5. Hypothesis Testing

Hypothsis Testing

• Need to define two opposing statements:
  1. Null hypothesis
  2. Alternative hypothesis
• Define critical value
  • One-tail critical region: p-value = 0.05
  • Two-tail critical regions: p-value = 0.05 / 2 = 0.025

Types of hypothesis testing

1. T-Test : A t-test is a type of inferential statistic which is used to determine if there is a significant difference between the means of two groups which may be related in certain features
   • One-sampled t-test (Whether the sample mean is statistically different from a known or hypothesized population mean)
     • Have 10 ages and you are checking whether avg age is 30 or not.
   • Two-sampled t-test (Compares the means of two independent groups)
     • Is there any association between week 1 and week 2 etc.
   • Paired-sampled t-test (Tests for a significant difference between 2 related variables)
     • H0 : means difference between two sample is 0
     • H1: mean difference between two sample is not 0
2. Z-Test (Sample size is greater than 30 and data points should be independent from each other)
3. ANOVA (F-Test) (Compare more than two groups at the same time)
   • The F distribution does not have any negative values because between and within group variability are always positive due to squaring each deviation

One-sampled T-test

from scipy import stats

import matplotlib.pyplot as plt

# Read in the brain_size.csv file and display a histogram  graph

data = pd.read_csv('brain_size.csv', sep=';', na_values=".")

print(data.head())

data['VIQ'].hist()

plt.show()

Unnamed: 0 Gender FSIQ VIQ PIQ Weight Height MRI_Count

0 1 Female 133 132 124 118.0 64.5 816932

1 2 Male 140 150 124 NaN 72.5 1001121

2 3 Male 139 123 150 143.0 73.3 1038437

3 4 Male 133 129 128 172.0 68.8 965353

4 5 Female 137 132 134 147.0 65.0 951545

# A two tail one sample t test on the VIQ with value 110

print(stats.ttest_1samp(data['VIQ'], 110))

# p-value = 0.533 which is more than (0.05/2 = 0.025), accept null hypothesis

Ttest_1sampResult(statistic=0.6293461053092635, pvalue=0.5327920500038907)

# A two tail one sample t test on the VIQ  with value 100

print(stats.ttest_1samp(data['VIQ'], 100))

# p-value = 0.002 which is less than (0.05/2 = 0.025), reject null hypothesis

Ttest_1sampResult(statistic=3.3074146385401786, pvalue=0.002030117404781822)

# One-tailed one sample t test, less than 130

m = 130

results = stats.ttest_1samp(data['VIQ'], m)

print(results)

alpha = 0.05

if (results[0] < 0) & (results[1]/2 < alpha):

    print("Reject null hypothesis, mean is less than {}".format(m))

else:

    print("Qccept null hypothesis")

Ttest_1sampResult(statistic=-4.726790961152567, pvalue=2.9541289074597695e-05)

Reject null hypothesis, mean is less than 130

# one-tailed one sample t test, greater than 80

m = 80

results = stats.ttest_1samp(data['VIQ'], m)

print(results)

alpha = 0.05

if (results[0] > 0) & (results[1]/2 < alpha):

    print("reject null hypothesis, mean is greater than {}".format(m))

else:

    print("accept null hypothesis")

Ttest_1sampResult(statistic=8.663551705002009, pvalue=1.2618678412297256e-10)

reject null hypothesis, mean is greater than 80

Two-sampled T-test

import scipy.stats as s # provides some additional predefined statistics functions.

import pandas as pd

import numpy as np

from math import *

from functools import reduce

df = pd.read_csv('p4_data1.tsv','\t')

print(df.head())

print(df.describe())

