S04a_Estimation&HyptothesisTesting.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Execute before using this notebook if using google colab\n",
    "\n",
    "kernel = str(get_ipython())\n",
    "\n",
    "if 'google.colab' in kernel:    \n",
    "    !wget https://raw.githubusercontent.com/fredzett/rmqa/master/utils.py -P local_modules -nc \n",
    "    !npx degit fredzett/rmqa/data data\n",
    "    import sys\n",
    "    sys.path.append('local_modules')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy.stats as stats\n",
    "from utils import Datasets, plot_density, plot_hist, plot_line\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Why estimate uncertainty?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In statistics / data analysis we are commonly estimating **parameters** of the population, i.e. we are estimating numerical characteristics such as the mean or the standard deviation, e.g. \n",
    "\n",
    "- what percentage of US voters will vote for Joe Biden?\n",
    "\n",
    "- how satisfied are our customers with our service?\n",
    "\n",
    "- what is the relation between body weight and exercise time?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Unfortunately, most of the time it is not feasible and/or to costly to determine these parameters directly from the population which is why we commonly deal with (representative) samples. We then estimate the parameter in question from our sample and treat it as an\n",
    "**estimator** for the true population mean. \n",
    "\n",
    "- a sample mean is then an estimator for the population mean \n",
    "\n",
    "- a sample standard deviation is then an estimator for the popoulation standard deviation\n",
    "\n",
    "- a sample proportion is then an estimator for the population proportion\n",
    "\n",
    "\n",
    " \n",
    "\n",
    "While it is easy to make such estimate it is crucial to understand the uncertainty and the possible margin of error of this estimate. \n",
    "\n",
    "There exist numerous examples (e.g. in journalism) where uncertainty around an estimator is not acknowledged and it is common to only state the point estimate (or summary statistic) instead of the plausible range in which the true parameter may lie. A good example is the news about unemployment figures. Many people don't realize that these numbers commonly are an estimate based on a sample. However, in stead of stating the range the point estimate is always communicated thereby treated as if certain. \n",
    "\n",
    "**In research we cannot be that slopy and need to understand and determine the uncertainty associated with our estimators.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How to estimate uncertainty?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once we have calculated a point estimate (e.g. the mean of a sample). How sure can we be that this estimate actually is close to the (true) mean of the population? After all, we don't have data about the enitre population, which is why we are **estimating** a statistic from a sample, instead of **calculating** a parameter from a population. \n",
    "\n",
    "Example:   \n",
    "Let's assume we conduct a randomized controlled trial to test the effectivenss of a new drug. Our trial yields that the average outcome differs between the treamtment and the control groups. \n",
    "\n",
    "How do we know that this difference is large enough to state that the groups are actually different, i.e. that the drug is actually effective?\n",
    "\n",
    "- Could this be due to chance?\n",
    "\n",
    "- Is the difference large enough that chance cannot explain the difference?\n",
    "\n",
    "- How large does the difference have to be that chance cannot explain the difference?\n",
    "\n",
    "\n",
    "We cannot assume that the point estimator actually matches the true parameter which is why we should calculate a range of values and a probability that the true parameter is within this range. \n",
    "\n",
    "There are two approaches to achieve this:\n",
    "\n",
    "1. using classical statistical concepts and probability theory\n",
    "\n",
    "2. using simulation and intuition\n",
    "\n",
    "In the following we will cover both approaches. Note that the first approach is usually covered in introductory statistics courses. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Estimating uncertainty"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Estimation of uncertainty using probability theory"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sampling distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From undergraduate statistics you may recall that a sampling distribution is the distribituion of a random sample based estimate (e.g. the mean or the standard deviation). It is the distribution of estimates when taking $n$ different samples."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "We will use (fake) population data on online spending for all family households in Little Town. The data contains:\n",
    "\n",
    "a. amount of online spending\n",
    "\n",
    "b. indicator if family has children (1 = yes, 0 = no)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = Datasets.fake_spent_children()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(29809.4372, 0.633, (5000, 2))"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(data[:,0]), np.mean(data[:,1]), data.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's assume we are not in possession of this large data set of the enitre population and instead we only have a sample of $n=50$. We are interested in two estimators:\n",
    "\n",
    "a. mean ($\\bar{x}$) online spending\n",
    "\n",
    "b. proportion ($p$) of families that has children"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below both parameters are estimated using a sample"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def take_sample(data, size):\n",
    "    total_rows = len(data)\n",
