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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Chapter 9
Examples and Exercises from Think Stats, 2nd Edition
http
Step1:  Hypothesis testing
The following is a version of thinkstats2.HypothesisTest with just the essential methods
Step2:  And here's an example that uses it to compute the p-value of an experiment where we toss a coin 250 times and get 140 heads.
Step3:  The p-value turns out to be about 7%, which is considered on the border of statistical significance.
Step4:  Permutation test
To compute the p-value of an observed difference in means, we can assume that there is no difference between the groups and generate simulated results by shuffling the data.
Step5:  Here's an example where we test the observed difference in pregnancy length for first babies and others.
Step6:  The p-value is about 17%, which means it is plausible that the observed difference is just the result of random sampling, and might not be generally true in the population.
Step7:  Here's the distrubution of the test statistic (the difference in means) over many simulated samples
Step8:  Under the null hypothesis, we often see differences bigger than the observed difference.
Step9:  If the hypothesis under test is that first babies come late, the appropriate test statistic is the raw difference between first babies and others, rather than the absolute value of the difference.  In that case, the p-value is smaller, because we are testing a more specific hypothesis.
Step10:  But in this example, the result is still not statistically significant.
Difference in standard deviation
In this framework, it is easy to use other test statistics.  For example, if we think the variance for first babies might be higher, we can run this test
Step11:  But that's not statistically significant either.
Testing correlation
To check whether an observed correlation is statistically significant, we can run a permutation test with a different test statistic.
Step12:  Here's an example testing the correlation between birth weight and mother's age.
Step13:  The reported p-value is 0, which means that in 1000 trials we didn't see a correlation, under the null hypothesis, that exceeded the observed correlation.  That means that the p-value is probably smaller than $1/1000$, but it is not actually 0.
To get a sense of how unexpected the observed value is under the null hypothesis, we can compare the actual correlation to the largest value we saw in the simulations.
Step14:  Testing proportions
Here's an example that tests whether the outcome of a rolling a six-sided die is suspicious, where the test statistic is the total absolute difference between the observed outcomes and the expected long-term averages.
Step15:  Here's an example using the data from the book
Step16:  The observed deviance from the expected values is not statistically significant.
By convention, it is more common to test data like this using the chi-squared statistic
Step17:  Using this test, we get a smaller p-value
Step18:  Taking this result at face value, we might consider the data statistically significant, but considering the results of both tests, I would not draw any strong conclusions.
Chi-square test of pregnancy length
Step19:  If we specifically test the deviations of first babies and others from the expected number of births in each week of pregnancy, the results are statistically significant with a very small p-value.  But at this point we have run so many tests, we should not be surprised to find at least one that seems significant.
Step21:  Power
Here's the function that estimates the probability of a non-significant p-value even is there really is a difference between the groups.
Step23:  In this example, the false negative rate is 70%, which means that the power of the test (probability of statistical significance if the actual difference is 0.078 weeks) is only 30%.
Exercises
Exercise
Step27:  Exercise
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Python Code:
from os.path import basename, exists
def download(url):
    filename = basename(url)
    if not exists(filename):
        from urllib.request import urlretrieve
        local, _ = urlretrieve(url, filename)
        print("Downloaded " + local)
download("https://github.com/AllenDowney/ThinkStats2/raw/master/code/thinkstats2.py")
download("https://github.com/AllenDowney/ThinkStats2/raw/master/code/thinkplot.py")
import numpy as np
import random
import thinkstats2
import thinkplot
Explanation: Chapter 9
Examples and Exercises from Think Stats, 2nd Edition
http://thinkstats2.com
Copyright 2016 Allen B. Downey
MIT License: https://opensource.org/licenses/MIT
End of explanation
class HypothesisTest(object):
    def __init__(self, data):
        self.data = data
        self.MakeModel()
        self.actual = self.TestStatistic(data)
    def PValue(self, iters=1000):
        self.test_stats = [self.TestStatistic(self.RunModel()) 
                           for _ in range(iters)]
        count = sum(1 for x in self.test_stats if x >= self.actual)
        return count / iters
    def TestStatistic(self, data):
        raise UnimplementedMethodException()
    def MakeModel(self):
        pass
    def RunModel(self):
        raise UnimplementedMethodException()
Explanation: Hypothesis testing
The following is a version of thinkstats2.HypothesisTest with just the essential methods:
End of explanation
class CoinTest(HypothesisTest):
    def TestStatistic(self, data):
        heads, tails = data
        test_stat = abs(heads - tails)
        return test_stat
    def RunModel(self):
        heads, tails = self.data
        n = heads + tails
        sample = [random.choice('HT') for _ in range(n)]
        hist = thinkstats2.Hist(sample)
        data = hist['H'], hist['T']
        return data
Explanation: And here's an example that uses it to compute the p-value of an experiment where we toss a coin 250 times and get 140 heads.
End of explanation
ct = CoinTest((140, 110))
pvalue = ct.PValue()
pvalue
Explanation: The p-value turns out to be about 7%, which is considered on the border of statistical significance.
End of explanation
class DiffMeansPermute(thinkstats2.HypothesisTest):
