10. Statistical inference#

10.1. Overview#

A typical data analysis task in practice is to draw conclusions about some unknown aspect of a population of interest based on observed data sampled from that population; we typically do not get data on the entire population. Data analysis questions regarding how summaries, patterns, trends, or relationships in a data set extend to the wider population are called inferential questions. This chapter will start with the fundamental ideas of sampling from populations and then introduce two common techniques in statistical inference: point estimation and interval estimation.

10.2. Chapter learning objectives#

By the end of the chapter, readers will be able to do the following:

  • Describe real-world examples of questions that can be answered with statistical inference.

  • Define common population parameters (e.g., mean, proportion, standard deviation) that are often estimated using sampled data, and estimate these from a sample.

  • Define the following statistical sampling terms: population, sample, population parameter, point estimate, and sampling distribution.

  • Explain the difference between a population parameter and a sample point estimate.

  • Use Python to draw random samples from a finite population.

  • Use Python to create a sampling distribution from a finite population.

  • Describe how sample size influences the sampling distribution.

  • Define bootstrapping.

  • Use Python to create a bootstrap distribution to approximate a sampling distribution.

  • Contrast the bootstrap and sampling distributions.

10.3. Why do we need sampling?#

We often need to understand how quantities we observe in a subset of data relate to the same quantities in the broader population. For example, suppose a retailer is considering selling iPhone accessories, and they want to estimate how big the market might be. Additionally, they want to strategize how they can market their products on North American college and university campuses. This retailer might formulate the following question:

What proportion of all undergraduate students in North America own an iPhone?

In the above question, we are interested in making a conclusion about all undergraduate students in North America; this is referred to as the population. In general, the population is the complete collection of individuals or cases we are interested in studying. Further, in the above question, we are interested in computing a quantity—the proportion of iPhone owners—based on the entire population. This proportion is referred to as a population parameter. In general, a population parameter is a numerical characteristic of the entire population. To compute this number in the example above, we would need to ask every single undergraduate in North America whether they own an iPhone. In practice, directly computing population parameters is often time-consuming and costly, and sometimes impossible.

A more practical approach would be to make measurements for a sample, i.e., a subset of individuals collected from the population. We can then compute a sample estimate—a numerical characteristic of the sample—that estimates the population parameter. For example, suppose we randomly selected ten undergraduate students across North America (the sample) and computed the proportion of those students who own an iPhone (the sample estimate). In that case, we might suspect that proportion is a reasonable estimate of the proportion of students who own an iPhone in the entire population. Fig. 10.1 illustrates this process. In general, the process of using a sample to make a conclusion about the broader population from which it is taken is referred to as statistical inference.

_images/population_vs_sample.png

Fig. 10.1 The process of using a sample from a broader population to obtain a point estimate of a population parameter. In this case, a sample of 10 individuals yielded 6 who own an iPhone, resulting in an estimated population proportion of 60% iPhone owners. The actual population proportion in this example illustration is 53.8%.#

Note that proportions are not the only kind of population parameter we might be interested in. For example, suppose an undergraduate student studying at the University of British Columbia in Canada is looking for an apartment to rent. They need to create a budget, so they want to know something about studio apartment rental prices in Vancouver, BC. This student might formulate the following question:

What is the average price-per-month of studio apartment rentals in Vancouver, Canada?

In this case, the population consists of all studio apartment rentals in Vancouver, and the population parameter is the average price-per-month. Here we used the average as a measure of the center to describe the “typical value” of studio apartment rental prices. But even within this one example, we could also be interested in many other population parameters. For instance, we know that not every studio apartment rental in Vancouver will have the same price per month. The student might be interested in how much monthly prices vary and want to find a measure of the rentals’ spread (or variability), such as the standard deviation. Or perhaps the student might be interested in the fraction of studio apartment rentals that cost more than $1000 per month. The question we want to answer will help us determine the parameter we want to estimate. If we were somehow able to observe the whole population of studio apartment rental offerings in Vancouver, we could compute each of these numbers exactly; therefore, these are all population parameters. There are many kinds of observations and population parameters that you will run into in practice, but in this chapter, we will focus on two settings:

  1. Using categorical observations to estimate the proportion of a category

  2. Using quantitative observations to estimate the average (or mean)

10.4. Sampling distributions#

10.4.1. Sampling distributions for proportions#

We will look at an example using data from Inside Airbnb [Cox, n.d.]. Airbnb is an online marketplace for arranging vacation rentals and places to stay. The data set contains listings for Vancouver, Canada, in September 2020. Our data includes an ID number, neighborhood, type of room, the number of people the rental accommodates, number of bathrooms, bedrooms, beds, and the price per night.