    "    rows_smpl = np.random.choice(range(total_rows), size=size, replace=False)\n",
    "    return data[rows_smpl]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(32197.22, 0.64)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smpl_size = 50\n",
    "smpl = take_sample(data, smpl_size)\n",
    "xbar, p = np.mean(smpl, axis=0)\n",
    "xbar, p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's repeat this for $1.000$ times to create a sampling distribution. The histograms below give an approximation of the sampling distributions of both estimators. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "sims = 10000\n",
    "estimates = np.empty((sims,2))\n",
    "for i in range(sims):\n",
    "    smpl = take_sample(data, smpl_size)\n",
    "    estimates[i] = np.mean(smpl, axis=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1606.1079136941726"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "means = estimates[:,0]\n",
    "np.std(means)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(29805.609024, 0.6341220000000001)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ps = estimates[:,1]\n",
    "np.mean(means), np.mean(ps)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 648x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_density(means, title=\"Density of sample mean spending \\nfrom $1.000$ random samples of size $50$ each\", xlabel=r\"Values for $\\bar{x}$\", ylabel=\"Probability density\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_density(ps, title=\"Density of sample proportion of children \\nfrom $1.000$ random samples of size $50$ each\", xlabel=r\"Values for $p$\", ylabel=\"Probability density\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Obviously, in practice we usually only have one sample (e.g. our sample of $5.000$ households). However, the above procedure illustrates that \n",
    "\n",
    "- in practice many different samples are possible\n",
    "\n",
    "- each sample therefore generates a different estimator (as proxy for the true parameter)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's investigate the above sampling distributions in more detail. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Expected value of $\\bar{X}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Th sampling distribution of $\\bar{X}$ is the probability distribution of all possible values of the sample mean (in our case online spent). Given we are commonly only in possesssion of one sample we are interested in the mean of all possible values, the **expected value**. \n",
    "\n",
    "The **expected value of $\\bar{X}$** is given by:\n",
    "\n",
    "$$E(\\bar{X}) = \\mu$$\n",
    "\n",
    "$\\text{where}$  \n",
    "$E(\\bar{X}) = \\text{the expected value of $\\bar{X}$}$  \n",
    "$\\mu = \\text{the mean of the population from which the sample is selected}$\n",
    "\n",
    "You may recall from previous statistical courses that this is called an **unbiased** estimator, as the expected value of a point estimator equals the population parameter."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Standard deviation of $\\bar{X}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It can be shown that whenever\n",
    "\n",
    "- the population is infinite or\n",
    "\n",
    "- the population is finite and the sample size is less than or equal to $5\\%$ of the population size ($\\frac{n}{N} \\leq 0.05)$\n",
    "\n",
    "the standard deviation of $\\bar{X}$ is given by:\n",
    "\n",
    "$$\\sigma_{\\bar{x}} = \\frac{\\sigma}{\\sqrt{n}}$$\n",
    "\n",
    "where\n",
    "\n",
    "$\\sigma_{\\bar{x}} = \\text{the standard deviation of $\\bar{X}$}$  \n",
    "$\\sigma = \\text{the standard deviation of the population}$  \n",
    "$n=\\text{the sample size}$  \n",
    "\n",
    "\n",
    "To avoid confusion and to make the distinction between standard deviation of the population / sample and the standard deviation of a point estimate we call the latter **standard error**.\n",
    "\n",
    "For the remainder of this course we assume that the population size is large and that the above condition holds.\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **standard error** for our example is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1619.9906005459177"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(data[:,0]) / np.sqrt(50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1619.9906005459177"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "std_population = np.std(data[:,0])\n",
    "se = std_population/np.sqrt(smpl_size)\n",
    "se"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which is fairly close to the standard devision of the sampled means"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1606.1079136941726"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(means)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Form of the sampling distribution of $\\bar{X}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's consider the following two cases:\n",
    "\n",
    "1. Population data is normally distributed\n",
    "\n",
    "2. Population data is not normally distributed\n",
    "\n",
    "\n",
    "In 1. the sampling distribution will also have a normal distribution.\n",
    "\n",
    "In 2. the **central limit theorem** applies, i.e. with large n the sampling distribution for $\\bar{X}$ also can be approximated with a normal distribution.\n",
    "\n",
    " \n",
    "> **Recall: Central limit theorem**  \n",
    "> when selecting random sample of size $n$ from a population, the sampling distribution of the sample mean $\\bar{X}$ is approximated by a normal distribution as the sample size becomes large. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's have a look at a few examples "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "N, n = 10_000, 100 # size population, size sample\n",
    "n_smpls = 100_000 # number of samples taken"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pop_dist = stats.binom(3,0.2).rvs(N)#stats.binom(2,2/6).rvs(N)\n",
