stats.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "3578d42e-98fc-4f56-83ac-9c0462fd5e96",
   "metadata": {},
   "source": [
    "# Hypothesis Testing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "862120aa-cf8f-4a9b-93d9-cb21cedff6c4",
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import feather\n",
    "\n",
    "from scipy.stats import norm\n",
    "from statsmodels.stats.proportion import proportions_ztest\n",
    "import pingouin\n",
    "from scipy.stats import chisquare\n",
    "\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "464351d0-33d4-4586-82e1-1549027fd9dc",
   "metadata": {},
   "outputs": [],
   "source": [
    "stack_overflow = pd.read_feather('stack_overflow.feather')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "7c44dcfe-9ff9-444c-9e8e-ff668d5b9caf",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "    .dataframe thead th {\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>respondent</th>\n",
       "      <th>main_branch</th>\n",
       "      <th>hobbyist</th>\n",
       "      <th>age</th>\n",
       "      <th>age_1st_code</th>\n",
       "      <th>age_first_code_cut</th>\n",
       "      <th>comp_freq</th>\n",
       "      <th>comp_total</th>\n",
       "      <th>converted_comp</th>\n",
       "      <th>country</th>\n",
       "      <th>...</th>\n",
       "      <th>survey_length</th>\n",
       "      <th>trans</th>\n",
       "      <th>undergrad_major</th>\n",
       "      <th>webframe_desire_next_year</th>\n",
       "      <th>webframe_worked_with</th>\n",
       "      <th>welcome_change</th>\n",
       "      <th>work_week_hrs</th>\n",
       "      <th>years_code</th>\n",
       "      <th>years_code_pro</th>\n",
       "      <th>age_cat</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>36.0</td>\n",
       "      <td>I am not primarily a developer, but I write co...</td>\n",
       "      <td>Yes</td>\n",
       "      <td>34.0</td>\n",
       "      <td>30.0</td>\n",
       "      <td>adult</td>\n",
       "      <td>Yearly</td>\n",
       "      <td>60000.0</td>\n",
       "      <td>77556.0</td>\n",
       "      <td>United Kingdom</td>\n",
       "      <td>...</td>\n",
       "      <td>Appropriate in length</td>\n",
       "      <td>No</td>\n",
       "      <td>Computer science, computer engineering, or sof...</td>\n",
       "      <td>Express;React.js</td>\n",
       "      <td>Express;React.js</td>\n",
       "      <td>Just as welcome now as I felt last year</td>\n",
       "      <td>40.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>At least 30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>47.0</td>\n",
       "      <td>I am a developer by profession</td>\n",
       "      <td>Yes</td>\n",
       "      <td>53.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>child</td>\n",
       "      <td>Yearly</td>\n",
       "      <td>58000.0</td>\n",
       "      <td>74970.0</td>\n",
       "      <td>United Kingdom</td>\n",
       "      <td>...</td>\n",
       "      <td>Appropriate in length</td>\n",
       "      <td>No</td>\n",
       "      <td>A natural science (such as biology, chemistry,...</td>\n",
       "      <td>Flask;Spring</td>\n",
       "      <td>Flask;Spring</td>\n",
       "      <td>Just as welcome now as I felt last year</td>\n",
       "      <td>40.0</td>\n",
       "      <td>43.0</td>\n",
       "      <td>28.0</td>\n",
       "      <td>At least 30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>69.0</td>\n",
       "      <td>I am a developer by profession</td>\n",
       "      <td>Yes</td>\n",
       "      <td>25.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>child</td>\n",
       "      <td>Yearly</td>\n",
       "      <td>550000.0</td>\n",
       "      <td>594539.0</td>\n",
       "      <td>France</td>\n",
       "      <td>...</td>\n",
       "      <td>Too short</td>\n",
       "      <td>No</td>\n",
       "      <td>Computer science, computer engineering, or sof...</td>\n",
       "      <td>Django;Flask</td>\n",
       "      <td>Django;Flask</td>\n",
       "      <td>Just as welcome now as I felt last year</td>\n",
       "      <td>40.0</td>\n",
       "      <td>13.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>Under 30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>125.0</td>\n",
       "      <td>I am not primarily a developer, but I write co...</td>\n",
       "      <td>Yes</td>\n",
       "      <td>41.0</td>\n",
       "      <td>30.0</td>\n",
       "      <td>adult</td>\n",
       "      <td>Monthly</td>\n",
       "      <td>200000.0</td>\n",