import pandas as pd

airbnb = pd.read_csv("data/listings.csv")
airbnb
id neighbourhood room_type accommodates bathrooms bedrooms beds price
0 1 Downtown Entire home/apt 5 2 baths 2 2 150.00
1 2 Downtown Eastside Entire home/apt 4 2 baths 2 2 132.00
2 3 West End Entire home/apt 2 1 bath 1 1 85.00
3 4 Kensington-Cedar Cottage Entire home/apt 2 1 bath 1 0 146.00
4 5 Kensington-Cedar Cottage Entire home/apt 4 1 bath 1 2 110.00
... ... ... ... ... ... ... ... ...
4589 4590 Downtown Eastside Entire home/apt 5 1 bath 1 1 99.00
4590 4591 Oakridge Private room 2 1.5 baths 1 1 42.51
4591 4592 Dunbar Southlands Private room 2 1.5 shared baths 1 1 53.29
4592 4593 West End Entire home/apt 4 1 bath 2 2 145.00
4593 4594 Shaughnessy Entire home/apt 6 1 bath 3 4 135.00

4594 rows × 8 columns

Suppose the city of Vancouver wants information about Airbnb rentals to help plan city bylaws, and they want to know how many Airbnb places are listed as entire homes and apartments (rather than as private or shared rooms). Therefore they may want to estimate the true proportion of all Airbnb listings where the room type is listed as “entire home or apartment.” Of course, we usually do not have access to the true population, but here let’s imagine (for learning purposes) that our data set represents the population of all Airbnb rental listings in Vancouver, Canada. We can find the proportion of listings for each room type by using the value_counts function with the normalize parameter as we did in previous chapters.

airbnb["room_type"].value_counts(normalize=True)

room_type
Entire home/apt    0.747497
Private room       0.246408
Shared room        0.005224
Hotel room         0.000871
Name: proportion, dtype: float64

We can see that the proportion of Entire home/apt listings in the data set is 0.747. This value, 0.747, is the population parameter. Remember, this parameter value is usually unknown in real data analysis problems, as it is typically not possible to make measurements for an entire population.

Instead, perhaps we can approximate it with a small subset of data! To investigate this idea, let’s try randomly selecting 40 listings (i.e., taking a random sample of size 40 from our population), and computing the proportion for that sample. We will use the sample method of the DataFrame object to take the sample. The argument n of sample is the size of the sample to take and since we are starting to use randomness here, we are also setting the random seed via numpy to make the results reproducible.

import numpy as np


np.random.seed(155)

airbnb.sample(n=40)["room_type"].value_counts(normalize=True)

room_type
Entire home/apt    0.725
Private room       0.250
Shared room        0.025
Name: proportion, dtype: float64

Here we see that the proportion of entire home/apartment listings in this random sample is 0.725. Wow—that’s close to our true population value! But remember, we computed the proportion using a random sample of size 40. This has two consequences. First, this value is only an estimate, i.e., our best guess of our population parameter using this sample. Given that we are estimating a single value here, we often refer to it as a point estimate. Second, since the sample was random, if we were to take another random sample of size 40 and compute the proportion for that sample, we would not get the same answer:

airbnb.sample(n=40)["room_type"].value_counts(normalize=True)

room_type
Entire home/apt    0.625
Private room       0.350
Shared room        0.025
Name: proportion, dtype: float64

Confirmed! We get a different value for our estimate this time. That means that our point estimate might be unreliable. Indeed, estimates vary from sample to sample due to sampling variability. But just how much should we expect the estimates of our random samples to vary? Or in other words, how much can we really trust our point estimate based on a single sample?

To understand this, we will simulate many samples (much more than just two) of size 40 from our population of listings and calculate the proportion of entire home/apartment listings in each sample. This simulation will create many sample proportions, which we can visualize using a histogram. The distribution of the estimate for all possible samples of a given size (which we commonly refer to as \(n\)) from a population is called a sampling distribution. The sampling distribution will help us see how much we would expect our sample proportions from this population to vary for samples of size 40.

We again use the sample to take samples of size 40 from our population of Airbnb listings. But this time we use a list comprehension to repeat the operation multiple times (as we did previously in Chapter 9). In this case we repeat the operation 20,000 times to obtain 20,000 samples of size 40. To make it clear which rows in the data frame come which of the 20,000 samples, we also add a column called replicate with this information using the assign function, introduced previously in Chapter 3. The call to concat concatenates all the 20,000 data frames returned from the list comprehension into a single big data frame.

samples = pd.concat([
    airbnb.sample(40).assign(replicate=n)
    for n in range(20_000)
])
samples
id neighbourhood room_type accommodates bathrooms bedrooms beds price replicate
605 606 Marpole Entire home/apt 3 1 bath 1 1 91.0 0
4579 4580 Downtown Entire home/apt 3 1 bath 1 2 160.0 0
1739 1740 West End Entire home/apt 2 1 bath 1 2 151.0 0
3904 3905 Oakridge Private room 6 1 private bath 3 3 185.0 0
1596 1597 Kitsilano Entire home/apt 1 1 bath 1 1 99.0 0
... ... ... ... ... ... ... ... ... ...
3060 3061 Hastings-Sunrise Private room 2 3.5 shared baths 1 1 78.0 19999
527 528 Kitsilano Private room 4 1 private bath 1 1 99.0 19999
1587 1588 Downtown Entire home/apt 6 1 bath 1 3 169.0 19999
3860 3861 Downtown Entire home/apt 3 1 bath 1 1 100.0 19999
2747 2748 Downtown Entire home/apt 3 1 bath 1 1 285.0 19999

800000 rows × 9 columns

Since the column replicate indicates the replicate/sample number, we can verify that we indeed seem to have 20,0000 samples starting at sample 0 and ending at sample 19,999.