    def TestStatistic(self, data):
        group1, group2 = data
        test_stat = abs(group1.mean() - group2.mean())
        return test_stat
    def MakeModel(self):
        group1, group2 = self.data
        self.n, self.m = len(group1), len(group2)
        self.pool = np.hstack((group1, group2))
    def RunModel(self):
        np.random.shuffle(self.pool)
        data = self.pool[:self.n], self.pool[self.n:]
        return data
Explanation: Permutation test
To compute the p-value of an observed difference in means, we can assume that there is no difference between the groups and generate simulated results by shuffling the data.
End of explanation
download("https://github.com/AllenDowney/ThinkStats2/raw/master/code/nsfg.py")
download("https://github.com/AllenDowney/ThinkStats2/raw/master/code/first.py")
download("https://github.com/AllenDowney/ThinkStats2/raw/master/code/2002FemPreg.dct")
download(
    "https://github.com/AllenDowney/ThinkStats2/raw/master/code/2002FemPreg.dat.gz"
)
import first
live, firsts, others = first.MakeFrames()
data = firsts.prglngth.values, others.prglngth.values
Explanation: Here's an example where we test the observed difference in pregnancy length for first babies and others.
End of explanation
ht = DiffMeansPermute(data)
pvalue = ht.PValue()
pvalue
Explanation: The p-value is about 17%, which means it is plausible that the observed difference is just the result of random sampling, and might not be generally true in the population.
End of explanation
ht.PlotCdf()
thinkplot.Config(xlabel='test statistic',
                   ylabel='CDF')
Explanation: Here's the distrubution of the test statistic (the difference in means) over many simulated samples:
End of explanation
class DiffMeansOneSided(DiffMeansPermute):
    def TestStatistic(self, data):
        group1, group2 = data
        test_stat = group1.mean() - group2.mean()
        return test_stat
Explanation: Under the null hypothesis, we often see differences bigger than the observed difference.
End of explanation
ht = DiffMeansOneSided(data)
pvalue = ht.PValue()
pvalue
Explanation: If the hypothesis under test is that first babies come late, the appropriate test statistic is the raw difference between first babies and others, rather than the absolute value of the difference.  In that case, the p-value is smaller, because we are testing a more specific hypothesis.
End of explanation
class DiffStdPermute(DiffMeansPermute):
    def TestStatistic(self, data):
        group1, group2 = data
        test_stat = group1.std() - group2.std()
        return test_stat
ht = DiffStdPermute(data)
pvalue = ht.PValue()
pvalue
Explanation: But in this example, the result is still not statistically significant.
Difference in standard deviation
In this framework, it is easy to use other test statistics.  For example, if we think the variance for first babies might be higher, we can run this test:
End of explanation
class CorrelationPermute(thinkstats2.HypothesisTest):
    def TestStatistic(self, data):
        xs, ys = data
        test_stat = abs(thinkstats2.Corr(xs, ys))
        return test_stat
    def RunModel(self):
        xs, ys = self.data
        xs = np.random.permutation(xs)
        return xs, ys
Explanation: But that's not statistically significant either.
Testing correlation
To check whether an observed correlation is statistically significant, we can run a permutation test with a different test statistic.
End of explanation
cleaned = live.dropna(subset=['agepreg', 'totalwgt_lb'])
data = cleaned.agepreg.values, cleaned.totalwgt_lb.values
ht = CorrelationPermute(data)
pvalue = ht.PValue()
pvalue
Explanation: Here's an example testing the correlation between birth weight and mother's age.
End of explanation
ht.actual, ht.MaxTestStat()
Explanation: The reported p-value is 0, which means that in 1000 trials we didn't see a correlation, under the null hypothesis, that exceeded the observed correlation.  That means that the p-value is probably smaller than $1/1000$, but it is not actually 0.
To get a sense of how unexpected the observed value is under the null hypothesis, we can compare the actual correlation to the largest value we saw in the simulations.
End of explanation
class DiceTest(thinkstats2.HypothesisTest):
    def TestStatistic(self, data):
        observed = data
        n = sum(observed)
        expected = np.ones(6) * n / 6
        test_stat = sum(abs(observed - expected))
        return test_stat
    def RunModel(self):
        n = sum(self.data)
        values = [1, 2, 3, 4, 5, 6]
        rolls = np.random.choice(values, n, replace=True)
        hist = thinkstats2.Hist(rolls)
        freqs = hist.Freqs(values)
        return freqs
Explanation: Testing proportions
Here's an example that tests whether the outcome of a rolling a six-sided die is suspicious, where the test statistic is the total absolute difference between the observed outcomes and the expected long-term averages.
End of explanation
data = [8, 9, 19, 5, 8, 11]
dt = DiceTest(data)
pvalue = dt.PValue(iters=10000)
pvalue
Explanation: Here's an example using the data from the book:
End of explanation
class DiceChiTest(DiceTest):
    def TestStatistic(self, data):
        observed = data
        n = sum(observed)
        expected = np.ones(6) * n / 6
        test_stat = sum((observed - expected)**2 / expected)
        return test_stat
Explanation: The observed deviance from the expected values is not statistically significant.
By convention, it is more common to test data like this using the chi-squared statistic:
End of explanation
dt = DiceChiTest(data)
pvalue = dt.PValue(iters=10000)
pvalue
Explanation: Using this test, we get a smaller p-value:
End of explanation
class PregLengthTest(thinkstats2.HypothesisTest):
    def MakeModel(self):
        firsts, others = self.data
        self.n = len(firsts)
        self.pool = np.hstack((firsts, others))
        pmf = thinkstats2.Pmf(self.pool)
        self.values = range(35, 44)
        self.expected_probs = np.array(pmf.Probs(self.values))
    def RunModel(self):
        np.random.shuffle(self.pool)
        data = self.pool[:self.n], self.pool[self.n:]
        return data
    