       "      <td>2000000.0</td>\n",
       "      <td>United States</td>\n",
       "      <td>...</td>\n",
       "      <td>Appropriate in length</td>\n",
       "      <td>No</td>\n",
       "      <td>None</td>\n",
       "      <td>None</td>\n",
       "      <td>None</td>\n",
       "      <td>Just as welcome now as I felt last year</td>\n",
       "      <td>40.0</td>\n",
       "      <td>11.0</td>\n",
       "      <td>11.0</td>\n",
       "      <td>At least 30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>147.0</td>\n",
       "      <td>I am not primarily a developer, but I write co...</td>\n",
       "      <td>No</td>\n",
       "      <td>28.0</td>\n",
       "      <td>15.0</td>\n",
       "      <td>adult</td>\n",
       "      <td>Yearly</td>\n",
       "      <td>50000.0</td>\n",
       "      <td>37816.0</td>\n",
       "      <td>Canada</td>\n",
       "      <td>...</td>\n",
       "      <td>Appropriate in length</td>\n",
       "      <td>No</td>\n",
       "      <td>Another engineering discipline (such as civil,...</td>\n",
       "      <td>None</td>\n",
       "      <td>Express;Flask</td>\n",
       "      <td>Just as welcome now as I felt last year</td>\n",
       "      <td>40.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>Under 30</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 63 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "   respondent                                        main_branch hobbyist  \\\n",
       "0        36.0  I am not primarily a developer, but I write co...      Yes   \n",
       "1        47.0                     I am a developer by profession      Yes   \n",
       "2        69.0                     I am a developer by profession      Yes   \n",
       "3       125.0  I am not primarily a developer, but I write co...      Yes   \n",
       "4       147.0  I am not primarily a developer, but I write co...       No   \n",
       "\n",
       "    age  age_1st_code age_first_code_cut comp_freq  comp_total  \\\n",
       "0  34.0          30.0              adult    Yearly     60000.0   \n",
       "1  53.0          10.0              child    Yearly     58000.0   \n",
       "2  25.0          12.0              child    Yearly    550000.0   \n",
       "3  41.0          30.0              adult   Monthly    200000.0   \n",
       "4  28.0          15.0              adult    Yearly     50000.0   \n",
       "\n",
       "   converted_comp         country  ...          survey_length trans  \\\n",
       "0         77556.0  United Kingdom  ...  Appropriate in length    No   \n",
       "1         74970.0  United Kingdom  ...  Appropriate in length    No   \n",
       "2        594539.0          France  ...              Too short    No   \n",
       "3       2000000.0   United States  ...  Appropriate in length    No   \n",
       "4         37816.0          Canada  ...  Appropriate in length    No   \n",
       "\n",
       "                                     undergrad_major  \\\n",
       "0  Computer science, computer engineering, or sof...   \n",
       "1  A natural science (such as biology, chemistry,...   \n",
       "2  Computer science, computer engineering, or sof...   \n",
       "3                                               None   \n",
       "4  Another engineering discipline (such as civil,...   \n",
       "\n",
       "  webframe_desire_next_year webframe_worked_with  \\\n",
       "0          Express;React.js     Express;React.js   \n",
       "1              Flask;Spring         Flask;Spring   \n",
       "2              Django;Flask         Django;Flask   \n",
       "3                      None                 None   \n",
       "4                      None        Express;Flask   \n",
       "\n",
       "                            welcome_change work_week_hrs years_code  \\\n",
       "0  Just as welcome now as I felt last year          40.0        4.0   \n",
       "1  Just as welcome now as I felt last year          40.0       43.0   \n",
       "2  Just as welcome now as I felt last year          40.0       13.0   \n",
       "3  Just as welcome now as I felt last year          40.0       11.0   \n",
       "4  Just as welcome now as I felt last year          40.0        5.0   \n",
       "\n",
       "  years_code_pro      age_cat  \n",
       "0            3.0  At least 30  \n",
       "1           28.0  At least 30  \n",
       "2            3.0     Under 30  \n",
       "3           11.0  At least 30  \n",
       "4            3.0     Under 30  \n",
       "\n",
