In [1]:
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import pandas as pd

Generating a permutation sample

In the video, you learned that permutation sampling is a great way to simulate the hypothesis that two variables have identical probability distributions. This is often a hypothesis you want to test, so in this exercise, you will write a function to generate a permutation sample from two data sets.

Remember, a permutation sample of two arrays having respectively n1 and n2 entries is constructed by concatenating the arrays together, scrambling the contents of the concatenated array, and then taking the first n1 entries as the permutation sample of the first array and the last n2 entries as the permutation sample of the second array.

In [7]:
array([4, 6, 1, 2, 3, 5])
In [9]:
def permutation_sample(data1, data2):
    """Generate a permutation sample from two data sets."""

    # Concatenate the data sets: data
    data = np.concatenate((data1, data2))

    # Permute the concatenated array: permuted_data
    permuted_data = np.random.permutation(data)

    # Split the permuted array into two: perm_sample_1, perm_sample_2
    perm_sample_1 = permuted_data[:len(data1)]
    perm_sample_2 = permuted_data[len(data1):]

    return perm_sample_1, perm_sample_2

Visualizing permutation sampling

To help see how permutation sampling works, in this exercise you will generate permutation samples and look at them graphically.

We will use the Sheffield Weather Station data again, this time considering the monthly rainfall in July (a dry month) and November (a wet month). We expect these might be differently distributed, so we will take permutation samples to see how their ECDFs would look if they were identically distributed.

The data are stored in the Numpy arrays rain_july and rain_november.

As a reminder, permutation_sample() has a function signature of permutation_sample(data_1, data_2) with a return value of permuted_data[:len(data_1)], permuted_data[len(data_1):], where permuted_data = np.random.permutation(np.concatenate((data_1, data_2))).

In [10]:
df = pd.read_fwf('./sheffield_weather_station.csv', skiprows=8)
print df.shape
(1602, 7)
yyyy mm tmax tmin af rain sun
0 1883 1 6.3 1.7 6 122.1 ---
1 1883 2 8.0 2.8 2 69.8 ---
2 1883 3 4.8 -1.6 23 29.6 ---
3 1883 4 12.2 3.8 2 74.0 ---
4 1883 5 14.7 6.2 0 31.2 ---
In [15]:
rain_july, rain_november = df[df['mm']==7]['rain'], df[df['mm']==11]['rain']
def ecdf(data):
    return np.sort(data), np.arange(1, len(data)+1)/float(len(data))
In [16]:
for _ in range(50):
    # Generate permutation samples
    perm_sample_1, perm_sample_2 = permutation_sample(
                                    rain_july, rain_november)

    # Compute ECDFs
    x_1, y_1 = ecdf(perm_sample_1)
    x_2, y_2 = ecdf(perm_sample_2)

    # Plot ECDFs of permutation sample
    _ = plt.plot(x_1, y_1, marker='.', linestyle='none',
                 color='red', alpha=0.02)
    _ = plt.plot(x_2, y_2, marker='.', linestyle='none',
                 color='blue', alpha=0.02)

# Create and plot ECDFs from original data
x_1, y_1 = ecdf(rain_july)
x_2, y_2 = ecdf(rain_november)
_ = plt.plot(x_1, y_1, marker='.', linestyle='none', color='red')
_ = plt.plot(x_2, y_2, marker='.', linestyle='none', color='blue')

# Label axes, set margin, and show plot
_ = plt.xlabel('monthly rainfall (mm)')
_ = plt.ylabel('ECDF')

Generating permutation replicates

As discussed in the video, a permutation replicate is a single value of a statistic computed from a permutation sample. As the draw_bs_reps() function you wrote in chapter 2 is useful for you to generate bootstrap replicates, it is useful to have a similar function, draw_perm_reps(), to generate permutation replicates. You will write this useful function in this exercise.

The function has call signature draw_perm_reps(data_1, data_2, func, size=1). Importantly, func must be a function that takes two arrays as arguments. In most circumstances, func will be a function you write yourself.

In [73]:
def draw_perm_reps(data_1, data_2, func, size=1):
    """Generate multiple permutation replicates."""