    "plot_density(pop_dist);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sample_dist(dist,n, n_smpls):\n",
    "    smpl = np.random.choice(dist, size=(n_smpls, n))\n",
    "    return np.mean(smpl, axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "smpl_dist = sample_dist(pop_dist, n, n_smpls)\n",
    "plot_density(smpl_dist);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Why do we care?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We care about the sample distribution and its properties because it provides useful information about the **difference between the sample mean and the population mean**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example:** let us assume that we take the sample mean online-spent as an approximation of the population mean online-spent. However, obviously we are only accepting this approximation if we it likely that the sampled mean is close to the true mean. \n",
    "\n",
    "If we are willing to accept a deviation of $1.000$ from the true mean, what is the probability that the sample mean lies within a range of $1.000$ of the population mean?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can answer this question using our sampling distributions of $\\bar{X}$ by calculating a z-score (i.e. standard normalizing our sampled values) and then use the CDF of the standard normal distribution to calculate probabilities:\n",
    "\n",
    "$$z = \\frac{(x \\pm \\bar{x})}{\\sigma_{\\bar{x}}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.6172875322011201"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xbar = np.mean(take_sample(data[:,0], 50))\n",
    "x = xbar - 1000 # lower bound; standard normal distribution is symetrically\n",
    "zscore = (x - xbar) / se\n",
    "zscore"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "p_low = stats.norm(0,1).cdf(zscore)\n",
    "p_up = stats.norm(0,1).cdf(-zscore)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7314774555036275"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_up"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.46295491100725505"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_up - p_low"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that the sample mean - given our sample size - has only a probability of $\\approx 50\\%$ to be within a range of $\\pm 1.000$ from the population mean."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can increase this probability by increaing the sample size given the **standard error** is a function of sample size $n$.\n",
    "\n",
    "Let's calculate standard error and $z$ for a sample size of 300 instead. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "661.3583932403976"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smpl_size_new = 300\n",
    "std_population = np.std(data[:,0])\n",
    "se = std_population/np.sqrt(smpl_size_new)\n",
    "se"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.5120394784745845"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zscore = (x - xbar) / se\n",
    "zscore"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8694761803235889"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_low = stats.norm(0,1).cdf(zscore)\n",
    "p_up = stats.norm(0,1).cdf(-zscore)\n",
    "p_up - p_low"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Increasing the sample size to 300 we increasd the probability of the sample mean being within a range of $\\pm 1.000$ of the population mean. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Note:** analysis of $p$ will be left as an (homework) excercise for you."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Interval estimation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given point estimators cannot be expected to be an exact value of the true population parameter we usually determine **interval estimates**. We do so by substracting and adding **margin of error** from the point estimator. \n",
    "\n",
    "Let's look into more detail how this is done."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Population standard deviation $\\sigma$ is known"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Although this is usually not the case let's assume that we can reasonably derive knowledge about the population standard deviation ($\\sigma$) from historical data. \n",
    "\n",
    "How can we then construct interval estimates?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Interval estimate of a population mean (when $\\sigma$ is known):**\n",
    "\n",
    "$$\\bar{x} \\pm z_{\\alpha/2}\\frac{\\sigma}{\\sqrt{n}}$$\n",
    "\n",
    "where\n",
    "\n",
    "$(1-\\alpha) = \\text{the confidence coefficient or level}$\n",
    "$z_{\\alpha/2} = \\text{the z value providing an area $\\alpha/2$ in the upper tail of the standard normal probability distribution}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**: we take a customer survey with $80$ customers to determine customer satisfaction. \n",
    "\n",
    "(Let's assume that customer satisfaction follows a normal distribution with $\\mu=80$ and $\\sigma = 12$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Generate population data (for demonstration purposes)\n",
    "mu, sig = 80, 12\n",
    "customers = stats.norm(mu,sig)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "79.10215386868747"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Take a sample and calculate the sample mean xbar\n",
    "n = 80\n",
    "smpl = customers.rvs(n)\n",
    "xbar = np.mean(smpl)#\n",
    "xbar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From our many previous customer surveys we \"know\" (i.e. we assume) that the standard deviation of customer satisfaction is 12. \n",
    "\n",
    "We want to be 95% sure that the interval estimate contains the true population mean (i.e. the true customer satisfaction score). This means that $\\alpha = 5\\%$ and we need to determine the $z_{\\alpha/2}$, i.e. the value of the standard normal distribution where only $2.5\\%$ ($\\alpha/2$) are above.\n",