       "[5 rows x 63 columns]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stack_overflow.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9d28281e-a9ea-4e9f-aa5f-c9b15fd2ae59",
   "metadata": {},
   "source": [
    "## Z-score\n",
    "$$z = \\frac{sample\\ statistic - hypoth.param.value}{standard\\ error}$$\n",
    "- standard error: standard deviation of the bootstrap distribution\n",
    "- sample statistic: sample proportion\n",
    "- hypoth.param.value: hypothesized progortion of population"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "40ee1e4b-e40e-4627-baf7-291b4d187fbd",
   "metadata": {},
   "source": [
    "### Z-score and p-value for one-sample proportion test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "8439489e-98d5-4b07-af9d-8e8d013e289e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Under 30       0.535604\n",
       "At least 30    0.464396\n",
       "Name: age_cat, dtype: float64"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stack_overflow['age_cat'].value_counts(normalize=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "5335a7d6-62e4-4865-85d7-b40a7ffdc623",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "significance level: 0.01, sample proportion: 0.5356037151702786\n"
     ]
    }
   ],
   "source": [
    "alpha = 0.01\n",
    "p_hat = (stack_overflow['age_cat']=='Under 30').mean()\n",
    "print(f'significance level: {alpha}, sample proportion: {p_hat}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "697174be-9749-4833-b8fc-49fbd31917fa",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hypothesized proportion of population: 0.5\n"
     ]
    }
   ],
   "source": [
    "p_0 = 0.5\n",
    "print(f'hypothesized proportion of population: {p_0}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "97c686f7-c5ec-4102-84e7-69c3ebf3b8ab",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sample size: 2261\n"
     ]
    }
   ],
   "source": [
    "n = len(stack_overflow)\n",
    "print(f'sample size: {n}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "457fffcc-1995-4298-9a0c-8633ff5db1b3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Z-score: 3.385911440783663\n"
     ]
    }
   ],
   "source": [
    "numerator = p_hat - p_0\n",
    "denominator = np.sqrt(p_0*(1-p_0)/n)\n",
    "z_score = numerator/denominator\n",
    "print(f'Z-score: {z_score}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "d359b6e3-94c0-44eb-9f06-63e1408151d0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p-value for right-tailed test: 0.00035471136840503625\n",
      "More participants under 30 than at least 30? True\n"
     ]
    }
   ],
   "source": [
    "p_value = 1- norm.cdf(z_score)\n",
    "print(f'p-value for right-tailed test: {p_value}')\n",
    "print(f'More participants under 30 than at least 30? {p_value < p_0}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "c21e1371-32c3-4dc2-b3f2-025d6dc6e943",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p-value for two-tailed test: 0.0007094227368100725\n",
      "Number of participants under 30 and at least 30 are not equal? True\n"
     ]
    }
   ],
   "source": [
    "p_value_two_tailed = 2*(1-norm.cdf(z_score)) # = norm.cdf(-z_score) + 1-norm.cdf(z_score)\n",
    "print(f'p-value for two-tailed test: {p_value_two_tailed}')\n",
    "print(f'Number of participants under 30 and at least 30 are not equal? {p_value_two_tailed < p_0}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "96d53c1b-827e-494b-b228-40feddbd5eeb",
   "metadata": {},
   "source": [
    "### Z-score and p-value for two-sample proportion test"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "65200f6c-b34b-4fc5-9654-4c41d7162aa9",
   "metadata": {},
   "source": [
    "#### Method 1: step-by-step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "4a627434-965b-4a62-b30a-5e9abe3e07d2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sample proportion: ≥30: 0.7733333333333333, <30: 0.8431048720066061\n"
     ]
    }
   ],
   "source": [
    "p_hats = stack_overflow.groupby('age_cat')['hobbyist'].value_counts(normalize=True)\n",
    "p_hat_at_least_30 = p_hats[(\"At least 30\", \"Yes\")]\n",
    "p_hat_under_30 = p_hats[(\"Under 30\", \"Yes\")]\n",
    "print(f'Sample proportion: ≥30: {p_hat_at_least_30}, <30: {p_hat_under_30}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "acc897e3-752f-4916-baed-96d20aae7cf9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sample size: ≥30: 1050, <30: 1211\n"
     ]
    }
   ],
   "source": [
    "n = stack_overflow.groupby('age_cat')['hobbyist'].count()\n",
    "n_at_least_30 = n[\"At least 30\"]\n",
    "n_under_30 = n[\"Under 30\"]\n",