    # Initialize array of replicates: perm_replicates
    perm_replicates = np.empty(size)

    for i in range(size):
        # Generate permutation sample
        perm_sample_1, perm_sample_2 = permutation_sample(data_1, data_2)

        # Compute the test statistic
        perm_replicates[i] = func(perm_sample_1, perm_sample_2)

    return perm_replicates

Look before you leap: EDA before hypothesis testing

Kleinteich and Gorb (Sci. Rep., 4, 5225, 2014) performed an interesting experiment with South American horned frogs. They held a plate connected to a force transducer, along with a bait fly, in front of them. They then measured the impact force and adhesive force of the frog's tongue when it struck the target.

Frog A is an adult and Frog B is a juvenile. The researchers measured the impact force of 20 strikes for each frog. In the next exercise, we will test the hypothesis that the two frogs have the same distribution of impact forces. But, remember, it is important to do EDA first! Let's make a bee swarm plot for the data. They are stored in a Pandas data frame, df, where column ID is the identity of the frog and column impact_force is the impact force in Newtons (N).

In [23]:
cat frog_tongue.csv|head -n 20

In [25]:
df = pd.read_csv('./frog_tongue.csv', skiprows=14)
date ID trial number impact force (mN) impact time (ms) impact force / body weight adhesive force (mN) time frog pulls on target (ms) adhesive force / body weight adhesive impulse (N-s) total contact area (mm2) contact area without mucus (mm2) contact area with mucus / contact area without mucus contact pressure (Pa) adhesive strength (Pa)
0 2013_02_26 I 3 1205 46 1.95 -785 884 1.27 -0.290 387 70 0.82 3117 -2030
1 2013_02_26 I 4 2527 44 4.08 -983 248 1.59 -0.181 101 94 0.07 24923 -9695
2 2013_03_01 I 1 1745 34 2.82 -850 211 1.37 -0.157 83 79 0.05 21020 -10239
3 2013_03_01 I 2 1556 41 2.51 -455 1025 0.74 -0.170 330 158 0.52 4718 -1381
4 2013_03_01 I 3 493 36 0.80 -974 499 1.57 -0.423 245 216 0.12 2012 -3975
In [35]:
# Make bee swarm plot
_ = sns.swarmplot(x='ID', y='impact force (mN)', data=df)

# Label axes
_ = plt.xlabel('frog')
_ = plt.ylabel('impact force (N)')

# Show the plot

Permutation test on frog data

The average strike force of Frog A was 0.71 Newtons (N), and that of Frog B was 0.42 N for a difference of 0.29 N. It is possible the frogs strike with the same force and this observed difference was by chance. You will compute the probability of getting at least a 0.29 N difference in mean strike force under the hypothesis that the distributions of strike forces for the two frogs are identical. We use a permutation test with a test statistic of the difference of means to test this hypothesis.

For your convenience, the data has been stored in the arrays force_a and force_b.

In [36]:
array(['I', 'II', 'III', 'IV'], dtype=object)
In [86]:
force_a, force_b = df[df['ID']=='II']['impact force (mN)'], df[df['ID']=='III']['impact force (mN)']
print force_a.shape, force_b.shape

force_a.mean(), force_b.mean()
(20,) (20,)
(707.35, 550.1)
In [85]:
def diff_of_means(data_1, data_2):
    """Difference in means of two arrays."""

    # The difference of means of data_1, data_2: diff
    diff = np.mean(data_1) - np.mean(data_2)

    return diff

# Compute difference of mean impact force from experiment: empirical_diff_means
empirical_diff_means = diff_of_means(force_a, force_b)

# Draw 10,000 permutation replicates: perm_replicates
perm_replicates = draw_perm_reps(force_a, force_b,
                                 diff_of_means, size=10000)

# Compute p-value: p
p = np.sum(perm_replicates >= empirical_diff_means) / float(len(perm_replicates))

# Print the result
print('p-value =', p)
('p-value =', 0.0587)

A one-sample bootstrap hypothesis test

Another juvenile frog was studied, Frog C, and you want to see if Frog B and Frog C have similar impact forces. Unfortunately, you do not have Frog C's impact forces available, but you know they have a mean of 0.55 N. Because you don't have the original data, you cannot do a permutation test, and you cannot assess the hypothesis that the forces from Frog B and Frog C come from the same distribution. You will therefore test another, less restrictive hypothesis: The mean strike force of Frog B is equal to that of Frog C.

To set up the bootstrap hypothesis test, you will take the mean as our test statistic. Remember, your goal is to calculate the probability of getting a mean impact force less than or equal to what was observed for Frog B if the hypothesis that the true mean of Frog B's impact forces is equal to that of Frog C is true. You first translate all of the data of Frog B such that the mean is 0.55 N. This involves adding the mean force of Frog C and subtracting the mean force of Frog B from each measurement of Frog B. This leaves other properties of Frog B's distribution, such as the variance, unchanged.