Now that we have obtained the samples, we need to compute the proportion of entire home/apartment listings in each sample. We first group the data by the replicate variable—to group the set of listings in each sample together—and then use value_counts with normalize=True to compute the proportion in each sample. Both the first and last few entries of the resulting data frame are printed below to show that we end up with 20,000 point estimates, one for each of the 20,000 samples.

(
    samples
    .groupby("replicate")
    ["room_type"]
    .value_counts(normalize=True)
)

replicate  room_type      
0          Entire home/apt    0.750
           Private room       0.250
1          Entire home/apt    0.775
           Private room       0.225
2          Entire home/apt    0.750
                              ...  
19998      Entire home/apt    0.700
           Private room       0.275
           Shared room        0.025
19999      Entire home/apt    0.750
           Private room       0.250
Name: proportion, Length: 44552, dtype: float64

The returned object is a series, and as we have previously learned we can use reset_index to change it to a data frame. However, there is one caveat here: when we use the value_counts function on a grouped series and try to reset_index we will end up with two columns with the same name and therefore get an error (in this case, room_type will occur twice). Fortunately, there is a simple solution: when we call reset_index, we can specify the name of the new column with the name parameter:

(
    samples
    .groupby("replicate")
    ["room_type"]
    .value_counts(normalize=True)
    .reset_index(name="sample_proportion")
)
replicate room_type sample_proportion
0 0 Entire home/apt 0.750
1 0 Private room 0.250
2 1 Entire home/apt 0.775
3 1 Private room 0.225
4 2 Entire home/apt 0.750
... ... ... ...
44547 19998 Entire home/apt 0.700
44548 19998 Private room 0.275
44549 19998 Shared room 0.025
44550 19999 Entire home/apt 0.750
44551 19999 Private room 0.250

44552 rows × 3 columns

Below we put everything together and also filter the data frame to keep only the room types that we are interested in.

sample_estimates = (
    samples
    .groupby("replicate")
    ["room_type"]
    .value_counts(normalize=True)
    .reset_index(name="sample_proportion")
)

sample_estimates = sample_estimates[sample_estimates["room_type"] == "Entire home/apt"]
sample_estimates
replicate room_type sample_proportion
0 0 Entire home/apt 0.750
2 1 Entire home/apt 0.775
4 2 Entire home/apt 0.750
6 3 Entire home/apt 0.675
8 4 Entire home/apt 0.725
... ... ... ...
44541 19995 Entire home/apt 0.775
44543 19996 Entire home/apt 0.750
44545 19997 Entire home/apt 0.725
44547 19998 Entire home/apt 0.700
44550 19999 Entire home/apt 0.750

20000 rows × 3 columns

We can now visualize the sampling distribution of sample proportions for samples of size 40 using a histogram in Fig. 10.2. Keep in mind: in the real world, we don’t have access to the full population. So we can’t take many samples and can’t actually construct or visualize the sampling distribution. We have created this particular example such that we do have access to the full population, which lets us visualize the sampling distribution directly for learning purposes.

sampling_distribution = alt.Chart(sample_estimates).mark_bar().encode(
    x=alt.X("sample_proportion")
        .bin(maxbins=20)
        .title("Sample proportions"),
    y=alt.Y("count()").title("Count"),
)

sampling_distribution

Fig. 10.2 Sampling distribution of the sample proportion for sample size 40.#

The sampling distribution in Fig. 10.2 appears to be bell-shaped, is roughly symmetric, and has one peak. It is centered around 0.75 and the sample proportions range from about 0.55 to about 0.95. In fact, we can calculate the mean of the sample proportions.

sample_estimates["sample_proportion"].mean()

0.74848375

We notice that the sample proportions are centered around the population proportion value, 0.748! In general, the mean of the sampling distribution should be equal to the population proportion. This is great news because it means that the sample proportion is neither an overestimate nor an underestimate of the population proportion. In other words, if you were to take many samples as we did above, there is no tendency towards over or underestimating the population proportion. In a real data analysis setting where you just have access to your single sample, this implies that you would suspect that your sample point estimate is roughly equally likely to be above or below the true population proportion.