    "print(f'Sample size: ≥30: {n_at_least_30}, <30: {n_under_30}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "6337d90e-1e05-4e9f-9f79-19c5e840adef",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Z-score: -4.223691463320559\n"
     ]
    }
   ],
   "source": [
    "# Calculate the pooled estimate of the population proportion\n",
    "p_hat = (n_at_least_30*p_hat_at_least_30 + n_under_30*p_hat_under_30)/(n_at_least_30+n_under_30)\n",
    "\n",
    "std_error = np.sqrt(p_hat*(1-p_hat)/n_at_least_30 + p_hat*(1-p_hat)/n_under_30)\n",
    "z_score = (p_hat_at_least_30 - p_hat_under_30)/std_error\n",
    "print(f'Z-score: {z_score}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "0bbc4e3b-bb7b-4d5a-93f3-60412e3a132c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p-value for two-tailed test: 1.9999759666985732\n"
     ]
    }
   ],
   "source": [
    "# Two-tailed test (\"not equal\")\n",
    "p_value_two_tailed = 2*(1-norm.cdf(z_score))\n",
    "print(f'p-value for two-tailed test: {p_value_two_tailed}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19655be5-2aa9-4be1-95b5-a7eaea75eeb4",
   "metadata": {},
   "source": [
    "#### Method 2: statsmodels.stats.proportion.proportion_ztest"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "057f6561-1899-4998-b468-850a29f09028",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "age_cat      hobbyist\n",
       "At least 30  Yes          812\n",
       "             No           238\n",
       "Under 30     Yes         1021\n",
       "             No           190\n",
       "Name: hobbyist, dtype: int64"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "age_by_hobbyist = stack_overflow.groupby('age_cat')['hobbyist'].value_counts()\n",
    "age_by_hobbyist"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "0c8ee6e3-b249-483b-ab90-226601134766",
   "metadata": {},
   "outputs": [],
   "source": [
    "success_counts = np.array([age_by_hobbyist[(\"At least 30\", \"Yes\")], age_by_hobbyist[(\"Under 30\", \"Yes\")]])\n",
    "n = np.array([sum(age_by_hobbyist[\"At least 30\"]), sum(age_by_hobbyist[\"Under 30\"])])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "af2b97bf-9e23-491f-8a54-249a12c890db",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Z-score: -4.223691463320559, p-value: 2.403330142685068e-05\n"
     ]
    }
   ],
   "source": [
    "stat, p_value = proportions_ztest(count=success_counts, nobs=n, alternative=\"two-sided\") # alternative = \"larger\"/\"two-sided\"/\"smaller\"\n",
    "print(f'Z-score: {stat}, p-value: {p_value}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2304eb08-eded-41fb-a444-c75c61d3c6f3",
   "metadata": {},
   "source": [
    "## Chi-squared test ($\\chi^2$ test)\n",
    "= test for independence of variables\n",
    "<br>$\\chi^2$: assuming independence, quantifies how far away the observed results are from the expected values\n",
    "$$\\chi^2 = Z^2$$\n",
    "$H_0$: independence\n",
    "<br>e.g. test if job satisfaction levels (5 categories) and age category (2 categories) are independent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "9b7a37f4-da45-4753-834d-9ee84e5427ca",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:xlabel='job_sat'>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# EDA: proportional stacked bar plot\n",
    "props = stack_overflow.groupby('job_sat')['age_cat'].value_counts(normalize=True) # proportion of age_cat grouped by job_sat\n",
    "wide_props = props.unstack() # unstack the result to be in wide format instead of long\n",
    "wide_props.plot(kind='bar', stacked=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "18032c01-bd05-4644-bd5a-4affbb2b1fff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>test</th>\n",
       "      <th>lambda</th>\n",
       "      <th>chi2</th>\n",
       "      <th>dof</th>\n",
       "      <th>pval</th>\n",
       "      <th>cramer</th>\n",
       "      <th>power</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>pearson</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>5.552373</td>\n",
       "      <td>4.0</td>\n",
       "      <td>0.235164</td>\n",
       "      <td>0.049555</td>\n",
       "      <td>0.437417</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>cressie-read</td>\n",
       "      <td>0.666667</td>\n",
       "      <td>5.554106</td>\n",
       "      <td>4.0</td>\n",
       "      <td>0.235014</td>\n",
       "      <td>0.049563</td>\n",
       "      <td>0.437545</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>log-likelihood</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>5.558529</td>\n",
       "      <td>4.0</td>\n",
       "      <td>0.234632</td>\n",