In [87]:
def bootstrap_replicate_1d(data, func):
    return func(np.random.choice(data, size=len(data)))

def draw_bs_reps(data, func, size=1):
    """Draw bootstrap replicates."""

    # Initialize array of replicates: bs_replicates
    bs_replicates = np.empty(size)

    # Generate replicates
    for i in range(size):
        bs_replicates[i] = bootstrap_replicate_1d(data, func)

    return bs_replicates
In [93]:
In [98]:
# Make an array of translated impact forces: translated_force_b
translated_force_b = force_b - np.mean(force_b) + 300

# Take bootstrap replicates of Frog B's translated impact forces: bs_replicates
bs_replicates = draw_bs_reps(translated_force_b, np.mean, 10000)

# Compute fraction of replicates that are less than the observed Frog B force: p
p = np.sum(bs_replicates <= np.mean(force_b)) / 10000.0

# Print the p-value
print('p = ', p)
('p = ', 1.0)

A bootstrap test for identical distributions

In the video, we looked at a one-sample test, but we can do two sample tests. We can even test the same hypothesis that we tested with a permutation test: that the Frog A and Frog B have identically distributed impact forces. To do this test on two arrays with n1 and n2 entries, we do a very similar procedure as a permutation test. We concatenate the arrays, generate a bootstrap sample from it, and take the first n1 entries of the bootstrap sample as belonging to the first data set and the last n2 as belonging to the second. We then compute the test statistic, e.g., the difference of means, to get a bootstrap replicate. The p-value is the number of bootstrap replicates for which the test statistic is less than what was observed.

Now, you will perform a bootstrap test of the hypothesis that Frog A and Frog B have identical distributions of impact forces using the difference of means test statistic.

The two arrays are available to you as force_a and force_b.

In [99]:
# Compute difference of mean impact force from experiment: empirical_diff_means
empirical_diff_means = diff_of_means(force_a, force_b)

# Concatenate forces: forces_concat
forces_concat = np.concatenate((force_a, force_b))

# Initialize bootstrap replicates: bs_replicates
bs_replicates = np.empty(10000)

for i in range(10000):
    # Generate bootstrap sample
    bs_sample = np.random.choice(forces_concat, size=len(forces_concat))

    # Compute replicate
    bs_replicates[i] = diff_of_means(bs_sample[:len(force_a)],

# Compute and print p-value: p
p = np.sum(bs_replicates >= empirical_diff_means) / float(len(bs_replicates))
print('p-value =', p)
('p-value =', 0.0575)

A two-sample bootstrap hypothesis test for difference of means.

You performed a one-sample bootstrap hypothesis test, which is impossible to do with permutation. Testing the hypothesis that two samples have the same distribution may be done with a bootstrap test, but a permutation test is preferred because it is more accurate (exact, in fact). But therein lies the limit of a permutation test; it is not very versatile. We now want to test the hypothesis that Frog A and Frog B have the same mean impact force, but not necessarily the same distribution. This, too, is impossible with a permutation test.

To do the two-sample bootstrap test, we shift both arrays to have the same mean, since we are simulating the hypothesis that their means are, in fact, equal. We then draw bootstrap samples out of the shifted arrays and compute the difference in means. This constitutes a bootstrap replicate, and we generate many of them. The p-value is the fraction of replicates with a difference in means greater than or equal to what was observed.

The objects forces_concat and empirical_diff_means are already in your namespace.

In [101]:
# Compute mean of all forces: mean_force
mean_force = np.mean(forces_concat)

# Generate shifted arrays
force_a_shifted = force_a - np.mean(force_a) + mean_force
force_b_shifted = force_b - np.mean(force_b) + mean_force

# Compute 10,000 bootstrap replicates from shifted arrays
bs_replicates_a = draw_bs_reps(force_a_shifted, np.mean, size=10000)
bs_replicates_b = draw_bs_reps(force_b_shifted, np.mean, size=10000)

# Get replicates of difference of means: bs_replicates
bs_replicates = bs_replicates_a - bs_replicates_b

# Compute and print p-value: p
p = np.sum(bs_replicates >= empirical_diff_means) / float(len(bs_replicates))
print('p-value =', p)
('p-value =', 0.058)