10.4.2. Sampling distributions for means#

In the previous section, our variable of interest—room_type—was categorical, and the population parameter was a proportion. As mentioned in the chapter introduction, there are many choices of the population parameter for each type of variable. What if we wanted to infer something about a population of quantitative variables instead? For instance, a traveler visiting Vancouver, Canada may wish to estimate the population mean (or average) price per night of Airbnb listings. Knowing the average could help them tell whether a particular listing is overpriced. We can visualize the population distribution of the price per night with a histogram.

population_distribution = alt.Chart(airbnb).mark_bar().encode(
    x=alt.X("price")
        .bin(maxbins=30)
        .title("Price per night (dollars)"),
    y=alt.Y("count()", title="Count"),
)

population_distribution

Fig. 10.3 Population distribution of price per night (dollars) for all Airbnb listings in Vancouver, Canada.#

In Fig. 10.3, we see that the population distribution has one peak. It is also skewed (i.e., is not symmetric): most of the listings are less than $250 per night, but a small number of listings cost much more, creating a long tail on the histogram’s right side. Along with visualizing the population, we can calculate the population mean, the average price per night for all the Airbnb listings.

airbnb["price"].mean()

154.5109773617762

The price per night of all Airbnb rentals in Vancouver, BC is $154.51, on average. This value is our population parameter since we are calculating it using the population data.

Now suppose we did not have access to the population data (which is usually the case!), yet we wanted to estimate the mean price per night. We could answer this question by taking a random sample of as many Airbnb listings as our time and resources allow. Let’s say we could do this for 40 listings. What would such a sample look like? Let’s take advantage of the fact that we do have access to the population data and simulate taking one random sample of 40 listings in Python, again using sample.

one_sample = airbnb.sample(n=40)

We can create a histogram to visualize the distribution of observations in the sample (Fig. 10.4), and calculate the mean of our sample.

sample_distribution = alt.Chart(one_sample).mark_bar().encode(
    x=alt.X("price")
        .bin(maxbins=30)
        .title("Price per night (dollars)"),
    y=alt.Y("count()").title("Count"),
)

sample_distribution

Fig. 10.4 Distribution of price per night (dollars) for sample of 40 Airbnb listings.#

one_sample["price"].mean()

153.48225

The average value of the sample of size 40 is $153.48. This number is a point estimate for the mean of the full population. Recall that the population mean was $154.51. So our estimate was fairly close to the population parameter: the mean was about 0.7% off. Note that we usually cannot compute the estimate’s accuracy in practice since we do not have access to the population parameter; if we did, we wouldn’t need to estimate it!

Also, recall from the previous section that the point estimate can vary; if we took another random sample from the population, our estimate’s value might change. So then, did we just get lucky with our point estimate above? How much does our estimate vary across different samples of size 40 in this example? Again, since we have access to the population, we can take many samples and plot the sampling distribution of sample means to get a sense for this variation. In this case, we’ll use the 20,000 samples of size 40 that we already stored in the samples variable. First we will calculate the sample mean for each replicate and then plot the sampling distribution of sample means for samples of size 40.

sample_estimates = (
    samples
    .groupby("replicate")
    ["price"]
    .mean()
    .reset_index()
    .rename(columns={"price": "mean_price"})
)
sample_estimates
replicate mean_price
0 0 187.00000
1 1 148.56075
2 2 165.50500
3 3 140.93925
4 4 139.14650
... ... ...
19995 19995 198.50000
19996 19996 192.66425
19997 19997 144.88600
19998 19998 146.08800
19999 19999 156.25000

20000 rows × 2 columns

sampling_distribution = alt.Chart(sample_estimates).mark_bar().encode(
    x=alt.X("mean_price")
        .bin(maxbins=30)
        .title("Sample mean price per night (dollars)"),
    y=alt.Y("count()").title("Count")
)

sampling_distribution

Fig. 10.5 Sampling distribution of the sample means for sample size of 40.#

In Fig. 10.5, the sampling distribution of the mean has one peak and is bell-shaped. Most of the estimates are between about $140 and $170; but there is a good fraction of cases outside this range (i.e., where the point estimate was not close to the population parameter). So it does indeed look like we were quite lucky when we estimated the population mean with only 0.7% error.

Let’s visualize the population distribution, distribution of the sample, and the sampling distribution on one plot to compare them in Fig. 10.6. Comparing these three distributions, the centers of the distributions are all around the same price (around $150). The original population distribution has a long right tail, and the sample distribution has a similar shape to that of the population distribution. However, the sampling distribution is not shaped like the population or sample distribution. Instead, it has a bell shape, and it has a lower spread than the population or sample distributions. The sample means vary less than the individual observations because there will be some high values and some small values in any random sample, which will keep the average from being too extreme.

Fig. 10.6 Comparison of population distribution, sample distribution, and sampling distribution.#

Given that there is quite a bit of variation in the sampling distribution of the sample mean—i.e., the point estimate that we obtain is not very reliable—is there any way to improve the estimate? One way to improve a point estimate is to take a larger sample. To illustrate what effect this has, we will take many samples of size 20, 50, 100, and 500, and plot the sampling distribution of the sample mean. We indicate the mean of the sampling distribution with a vertical line.