       "      <td>0.049583</td>\n",
       "      <td>0.437871</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>freeman-tukey</td>\n",
       "      <td>-0.500000</td>\n",
       "      <td>5.562688</td>\n",
       "      <td>4.0</td>\n",
       "      <td>0.234274</td>\n",
       "      <td>0.049601</td>\n",
       "      <td>0.438178</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>mod-log-likelihood</td>\n",
       "      <td>-1.000000</td>\n",
       "      <td>5.567570</td>\n",
       "      <td>4.0</td>\n",
       "      <td>0.233854</td>\n",
       "      <td>0.049623</td>\n",
       "      <td>0.438538</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>neyman</td>\n",
       "      <td>-2.000000</td>\n",
       "      <td>5.579519</td>\n",
       "      <td>4.0</td>\n",
       "      <td>0.232828</td>\n",
       "      <td>0.049676</td>\n",
       "      <td>0.439419</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                 test    lambda      chi2  dof      pval    cramer     power\n",
       "0             pearson  1.000000  5.552373  4.0  0.235164  0.049555  0.437417\n",
       "1        cressie-read  0.666667  5.554106  4.0  0.235014  0.049563  0.437545\n",
       "2      log-likelihood  0.000000  5.558529  4.0  0.234632  0.049583  0.437871\n",
       "3       freeman-tukey -0.500000  5.562688  4.0  0.234274  0.049601  0.438178\n",
       "4  mod-log-likelihood -1.000000  5.567570  4.0  0.233854  0.049623  0.438538\n",
       "5              neyman -2.000000  5.579519  4.0  0.232828  0.049676  0.439419"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# chi2 of age_cat vs. job_sat\n",
    "expected, observed, stats = pingouin.chi2_independence(data=stack_overflow, \n",
    "                                                       x='job_sat', \n",
    "                                                       y='age_cat',\n",
    "                                                       correction=False)\n",
    "stats"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8445cdb5-31ec-41bc-b14b-d6bdb0a01dc5",
   "metadata": {},
   "source": [
    "> <i>correction</i>: specifies whether or not to apply Yate's continuity correction, typically applied when the sample size is small and degrees of freedom is 1. In this case since each group has relatively large records, we don't need it here"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "33c47eca-c2b8-4df5-9ace-f2341f3b113d",
   "metadata": {},
   "source": [
    "> $dof = (No. of\\ response\\ categories - 1) * (No. of\\ explanatory\\ categories - 1)$\n",
    "<br>= (2-1) * (5-1) = 4"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "45e3f3d3-51a7-46bc-9301-da5eb8a52519",
   "metadata": {},
   "source": [
    "#### Swap the repsonse and explanatory variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "b09462f8-0ed3-4c79-9f35-5a6064c5319b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:xlabel='age_cat'>"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# EDA \n",
    "props = stack_overflow.groupby('age_cat')['job_sat'].value_counts(normalize=True)\n",
    "wide_props = props.unstack()\n",
    "wide_props.plot(kind='bar', stacked=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "ac3e912a-6279-4888-be0e-27901e6dd286",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>test</th>\n",
       "      <th>lambda</th>\n",
       "      <th>chi2</th>\n",
       "      <th>dof</th>\n",
       "      <th>pval</th>\n",
       "      <th>cramer</th>\n",
       "      <th>power</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>pearson</td>\n",
       "      <td>1.0</td>\n",
       "      <td>5.552373</td>\n",
       "      <td>4.0</td>\n",
       "      <td>0.235164</td>\n",
       "      <td>0.049555</td>\n",
       "      <td>0.437417</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      test  lambda      chi2  dof      pval    cramer     power\n",
       "0  pearson     1.0  5.552373  4.0  0.235164  0.049555  0.437417"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# chi2 of job_sat vs. age_cat\n",
    "expected, observed, stats = pingouin.chi2_independence(data=stack_overflow, \n",
    "                                                       x='age_cat', \n",
    "                                                       y='job_sat')\n",
    "stats[stats['test']=='pearson']"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a25aef3-dad1-4bc8-890e-bddcc3898725",
   "metadata": {},
   "source": [
    "> After swapping variables, the stats of chi2 test is <b>identical</b>\n",
