Statistics for Data Science/Day 18 Statistical Hypothesis Tests.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Participant**\n",
    "\n",
    "Title= \"Dr\"\n",
    "\n",
    "Name= \"Muhammad Arslan\"\n",
    "\n",
    "email = \"arslanmuhammad281@yahoo.com\"\n",
    "\n",
    "whatsapp = \"009230642042\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# In this session i will try to do following Tests\n",
    "## Statical hypotsis tests\n",
    "### Normality Tests\n",
    "- Shapiro-Wilk Test\n",
    "- D’Agostino’s K^2 Test\n",
    "- Anderson-Darling Test\n",
    "\n",
    "### Correlation Tests\n",
    "- Pearson’s Correlation Coefficient\n",
    "- Spearman’s Rank Correlation\n",
    "- Kendall’s Rank Correlation\n",
    "- Chi-Squared Test\n",
    "\n",
    "### Stationary Tests\n",
    "- Augmented Dickey-Fuller\n",
    "- Kwiatkowski-Phillips-Schmidt-Shin\n",
    "\n",
    "### Parametric Statistical Hypothesis Tests\n",
    "- Student’s t-test\n",
    "- Paired Student’s t-test\n",
    "- Analysis of Variance Test (ANOVA)\n",
    "- Repeated Measures ANOVA Test\n",
    "\n",
    "### Nonparametric Statistical Hypothesis Tests\n",
    "- Mann-Whitney U Test\n",
    "- Wilcoxon Signed-Rank Test\n",
    "- Kruskal-Wallis H Test\n",
    "- Friedman Test "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Normality Tests\n",
    "This section lists statistical tests that you can use to check if your data has a Gaussian distribution.\n",
    "\n",
    "### Shapiro-Wilk Test\n",
    "- Tests whether a data sample has a Gaussian distribution.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in each sample are independent and identically distributed (iid).\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the sample has a Gaussian distribution.  \n",
    "- H1: the sample does not have a Gaussian distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# importing the required module\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=0.895, p=0.193\n",
      "Probably Gaussian\n"
     ]
    }
   ],
   "source": [
    "# Example of the Shapiro-Wilk Normality Test\n",
    "from scipy.stats import shapiro\n",
    "data = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
    "stat, p = shapiro(data)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably Gaussian')\n",
    "else:\n",
    "\tprint('Probably not Gaussian')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### D’Agostino’s K^2 Test\n",
    "Tests whether a data sample has a Gaussian distribution.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in each sample are independent and identically distributed (iid).\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the sample has a Gaussian distribution.  \n",
    "- H1: the sample does not have a Gaussian distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=3.392, p=0.183\n",
      "Probably Gaussian\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\Arsl\\anaconda3\\lib\\site-packages\\scipy\\stats\\stats.py:1541: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=10\n",
      "  warnings.warn(\"kurtosistest only valid for n>=20 ... continuing \"\n"
     ]
    }
   ],
   "source": [
    "# Example of the D'Agostino's K^2 Normality Test\n",
    "from scipy.stats import normaltest\n",
    "data = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
    "stat, p = normaltest(data)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably Gaussian')\n",
    "else:\n",
    "\tprint('Probably not Gaussian')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Anderson-Darling Test\n",
    "Tests whether a data sample has a Gaussian distribution.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in each sample are independent and identically distributed (iid).\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the sample has a Gaussian distribution.  \n",
    "- H1: the sample does not have a Gaussian distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=0.424\n",
      "Probably Gaussian at the 15.0% level\n",
      "Probably Gaussian at the 10.0% level\n",
      "Probably Gaussian at the 5.0% level\n",
      "Probably Gaussian at the 2.5% level\n",
      "Probably Gaussian at the 1.0% level\n"
     ]
    }
   ],
   "source": [
    "# Example of the Anderson-Darling Normality Test\n",
    "from scipy.stats import anderson\n",
    "data = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
    "result = anderson(data)\n",
    "print('stat=%.3f' % (result.statistic))\n",
    "for i in range(len(result.critical_values)):\n",
    "\tsl, cv = result.significance_level[i], result.critical_values[i]\n",
    "\tif result.statistic < cv:\n",
    "\t\tprint('Probably Gaussian at the %.1f%% level' % (sl))\n",
    "\telse:\n",
    "\t\tprint('Probably not Gaussian at the %.1f%% level' % (sl))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  Correlation Tests\n",
    "This section lists statistical tests that you can use to check if two samples are related.\n",
    "\n",
    "### Pearson’s Correlation Coefficient\n",
    "Tests whether two samples have a linear relationship.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in each sample are independent and identically distributed (iid).\n",
    "- Observations in each sample are normally distributed.\n",
    "- Observations in each sample have the same variance.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the two samples are independent.\n",