Fig. 10.7 Comparison of sampling distributions, with mean highlighted as a vertical line.#

Based on the visualization in Fig. 10.7, three points about the sample mean become clear:

  1. The mean of the sample mean (across samples) is equal to the population mean. In other words, the sampling distribution is centered at the population mean.

  2. Increasing the size of the sample decreases the spread (i.e., the variability) of the sampling distribution. Therefore, a larger sample size results in a more reliable point estimate of the population parameter.

  3. The distribution of the sample mean is roughly bell-shaped.

Note

You might notice that in the n = 20 case in Fig. 10.7, the distribution is not quite bell-shaped. There is a bit of skew towards the right! You might also notice that in the n = 50 case and larger, that skew seems to disappear. In general, the sampling distribution—for both means and proportions—only becomes bell-shaped once the sample size is large enough. How large is “large enough?” Unfortunately, it depends entirely on the problem at hand. But as a rule of thumb, often a sample size of at least 20 will suffice.

10.4.3. Summary#

  1. A point estimate is a single value computed using a sample from a population (e.g., a mean or proportion).

  2. The sampling distribution of an estimate is the distribution of the estimate for all possible samples of a fixed size from the same population.

  3. The shape of the sampling distribution is usually bell-shaped with one peak and centered at the population mean or proportion.

  4. The spread of the sampling distribution is related to the sample size. As the sample size increases, the spread of the sampling distribution decreases.

10.5. Bootstrapping#

10.5.1. Overview#

Why all this emphasis on sampling distributions?

We saw in the previous section that we could compute a point estimate of a population parameter using a sample of observations from the population. And since we constructed examples where we had access to the population, we could evaluate how accurate the estimate was, and even get a sense of how much the estimate would vary for different samples from the population. But in real data analysis settings, we usually have just one sample from our population and do not have access to the population itself. Therefore we cannot construct the sampling distribution as we did in the previous section. And as we saw, our sample estimate’s value can vary significantly from the population parameter. So reporting the point estimate from a single sample alone may not be enough. We also need to report some notion of uncertainty in the value of the point estimate.

Unfortunately, we cannot construct the exact sampling distribution without full access to the population. However, if we could somehow approximate what the sampling distribution would look like for a sample, we could use that approximation to then report how uncertain our sample point estimate is (as we did above with the exact sampling distribution). There are several methods to accomplish this; in this book, we will use the bootstrap. We will discuss interval estimation and construct confidence intervals using just a single sample from a population. A confidence interval is a range of plausible values for our population parameter.

Here is the key idea. First, if you take a big enough sample, it looks like the population. Notice the histograms’ shapes for samples of different sizes taken from the population in Fig. 10.8. We see that the sample’s distribution looks like that of the population for a large enough sample.

Fig. 10.8 Comparison of samples of different sizes from the population.#

In the previous section, we took many samples of the same size from our population to get a sense of the variability of a sample estimate. But if our sample is big enough that it looks like our population, we can pretend that our sample is the population, and take more samples (with replacement) of the same size from it instead! This very clever technique is called the bootstrap. Note that by taking many samples from our single, observed sample, we do not obtain the true sampling distribution, but rather an approximation that we call the bootstrap distribution.

Note

We must sample with replacement when using the bootstrap. Otherwise, if we had a sample of size \(n\), and obtained a sample from it of size \(n\) without replacement, it would just return our original sample!

This section will explore how to create a bootstrap distribution from a single sample using Python. The process is visualized in Fig. 10.9. For a sample of size \(n\), you would do the following:

  1. Randomly select an observation from the original sample, which was drawn from the population.

  2. Record the observation’s value.

  3. Replace that observation.

  4. Repeat steps 1–3 (sampling with replacement) until you have \(n\) observations, which form a bootstrap sample.

  5. Calculate the bootstrap point estimate (e.g., mean, median, proportion, slope, etc.) of the \(n\) observations in your bootstrap sample.

  6. Repeat steps 1–5 many times to create a distribution of point estimates (the bootstrap distribution).

  7. Calculate the plausible range of values around our observed point estimate.

_images/intro-bootstrap.jpeg

Fig. 10.9 Overview of the bootstrap process.#

10.5.2. Bootstrapping in Python#

Let’s continue working with our Airbnb example to illustrate how we might create and use a bootstrap distribution using just a single sample from the population. Once again, suppose we are interested in estimating the population mean price per night of all Airbnb listings in Vancouver, Canada, using a single sample size of 40. Recall our point estimate was $153.48. The histogram of prices in the sample is displayed in Fig. 10.10.