    "><br> So we should rephrase the question as: <b>are the variables X and Y independent?</b> since the order doesn't matter"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d38053b9-4487-4eee-84fa-6a3917caf339",
   "metadata": {},
   "source": [
    "## Goodness of fit test (one-sample Chi-square test)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "b4d25ac7-56f6-47a7-b8f3-990cd34e4f06",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>purple_link</th>\n",
       "      <th>n</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Hello, old friend</td>\n",
       "      <td>1225</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Indifferent</td>\n",
       "      <td>405</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Amused</td>\n",
       "      <td>368</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Annoyed</td>\n",
       "      <td>263</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         purple_link     n\n",
       "0  Hello, old friend  1225\n",
       "1        Indifferent   405\n",
       "2             Amused   368\n",
       "3            Annoyed   263"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "purple_link_counts = stack_overflow['purple_link'].value_counts()\n",
    "purple_link_counts = purple_link_counts.rename_axis('purple_link').reset_index(name='n')\n",
    "purple_link_counts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "3c70f524-259b-49d3-ad64-ec752d480586",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>purple_link</th>\n",
       "      <th>prop</th>\n",
       "      <th>n</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Hello, old friend</td>\n",
       "      <td>0.500000</td>\n",
       "      <td>1130.500000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Amused</td>\n",
       "      <td>0.166667</td>\n",
       "      <td>376.833333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Indifferent</td>\n",
       "      <td>0.166667</td>\n",
       "      <td>376.833333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Annoyed</td>\n",
       "      <td>0.166667</td>\n",
       "      <td>376.833333</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         purple_link      prop            n\n",
       "0  Hello, old friend  0.500000  1130.500000\n",
       "1             Amused  0.166667   376.833333\n",
       "2        Indifferent  0.166667   376.833333\n",
       "3            Annoyed  0.166667   376.833333"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hypothesized = pd.DataFrame({'purple_link': ['Hello, old friend', 'Amused', 'Indifferent', 'Annoyed'],\n",
    "                            'prop': [1/2, 1/6, 1/6, 1/6]})\n",
    "hypothesized['n'] = hypothesized['prop']*len(stack_overflow)\n",
    "hypothesized"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "fca80974-b1b5-4b51-99e2-327ea63d7eec",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# EDA\n",
    "plt.bar(purple_link_counts['purple_link'], purple_link_counts['n'], color=\"yellow\", alpha=0.5)\n",
    "plt.scatter(hypothesized['purple_link'], hypothesized['n'], color=\"purple\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "178b3d75-8084-4aa1-90da-120441d18f32",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Power_divergenceResult(statistic=44.59840778416629, pvalue=1.1261810719413759e-09)\n"
     ]
    }
   ],
   "source": [
    "# Goodness of fit test\n",
    "gof_test = chisquare(f_obs = purple_link_counts['n'], f_exp=hypothesized['n']) # alpha=0.01\n",
    "print(gof_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9788c5df-9b2a-488f-9e3a-029262843c8d",
   "metadata": {},
   "source": [
    "> The p-value is much smaller than significance level, so the sample distribution of proportions is different from the hypothesized distribution of proportions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a38d8cfd-3092-4c59-b448-3d8117ec7dc4",
   "metadata": {},
   "source": [
    "## Wilcoxon-Mann-Whitney test (Mann Whitney U test)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "8726d226-b48c-479c-bb2b-e08dd7aa7014",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>converted_comp</th>\n",
       "      <th>age_first_code_cut</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>77556.0</td>\n",
       "      <td>adult</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>74970.0</td>\n",
       "      <td>child</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>594539.0</td>\n",
       "      <td>child</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>2000000.0</td>\n",
       "      <td>adult</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>37816.0</td>\n",
