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Z-критерий для двух долей





In [1]:



    


import numpy as np
import pandas as pd

import scipy
from statsmodels.stats.weightstats import *
from statsmodels.stats.proportion import proportion_confint

Загрузка данных





In [2]:



    


data = pd.read_csv('banner_click_stat.txt', header = None, sep = '\t')
data.columns = ['banner_a', 'banner_b']



    











In [3]:



    


data.head()



    


















    
Out[3]:










  
    

      
      banner_a
      banner_b
    

  
  
    

      0
      0
      0
    

    

      1
      1
      1
    

    

      2
      0
      0
    

    

      3
      0
      0
    

    

      4
      0
      0
    

  




















In [4]:



    


data.describe()



    


















    
Out[4]:










  
    

      
      banner_a
      banner_b
    

  
  
    

      count
      1000.000000
      1000.000000
    

    

      mean
      0.037000
      0.053000
    

    

      std
      0.188856
      0.224146
    

    

      min
      0.000000
      0.000000
    

    

      25%
      0.000000
      0.000000
    

    

      50%
      0.000000
      0.000000
    

    

      75%
      0.000000
      0.000000
    

    

      max
      1.000000
      1.000000

Интервальные оценки долей

$$\frac1{ 1 + \frac{z^2}{n} } \left( \hat{p} + \frac{z^2}{2n} \pm z \sqrt{ \frac{ \hat{p}\left(1-\hat{p}\right)}{n} + \frac{z^2}{4n^2} } \right), \;\; z \equiv z_{1-\frac{\alpha}{2}}$$





In [5]:



    


conf_interval_banner_a = proportion_confint(sum(data.banner_a), 
                                            data.shape[0],
                                            method = 'wilson')
conf_interval_banner_b = proportion_confint(sum(data.banner_b), 
                                            data.shape[0],
                                            method = 'wilson')



    











In [6]:



    


print '95%% confidence interval for a click probability, banner a: [%f, %f]' % conf_interval_banner_a
print '95%% confidence interval for a click probability, banner b [%f, %f]' % conf_interval_banner_b



    


















    






interval for banner a [0.026961, 0.050582]
interval for banner b [0.040747, 0.068675]

Z-критерий для разности долей (независимые выборки)

$X_1$ $X_2$
1 a b
0 c d
$\sum$ $n_1$ $n_2$
$$ \hat{p}_1 = \frac{a}{n_1}$$$$ \hat{p}_2 = \frac{b}{n_2}$$$$\text{Доверительный интервал для }p_1 - p_2\colon \;\; \hat{p}_1 - \hat{p}_2 \pm z_{1-\frac{\alpha}{2}}\sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$$$$Z-статистика: Z({X_1, X_2}) = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{P(1 - P)(\frac{1}{n_1} + \frac{1}{n_2})}}$$$$P = \frac{\hat{p}_1{n_1} + \hat{p}_2{n_2}}{{n_1} + {n_2}} $$ 




In [7]:



    


def proportions_diff_confint_ind(sample1, sample2, alpha = 0.05):    
    z = scipy.stats.norm.ppf(1 - alpha / 2.)
    
    p1 = float(sum(sample1)) / len(sample1)
    p2 = float(sum(sample2)) / len(sample2)
    
    left_boundary = (p1 - p2) - z * np.sqrt(p1 * (1 - p1)/ len(sample1) + p2 * (1 - p2)/ len(sample2))
    right_boundary = (p1 - p2) + z * np.sqrt(p1 * (1 - p1)/ len(sample1) + p2 * (1 - p2)/ len(sample2))
    
    return (left_boundary, right_boundary)



    











In [8]:



    


def proportions_diff_z_stat_ind(sample1, sample2):
    n1 = len(sample1)
    n2 = len(sample2)
    
    p1 = float(sum(sample1)) / n1
    p2 = float(sum(sample2)) / n2 
    P = float(p1*n1 + p2*n2) / (n1 + n2)
    
    return (p1 - p2) / np.sqrt(P * (1 - P) * (1. / n1 + 1. / n2))