    "- H1: there is a dependency between the samples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=0.688, p=0.028\n",
      "Probably dependent\n"
     ]
    }
   ],
   "source": [
    "# Example of the Pearson's Correlation test\n",
    "from scipy.stats import pearsonr\n",
    "data1 = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
    "data2 = [0.353, 3.517, 0.125, -7.545, -0.555, -1.536, 3.350, -1.578, -3.537, -1.579]\n",
    "stat, p = pearsonr(data1, data2)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably independent')\n",
    "else:\n",
    "\tprint('Probably dependent')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Spearman’s Rank Correlation\n",
    "Tests whether two samples have a monotonic relationship.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in each sample are independent and identically distributed (iid).\n",
    "- Observations in each sample can be ranked.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the two samples are independent.\n",
    "- H1: there is a dependency between the samples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=0.855, p=0.002\n",
      "Probably dependent\n"
     ]
    }
   ],
   "source": [
    "# Example of the Spearman's Rank Correlation Test\n",
    "from scipy.stats import spearmanr\n",
    "data1 = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
    "data2 = [0.353, 3.517, 0.125, -7.545, -0.555, -1.536, 3.350, -1.578, -3.537, -1.579]\n",
    "stat, p = spearmanr(data1, data2)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably independent')\n",
    "else:\n",
    "\tprint('Probably dependent')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Kendall’s Rank Correlation\n",
    "Tests whether two samples have a monotonic relationship.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- bservations in each sample are independent and identically distributed (iid).\n",
    "- Observations in each sample can be ranked.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the two samples are independent.\n",
    "- H1: there is a dependency between the samples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=0.733, p=0.002\n",
      "Probably dependent\n"
     ]
    }
   ],
   "source": [
    "# Example of the Kendall's Rank Correlation Test\n",
    "from scipy.stats import kendalltau\n",
    "data1 = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
    "data2 = [0.353, 3.517, 0.125, -7.545, -0.555, -1.536, 3.350, -1.578, -3.537, -1.579]\n",
    "stat, p = kendalltau(data1, data2)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably independent')\n",
    "else:\n",
    "\tprint('Probably dependent')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Chi-Squared Test\n",
    "Tests whether two categorical variables are related or independent.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations used in the calculation of the contingency table are independent.\n",
    "- 25 or more examples in each cell of the contingency table.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the two samples are independent.\n",
    "- H1: there is a dependency between the samples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=0.272, p=0.873\n",
      "Probably independent\n"
     ]
    }
   ],
   "source": [
    "# Example of the Chi-Squared Test\n",
    "from scipy.stats import chi2_contingency\n",
    "table = [[10, 20, 30],[6,  9,  17]]\n",
    "stat, p, dof, expected = chi2_contingency(table)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably independent')\n",
    "else:\n",
    "\tprint('Probably dependent')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  Stationary Tests\n",
    "This section lists statistical tests that you can use to check if a time series is stationary or not.\n",
    "\n",
    "### Augmented Dickey-Fuller Unit Root Test\n",
    "Tests whether a time series has a unit root, e.g. has a trend or more generally is autoregressive.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in are temporally ordered.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: a unit root is present (series is non-stationary).\n",
    "- H1: a unit root is not present (series is stationary)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=0.992, p=0.994\n",
      "Probably not Stationary\n"
     ]
    }
   ],
   "source": [
    "# Example of the Augmented Dickey-Fuller unit root test\n",
    "from statsmodels.tsa.stattools import adfuller\n",
    "data = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
    "stat, p, lags, obs, crit, t = adfuller(data)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably not Stationary')\n",
    "else:\n",
    "\tprint('Probably Stationary')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Kwiatkowski-Phillips-Schmidt-Shin\n",
    "Tests whether a time series is trend stationary or not.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in are temporally ordered.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the time series is trend-stationary.\n",
    "- H1: the time series is not trend-stationary."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=0.410, p=0.073\n",
