%matplotlib inline
from __future__ import print_function, division

import numpy
import scipy.stats

import matplotlib.pyplot as pyplot

from ipywidgets import interact, interactive, fixed
import ipywidgets as widgets

# seed the random number generator so we all get the same results
numpy.random.seed(17)

# some nice colors from http://colorbrewer2.org/
COLOR1 = '#7fc97f'
COLOR2 = '#beaed4'
COLOR3 = '#fdc086'
COLOR4 = '#ffff99'
COLOR5 = '#386cb0'



"""
Explanation: Effect Size
Examples and exercises for a tutorial on statistical inference.
Copyright 2016 Allen Downey
License: Creative Commons Attribution 4.0 International
End of explanation
"""


mu1, sig1 = 178, 7.7
male_height = scipy.stats.norm(mu1, sig1)

mu2, sig2 = 163, 7.3
female_height = scipy.stats.norm(mu2, sig2)

"""
Explanation: Part One
To explore statistics that quantify effect size, we'll look at the difference in height between men and women.  I used data from the Behavioral Risk Factor Surveillance System (BRFSS) to estimate the mean and standard deviation of height in cm for adult women and men in the U.S.
I'll use scipy.stats.norm to represent the distributions.  The result is an rv object (which stands for random variable).
End of explanation
"""


def eval_pdf(rv, num=4):
    mean, std = rv.mean(), rv.std()
    xs = numpy.linspace(mean - num*std, mean + num*std, 100)
    ys = rv.pdf(xs)
    return xs, ys

"""
Explanation: The following function evaluates the normal (Gaussian) probability density function (PDF) within 4 standard deviations of the mean.  It takes and rv object and returns a pair of NumPy arrays.
End of explanation
"""


xs, ys = eval_pdf(male_height)
pyplot.plot(xs, ys, label='male', linewidth=4, color=COLOR2)

xs, ys = eval_pdf(female_height)
pyplot.plot(xs, ys, label='female', linewidth=4, color=COLOR3)
pyplot.xlabel('height (cm)')
None

"""
Explanation: Here's what the two distributions look like.
End of explanation
"""


male_sample = male_height.rvs(1000)

female_sample = female_height.rvs(1000)

"""
Explanation: Let's assume for now that those are the true distributions for the population.
I'll use rvs to generate random samples from the population distributions.  Note that these are totally random, totally representative samples, with no measurement error!
End of explanation
"""


mean1, std1 = male_sample.mean(), male_sample.std()
mean1, std1

"""
Explanation: Both samples are NumPy arrays.  Now we can compute sample statistics like the mean and standard deviation.
End of explanation
"""


mean2, std2 = female_sample.mean(), female_sample.std()
mean2, std2

"""
Explanation: The sample mean is close to the population mean, but not exact, as expected.
End of explanation
"""


difference_in_means = male_sample.mean() - female_sample.mean()
difference_in_means # in cm

"""
Explanation: And the results are similar for the female sample.
Now, there are many ways to describe the magnitude of the difference between these distributions.  An obvious one is the difference in the means:
End of explanation
"""


# Solution goes here
(male_sample.mean()/ len(male_sample)) - (female_sample.mean() / len(female_sample))

"""
Explanation: On average, men are 14--15 centimeters taller.  For some applications, that would be a good way to describe the difference, but there are a few problems:


Without knowing more about the distributions (like the standard deviations) it's hard to interpret whether a difference like 15 cm is a lot or not.


The magnitude of the difference depends on the units of measure, making it hard to compare across different studies.


