```
In [1]:
```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
import first
# seed the random number generator so we all get the same results
numpy.random.seed(19)
# some nicer colors from http://colorbrewer2.org/
COLOR1 = '#7fc97f'
COLOR2 = '#beaed4'
COLOR3 = '#fdc086'
COLOR4 = '#ffff99'
COLOR5 = '#386cb0'
%matplotlib inline

*Think Stats* is first babies compared with others. The `first`

module provides code to read the data into three pandas Dataframes.

```
In [2]:
```live, firsts, others = first.MakeFrames()

```
In [3]:
```def TestStatistic(data):
group1, group2 = data
test_stat = abs(group1.mean() - group2.mean())
return test_stat

```
In [4]:
```group1 = firsts.prglngth
group2 = others.prglngth

The actual difference in the means is 0.078 weeks, which is only 13 hours.

```
In [5]:
```actual = TestStatistic((group1, group2))
actual

```
Out[5]:
```

```
In [6]:
```n, m = len(group1), len(group2)
pool = numpy.hstack((group1, group2))

```
In [7]:
```def RunModel():
numpy.random.shuffle(pool)
data = pool[:n], pool[n:]
return data

The result of running the model is two NumPy arrays with the shuffled pregnancy lengths:

```
In [8]:
```RunModel()

```
Out[8]:
```

Then we compute the same test statistic using the simulated data:

```
In [9]:
```TestStatistic(RunModel())

```
Out[9]:
```

```
In [10]:
```test_stats = numpy.array([TestStatistic(RunModel()) for i in range(1000)])
test_stats.shape

```
Out[10]:
```

```
In [11]:
```def VertLine(x):
"""Draws a vertical line at x."""
pyplot.plot([x, x], [0, 300], linewidth=3, color='0.8')
VertLine(actual)
pyplot.hist(test_stats, color=COLOR5)
pyplot.xlabel('difference in means')
pyplot.ylabel('count')
None

```
```

```
In [12]:
```pvalue = sum(test_stats >= actual) / len(test_stats)
pvalue

```
Out[12]:
```

In this case the result is about 15%, which means that even if there is no difference between the groups, it is plausible that we could see a sample difference as big as 0.078 weeks.

We conclude that the apparent effect might be due to chance, so we are not confident that it would appear in the general population, or in another sample from the same population.

```
In [13]:
```class HypothesisTest(object):
"""Represents a hypothesis test."""
def __init__(self, data):
"""Initializes.
data: data in whatever form is relevant
"""
self.data = data
self.MakeModel()
self.actual = self.TestStatistic(data)
self.test_stats = None
def PValue(self, iters=1000):
"""Computes the distribution of the test statistic and p-value.
iters: number of iterations
returns: float p-value
"""
self.test_stats = numpy.array([self.TestStatistic(self.RunModel())
for _ in range(iters)])
count = sum(self.test_stats >= self.actual)
return count / iters
def MaxTestStat(self):
"""Returns the largest test statistic seen during simulations.
"""
return max(self.test_stats)
def PlotHist(self, label=None):
"""Draws a Cdf with vertical lines at the observed test stat.
"""
def VertLine(x):
"""Draws a vertical line at x."""
pyplot.plot([x, x], [0, max(ys)], linewidth=3, color='0.8')
ys, xs, patches = pyplot.hist(ht.test_stats, color=COLOR4)
VertLine(self.actual)
pyplot.xlabel('test statistic')
pyplot.ylabel('count')
def TestStatistic(self, data):
"""Computes the test statistic.
data: data in whatever form is relevant
"""
raise UnimplementedMethodException()
def MakeModel(self):
"""Build a model of the null hypothesis.
"""
pass
def RunModel(self):
"""Run the model of the null hypothesis.
returns: simulated data
"""
raise UnimplementedMethodException()

`HypothesisTest`

is an abstract parent class that encodes the template. Child classes fill in the missing methods. For example, here's the test from the previous section.

```
In [14]:
```class DiffMeansPermute(HypothesisTest):
"""Tests a difference in means by permutation."""
def TestStatistic(self, data):
"""Computes the test statistic.
data: data in whatever form is relevant
"""
group1, group2 = data
test_stat = abs(group1.mean() - group2.mean())
return test_stat
def MakeModel(self):
"""Build a model of the null hypothesis.
"""
group1, group2 = self.data
self.n, self.m = len(group1), len(group2)
self.pool = numpy.hstack((group1, group2))
def RunModel(self):
"""Run the model of the null hypothesis.
returns: simulated data
"""
numpy.random.shuffle(self.pool)
data = self.pool[:self.n], self.pool[self.n:]
return data

Now we can run the test by instantiating a DiffMeansPermute object:

```
In [15]:
```data = (firsts.prglngth, others.prglngth)
ht = DiffMeansPermute(data)
p_value = ht.PValue(iters=1000)
print('\nmeans permute pregnancy length')
print('p-value =', p_value)
print('actual =', ht.actual)
print('ts max =', ht.MaxTestStat())

```
```

And we can plot the sampling distribution of the test statistic under the null hypothesis.

```
In [16]:
```ht.PlotHist()

```
```

`DiffStdPermute`

that extends `DiffMeansPermute`

and overrides `TestStatistic`

to compute the difference in standard deviations.

```
In [17]:
```class DiffStdPermute(DiffMeansPermute):
"""Tests a difference in means by permutation."""
def TestStatistic(self, data):
"""Computes the test statistic.
data: data in whatever form is relevant
"""
group1, group2 = data
test_stat = abs(group1.std() - group2.std())
return test_stat
data = (firsts.prglngth, others.prglngth)
ht = DiffStdPermute(data)
p_value = ht.PValue(iters=1000)
print('\nstd permute pregnancy length')
print('p-value =', p_value)
print('actual =', ht.actual)
print('ts max =', ht.MaxTestStat())

```
```

```
In [18]:
```data = (firsts.totalwgt_lb.dropna(), others.totalwgt_lb.dropna())
ht = DiffMeansPermute(data)
p_value = ht.PValue(iters=1000)
print('\nmeans permute birthweight')
print('p-value =', p_value)
print('actual =', ht.actual)
print('ts max =', ht.MaxTestStat())

```
```

In this case, after 1000 attempts, we never see a sample difference as big as the observed difference, so we conclude that the apparent effect is unlikely under the null hypothesis. Under normal circumstances, we can also make the inference that the apparent effect is unlikely to be caused by random sampling.

One final note: in this case I would report that the p-value is less than 1/1000 or 0.001. I would not report that p=0, because the apparent effect is not impossible under the null hypothesis; just unlikely.

```
In [18]:
```