from __future__ import print_function, division import numpy import scipy.stats import matplotlib.pyplot as pyplot from IPython.html.widgets import interact, fixed from IPython.html import widgets # seed the random number generator so we all get the same results numpy.random.seed(18) # some nicer colors from http://colorbrewer2.org/ COLOR1 = '#7fc97f' COLOR2 = '#beaed4' COLOR3 = '#fdc086' COLOR4 = '#ffff99' COLOR5 = '#386cb0' %matplotlib inline
Suppose we want to estimate the average weight of men and women in the U.S.
And we want to quantify the uncertainty of the estimate.
One approach is to simulate many experiments and see how much the results vary from one experiment to the next.
I'll start with the unrealistic assumption that we know the actual distribution of weights in the population. Then I'll show how to solve the problem without that assumption.
Based on data from the BRFSS, I found that the distribution of weight in kg for women in the U.S. is well modeled by a lognormal distribution with the following parameters:
weight = scipy.stats.lognorm(0.23, 0, 70.8) weight.mean(), weight.std()
Here's what that distribution looks like:
xs = numpy.linspace(20, 160, 100) ys = weight.pdf(xs) pyplot.plot(xs, ys, linewidth=4, color=COLOR1) pyplot.xlabel('weight (kg)') pyplot.ylabel('PDF') None
make_sample draws a random sample from this distribution. The result is a NumPy array.
def make_sample(n=100): sample = weight.rvs(n) return sample
Here's an example with
n=100. The mean and std of the sample are close to the mean and std of the population, but not exact.
sample = make_sample(n=100) sample.mean(), sample.std()
We want to estimate the average weight in the population, so the "sample statistic" we'll use is the mean:
def sample_stat(sample): return sample.mean()
One iteration of "the experiment" is to collect a sample of 100 women and compute their average weight.
We can simulate running this experiment many times, and collect a list of sample statistics. The result is a NumPy array.
def compute_sample_statistics(n=100, iters=1000): stats = [sample_stat(make_sample(n)) for i in range(iters)] return numpy.array(stats)
The next line runs the simulation 1000 times and puts the results in
sample_means = compute_sample_statistics(n=100, iters=1000)
Let's look at the distribution of the sample means. This distribution shows how much the results vary from one experiment to the next.
Remember that this distribution is not the same as the distribution of weight in the population. This is the distribution of results across repeated imaginary experiments.
pyplot.hist(sample_means, color=COLOR5) pyplot.xlabel('sample mean (n=100)') pyplot.ylabel('count') None
The mean of the sample means is close to the actual population mean, which is nice, but not actually the important part.
The standard deviation of the sample means quantifies the variability from one experiment to the next, and reflects the precision of the estimate.
This quantity is called the "standard error".
std_err = sample_means.std() std_err
We can also use the distribution of sample means to compute a "90% confidence interval", which contains 90% of the experimental results:
conf_int = numpy.percentile(sample_means, [5, 95]) conf_int
array([ 69.92149384, 75.40866638])
The following function takes an array of sample statistics and prints the SE and CI:
def summarize_sampling_distribution(sample_stats): print('SE', sample_stats.std()) print('90% CI', numpy.percentile(sample_stats, [5, 95]))
And here's what that looks like:
SE 1.6355262477 90% CI [ 69.92149384 75.40866638]
Now we'd like to see what happens as we vary the sample size,
n. The following function takes
n, runs 1000 simulated experiments, and summarizes the results.
def plot_sample_stats(n, xlim=None): sample_stats = compute_sample_statistics(n, iters=1000) summarize_sampling_distribution(sample_stats) pyplot.hist(sample_stats, color=COLOR2) pyplot.xlabel('sample statistic') pyplot.xlim(xlim)
Here's a test run with
SE 1.71202891175 90% CI [ 69.96057332 75.58582662]
Now we can use
interact to run
plot_sample_stats with different values of
xlim sets the limits of the x-axis so the figure doesn't get rescaled as we vary
def sample_stat(sample): return sample.mean() slider = widgets.IntSliderWidget(min=10, max=1000, value=100) interact(plot_sample_stats, n=slider, xlim=fixed([55, 95])) None
SE 1.71314776815 90% CI [ 69.99274896 75.65943185]
This framework works with any other quantity we want to estimate. By changing
sample_stat, you can compute the SE and CI for any sample statistic.
As an exercise, fill in
sample_stat below with any of these statistics:
NumPy array methods you might find useful include
Depending on the results, you might want to adjust
def sample_stat(sample): # TODO: replace the following line with another sample statistic return sample.mean() slider = widgets.IntSliderWidget(min=10, max=1000, value=100) interact(plot_sample_stats, n=slider, xlim=fixed([0, 100])) None
SE 1.67195986148 90% CI [ 69.82954731 75.32184298]
So far we have shown that if we know the actual distribution of the population, we can compute the sampling distribution for any sample statistic, and from that we can compute SE and CI.
But in real life we don't know the actual distribution of the population. If we did, we wouldn't need to estimate it!
In real life, we use the sample to build a model of the population distribution, then use the model to generate the sampling distribution. A simple and popular way to do that is "resampling," which means we use the sample itself as a model of the population distribution and draw samples from it.
