence interval (90%, 95%, 99%)? Explain

  • If we want to be very certain that we capture the population parameter, should we use a wider or a narrower interval? What drawbacks are associated with using a wider interval?

Exercise 10

  • If your sample size \(n\) is increased, what impact do you think this will have on accuracy and precision?

  • What about if you increase the number of bootstrapped replicates?

Exercise 11

Your professor is interested in calculating the average amount of time CofI students spend doing homework.

  • If he collects a set of data and asks 100 students to compute 95% confidence intervals from that data, how many of those would you expect to contain the true average?

  • If, instead, he has each of those 100 students collect their own data and compute 95% confidence intervals from their own data, how many would you expect to contain the true average?

Exercise 12

Write down your research question in words then state the null and alternative hypotheses in both words and mathematical notation. You can use dollar signs to engage “math mode”.

\[ H_0: \]

Exercise 13

Generate null distribution using permutation

Fill in the code, then set eval: true .

n = 100
set.seed(212)

null_dist <- _____ |>
  specify(______) |>
  hypothesize(null = "independence") |>
  generate(reps = _____, type = "permute") |>
  fit()

Exercise 14

Visualize the null distribution. Does your slope seem unusual if we assume the null hypothesis is the truth?

# Code for histogram of null distribution

Exercise 15

Fill in the code below to compute the p-value. Have your reporter write the value on the board.

# get observed fit 
observed_fit <- _____ |>
  specify(____) |>
  fit()

# calculate p-value
get_p_value(
  ____,
  obs_stat = ____,
  direction = "two-sided"
)

Exercise 16

  • Do larger or smaller p-values provide evidence for the alternative hypothesis?

  • Do larger or smaller p-values provide evidence for your research question?

  • Interpret your p-value in the context of the problem. Do you think your data provides strong evidence for your research question?

Important

To submit the AE:

Render the document to produce the HTML with all of your work from today’s class.

The driver for your group should upload your .qmd and .html files to the Canvas assignment.