    "\n",
    "We are looking for the value $z$ where `stats.norm(0,1).cdf(z)` $\\approx 2.5\\%$. This could be done using trial and error. However, it is easier to use the `stats.norm.ppf` function which does exactly this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.959963984540054"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ci = .95\n",
    "alpha = 1 - ci\n",
    "z = stats.norm(0,1).ppf(alpha/2)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([76.47258625, 81.73172149])"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "se = sig/np.sqrt(n)\n",
    "margin =  z*se\n",
    "interval = xbar +  np.array((margin, -margin))\n",
    "interval"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With this interval we are 95% sure that the interval contains the true population mean. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Population standard deviation $\\sigma$ is unknown"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In many real life cases we don't know what the population standard deviation is. If we knew we mostly wouldn't need to calculate estimates in the first place. If $\\sigma$ is unknown we must use the **sample standard deviation ($s$)**. \n",
    "\n",
    "In this case the margin of error and the interval estimate is based on the probability distribution known as the **$t$ distribution**.\n",
    "\n",
    "\n",
    "The t distribution is similar to the normal distribution. A specific t distribution depends on one parameter, the **degree of freedom**. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "N = 100\n",
    "df = 1 # The more degree of freedom the more t-distribution converges to normal distribution\n",
    "tdist = stats.t(df=df)\n",
    "norm = stats.norm()\n",
    "x = np.linspace(-4,4,N)\n",
    "_, ax = plot_line(x, [tdist.pdf(x), norm.pdf(x)],title=f\"t-Distribution (df = {df}) \\nvs.\\n Standard normal distribution\", zero_origin=False);\n",
    "ax.legend([\"t-Distribution\", \"Standard Normal Distribution\"]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With higher degrees of freedom the t-distribution can be approximated by the normal distribtution. Which can be shown when calculating the t and z values repectively. As a rule of thumb we can assume that with a degree of freedom of $\\geq 100$ the standard normal z value provides a good approximation to the t value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1.9623390808264078, -1.9599639845400545)"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = 1000\n",
    "alpha = 0.025\n",
    "t = stats.t(df=df)\n",
    "t.ppf(alpha), stats.norm().ppf(alpha)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "def show_dist(a=0.25, dist=stats.norm(), ax=None):\n",
    "    # Helper function to show a distribution\n",
    "    low, high = -4, 4\n",
    "    x = np.linspace(low,high,1000)\n",
    "    y = dist.pdf(x)\n",
    "    tvalue = t.ppf(0.025)\n",
    "    xlow = np.linspace(low, tvalue)\n",
    "    xhigh = np.linspace(-tvalue,high)\n",
    "    if ax==None: fig, ax = plt.subplots(figsize=(9,7))\n",
    "    ax.plot(x,y)\n",
    "    ax.fill_between(xlow,dist.pdf(xlow), color=\"red\", alpha=0.4)\n",
    "    ax.fill_between(xhigh,dist.pdf(xhigh), color=\"red\", alpha=0.4)\n",
    "    ax.spines[\"right\"].set_visible(False)\n",
    "    ax.spines[\"top\"].set_visible(False)\n",
    "    ax.spines[\"bottom\"].set_position(\"zero\")\n",
    "    return ax"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fd588fe9a90>"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "df = 1\n",
    "ax = show_dist();\n",
    "show_dist(dist=stats.t(df),ax=ax); # \n",
    "ax.legend([\"Standard normal distribution\", \"t distribution\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How do we calculate interval estimates?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Interval estimate of a population mean (when $\\sigma$ is unknown):**\n",
    "\n",
    "$$\\bar{x} \\pm t_{\\alpha/2}\\frac{s}{\\sqrt{n}}$$\n",
    "\n",
    "where\n",
    "\n",
    "$s = \\text{the sample standard deviation}$  \n",
    "$(1-\\alpha) = \\text{the confidence coefficient or level}$  \n",
    "$t_{\\alpha/2} = \\text{the t value providing an area $\\alpha/2$ in the upper tail of the (student's) t probability distribution}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "We have conducted a study with $70$ University sutdents to estimate the mean credit card debt for a population of University students. Given we don't know anything about the population standard deviation we have to work with our sample.  \n",
    "\n",
    "- we compute mean ($\\bar{x}$) and standard deviation ($s$) of the sample ($n = 70$)  \n",
    "- we then compute the margin of error and the interval estimates to determine a range of values that include the true (population) mean with a certainty of $90\\%$\n",
    "\n",
    "(Note: the sample data is fake data created using np.random)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [],
   "source": [
    "smpl = Datasets.credit_card_debt(n=70)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(9995.962718784866, 3843.978816977805)"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xbar, s = np.mean(smpl), np.std(smpl,ddof=1)\n",
    "xbar, s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.667238548542593"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha = .1\n",
    "n = len(smpl)\n",
    "df = n - 1\n",
    "t = stats.t(df).ppf(alpha/2)\n",
    "t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 9229.96106217, 10761.9643754 ])"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "error_margin = t*s/np.sqrt(n)\n",
    "ci = xbar + np.array((error_margin, -error_margin))\n",
