{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to mathematical statistics \n",
    "\n",
    "Welcome to the lecture 12 in 02403\n",
    "\n",
    "During the lectures we will present both slides and notebooks. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "import scipy.stats as stats\n",
    "import statsmodels.api as sm\n",
    "import statsmodels.formula.api as smf\n",
    "import statsmodels.stats.power as smp\n",
    "import statsmodels.stats.proportion as smprop"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example: CI for exponential rate or mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "wait_times = np.array([32.6, 1.6, 42.1, 29.2, 53.4, 79.3, 2.3, 4.7, 13.6, 2.0])\n",
    "\n",
    "plt.hist(wait_times)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We think the underlying distribution is an exponential distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "26.080000000000002\n"
     ]
    }
   ],
   "source": [
    "# lets compute the mean from the data: \n",
    "print(wait_times.mean())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.03834355828220859\n"
     ]
    }
   ],
   "source": [
    "# calculate the average rate:\n",
    "lamb = 1/wait_times.mean()\n",
    "print(lamb)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Parametric bootstrap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "array([[ 2.96495993,  4.43628773, 24.48763239, ..., 16.73287127,\n",
       "        15.94850856, 21.62056837],\n",
       "       [ 9.04572614, 58.89290713, 33.33033185, ...,  0.53766852,\n",
       "         9.19729711, 25.62373394],\n",
       "       [11.1425948 , 10.781019  , 95.7625212 , ...,  1.69305451,\n",
       "        22.98697186,  8.08183043],\n",
       "       ...,\n",
       "       [ 1.10549793,  5.73772725, 15.63935791, ..., 14.34766344,\n",
       "        13.23584009, 78.22722848],\n",
       "       [ 7.83029244, 13.21329698, 13.71433202, ..., 17.84169249,\n",
       "         1.30279392, 16.05568805],\n",
       "       [42.7090726 ,  3.28390998, 10.9779512 , ..., 19.01209514,\n",
       "        39.59550638, 13.91474392]])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Now do 100000 simulations\n",
    "k = 100000\n",
    "sim_wait_times = stats.expon.rvs(size=(k,10), scale=1/lamb)\n",
    "\n",
    "# for each simulation we calculate the mean:\n",
    "sim_means = sim_wait_times.mean(axis=1)\n",
    "\n",
    "# now we plot a histogram of all the (100000) mean values\n",
    "plt.hist(sim_means, density=True, bins=30)\n",
    "plt.show()\n",
    "sim_wait_times"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For each simulation we had n = 10. \n",
    "\n",
    "This is not very large and CLT does not apply - we also see in the plot above that the distribution of the means does not look like a normal distribution\n",
    "\n",
    "(according to the CLT it should approach a normal distribution as n increases - try this yourself. How large do YOU think n should be? Do you agree that maybe n>30 is not always enough?)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[12.48394811 44.46039936]\n"
     ]
    }
   ],
   "source": [
    "# From simulated means we can find the 95% CI for the mean:\n",
    "CI = np.percentile(sim_means, [2.5, 97.5], method=\"averaged_inverted_cdf\")\n",
    "print(CI)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We just made a simulation based confidence interval for the mean :-)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Without distribution assumption"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[53.4 79.3 53.4 79.3 79.3]\n",
      " [ 2.3  2.3 79.3 13.6 42.1]\n",
      " [ 2.3 32.6 13.6  2.   1.6]\n",
      " [29.2 53.4 13.6  4.7  2. ]\n",
      " [ 2.   2.3 32.6 79.3 79.3]\n",
      " [32.6  2.  32.6 13.6  2. ]\n",
      " [79.3  4.7 53.4 13.6 53.4]\n",
      " [ 2.3 53.4 13.6 79.3 79.3]\n",
      " [79.3 42.1  2.3 29.2 13.6]\n",
      " [42.1 79.3 42.1 13.6 53.4]]\n"
     ]
    }
   ],
   "source": [
    "# First lets try to simulate 5 more samples by re-sampling the original data:\n",
    "\n",
    "sim_data = np.random.choice(wait_times,size=(len(wait_times), 5))\n",
