Rent in Edinburgh

Sample

Fifteen 3 BR flats in Edinburgh were randomly selected on rightmove.co.uk.

library(tidyverse)
edi_3br <- read_csv2("data/edi-3br.csv") # ; separated

## # A tibble: 15 × 4
##   flat_id  rent title                       address              
##   <chr>   <dbl> <chr>                       <chr>                
## 1 flat_01   825 3 bedroom apartment to rent Burnhead Grove, Edin…
## 2 flat_02  2400 3 bedroom flat to rent      Simpson Loan, Quarte…
## 3 flat_03  1900 3 bedroom flat to rent      FETTES ROW, NEW TOWN…
## 4 flat_04  1500 3 bedroom apartment to rent Eyre Crescent, Edinb…
## 5 flat_05  3250 3 bedroom flat to rent      Walker Street, Edinb…
## 6 flat_06  2145 3 bedroom flat to rent      George Street, City …
## # … with 9 more rows

Observed sample

Observed sample

Sample mean ≈ £1895 😱

Bootstrap population

Generated assuming there are more flats like the ones in the observed sample... Population mean = ❓

Bootstrapping scheme

1. Take a bootstrap sample - a random sample taken with replacement from the original sample, of the same size as the original sample
2. Calculate the bootstrap statistic - a statistic such as mean, median, proportion, slope, etc. computed on the bootstrap samples
3. Repeat steps (1) and (2) many times to create a bootstrap distribution - a distribution of bootstrap statistics
4. Calculate the bounds of the XX% confidence interval as the middle XX% of the bootstrap distribution

Generate bootstrap means

edi_3br %>%
  # specify the variable of interest
  specify(response = rent)

Generate bootstrap means

edi_3br %>%
  # specify the variable of interest
  specify(response = rent)
  # generate 15000 bootstrap samples
  generate(reps = 15000, type = "bootstrap")

Generate bootstrap means

edi_3br %>%
  # specify the variable of interest
  specify(response = rent)
  # generate 15000 bootstrap samples
  generate(reps = 15000, type = "bootstrap")
  # calculate the mean of each bootstrap sample
  calculate(stat = "mean")

Generate bootstrap means

# save resulting bootstrap distribution
boot_df <- edi_3br %>%
  # specify the variable of interest
  specify(response = rent) %>% 
  # generate 15000 bootstrap samples
  generate(reps = 15000, type = "bootstrap") %>% 
  # calculate the mean of each bootstrap sample
  calculate(stat = "mean")

The bootstrap sample

boot_df

## Response: rent (numeric)
## # A tibble: 15,000 × 2
##   replicate  stat
##       <int> <dbl>
## 1         1 1793.
## 2         2 1938.
## 3         3 2175 
## 4         4 2159.
## 5         5 2084 
## 6         6 1761 
## # … with 14,994 more rows

Visualize the bootstrap distribution

ggplot(data = boot_df, mapping = aes(x = stat)) +
  geom_histogram(binwidth = 100) +
  labs(title = "Bootstrap distribution of means")

Calculate the confidence interval

A 95% confidence interval is bounded by the middle 95% of the bootstrap distribution.

boot_df %>%
  summarize(lower = quantile(stat, 0.025),
            upper = quantile(stat, 0.975))

## # A tibble: 1 × 2
##   lower upper
##   <dbl> <dbl>
## 1 1603. 2213.

Visualize the confidence interval

Interpret the confidence interval

Confidence level

We are 95% confident that ...

• Suppose we took many samples from the original population and built a 95% confidence interval based on each sample.
• Then about 95% of those intervals would contain the true population parameter.

Commonly used confidence levels

Precision vs. accuracy

Changing confidence level

edi_3br %>%
  specify(response = rent) %>% 
  generate(reps = 15000, type = "bootstrap") %>% 
  calculate(stat = "mean") %>%
  summarize(lower = quantile(stat, 0.025),
            upper = quantile(stat, 0.975))

Recap

• Sample statistic $\ne$ population parameter, but if the sample is good, it can be a good estimate
• We report the estimate with a confidence interval, and the width of this interval depends on the variability of sample statistics from different samples from the population
• Since we can't continue sampling from the population, we bootstrap from the one sample we have to estimate sampling variability
• We can do this for any sample statistic:
  • For a mean: calculate(stat = "mean")
  • For a median: calculate(stat = "median")
  • Learn about calculating bootstrap intervals for other statistics in your homework