    "ci"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With a probability of $90\\%$ the true population mean of University student's credit card debt lies within the above range. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that for a rather small sample size of $n = 70$ the difference between using the standard normal distribution and the t-distribution makes a difference. The interval estimate using the standard normal distribution is tighter. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ci(data, alpha=0.05, dist=\"norm\"):\n",
    "    '''\n",
    "    Calculate confidence interval for sample using standard normal distribution\n",
    "    '''\n",
    "    n = len(data)\n",
    "    s = np.std(data, ddof=1)\n",
    "    \n",
    "    if dist == \"norm\": \n",
    "        z = stats.norm(0,1).ppf(alpha/2)\n",
    "    elif dist == \"t\": \n",
    "        z = stats.t(len(data)-1).ppf(alpha/2)\n",
    "    else:\n",
    "        raise AttributeError(\"Dist must be either 'norm' or 't'\")\n",
    "    \n",
    "    error_margin = z*s/np.sqrt(n)\n",
    "    \n",
    "    return np.mean(data) + np.array((error_margin, -error_margin))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 9240.24566544, 10751.67977213]),\n",
       " array([ 9229.96106217, 10761.9643754 ]))"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ci(smpl, alpha=0.1,dist=\"norm\"), ci(smpl, alpha=0.1, dist=\"t\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This difference becomes negigible with a larger data set. Below you can see the difference using a sample size of $n = 500$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 9837.60538399, 10403.73801795]),\n",
       " array([ 9837.0788928 , 10404.26450914]))"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "smpl = Datasets.credit_card_debt(n=500)\n",
    "ci(smpl, alpha=0.1, dist=\"norm\"), ci(smpl, alpha=0.1, dist=\"t\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bootstrapping"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So far we have been using probabilities, theory and assumptions about distributions of populations and samples to calculate estimation intervals. \n",
    "\n",
    "It turns out there is another (for many people more intuitive way) of calculating the above estimation intervals: **bootstrapping**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Intuition for bootstrapping algorithm(s):**\n",
    "\n",
    "In theory we could see how much our sample estimator deviates from the true population parameter if we were able to draw many different sample from the population. We could then see how the sample statistic fluctuates. \n",
    "\n",
    "Bootstraping is build on the assumption that the sample should be a good approximation for the population (otherwise the sampling should be reconsidered because it is e.g. not representative for the population). \n",
    "\n",
    "Therefore, it is reasonable to create new samples by drawing from our sample. \n",
    "\n",
    "In doing so we can see how our estimate changes with different samples and draw conclusions from this. It turns out that this simple approach works surpringly well in many contexts. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Algorithm for Nonparametric bootstrap:**\n",
    "\n",
    "Given your sample data ${\\{z_i\\}_{i=1}^n}$\n",
    "\n",
    "For $b = 1 \\cdots B$:\n",
    "\n",
    "- resample *with replacement* $n$ observations ${\\{z_i^{b}\\}_{i=1}^n}$\n",
    "\n",
    "- calculate your estimate ${\\hat{\\theta}_b}$, e.g. the mean, using the resampled set of data\n",
    "\n",
    "then ${\\{\\hat{\\theta}_b\\}_{b=1}^B}$ is  an approximation for the sampling distribution for $\\hat{\\theta}$. \n",
    "\n",
    "Given the sample distribution we can then calculate the standard error (by calculating the standard deviation of the sample) and hence can calculate the estimate intervals."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at our (fake) credit debt example again.  Let's say we have a sample of $n=500$ students and we want to estimate the mean credit debt of (unknown student) population. We want to make sure that our interval estimate with a $90\\%$ probabiltiy contains the population mean. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Traditional approach from previous chapter**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 9837.60538399, 10403.73801795])"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 500\n",
    "smpl_original = Datasets.credit_card_debt(n)\n",
    "alpha = 0.1\n",
    "ci(smpl_original,alpha, dist=\"norm\" ) # Note we use standard normal distribution given the large sample size"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Bootstrapp approach:**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bootstrap(smpl, sims): \n",
    "    size_smpl = len(smpl) # size for samples drawn from original sample\n",
    "\n",
    "    estimates = np.empty(sims)\n",
    "    for i in range(sims):\n",
    "        new_smpl = np.random.choice(smpl, size=size_smpl, replace=True) # Draw new sample form orginal sample\n",
    "        theta = np.mean(new_smpl) # calculate estimate we want (here: mean)\n",
    "        estimates[i] = theta # append to list of estimates (to save distribution)\n",
    "    \n",
    "    return estimates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_sims = 10_000\n",
    "estimates = bootstrap(smpl_original, n_sims)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_density(estimates, title=\"Sample distribution for mean estimate\", xlabel=\"Mean credit debt\", ylabel=\"density\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use this distribution to calculate statistics, e.g. the standard error of our estimate by calculating the standard deviation of estimates. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [],
   "source": [
    "xbar = np.mean(estimates) # mean of our estimates (i.e. our point estimate of the true parameter)\n",
    "se = np.std(estimates) # standard error of our point estimate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [],
   "source": [