    "print(sim_data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[11.71 42.44]\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Now simulate MANY more samples (by re-sapling the original data many times):\n",
    "k = 100000\n",
    "sim_data = np.random.choice(wait_times,size=(len(wait_times), k))\n",
    "\n",
    "# calculate mean of each sample:\n",
    "sim_means = sim_data.mean(axis=0)\n",
    "\n",
    "# caluclate the 95% CI from the samples:\n",
    "CI_nonpar = np.percentile(sim_means, [2.5, 97.5], method=\"averaged_inverted_cdf\")\n",
    "print(CI_nonpar)\n",
    "\n",
    "# always visualise :-)\n",
    "plt.hist(sim_means, density=True,bins=30)\n",
    "plt.plot([CI[0], CI[0]], [0,.05], '--', color=\"black\")\n",
    "plt.plot([CI[1], CI[1]], [0,.05], '--', color=\"black\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([12.48394811, 44.46039936])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CI"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example: Simulation of difference between two samples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Callcenter - now from two different days. \n",
    "# We want to know if there is a significant difference in the average \n",
    "# waiting time between calls for the two days. \n",
    "\n",
    "# Data day 1\n",
    "x = np.array([32.6, 1.6, 42.1, 29.2, 53.4, 79.3, 2.3 , 4.7, 13.6, 2.0])\n",
    "n1 = len(x)\n",
    "\n",
    "# Data day 2\n",
    "y = np.array([9.6, 22.2, 52.5, 12.6, 33.0, 15.2, 76.6, 36.3, 110.2, 18.0, 62.4, 10.3])\n",
    "n2 = len(y)\n",
    "\n",
    "# always visualise :-)\n",
    "plt.hist(x, alpha=.5)\n",
    "plt.hist(y, alpha=.5)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will assume that both samples come from underlying exponential distributions (but with different means)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# simulate k samples of size n1 and n2\n",
    "\n",
    "k = 100000\n",
    "\n",
    "x_sim = stats.expon.rvs(size=(n1,k), scale=x.mean())\n",
    "y_sim = stats.expon.rvs(size=(n2,k), scale=y.mean())\n",
    "\n",
    "x_means = x_sim.mean(axis=0)\n",
    "y_means = y_sim.mean(axis=0)\n",
    "\n",
    "diffs = x_means - y_means\n",
    "\n",
    "plt.hist(diffs, bins=30)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-40.53449269  13.91135617]\n"
     ]
    }
   ],
   "source": [
    "# find 95% confidence interval for the difference\n",
    "CI = np.percentile(diffs, [2.5, 97.5], method=\"averaged_inverted_cdf\")\n",
    "print(CI)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Non parametric"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-35.47166667  10.7       ]\n"
     ]
    }
   ],
   "source": [
    "# Make simulations:\n",
    "k = 100000\n",
    "\n",
    "sim_x = np.random.choice(x,size=(len(x),k))\n",
    "sim_y = np.random.choice(y,size=(len(y),k))\n",
    "\n",
    "# calculate difference of means in simulated data:\n",
    "sim_mean_dif = sim_x.mean(axis=0) - sim_y.mean(axis=0)\n",
    "\n",
    "# calculate the 95% CI:\n",
    "CI = np.percentile(sim_mean_dif, [2.5, 97.5], method=\"averaged_inverted_cdf\")\n",
    "print(CI)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inference for proportions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### Example: Normal approximation of the binomial distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 2000x400 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "p = 1/2\n",
    "\n",
    "fig, axs = plt.subplots(1, 4, figsize=(20,4))\n",
    "\n",
    "# Plot binomial distribution for n = 10\n",
    "n = 10\n",
    "axs[0].bar(np.arange(0, n+1, 1), stats.binom.pmf(k=np.arange(0,n+1,1), n=n, p=p), width=0.2, color='red')\n",
    "\n",
    "# Plot binomial distribution for n = 20\n",
    "n = 20\n",
    "axs[1].bar(np.arange(0, n+1, 1), stats.binom.pmf(k=np.arange(0,n+1,1), n=n, p=p), width=0.3, color='red')\n",
    "\n",
    "# Plot binomial distribution for n = 30\n",
    "n = 30\n",
    "axs[2].bar(np.arange(0, n+1, 1), stats.binom.pmf(k=np.arange(0,n+1,1), n=n, p=p), width=0.4, color='red')\n",
    "\n",
    "# Plot binomial distribution for n = 40\n",
    "n = 40\n",