one_sample
id neighbourhood room_type accommodates bathrooms bedrooms beds price
4025 4026 Renfrew-Collingwood Private room 1 1 shared bath 1 1 40.00
1977 1978 Fairview Private room 1 2 baths 3 1 70.00
4008 4009 Downtown Entire home/apt 4 1 bath 1 1 269.00
1543 1544 Kensington-Cedar Cottage Entire home/apt 6 2 baths 3 5 320.00
3350 3351 Downtown Entire home/apt 2 1 bath 1 1 140.00
804 805 Mount Pleasant Private room 2 1 private bath 1 1 77.00
2286 2287 Marpole Entire home/apt 4 1 bath 2 2 105.00
1010 1011 Strathcona Entire home/apt 2 1 bath 1 1 120.00
1878 1879 Fairview Private room 2 1 private bath 1 1 175.00
1644 1645 Downtown Entire home/apt 4 2 baths 2 1 150.00
4579 4580 Downtown Entire home/apt 3 1 bath 1 2 160.00
2771 2772 Dunbar Southlands Entire home/apt 5 1 bath 2 2 130.00
4151 4152 Kitsilano Entire home/apt 2 1 bath 1 1 289.00
4495 4496 Riley Park Entire home/apt 2 1 bath 1 1 115.00
1308 1309 Riley Park Entire home/apt 2 1 bath 1 1 105.00
2246 2247 Mount Pleasant Entire home/apt 4 1 bath 1 2 105.00
2335 2336 Kensington-Cedar Cottage Entire home/apt 3 1 bath 2 2 85.00
4059 4060 Downtown Entire home/apt 5 2 baths 2 3 250.00
1280 1281 Hastings-Sunrise Entire home/apt 6 1 bath 2 3 100.00
4324 4325 Renfrew-Collingwood Private room 1 1 shared bath 1 0 25.00
3403 3404 Arbutus Ridge Entire home/apt 15 4 baths 6 8 664.00
1729 1730 Renfrew-Collingwood Entire home/apt 6 1 bath 2 3 93.00
3722 3723 Downtown Entire home/apt 4 1 bath 3 6 275.00
241 242 Hastings-Sunrise Private room 2 1 shared bath 1 1 50.00
3955 3956 Dunbar Southlands Private room 2 1.5 shared baths 1 1 60.00
1042 1043 Kitsilano Private room 1 1 shared bath 1 1 66.00
649 650 Sunset Entire home/apt 5 1 bath 2 2 196.00
1995 1996 Riley Park Private room 2 1 shared bath 1 1 70.00
363 364 Kensington-Cedar Cottage Entire home/apt 3 1 bath 1 2 120.00
1783 1784 Downtown Eastside Private room 2 1 private bath 1 1 60.00
805 806 Mount Pleasant Entire home/apt 4 1 bath 2 2 150.00
254 255 Downtown Entire home/apt 4 1.5 baths 2 2 300.00
3365 3366 Sunset Private room 1 1 private bath 1 1 100.00
4562 4563 Downtown Private room 1 1 shared bath 1 1 64.29
2124 2125 Downtown Entire home/apt 4 1 bath 2 3 200.00
18 19 Downtown Entire home/apt 4 2 baths 2 2 200.00
1997 1998 Downtown Entire home/apt 6 2 baths 3 3 257.00
4329 4330 West End Entire home/apt 2 1 bath 1 0 92.00
3408 3409 Downtown Entire home/apt 2 2 baths 2 2 189.00
635 636 Grandview-Woodland Entire home/apt 6 2 baths 1 2 103.00 
one_sample_dist = alt.Chart(one_sample).mark_bar().encode(
    x=alt.X("price")
        .bin(maxbins=30)
        .title("Price per night (dollars)"),
    y=alt.Y("count()").title("Count"),
)

one_sample_dist

Fig. 10.10 Histogram of price per night (dollars) for one sample of size 40.#

The histogram for the sample is skewed, with a few observations out to the right. The mean of the sample is $153.48. Remember, in practice, we usually only have this one sample from the population. So this sample and estimate are the only data we can work with.

We now perform steps 1–5 listed above to generate a single bootstrap sample in Python and calculate a point estimate from that bootstrap sample. We will continue using the sample function of our dataframe, Critically, note that we now set frac=1 (“fraction”) to indicate that we want to draw as many samples as there are rows in the dataframe (we could also have set n=40 but then we would need to manually keep track of how many rows there are). Since we need to sample with replacement when bootstrapping, we change the replace parameter to True.

boot1 = one_sample.sample(frac=1, replace=True)
boot1_dist = alt.Chart(boot1).mark_bar().encode(
    x=alt.X("price")
        .bin(maxbins=30)
        .title("Price per night (dollars)"),
    y=alt.Y("count()", title="Count"),
)

boot1_dist

Fig. 10.11 Bootstrap distribution.#

boot1["price"].mean()

132.65

Notice in Fig. 10.11 that the histogram of our bootstrap sample has a similar shape to the original sample histogram. Though the shapes of the distributions are similar, they are not identical. You’ll also notice that the original sample mean and the bootstrap sample mean differ. How might that happen? Remember that we are sampling with replacement from the original sample, so we don’t end up with the same sample values again. We are pretending that our single sample is close to the population, and we are trying to mimic drawing another sample from the population by drawing one from our original sample.