    "z = stats.norm().ppf(alpha/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 9835.7816785 , 10403.99657339])"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xbar + np.array((z,-z))*se"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 9839.9304876 , 10408.22327707])"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.quantile(estimates, [0.05,0.95])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that the bootstrap algorithm is an extremely powerfull tool to estimate uncertainties associated with an estimator. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Determining estimation intervals for proportions**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In our very first example we have used online spent data and calculated two estimators:\n",
    "\n",
    "1. mean of online spent\n",
    "\n",
    "2. proportion of people having children. \n",
    "\n",
    "We then looked at 1. to show how to calculate interval estimates; in particular we needed to calculate the standard error for the mean to do this. \n",
    "\n",
    "In order to calculate the standard error for the proportion, you may recall from undergrad statistics, we need to apply a different formula (because we are looking to calculate the standard deviation from a binomial distributed variable). \n",
    "\n",
    "the standard deviation of $p$ is given by:\n",
    "\n",
    "$$\\sigma_{p} = \\sqrt{\\frac{(p(1-p)}{n}}$$\n",
    "\n",
    "where\n",
    "\n",
    "$p = \\text{proportion}$   \n",
    "$n = \\text{the sample size}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using the bootsrap approch we can calculate interval estimates for the proportion directly without knowledge about the above formular given we approximate the sample distribution and can thus determine the required statistics directly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = Datasets.fake_spent_children()\n",
    "children = data[:,1]\n",
    "p = np.mean(children)\n",
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = np.mean(children)\n",
    "se = np.sqrt((p*(1-p)/n))\n",
    "z = 1.645 # proxy for stats.norm.ppf(0.95)\n",
    "p + z*np.array((-se,se))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_sims = 10_000\n",
    "estimators = bootstrap(children,n_sims )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "se = np.std(estimators)\n",
    "low, high = p -1.645*np.array((se,-se))\n",
    "low, high"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "_, ax = plot_density(estimators, title=\"Sample distribution for proportion of households with children ($p$)\\n(confidence limits determined using bootstrap approach)\");\n",
    "ax.vlines(low, 0, 20, linestyle=\"--\")\n",
    "ax.vlines(high, 0, 20, linestyle=\"--\")\n",
    "ax.annotate(f\"Lower limit {low:.4f}\",(low, 22),ha=\"right\");\n",
    "ax.annotate(f\"Upper limit {high:.4f}\",(high, 22),ha=\"left\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Determine confidence interval for linear regression coefficient**\n",
    "\n",
    "We can use the bootstrap algorithm for model parameters such as regression coefficients as well. This shows the flexibility and power of using bootstrap analysis. \n",
    "\n",
    "We will use **advertising** data from [ISLR](http://faculty.marshall.usc.edu/gareth-james/ISL/data.html) for our next example. \n",
    "\n",
    "For our example the data contains\n",
    "\n",
    "- data on TV advertising spent\n",
    "\n",
    "- sales data\n",
    "\n",
    "We run a simle (linear) regression model of the following type:\n",
    "\n",
    "$$\\hat{y} \\approx \\beta_0 + \\beta_1x$$\n",
    "\n",
    "where:\n",
    "\n",
    "$\\hat{y} = $ sales data  \n",
    "$x = $ tv advertising spent\n",
    "\n",
    "We want to understand if there is a relation between money spent on tv advertising and generated sales. If not, we should stop or alter tv advertisment. In order to determine the relation we estimate $\\beta_1$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = Datasets.advertising() \n",
    "tv, sales = data[:,0], data[:,1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can easily calculate coefficients and additional output for regression analysis using the below \"helper\" function which uses special statistical packages for these purposes (we will introduce them in next lecture)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.612</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.610</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   312.1</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Fri, 06 Nov 2020</td> <th>  Prob (F-statistic):</th> <td>1.47e-42</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>20:03:36</td>     <th>  Log-Likelihood:    </th> <td> -519.05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>   200</td>      <th>  AIC:               </th> <td>   1042.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>   198</td>      <th>  BIC:               </th> <td>   1049.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     1</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "    <td></td>       <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th> <td>    7.0326</td> <td>    0.458</td> <td>   15.360</td> <td> 0.000</td> <td>    6.130</td> <td>    7.935</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td>    0.0475</td> <td>    0.003</td> <td>   17.668</td> <td> 0.000</td> <td>    0.042</td> <td>    0.053</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 0.531</td> <th>  Durbin-Watson:     </th> <td>   1.935</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.767</td> <th>  Jarque-Bera (JB):  </th> <td>   0.669</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.089</td> <th>  Prob(JB):          </th> <td>   0.716</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 2.779</td> <th>  Cond. No.          </th> <td>    338.</td>\n",
       "</tr>\n",
       "</table><br/><br/>Warnings:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       0.612\n",
       "Model:                            OLS   Adj. R-squared:                  0.610\n",