    











In [9]:



    


def proportions_diff_z_test(z_stat, alternative = 'two-sided'):
    if alternative not in ('two-sided', 'less', 'greater'):
        raise ValueError("alternative not recognized\n"
                         "should be 'two-sided', 'less' or 'greater'")
    
    if alternative == 'two-sided':
        return 2 * (1 - scipy.stats.norm.cdf(np.abs(z_stat)))
    
    if alternative == 'less':
        return scipy.stats.norm.cdf(z_stat)

    if alternative == 'greater':
        return 1 - scipy.stats.norm.cdf(z_stat)



    











In [10]:



    


print "95%% confidence interval for a difference between proportions: [%f, %f]" %\
      proportions_diff_confint_ind(data.banner_a, data.banner_b)



    


















    






confidence interval: [-0.034157, 0.002157]

















In [11]:



    


print "p-value: %f" % proportions_diff_z_test(proportions_diff_z_stat_ind(data.banner_a, data.banner_b))



    


















    






p_value: 0.084379

















In [12]:



    


print "p-value: %f" % proportions_diff_z_test(proportions_diff_z_stat_ind(data.banner_a, data.banner_b), 'less')



    


















    






p_value: 0.042189

Z-критерий для разности долей (связанные выборки)

$X_1$ \ $X_2$ 1 0 $\sum$
1 e f e + f
0 g h g + h
$\sum$ e + g f + h n
$$ \hat{p}_1 = \frac{e + f}{n}$$$$ \hat{p}_2 = \frac{e + g}{n}$$$$ \hat{p}_1 - \hat{p}_2 = \frac{f - g}{n}$$$$\text{Доверительный интервал для }p_1 - p_2\colon \;\; \frac{f - g}{n} \pm z_{1-\frac{\alpha}{2}}\sqrt{\frac{f + g}{n^2} - \frac{(f - g)^2}{n^3}}$$$$Z-статистика: Z({X_1, X_2}) = \frac{f - g}{\sqrt{f + g - \frac{(f-g)^2}{n}}}$$ 




In [13]:



    


def proportions_diff_confint_rel(sample1, sample2, alpha = 0.05):
    z = scipy.stats.norm.ppf(1 - alpha / 2.)
    sample = zip(sample1, sample2)
    n = len(sample)
        
    f = sum([1 if (x[0] == 1 and x[1] == 0) else 0 for x in sample])
    g = sum([1 if (x[0] == 0 and x[1] == 1) else 0 for x in sample])
    
    left_boundary = float(f - g) / n  - z * np.sqrt(float((f + g)) / n**2 - float((f - g)**2) / n**3)
    right_boundary = float(f - g) / n  + z * np.sqrt(float((f + g)) / n**2 - float((f - g)**2) / n**3)
    return (left_boundary, right_boundary)



    











In [14]:



    


def proportions_diff_z_stat_rel(sample1, sample2):
    sample = zip(sample1, sample2)
    n = len(sample)
    
    f = sum([1 if (x[0] == 1 and x[1] == 0) else 0 for x in sample])
    g = sum([1 if (x[0] == 0 and x[1] == 1) else 0 for x in sample])
    
    return float(f - g) / np.sqrt(f + g - float((f - g)**2) / n )



    











In [15]:



    


print "95%% confidence interval for a difference between proportions: [%f, %f]" \
      % proportions_diff_confint_rel(data.banner_a, data.banner_b)



    


















    






confidence interval: [-0.026689, -0.005311]

















In [22]:



    


print "p-value: %f" % proportions_diff_z_test(proportions_diff_z_stat_rel(data.banner_a, data.banner_b))



    


















    






p_value: 0.003349

















In [23]:



    


print "p-value: %f" % proportions_diff_z_test(proportions_diff_z_stat_rel(data.banner_a, data.banner_b), 'less')



    


















    






p_value: 0.001675

Content source: maxis42/ML-DA-Coursera-Yandex-MIPT

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