       "      <td>adult</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   converted_comp age_first_code_cut\n",
       "0         77556.0              adult\n",
       "1         74970.0              child\n",
       "2        594539.0              child\n",
       "3       2000000.0              adult\n",
       "4         37816.0              adult"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "age_vs_comp = stack_overflow[['converted_comp', 'age_first_code_cut']]\n",
    "age_vs_comp.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "b4aeb8d5-f695-4ac3-82c3-5742809ef63d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>age_first_code_cut</th>\n",
       "      <th>adult</th>\n",
       "      <th>child</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>77556.0</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>NaN</td>\n",
       "      <td>74970.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>NaN</td>\n",
       "      <td>594539.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>2000000.0</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>37816.0</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "age_first_code_cut      adult     child\n",
       "0                     77556.0       NaN\n",
       "1                         NaN   74970.0\n",
       "2                         NaN  594539.0\n",
       "3                   2000000.0       NaN\n",
       "4                     37816.0       NaN"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Convert df from long to wide\n",
    "age_vs_comp_wide = age_vs_comp.pivot(columns='age_first_code_cut',\n",
    "                                     values='converted_comp')\n",
    "age_vs_comp_wide.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "bdb01248-cd1e-4429-bbcc-8d16cd978d5a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>U-val</th>\n",
       "      <th>alternative</th>\n",
       "      <th>p-val</th>\n",
       "      <th>RBC</th>\n",
       "      <th>CLES</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>MWU</th>\n",
       "      <td>744365.5</td>\n",
       "      <td>greater</td>\n",
       "      <td>1.902723e-19</td>\n",
       "      <td>-0.222516</td>\n",
       "      <td>0.611258</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        U-val alternative         p-val       RBC      CLES\n",
       "MWU  744365.5     greater  1.902723e-19 -0.222516  0.611258"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha = 0.01\n",
    "mwu_test = pingouin.mwu(x=age_vs_comp_wide['child'],\n",
    "                        y=age_vs_comp_wide['adult'],\n",
    "                        alternative='greater')\n",
    "mwu_test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "b170a041-46dd-40e1-9f03-f7faacfedabe",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Reject null hypothesis? True\n"
     ]
    }
   ],
   "source": [
    "print(f\"Reject null hypothesis? {mwu_test['p-val'][0]<alpha}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31cf4b67-f2aa-4c68-837f-2e47317eb5ff",
   "metadata": {},
   "source": [
    "## Kruskal-Wallis test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "852b61b2-2a1a-4c61-b487-7d33dfc66662",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Source</th>\n",
       "      <th>ddof1</th>\n",
       "      <th>H</th>\n",
       "      <th>p-unc</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Kruskal</th>\n",
       "      <td>job_sat</td>\n",
       "      <td>4</td>\n",
       "      <td>72.814939</td>\n",
       "      <td>5.772915e-15</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          Source  ddof1          H         p-unc\n",
       "Kruskal  job_sat      4  72.814939  5.772915e-15"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha = 0.01\n",
    "kruskal_test = pingouin.kruskal(data=stack_overflow,\n",
    "                                dv='converted_comp',\n",
    "                                between='job_sat')\n",
    "kruskal_test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "5fa60049-aab4-455a-9e62-1ca27c94a6c7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "At least one of the mean compensation totals is different than the others across job satisfaction groups? True\n"
     ]
    }
   ],
   "source": [
    "print(f\"At least one of the mean compensation totals is different than the others across job satisfaction groups? {kruskal_test['p-unc'][0]<alpha}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "bbe57a45-e7b8-4f68-a2b9-f0d81bf1172a",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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