Let’s now take 20,000 bootstrap samples from the original sample (one_sample) and calculate the means for each of those replicates. Recall that this assumes that one_sample looks like our original population; but since we do not have access to the population itself, this is often the best we can do. Note that here we break the list comprehension over multiple lines so that it is easier to read.

boot20000 = pd.concat([
    one_sample.sample(frac=1, replace=True).assign(replicate=n)
    for n in range(20_000)
])
boot20000
id neighbourhood room_type accommodates bathrooms bedrooms beds price replicate
2286 2287 Marpole Entire home/apt 4 1 bath 2 2 105.0 0
254 255 Downtown Entire home/apt 4 1.5 baths 2 2 300.0 0
2246 2247 Mount Pleasant Entire home/apt 4 1 bath 1 2 105.0 0
4579 4580 Downtown Entire home/apt 3 1 bath 1 2 160.0 0
4495 4496 Riley Park Entire home/apt 2 1 bath 1 1 115.0 0
... ... ... ... ... ... ... ... ... ...
1997 1998 Downtown Entire home/apt 6 2 baths 3 3 257.0 19999
241 242 Hastings-Sunrise Private room 2 1 shared bath 1 1 50.0 19999
1878 1879 Fairview Private room 2 1 private bath 1 1 175.0 19999
1783 1784 Downtown Eastside Private room 2 1 private bath 1 1 60.0 19999
4495 4496 Riley Park Entire home/apt 2 1 bath 1 1 115.0 19999

800000 rows × 9 columns

Let’s take a look at the histograms of the first six replicates of our bootstrap samples.

six_bootstrap_samples = boot20000.query("replicate < 6")
six_bootstrap_fig = alt.Chart(six_bootstrap_samples, height=150).mark_bar().encode(
    x=alt.X("price")
        .bin(maxbins=20)
        .title("Price per night (dollars)"),
    y=alt.Y("count()").title("Count")
).facet(
    "replicate:N",  # Recall that `:N` converts the variable to a categorical type
    columns=2
)
six_bootstrap_fig

Fig. 10.12 Histograms of the first six replicates of the bootstrap samples.#

We see in Fig. 10.12 how the distributions of the bootstrap samples differ. If we calculate the sample mean for each of these six samples, we can see that these are also different between samples. To compute the mean for each sample, we first group by the “replicate” which is the column containing the sample/replicate number. Then we compute the mean of the price column and rename it to mean_price for it to be more descriptive. Finally we use reset_index to get the replicate values back as a column in the dataframe.

(
    six_bootstrap_samples
    .groupby("replicate")
    ["price"]
    .mean()
    .reset_index()
    .rename(columns={"price": "mean_price"})
)
replicate mean_price
0 0 155.67175
1 1 154.42500
2 2 149.35000
3 3 169.13225
4 4 179.79675
5 5 188.28225

The distributions and the means differ between the bootstrapped samples because we are sampling with replacement. If we instead would have sampled without replacement, we would end up with the exact same values in the sample each time.

We will now calculate point estimates of the mean for our 20,000 bootstrap samples and generate a bootstrap distribution of these point estimates. The bootstrap distribution (Fig. 10.13) suggests how we might expect our point estimate to behave if we take multiple samples.

boot20000_means = (
    boot20000
    .groupby("replicate")
    ["price"]
    .mean()
    .reset_index()
    .rename(columns={"price": "mean_price"})
)

boot20000_means
replicate mean_price
0 0 155.67175
1 1 154.42500
2 2 149.35000
3 3 169.13225
4 4 179.79675
... ... ...
19995 19995 159.29675
19996 19996 137.20000
19997 19997 136.55725
19998 19998 161.93950
19999 19999 170.22500

20000 rows × 2 columns

boot_est_dist = alt.Chart(boot20000_means).mark_bar().encode(
    x=alt.X("mean_price")
        .bin(maxbins=20)
        .title("Sample mean price per night (dollars)"),
    y=alt.Y("count()").title("Count"),
)

boot_est_dist

Fig. 10.13 Distribution of the bootstrap sample means.#

Let’s compare the bootstrap distribution—which we construct by taking many samples from our original sample of size 40—with the true sampling distribution—which corresponds to taking many samples from the population.

Fig. 10.14 Comparison of the distribution of the bootstrap sample means and sampling distribution.#

There are two essential points that we can take away from Fig. 10.14. First, the shape and spread of the true sampling distribution and the bootstrap distribution are similar; the bootstrap distribution lets us get a sense of the point estimate’s variability. The second important point is that the means of these two distributions are slightly different. The sampling distribution is centered at $154.51, the population mean value. However, the bootstrap distribution is centered at the original sample’s mean price per night, $153.48. Because we are resampling from the original sample repeatedly, we see that the bootstrap distribution is centered at the original sample’s mean value (unlike the sampling distribution of the sample mean, which is centered at the population parameter value).