Before we go on, I want to collect some of the code from Part One and organize it as a class. This class represents a framework for computing sampling distributions.
class Resampler(object): """Represents a framework for computing sampling distributions.""" def __init__(self, sample, xlim=None): """Stores the actual sample.""" self.sample = sample self.n = len(sample) self.xlim = xlim def resample(self): """Generates a new sample by choosing from the original sample with replacement. """ new_sample = numpy.random.choice(self.sample, self.n, replace=True) return new_sample def sample_stat(self, sample): """Computes a sample statistic using the original sample or a simulated sample. """ return sample.mean() def compute_sample_statistics(self, iters=1000): """Simulates many experiments and collects the resulting sample statistics. """ stats = [self.sample_stat(self.resample()) for i in range(iters)] return numpy.array(stats) def plot_sample_stats(self): """Runs simulated experiments and summarizes the results. """ sample_stats = self.compute_sample_statistics() summarize_sampling_distribution(sample_stats) pyplot.hist(sample_stats, color=COLOR2) pyplot.xlabel('sample statistic') pyplot.xlim(self.xlim)
The following function instantiates a
Resampler and runs it.
def plot_resampled_stats(n=100): sample = weight.rvs(n) resampler = Resampler(sample, xlim=[55, 95]) resampler.plot_sample_stats()
Here's a test run with
SE 1.72606450921 90% CI [ 71.35648645 76.82647135]
Now we can use
plot_resampled_stats in an interaction:
slider = widgets.IntSliderWidget(min=10, max=1000, value=100) interact(plot_resampled_stats, n=slider, xlim=fixed([1, 15])) None
SE 1.67407589545 90% CI [ 69.60129748 75.13161693]
Exercise: write a new class called
StdResampler that inherits from
Resampler and overrides
sample_stat so it computes the standard deviation of the resampled data.
class StdResampler(Resampler): """Computes the sampling distribution of the standard deviation.""" def sample_stat(self, sample): """Computes a sample statistic using the original sample or a simulated sample. """ return sample.std()
Test your code using the cell below:
def plot_resampled_stats(n=100): sample = weight.rvs(n) resampler = StdResampler(sample, xlim=[0, 100]) resampler.plot_sample_stats() plot_resampled_stats()
SE 1.30056137605 90% CI [ 13.70615766 18.05008376]
StdResampler is working, you should be able to interact with it:
slider = widgets.IntSliderWidget(min=10, max=1000, value=100) interact(plot_resampled_stats, n=slider) None
SE 1.29095098626 90% CI [ 15.13442137 19.27452588]
We can extend this framework to compute SE and CI for a difference in means.
For example, men are heavier than women on average. Here's the women's distribution again (from BRFSS data):
female_weight = scipy.stats.lognorm(0.23, 0, 70.8) female_weight.mean(), female_weight.std()
And here's the men's distribution:
male_weight = scipy.stats.lognorm(0.20, 0, 87.3) male_weight.mean(), male_weight.std()
I'll simulate a sample of 100 men and 100 women:
female_sample = female_weight.rvs(100) male_sample = male_weight.rvs(100)
The difference in means should be about 17 kg, but will vary from one random sample to the next:
male_sample.mean() - female_sample.mean()
Here's the function that computes Cohen's $d$ again:
def CohenEffectSize(group1, group2): """Compute Cohen's d. group1: Series or NumPy array group2: Series or NumPy array returns: float """ diff = group1.mean() - group2.mean() n1, n2 = len(group1), len(group2) var1 = group1.var() var2 = group2.var() pooled_var = (n1 * var1 + n2 * var2) / (n1 + n2) d = diff / numpy.sqrt(pooled_var) return d
The difference in weight between men and women is about 1 standard deviation:
Now we can write a version of the
Resampler that computes the sampling distribution of $d$.
class CohenResampler(Resampler): def __init__(self, group1, group2, xlim=None): self.group1 = group1 self.group2 = group2 self.xlim = xlim def resample(self): group1 = numpy.random.choice(self.group1, len(self.group1), replace=True) group2 = numpy.random.choice(self.group2, len(self.group2), replace=True) return group1, group2 def sample_stat(self, groups): group1, group2 = groups return CohenEffectSize(group1, group2) # NOTE: The following functions are the same as the ones in Resampler, # so I could just inherit them, but I'm including them for readability def compute_sample_statistics(self, iters=1000): stats = [self.sample_stat(self.resample()) for i in range(iters)] return numpy.array(stats) def plot_sample_stats(self): sample_stats = self.compute_sample_statistics() summarize_sampling_distribution(sample_stats) pyplot.hist(sample_stats, color=COLOR2) pyplot.xlabel('sample statistic') pyplot.xlim(self.xlim)
Now we can instantiate a
CohenResampler and plot the sampling distribution.
resampler = CohenResampler(male_sample, female_sample) resampler.plot_sample_stats()
SE 0.160707033098 90% CI [ 0.6808076 1.1974013]
This example demonstrates an advantage of the computational framework over mathematical analysis. Statistics like Cohen's $d$, which is the ratio of other statistics, are relatively difficult to analyze. But with a computational approach, all sample statistics are equally "easy".
One note on vocabulary: what I am calling "resampling" here is a specific kind of resampling called "bootstrapping". Other techniques that are also considering resampling include permutation tests, which we'll see in the next section, and "jackknife" resampling. You can read more at http://en.wikipedia.org/wiki/Resampling_(statistics).