    "axs[3].bar(np.arange(0, n+1, 1), stats.binom.pmf(k=np.arange(0,n+1,1), n=n, p=p), width=0.5, color='red')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 2000x400 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Lets plot some binomial distributions with p = 0.10 and an increasing number of observations (n)\n",
    "\n",
    "fig, axs = plt.subplots(1, 4, figsize=(20,4))\n",
    "\n",
    "p = 1/10\n",
    "\n",
    "# Again for n = 10,20,30,40\n",
    "n = 10\n",
    "axs[0].bar(np.arange(0, n+1, 1), stats.binom.pmf(k=np.arange(0,n+1,1), n=n, p=p), width=0.2, color='red')\n",
    "n = 20\n",
    "axs[1].bar(np.arange(0, n+1, 1), stats.binom.pmf(k=np.arange(0,n+1,1), n=n, p=p), width=0.3, color='red')\n",
    "n = 30\n",
    "axs[2].bar(np.arange(0, n+1, 1), stats.binom.pmf(k=np.arange(0,n+1,1), n=n, p=p), width=0.4, color='red')\n",
    "n = 40\n",
    "axs[3].bar(np.arange(0, n+1, 1), stats.binom.pmf(k=np.arange(0,n+1,1), n=n, p=p), width=0.5, color='red')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sample size determination and marginn of error (from pole)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.02463853355540394)"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 0.5\n",
    "z = stats.norm.ppf(0.975)\n",
    "n = 1582\n",
    "z * np.sqrt(p*(1-p)/n) #ME"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(9603.647051735312)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ME = 0.01\n",
    "p * (1 - p)*(z / ME)**2 ## Sample size"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example: Confidence interval for one porportion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.7698412698412699\n"
     ]
    }
   ],
   "source": [
    "n = 252  # total number of people in the sample\n",
    "y = n-58   # number of left-handed in the sample\n",
    "\n",
    "p_hat = y/n\n",
    "print(p_hat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. **Calculating Standard Error**:  Using the sample proportion calculated, what is the standard error of the proportion for left-handed individuals in this sample?\n",
    "\n",
    "$ \\sigma_{\\hat{p}} $ is the standard error of the sample proportion, calculated as $ \\sigma_{\\hat{p}} = \\sqrt{\\frac{\\hat{p}(1 - \\hat{p})}{n}} $."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.026516368790824203\n"
     ]
    }
   ],
   "source": [
    "# compute the standard error\n",
    "se_p_hat = np.sqrt(p_hat*(1-p_hat)/n)\n",
    "print(se_p_hat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. **Calculating Confidence Interval** (assuming normal approximation):  Assuming a normal approximation, what is the 95% confidence interval for the proportion of left-handed individuals in the population, based on this sample?\n",
    "\n",
    "$\\hat{p} \\pm z_{1-\\alpha/2} \\sigma_{\\hat{p}}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[np.float64(0.7178691870112545), np.float64(0.8218133526712853)]\n"
     ]
    }
   ],
   "source": [
    "# compute confidence interval using normal-approximation\n",
    "print([p_hat - 1.96*se_p_hat, p_hat + 1.96*se_p_hat])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hypothesis test "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(y-1/6*n)/np.sqrt(n*1/6*5/5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example: Contraceptive pills and the risk of blood clots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Group using birth control pills:\n",
    "y1 = 23\n",
    "n1 = 23 + 34\n",
    "p1 = y1/n1\n",
    "print(p1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Group not using birth control pills (control group):\n",
    "y2 = 35\n",
    "n2 = 35 + 132\n",
    "p2 = y2/n2\n",
    "print(p2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# difference between groups:\n",
    "diff = p1-p2\n",
    "print(diff)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# confidence interval for diff:\n",
    "se_diff = np.sqrt(p1*(1-p1)/n1 + p2*(1-p2)/n2)\n",
    "\n",
    "print([diff - 1.96*se_diff, diff + 1.96*se_diff])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "###### Test for equal proportions in the two groups:\n",
    "# We saw from the interval above that 0.5 was not in the interval. So what do we expect here?\n",