    def TestStatistic(self, data):
        firsts, others = data
        stat = self.ChiSquared(firsts) + self.ChiSquared(others)
        return stat
    def ChiSquared(self, lengths):
        hist = thinkstats2.Hist(lengths)
        observed = np.array(hist.Freqs(self.values))
        expected = self.expected_probs * len(lengths)
        stat = sum((observed - expected)**2 / expected)
        return stat
Explanation: Taking this result at face value, we might consider the data statistically significant, but considering the results of both tests, I would not draw any strong conclusions.
Chi-square test of pregnancy length
End of explanation
data = firsts.prglngth.values, others.prglngth.values
ht = PregLengthTest(data)
p_value = ht.PValue()
print('p-value =', p_value)
print('actual =', ht.actual)
print('ts max =', ht.MaxTestStat())
Explanation: If we specifically test the deviations of first babies and others from the expected number of births in each week of pregnancy, the results are statistically significant with a very small p-value.  But at this point we have run so many tests, we should not be surprised to find at least one that seems significant.
End of explanation
def FalseNegRate(data, num_runs=1000):
    Computes the chance of a false negative based on resampling.
    data: pair of sequences
    num_runs: how many experiments to simulate
    returns: float false negative rate
    
    group1, group2 = data
    count = 0
    for i in range(num_runs):
        sample1 = thinkstats2.Resample(group1)
        sample2 = thinkstats2.Resample(group2)
        ht = DiffMeansPermute((sample1, sample2))
        p_value = ht.PValue(iters=101)
        if p_value > 0.05:
            count += 1
    return count / num_runs
neg_rate = FalseNegRate(data)
neg_rate
Explanation: Power
Here's the function that estimates the probability of a non-significant p-value even is there really is a difference between the groups.
End of explanation
# Solution
def RunTests(live, iters=1000):
    Runs the tests from Chapter 9 with a subset of the data.
    live: DataFrame
    iters: how many iterations to run
    