       "Method:                 Least Squares   F-statistic:                     312.1\n",
       "Date:                Fri, 06 Nov 2020   Prob (F-statistic):           1.47e-42\n",
       "Time:                        20:03:36   Log-Likelihood:                -519.05\n",
       "No. Observations:                 200   AIC:                             1042.\n",
       "Df Residuals:                     198   BIC:                             1049.\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "const          7.0326      0.458     15.360      0.000       6.130       7.935\n",
       "x1             0.0475      0.003     17.668      0.000       0.042       0.053\n",
       "==============================================================================\n",
       "Omnibus:                        0.531   Durbin-Watson:                   1.935\n",
       "Prob(Omnibus):                  0.767   Jarque-Bera (JB):                0.669\n",
       "Skew:                          -0.089   Prob(JB):                        0.716\n",
       "Kurtosis:                       2.779   Cond. No.                         338.\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from utils import lreg_summary\n",
    "\n",
    "lreg_summary(tv,sales)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that $\\beta_1$ (denoted $x_1$) is $0.0475$ with a interval estimates of $0.042$ to $0.053$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Let's try to recalculate the confidence intervals for $x1$ using a bootstrap approach.**\n",
    "\n",
    "Our estimator in this case is not the mean or a proportion but $\\beta_1$ which - you may recal from undergraduate statistics - can be calculated as follows:\n",
    "\n",
    "$$\\hat{\\beta}_1 = \\frac{\\sum_{i=1}^n (x_i - \\bar{x})(y_i-\\bar{y})}{\\sum_{i=1}^n (x_i - \\bar{x})^2}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will have to do the following:\n",
    "\n",
    "1. implement a function that estimates $\\beta_1$\n",
    "\n",
    "2. bootstrap $beta_1$ repeatedly to generate a distribution of sampled estimates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 1. create function for beta1\n",
    "def beta1(X,Y):\n",
    "    return np.sum((X - np.mean(X))*(Y-np.mean(Y))) / np.sum((X - np.mean(X))**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 2. bootstrap\n",
    "n = len(data)\n",
    "sims = 5000\n",
    "betas = np.empty(sims)\n",
    "for i in range(sims):\n",
    "    rows = np.random.choice(range(n),n)\n",
    "    X,Y = data[rows,0], data[rows,1]\n",
    "    betas[i] = beta1(X,Y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plot_density(betas);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Calculate confidence interval from sample distribution\n",
    "alpha = 0.05\n",
    "z = stats.norm.ppf(alpha/2) # alpha is 0.05\n",
    "np.mean(betas) + np.array((z,-z))*np.std(betas)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We were able to confirm the confidence intervals (although you may not recall how to actually calculate them) by using a bootstrap approach. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "While for our examples most statistical package have \"out of the box\" functionality, this approach gives us confidence to calculate uncertainty ranges even when using complex models or estimators where no out of the box functionality might be available."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Conclusion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Advantage**\n",
    "\n",
    "- does not make strong assumptions about population or sample distributions\n",
    "\n",
    "- often theoretical sampling distributions are not normally distributed\n",
    "\n",
    "- is to understand without knowledge in probability theory / about probability distributions\n",
    "\n",
    "- no knowledge about formulas for standard errors required \n",
    "\n",
    "**Disadvantage**\n",
    "\n",
    "- not feasible for large samples because of computational effort \n",
    "\n",
    "- computational effort when sometimes a simple formular yields an answer"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hypothesis testing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Please refer to Imai (2017), p. 342 to 362 for background information on hypothesis testing. It is generally assumed that you are familiar with hypothesis testing. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Examples using python"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below are some examples of how to calculate simple t-tests. We will not cover this in detail as these standard statistics are calculated by statistical packages we will be using when calculating more difficult models. \n",
    ":\n",
    "To get an impression how to conduct such tests in python see below"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Two-tailed test (one sample)**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example: It is generally assumed that stock returns on average have a daily return of $0$. \n",
    "\n",
    "Let's build our hypothesis:\n",
    "\n",
    "\\begin{equation} \\label{eq1}\n",
    "\\begin{split}\n",
    "& H_0: \\mu = 0 \\\\\n",
    "& H_1: \\mu \\neq 0 \n",
    "\\end{split}\n",
    "\\end{equation}\n",
    "\n",
    "where:\n",
    "\n",
    "$\\mu = $ daily stock return\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's do this in numpy from scratch. Recall that the test statistic is given by:\n",
    "\n",
    "$$z = \\frac{\\bar{x}-\\mu_0}{s/\\sqrt{n}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Load Dax 30 daily data\n",
    "dax = Datasets.dax_daily() # prices\n",
    "ret = dax[1:] / dax[:-1] - 1 # calculate returns\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mu = 0 # our assumed population mean of daily returns\n",
    "xbar = np.mean(ret)\n",
    "n = len(ret)\n",
    "se = np.std(ret,ddof=1) / np.sqrt(n)\n",
    "\n",
    "\n",
    "z = (xbar - mu) / se\n",