      "Probably Stationary\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\Arsl\\anaconda3\\lib\\site-packages\\statsmodels\\tsa\\stattools.py:1875: FutureWarning: The behavior of using nlags=None will change in release 0.13.Currently nlags=None is the same as nlags=\"legacy\", and so a sample-size lag length is used. After the next release, the default will change to be the same as nlags=\"auto\" which uses an automatic lag length selection method. To silence this warning, either use \"auto\" or \"legacy\"\n",
      "  warnings.warn(msg, FutureWarning)\n"
     ]
    }
   ],
   "source": [
    "# Example of the Kwiatkowski-Phillips-Schmidt-Shin test\n",
    "from statsmodels.tsa.stattools import kpss\n",
    "data = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
    "stat, p, lags, crit = kpss(data)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably Stationary')\n",
    "else:\n",
    "\tprint('Probably not Stationary')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Parametric Statistical Hypothesis Tests\n",
    "This section lists statistical tests that you can use to compare data samples.\n",
    "\n",
    "### Student’s t-test\n",
    "Tests whether the means of two independent samples are significantly different.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in each sample are independent and identically distributed (iid).\n",
    "- Observations in each sample are normally distributed.\n",
    "- Observations in each sample have the same variance.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the means of the samples are equal.\n",
    "- H1: the means of the samples are unequal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=-0.334, p=0.746\n",
      "Probably the same distribution\n"
     ]
    }
   ],
   "source": [
    "# Example of the Paired Student's t-test\n",
    "from scipy.stats import ttest_rel\n",
    "data1 = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
    "data2 = [1.142, -0.432, -0.938, -0.729, -0.846, -0.157, 0.500, 1.183, -1.075, -0.169]\n",
    "stat, p = ttest_rel(data1, data2)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably the same distribution')\n",
    "else:\n",
    "\tprint('Probably different distributions')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Paired Student’s t-test\n",
    "Tests whether the means of two paired samples are significantly different.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in each sample are independent and identically distributed (iid).\n",
    "- Observations in each sample are normally distributed.\n",
    "- Observations in each sample have the same variance.\n",
    "- Observations across each sample are paired.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the means of the samples are equal.\n",
    "- H1: the means of the samples are unequal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=-0.334, p=0.746\n",
      "Probably the same distribution\n"
     ]
    }
   ],
   "source": [
    "# Example of the Paired Student's t-test\n",
    "from scipy.stats import ttest_rel\n",
    "data1 = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
    "data2 = [1.142, -0.432, -0.938, -0.729, -0.846, -0.157, 0.500, 1.183, -1.075, -0.169]\n",
    "stat, p = ttest_rel(data1, data2)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably the same distribution')\n",
    "else:\n",
    "\tprint('Probably different distributions')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Analysis of Variance Test (ANOVA)\n",
    "Tests whether the means of two or more independent samples are significantly different.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in each sample are independent and identically distributed (iid).\n",
    "- Observations in each sample are normally distributed.\n",
    "- Observations in each sample have the same variance.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the means of the samples are equal.\n",
    "- H1: one or more of the means of the samples are unequal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=0.096, p=0.908\n",
      "Probably the same distribution\n"
     ]
    }
   ],
   "source": [
    "# Example of the Analysis of Variance Test\n",
    "from scipy.stats import f_oneway\n",
    "data1 = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
    "data2 = [1.142, -0.432, -0.938, -0.729, -0.846, -0.157, 0.500, 1.183, -1.075, -0.169]\n",
    "data3 = [-0.208, 0.696, 0.928, -1.148, -0.213, 0.229, 0.137, 0.269, -0.870, -1.204]\n",
    "stat, p = f_oneway(data1, data2, data3)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably the same distribution')\n",
    "else:\n",
    "\tprint('Probably different distributions')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Repeated Measures ANOVA Test\n",
    "Tests whether the means of two or more paired samples are significantly different.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in each sample are independent and identically distributed (iid).\n",
    "- Observations in each sample are normally distributed.\n",
    "- Observations in each sample have the same variance.\n",
    "- Observations across each sample are paired.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the means of the samples are equal.\n",
    "- H1: one or more of the means of the samples are unequal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "              Anova\n",
      "==================================\n",
      "     F Value Num DF  Den DF Pr > F\n",
      "----------------------------------\n",
      "drug 24.7589 3.0000 12.0000 0.0000\n",
      "==================================\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Example of the repeated measures ANOVA Test\n",