Fig. 10.15 summarizes the bootstrapping process. The idea here is that we can use this distribution of bootstrap sample means to approximate the sampling distribution of the sample means when we only have one sample. Since the bootstrap distribution pretty well approximates the sampling distribution spread, we can use the bootstrap spread to help us develop a plausible range for our population parameter along with our estimate!

_images/11-bootstrapping7-1.png

Fig. 10.15 Summary of bootstrapping process.#

10.5.3. Using the bootstrap to calculate a plausible range#

Now that we have constructed our bootstrap distribution, let’s use it to create an approximate 95% percentile bootstrap confidence interval. A confidence interval is a range of plausible values for the population parameter. We will find the range of values covering the middle 95% of the bootstrap distribution, giving us a 95% confidence interval. You may be wondering, what does “95% confidence” mean? If we took 100 random samples and calculated 100 95% confidence intervals, then about 95% of the ranges would capture the population parameter’s value. Note there’s nothing special about 95%. We could have used other levels, such as 90% or 99%. There is a balance between our level of confidence and precision. A higher confidence level corresponds to a wider range of the interval, and a lower confidence level corresponds to a narrower range. Therefore the level we choose is based on what chance we are willing to take of being wrong based on the implications of being wrong for our application. In general, we choose confidence levels to be comfortable with our level of uncertainty but not so strict that the interval is unhelpful. For instance, if our decision impacts human life and the implications of being wrong are deadly, we may want to be very confident and choose a higher confidence level.

To calculate a 95% percentile bootstrap confidence interval, we will do the following:

  1. Arrange the observations in the bootstrap distribution in ascending order.

  2. Find the value such that 2.5% of observations fall below it (the 2.5% percentile). Use that value as the lower bound of the interval.

  3. Find the value such that 97.5% of observations fall below it (the 97.5% percentile). Use that value as the upper bound of the interval.

To do this in Python, we can use the quantile function of our DataFrame. Quantiles are expressed in proportions rather than percentages, so the 2.5th and 97.5th percentiles would be the 0.025 and 0.975 quantiles, respectively.

ci_bounds = boot20000_means["mean_price"].quantile([0.025, 0.975])
ci_bounds

0.025    121.607069
0.975    191.525362
Name: mean_price, dtype: float64

Our interval, $121.61 to $191.53, captures the middle 95% of the sample mean prices in the bootstrap distribution. We can visualize the interval on our distribution in Fig. 10.16.

Fig. 10.16 Distribution of the bootstrap sample means with percentile lower and upper bounds.#

To finish our estimation of the population parameter, we would report the point estimate and our confidence interval’s lower and upper bounds. Here the sample mean price-per-night of 40 Airbnb listings was $153.48, and we are 95% “confident” that the true population mean price-per-night for all Airbnb listings in Vancouver is between $121.61 and $191.53. Notice that our interval does indeed contain the true population mean value, $154.51! However, in practice, we would not know whether our interval captured the population parameter or not because we usually only have a single sample, not the entire population. This is the best we can do when we only have one sample!

This chapter is only the beginning of the journey into statistical inference. We can extend the concepts learned here to do much more than report point estimates and confidence intervals, such as testing for real differences between populations, tests for associations between variables, and so much more. We have just scratched the surface of statistical inference; however, the material presented here will serve as the foundation for more advanced statistical techniques you may learn about in the future!

10.6. Exercises#

Practice exercises for the material covered in this chapter can be found in the accompanying worksheets repository in the two “Statistical inference” rows. You can launch an interactive version of each worksheet in your browser by clicking the “launch binder” button. You can also preview a non-interactive version of each worksheet by clicking “view worksheet.” If you instead decide to download the worksheets and run them on your own machine, make sure to follow the instructions for computer setup found in Chapter 13. This will ensure that the automated feedback and guidance that the worksheets provide will function as intended.

10.7. Additional resources#

  • Chapters 4 to 7 of OpenIntro Statistics [Diez et al., 2019] provide a good next step in learning about inference. Although it is still certainly an introductory text, things get a bit more mathematical here. Depending on your background, you may actually want to start going through Chapters 1 to 3 first, where you will learn some fundamental concepts in probability theory. Although it may seem like a diversion, probability theory is the language of statistics; if you have a solid grasp of probability, more advanced statistics will come naturally to you!

10.8. References#

Coxd.

Murray Cox. Inside Airbnb. n.d. URL: http://insideairbnb.com/ (visited on 2020-09-01).

DcCetinkayaRB19

David Diez, Mine Çetinkaya-Rundel, and Christopher Barr. OpenIntro Statistics. OpenIntro, Inc., 2019. URL: https://openintro.org/book/os/.