There are a number of ways to quantify the difference between distributions.  A simple option is to express the difference as a percentage of the mean.
Exercise 1: what is the relative difference in means, expressed as a percentage?
End of explanation
"""


simple_thresh = (mean1 + mean2) / 2
simple_thresh

"""
Explanation: STOP HERE: We'll regroup and discuss before you move on.
Part Two
An alternative way to express the difference between distributions is to see how much they overlap.  To define overlap, we choose a threshold between the two means.  The simple threshold is the midpoint between the means:
End of explanation
"""


thresh = (std1 * mean2 + std2 * mean1) / (std1 + std2)
thresh

"""
Explanation: A better, but slightly more complicated threshold is the place where the PDFs cross.
End of explanation
"""


male_below_thresh = sum(male_sample < thresh)
male_below_thresh

"""
Explanation: In this example, there's not much difference between the two thresholds.
Now we can count how many men are below the threshold:
End of explanation
"""


female_above_thresh = sum(female_sample > thresh)
female_above_thresh

"""
Explanation: And how many women are above it:
End of explanation
"""


overlap = male_below_thresh / len(male_sample) + female_above_thresh / len(female_sample)
overlap

"""
Explanation: The "overlap" is the total area under the curves that ends up on the wrong side of the threshold.
End of explanation
"""


misclassification_rate = overlap / 2
misclassification_rate

"""
Explanation: Or in more practical terms, you might report the fraction of people who would be misclassified if you tried to use height to guess sex:
End of explanation
"""


# Solution goes here
sum(male_sample > female_sample) / len(male_sample)

"""
Explanation: Another way to quantify the difference between distributions is what's called "probability of superiority", which is a problematic term, but in this context it's the probability that a randomly-chosen man is taller than a randomly-chosen woman.
Exercise 2:  Suppose I choose a man and a woman at random.  What is the probability that the man is taller?
End of explanation
"""


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

"""
Explanation: Overlap (or misclassification rate) and "probability of superiority" have two good properties:


As probabilities, they don't depend on units of measure, so they are comparable between studies.


They are expressed in operational terms, so a reader has a sense of what practical effect the difference makes.


Cohen's d
There is one other common way to express the difference between distributions.  Cohen's $d$ is the difference in means, standardized by dividing by the standard deviation.  Here's a function that computes it:
End of explanation
"""


CohenEffectSize(male_sample, female_sample)

"""
Explanation: Computing the denominator is a little complicated; in fact, people have proposed several ways to do it.  This implementation uses the "pooled standard deviation", which is a weighted average of the standard deviations of the two groups.
And here's the result for the difference in height between men and women.
End of explanation
"""


def overlap_superiority(control, treatment, n=1000):
    """Estimates overlap and superiority based on a sample.
    
    control: scipy.stats rv object
    treatment: scipy.stats rv object
    n: sample size
    """
    control_sample = control.rvs(n)
    treatment_sample = treatment.rvs(n)
    thresh = (control.mean() + treatment.mean()) / 2
    
    control_above = sum(control_sample > thresh)
    treatment_below = sum(treatment_sample < thresh)
    overlap = (control_above + treatment_below) / n
    
    superiority = sum(x > y for x, y in zip(treatment_sample, control_sample)) / n
    return overlap, superiority

"""
Explanation: Most people don't have a good sense of how big $d=1.9$ is, so let's make a visualization to get calibrated.
Here's a function that encapsulates the code we already saw for computing overlap and probability of superiority.
End of explanation
"""


def plot_pdfs(cohen_d=2):
    """Plot PDFs for distributions that differ by some number of stds.
    
    cohen_d: number of standard deviations between the means
    """
    control = scipy.stats.norm(0, 1)
    treatment = scipy.stats.norm(cohen_d, 1)
    xs, ys = eval_pdf(control)
    pyplot.fill_between(xs, ys, label='control', color=COLOR3, alpha=0.7)

    xs, ys = eval_pdf(treatment)
    pyplot.fill_between(xs, ys, label='treatment', color=COLOR2, alpha=0.7)
    
    o, s = overlap_superiority(control, treatment)
    print('overlap', o)
    print('superiority', s)

"""
Explanation: Here's the function that takes Cohen's $d$, plots normal distributions with the given effect size, and prints their overlap and superiority.
End of explanation
"""


plot_pdfs(2)

"""
Explanation: Here's an example that demonstrates the function:
End of explanation
"""


slider = widgets.FloatSlider(min=0, max=4, value=2)
interact(plot_pdfs, cohen_d=slider)
None

"""
Explanation: And an interactive widget you can use to visualize what different values of $d$ mean:
End of explanation
"""