    "\n",
    "z_obs,p_value = smprop.proportions_ztest(count = [23, 35], nobs = [57, 167], value=0, prop_var=0)\n",
    "print(z_obs, p_value)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Hypothesis Test for Two Proportions**  \n",
    "When comparing two proportions (shown here for a two-sided alternative hypothesis):\n",
    "\n",
    "$\n",
    "H_0 : \\; p_1 = p_2,\n",
    "$\n",
    "$\n",
    "H_1 : \\; p_1 \\neq p_2.\n",
    "$\n",
    "\n",
    "**Use the test statistic**\n",
    "\n",
    "$\n",
    "z_{\\text{obs}} = \\frac{\\hat{p}_1 - \\hat{p}_2}{\\sqrt{\\hat{p}(1 - \\hat{p})\\left(\\frac{1}{n_1} + \\frac{1}{n_2}\\right)}},\n",
    "$\n",
    "where $\\hat{p} = \\frac{x_1 + x_2}{n_1 + n_2}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# *Manual* calculations for the same test: \n",
    "p_pooled = (y1+y2)/(n1+n2)\n",
    "print(\"p_hat or p_pooled:\", p_pooled)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# test statistic\n",
    "z_obs = diff / np.sqrt(p_pooled*(1-p_pooled)*(1/n1 + 1/n2))\n",
    "print(\"Test statistic or z_obs:\", z_obs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# p-value\n",
    "print(\"p-value:\", 2 * stats.norm.cdf(-z_obs, loc=0, scale=1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example: Contraceptive pills with $\\chi^2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The data in a table:\n",
    "table_data = np.array([[23,35],[34,132]])\n",
    "print(table_data)\n",
    "pill_study = pd.DataFrame(table_data, index=['Blood Clot', 'No Clot'], columns=['Pill', 'No pill'])\n",
    "# With pandas we can make a better table:\n",
    "display(pill_study)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# this function can take either a pandas table or the data (so both table_data and pill_study)\n",
    "chi2, p_val, dof, (expected) = stats.chi2_contingency(pill_study, correction=False)\n",
    "# returns test statistic, p-value, degrees of freedom, and expected frequencies"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(expected) # expected frequencies under the null hypothesis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(\"Chi-square test statistic:\", chi2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(\"P-value:\", p_val)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(dof)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example: Candidate votes over time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# First put data into a pandas dataframe\n",
    "poll = np.array([[79, 91, 93], [84, 66, 60], [37, 43, 47]])\n",
    "print(poll)\n",
    "poll_df = pd.DataFrame(poll, index=['Cand1', 'Cand2', 'Undecided'], columns = ['4 weeks', '2 weeks', '1 week'])\n",
    "display(poll_df)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Row 1: votes for Candidate 1 (4, 2 and 1 week(s) before the election) <br>\n",
    "Row 1: votes for Candidate 2 (4, 2 and 1 week(s) before the election) <br>\n",
    "Row 1: undecided votes       (4, 2 and 1 week(s) before the election) <br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# calculate total number of people asked at every sample / timepoint:\n",
    "print(np.sum(poll, axis=0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# total number for each candidate across all three timepoints:\n",
    "print(np.sum(poll, axis=1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the overall distribution of votes. \n",
    "\n",
    "We want to know if the distributions of votes within each timepoint (sample) differs significantly from the overall distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Now do the chi2 test:\n",
    "# Again, we can use either the data or the pandas dataframe as input \n",
    "chi2, p_val, dof, expected = stats.chi2_contingency(poll, correction=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(expected) # Expected under the assumption that the null hypothesis is true (all distributions are the same)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(chi2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(p_val)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(dof)"
   ]
  }
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