Consumer Before After

0 1 3.15 3.42

1 2 2.33 2.70

2 3 2.95 3.20

3 4 2.15 2.50

4 5 3.33 3.40

Consumer Before After

count 12.000000 12.000000 12.000000

mean 6.500000 2.752500 2.930833

std 3.605551 0.643529 0.559114

min 1.000000 1.950000 2.100000

25% 3.750000 2.150000 2.475000

50% 6.500000 2.665000 2.950000

75% 9.250000 3.245000 3.405000

max 12.000000 3.950000 3.800000

# Two-Sampled T-test

results = s.ttest_ind(df['Before'], df['After'])

print(results)

# pvalue = 0.48 which is greater than 0.05, accept null hypothesis

alpha = 0.05

if (results[0] > 0) & (results[1] < alpha/2):

    print("reject null hypothesis")

else:

    print("accept null hypothesis")

Ttest_indResult(statistic=-0.7246599369194594, pvalue=0.47629775671675556)

accept null hypothesis

Pair Sampled T-test

import pandas as pd

from scipy import stats

from statsmodels.stats import weightstats as stests

df = pd.read_csv('blood_pressure.csv')

print(df[['bp_before','bp_after']].describe())

print(df.head(5))

bp_before bp_after

count 120.000000 120.000000

mean 156.450000 151.358333

std 11.389845 14.177622

min 138.000000 125.000000

25% 147.000000 140.750000

50% 154.500000 149.500000

75% 164.000000 161.000000

max 185.000000 185.000000

patient sex agegrp bp_before bp_after

0 1 Male 30-45 143 153

1 2 Male 30-45 163 170

2 3 Male 30-45 153 168

3 4 Male 30-45 153 142

4 5 Male 30-45 146 141

# Pair Sampled t test

results = stats.ttest_rel(df['bp_before'], df['bp_after']) #Calculate t-test on two related samples

print(results)

# pvalue= 0.0011 which is less than 0.05, reject null hypothesis

alpha = 0.05

if (results[0] > 0) & (results[1] < alpha/2):

    print("reject null hypothesis")

else:

    print("accept null hypothesis")

Ttest_relResult(statistic=3.3371870510833657, pvalue=0.0011297914644840823)

reject null hypothesis

Z-Test

import pandas as pd

from scipy import stats

from statsmodels.stats import weightstats as stests

df = pd.read_csv('blood_pressure.csv')

print(df.head())

print(df[['bp_before', 'bp_after']].describe())

patient sex agegrp bp_before bp_after

0 1 Male 30-45 143 153

1 2 Male 30-45 163 170

2 3 Male 30-45 153 168

3 4 Male 30-45 153 142

4 5 Male 30-45 146 141

bp_before bp_after

count 120.000000 120.000000

mean 156.450000 151.358333

std 11.389845 14.177622

min 138.000000 125.000000

25% 147.000000 140.750000

50% 154.500000 149.500000

75% 164.000000 161.000000

max 185.000000 185.000000

ztest, pval1 = stests.ztest(df['bp_before'], x2=df['bp_after'], value=0, alternative='two-sided')

print(float(pval1))

# # pvalue = 0.0022 which is lesser than 0.05, reject null hypothesis

0.002162306611369422

ANOVA (F-test)

import scipy.stats as s # provides some additional predefined statistics functions.

import pandas as pd

import numpy as np

from math import *

from functools import reduce

df = pd.read_csv('p4_data1.tsv','\t')

print(df.head())

Consumer Before After

0 1 3.15 3.42

1 2 2.33 2.70

2 3 2.95 3.20

3 4 2.15 2.50

4 5 3.33 3.40

var_before = df['Before'].var()

var_after = df['After'].var()

print("Var Before: ", var_before)

print("Var After: ", var_after)

f = var_before / var_after if (var_before > var_after) else (var_after / var_before)

print(f)

Var Before: 0.41412954545454556

Var After: 0.31260833333333327

1.3247552969516028

p_value_var_test = s.f.sf(f, df['Before'].count() -1 , df['After'].count() - 1) # same as 1 - cdf (Cumulative distribution function)

print(p_value_var_test)

# pvalue = 0.32 which is greater than 0.05, accept null hypothesis

0.32449424960699924