R: Bootstrap confidence intervals

Mark NeedhamJuly 22nd, 2015Last Updated: July 21st, 2015

1 31 2 minutes read

I recently came across an interesting post on Julia Evans’ blog showing how to generate a bigger set of data points by sampling the small set of data points that we actually have using bootstrapping. Julia’s examples are all in Python so I thought it’d be a fun exercise to translate them into R.

We’re doing the bootstrapping to simulate the number of no-shows for a flight so we can work out how many seats we can overbook the plane by.

We start out with a small sample of no-shows and work off the assumption that it’s ok to kick someone off a flight 5% of the time. Let’s work out how many people that’d be for our initial sample:

> data = c(0, 1, 3, 2, 8, 2, 3, 4)
> quantile(data, 0.05)
  5% 
0.35

0.35 people! That’s not a particularly useful result so we’re going to resample the initial data set 10,000 times, taking the 5%ile each time and see if we come up with something better:

We’re going to use the sample function with replacement to generate our resamples:

> sample(data, replace = TRUE)
[1] 0 3 2 8 8 0 8 0
> sample(data, replace = TRUE)
[1] 2 2 4 3 4 4 2 2

Now let’s write a function to do that multiple times:

library(ggplot)
 
bootstrap_5th_percentile = function(data, n_bootstraps) {
  return(sapply(1:n_bootstraps, 
                function(iteration) quantile(sample(data, replace = TRUE), 0.05)))
}
 
values = bootstrap_5th_percentile(data, 10000)
 
ggplot(aes(x = value), data = data.frame(value = values)) + geom_histogram(binwidth=0.25)

So this visualisation is telling us that we can oversell by 0-2 people but we don’t know an exact number.

Let’s try the same exercise but with a bigger initial data set of 1,000 values rather than just 8. First we’ll generate a distribution (with a mean of 5 and standard deviation of 2) and visualise it:

library(dplyr)
 
df = data.frame(value = rnorm(1000,5, 2))
df = df %>% filter(value >= 0) %>% mutate(value = as.integer(round(value)))
ggplot(aes(x = value), data = df) + geom_histogram(binwidth=1)

Our distribution seems to have a lot more values around 4 & 5 whereas the Python version has a flatter distribution – I’m not sure why that is so if you have any ideas let me know. In any case, let’s check the 5%ile for this data set:

> quantile(df$value, 0.05)
5% 
 2

Cool! Now at least we have an integer value rather than the 0.35 we got earlier. Finally let’s do some bootstrapping over our new distribution and see what 5%ile we come up with:

resampled = bootstrap_5th_percentile(df$value, 10000)
byValue = data.frame(value = resampled) %>% count(value)
 
> byValue
Source: local data frame [3 x 2]
 
  value    n
1   1.0    3
2   1.7    2
3   2.0 9995
 
ggplot(aes(x = value, y = n), data = byValue) + geom_bar(stat = "identity")

‘2’ is by far the most popular 5%ile here although it seems weighted more towards that value than with Julia’s Python version, which I imagine is because we seem to have sampled from a slightly different distribution.

Reference:

R: Bootstrap confidence intervals from our JCG partner Mark Needham at the Mark Needham Blog blog.