    n = len(live)
    firsts = live[live.birthord == 1]
    others = live[live.birthord != 1]
    # compare pregnancy lengths
    data = firsts.prglngth.values, others.prglngth.values
    ht = DiffMeansPermute(data)
    p1 = ht.PValue(iters=iters)
    data = (firsts.totalwgt_lb.dropna().values,
            others.totalwgt_lb.dropna().values)
    ht = DiffMeansPermute(data)
    p2 = ht.PValue(iters=iters)
    # test correlation
    live2 = live.dropna(subset=['agepreg', 'totalwgt_lb'])
    data = live2.agepreg.values, live2.totalwgt_lb.values
    ht = CorrelationPermute(data)
    p3 = ht.PValue(iters=iters)
    # compare pregnancy lengths (chi-squared)
    data = firsts.prglngth.values, others.prglngth.values
    ht = PregLengthTest(data)
    p4 = ht.PValue(iters=iters)
    print('%d\t%0.2f\t%0.2f\t%0.2f\t%0.2f' % (n, p1, p2, p3, p4))
# Solution
n = len(live)
for _ in range(7):
    sample = thinkstats2.SampleRows(live, n)
    RunTests(sample)
    n //= 2
# Solution
# My results:
# test1: difference in mean pregnancy length
# test2: difference in mean birth weight
# test3: correlation of mother's age and birth weight
# test4: chi-square test of pregnancy length
# n       test1   test2   test2   test4
# 9148	0.16	0.00	0.00	0.00
# 4574	0.10	0.01	0.00	0.00
# 2287	0.25	0.06	0.00	0.00
# 1143	0.24	0.03	0.39	0.03
# 571	0.81	0.00	0.04	0.04
# 285	0.57	0.41	0.48	0.83
# 142	0.45	0.08	0.60	0.04
# Conclusion: As expected, tests that are positive with large sample
# sizes become negative as we take away data.  But the pattern is
# erratic, with some positive tests even at small sample sizes.
Explanation: In this example, the false negative rate is 70%, which means that the power of the test (probability of statistical significance if the actual difference is 0.078 weeks) is only 30%.
Exercises
Exercise: As sample size increases, the power of a hypothesis test increases, which means it is more likely to be positive if the effect is real. Conversely, as sample size decreases, the test is less likely to be positive even if the effect is real.
To investigate this behavior, run the tests in this chapter with different subsets of the NSFG data. You can use thinkstats2.SampleRows to select a random subset of the rows in a DataFrame.
What happens to the p-values of these tests as sample size decreases? What is the smallest sample size that yields a positive test?
End of explanation
# Solution
class DiffMeansResample(DiffMeansPermute):
    Tests a difference in means using resampling.
    
    def RunModel(self):
        Run the model of the null hypothesis.
        returns: simulated data
        
        group1 = np.random.choice(self.pool, self.n, replace=True)
        group2 = np.random.choice(self.pool, self.m, replace=True)
        return group1, group2
  
# Solution
def RunResampleTest(firsts, others):
    Tests differences in means by resampling.
    firsts: DataFrame
    others: DataFrame
    
    data = firsts.prglngth.values, others.prglngth.values
    ht = DiffMeansResample(data)
    p_value = ht.PValue(iters=10000)
    print('\ndiff means resample preglength')
    print('p-value =', p_value)
    print('actual =', ht.actual)
    print('ts max =', ht.MaxTestStat())
    data = (firsts.totalwgt_lb.dropna().values,
            others.totalwgt_lb.dropna().values)
    ht = DiffMeansPermute(data)
    p_value = ht.PValue(iters=10000)
    print('\ndiff means resample birthweight')
    print('p-value =', p_value)
    print('actual =', ht.actual)
    print('ts max =', ht.MaxTestStat())
# Solution
RunResampleTest(firsts, others)
# Solution
# Conclusions: Using resampling instead of permutation has very
# little effect on the results.
# The two models are based on slightly difference assumptions, and in
# this example there is no compelling reason to choose one or the other.
# But in general p-values depend on the choice of the null hypothesis;
# different models can yield very different results.
Explanation: Exercise: In Section 9.3, we simulated the null hypothesis by permutation; that is, we treated the observed values as if they represented the entire population, and randomly assigned the members of the population to the two groups.
An alternative is to use the sample to estimate the distribution for the population, then draw a random sample from that distribution. This process is called resampling. There are several ways to implement resampling, but one of the simplest is to draw a sample with replacement from the observed values, as in Section 9.10.
Write a class named DiffMeansResample that inherits from DiffMeansPermute and overrides RunModel to implement resampling, rather than permutation.
Use this model to test the differences in pregnancy length and birth weight. How much does the model affect the results?
End of explanation
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