    "pvalue = (1-stats.t(len(ret) -1).cdf(t))*2 # multiply by two because it is a two-sided t-test\n",
    "z, pvalue"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This can also be done using `scipy.stats.ttest_..`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = 3\n",
    "n = 10\n",
    "s / np.sqrt(n) == np.sqrt(s**2 / n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "tvalue, pvalue = stats.ttest_1samp(ret, mu) # t-statistic and pvalue\n",
    "tvalue, pvalue"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Two-Sample Test**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example: let's consider a medical trial where we have to groups the trial group (receiving a new medicine) and a control group (receiving a placebo). We now want to understand of the two groups differ regarding a specific parameter. \n",
    "\n",
    "Let's build our hypothesis:\n",
    "\n",
    "\\begin{equation} \\label{eq1}\n",
    "\\begin{split}\n",
    "& H_0: \\mu_0 = \\mu_1 \\\\\n",
    "& H_1: \\mu_0 \\neq \\mu_1 \n",
    "\\end{split}\n",
    "\\end{equation}\n",
    "\n",
    "where:\n",
    "\n",
    "$\\mu = $ is e.g. the mean of a specific medical parameter (e.g. blood sugar). \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's implement this in python. Let's create two groups with random data both $\\sim$ $N(100,10)$. A two-sample test should yield that both groups do not differ this is $H_0$ cannot be rejected. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "trial_group = stats.norm(100,10).rvs(1000)\n",
    "control_group = stats.norm(100,10).rvs(1000)\n",
    "stats.ttest_ind(trial_group, control_group)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that this is indeed the case given the pvalue of $69\\%$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Caution with p-values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In 2016 an article in *The American Statistician* ([here](https://amstat.tandfonline.com/doi/full/10.1080/00031305.2016.1154108)) criticized the \n",
    "\n",
    "\"[...] widespread use of “statistical significance” (generally interpreted as “p 0.05”) as a license for making a claim of a scientific finding (or implied truth) leads to considerable distortion of the scientific process.\"\n",
    "\n",
    "Unfortunately, this practice is still quite common in research. It should be noted that research does not mean \"finding effects with a p-value below 5%\". While there is nothing wrong or false about the concept of p-values it is sometimes misused. \n",
    "\n",
    "This is due to several reasons:\n",
    "\n",
    "a. p value tends to decrease with largers samples, i.e. the larger the sample the higher the probability of finding a \"significant effect\" (ceteris paribus)\n",
    "\n",
    "b. focus is too much on identifying signficiance while confirming the null hypothesis should actually be considered as a finding as well. Instead behaviors termed \"p-hacking\" becomes common ([see here](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.590.4360&rep=rep1&type=pdf))\n",
    "\n",
    "c. focus on one value is misleading; instead at least ranges should be reported; at best additional tests and indication for confirmation of research hypotheses should be provided. \n",
    "\n",
    "d. statistical significance is often confused with practical relevance \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Why p value may be misleading given a large sample size**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Strategy of p value hacking is dangerous and may lead to wrong research findings. \n",
    "\n",
    "In the following we will see that larger samples may lead to statistical effects although the practical relevance of theses statistical significant findings are practically irrelevant. \n",
    "\n",
    "Let's consider the following example (from Daniela Witten): \n",
    "\n",
    "- we are testing whether a mean of a random varialbe $X \\sim N(0,1)$ is equal to $0$\n",
    "\n",
    "- we draw a sample and find that the mean ($\\bar{x}$) is slightly different from 0 (e.g. 0.000001)\n",
    "\n",
    "- Practically we can conclude for (almost all) practical research hypothesis that $H_0$ can be confirmed and that the mean of a sample is $0$. \n",
    "\n",
    "- However, depending on the size of your sample correctly calculated p-value of your estimator will actually indicate rejection of said hypothesis. \n",
    "\n",
    "Let's look at an example:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given a large sample size we can use the normal distribution and the z-value to test for significance. \n",
    "\n",
    "Recall that the test statistic is given by:\n",
    "\n",
    "$$z = \\frac{\\bar{x}-\\mu_0}{s/\\sqrt{n}}$$\n",
    "\n",
    "As per our assumption \n",
    "\n",
    "- $s$ will be 1\n",
    "\n",
    "- $\\mu_0$ will be zero\n",
    "\n",
    "This gives us:\n",
    "\n",
    "$$z = \\bar{x}*\\sqrt{n} = 0.000001 * \\sqrt{n}$$\n",
    "\n",
    "If the sample size is big (e.g. $10^6$) - which in the age of big data cannot be considered an exception anymore - the z value will become very big (yielding a low p-value) and leads to the conclusion that $H_0$ needs to be rejected. \n",
    "\n",
    "However, for all practical reasons $\\bar{x}$ is very close to zero and we only observe this finding given the large sample size. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 10e6\n",
    "xbar = 0.000001\n",
    "zvalue = n*xbar\n",
    "zvalue"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pvalue = 1 - stats.norm(0,1).cdf(zvalue)\n",
    "pvalue"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This does not mean that p-values are wrong. Statistically speaking the result is correct. However, it should be noted that with large sample size we are to see statistical effects that may in fact no indicate practical findings given the deviance from $H_0$ is negligible.  "
   ]
  }
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