    "#create data\n",
    "df = pd.DataFrame({'patient': np.repeat([1, 2, 3, 4, 5], 4),\n",
    "                   'drug': np.tile([1, 2, 3, 4], 5),\n",
    "                   'response': [30, 28, 16, 34,\n",
    "                                14, 18, 10, 22,\n",
    "                                24, 20, 18, 30,\n",
    "                                38, 34, 20, 44, \n",
    "                                26, 28, 14, 30]})\n",
    "                            \n",
    "#perform the repeated measures ANOVA\n",
    "print(AnovaRM(data=df, depvar='response', subject='patient', within=['drug']).fit())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Nonparametric Statistical Hypothesis Tests\n",
    "### Mann-Whitney U Test\n",
    "Tests whether the distributions of two independent samples are equal or not.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in each sample are independent and identically distributed (iid).\n",
    "- Observations in each sample can be ranked.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the distributions of both samples are equal.\n",
    "- H1: the distributions of both samples are not equal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=40.000, p=0.473\n",
      "Probably the same distribution\n"
     ]
    }
   ],
   "source": [
    "# Example of the Mann-Whitney U Test\n",
    "from scipy.stats import mannwhitneyu\n",
    "data1 = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
    "data2 = [1.142, -0.432, -0.938, -0.729, -0.846, -0.157, 0.500, 1.183, -1.075, -0.169]\n",
    "stat, p = mannwhitneyu(data1, data2)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably the same distribution')\n",
    "else:\n",
    "\tprint('Probably different distributions')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Wilcoxon Signed-Rank Test\n",
    "Tests whether the distributions of two paired samples are equal or not.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in each sample are independent and identically distributed (iid).\n",
    "- Observations in each sample can be ranked.\n",
    "- Observations across each sample are paired.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the distributions of both samples are equal.\n",
    "- H1: the distributions of both samples are not equal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=21.000, p=0.557\n",
      "Probably the same distribution\n"
     ]
    }
   ],
   "source": [
    "# Example of the Wilcoxon Signed-Rank Test\n",
    "from scipy.stats import wilcoxon\n",
    "data1 = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
    "data2 = [1.142, -0.432, -0.938, -0.729, -0.846, -0.157, 0.500, 1.183, -1.075, -0.169]\n",
    "stat, p = wilcoxon(data1, data2)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably the same distribution')\n",
    "else:\n",
    "\tprint('Probably different distributions')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Kruskal-Wallis H Test\n",
    "Tests whether the distributions of two or more independent samples are equal or not.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in each sample are independent and identically distributed (iid).\n",
    "- Observations in each sample can be ranked.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the distributions of all samples are equal.\n",
    "- H1: the distributions of one or more samples are not equal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=0.571, p=0.450\n",
      "Probably the same distribution\n"
     ]
    }
   ],
   "source": [
    "# Example of the Kruskal-Wallis H Test\n",
    "from scipy.stats import kruskal\n",
    "data1 = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
    "data2 = [1.142, -0.432, -0.938, -0.729, -0.846, -0.157, 0.500, 1.183, -1.075, -0.169]\n",
    "stat, p = kruskal(data1, data2)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably the same distribution')\n",
    "else:\n",
    "\tprint('Probably different distributions')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Friedman Test\n",
    "Tests whether the distributions of two or more paired samples are equal or not.\n",
    "\n",
    "#### Assumptions\n",
    "\n",
    "- Observations in each sample are independent and identically distributed (iid).\n",
    "- Observations in each sample can be ranked.\n",
    "- Observations across each sample are paired.\n",
    "\n",
    "#### Interpretation\n",
    "\n",
    "- H0: the distributions of all samples are equal.\n",
    "- H1: the distributions of one or more samples are not equal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "stat=0.800, p=0.670\n",
      "Probably the same distribution\n"
     ]
    }
   ],
   "source": [
    "# Example of the Friedman Test\n",
    "from scipy.stats import friedmanchisquare\n",
    "data1 = [0.873, 2.817, 0.121, -0.945, -0.055, -1.436, 0.360, -1.478, -1.637, -1.869]\n",
    "data2 = [1.142, -0.432, -0.938, -0.729, -0.846, -0.157, 0.500, 1.183, -1.075, -0.169]\n",
    "data3 = [-0.208, 0.696, 0.928, -1.148, -0.213, 0.229, 0.137, 0.269, -0.870, -1.204]\n",
    "stat, p = friedmanchisquare(data1, data2, data3)\n",
    "print('stat=%.3f, p=%.3f' % (stat, p))\n",
    "if p > 0.05:\n",
    "\tprint('Probably the same distribution')\n",
    "else:\n",
    "\tprint('Probably different distributions')"
   ]
  }
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