p16i/gist:622beb58d06c649e082a173993998c0e

## gistfile1.txt
{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "Stats-W6-Hypothesis-Testing",
      "provenance": [],
      "collapsed_sections": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
  },
  "cells": [
    {
      "cell_type": "code",
      "metadata": {
        "id": "WOKdiZY_AwLT",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "import numpy as np\n",
        "from scipy.stats import hypergeom\n",
        "from matplotlib import pyplot as plt"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "v7TvEOTIA6uZ",
        "colab_type": "text"
      },
      "source": [
        "# Fisher's Exact Test\n",
        "\n",
        "Consider the lady drinking tea situation. The lady claims that she can distinguish whether tea is made by pouring milk first or vice versa.\n",
        " \n",
        "Consider the contingency table $T$ below.\n",
        "\n",
        "|Predicted / Actual | Tea first | Milk first | # |\n",
        "|-------------------:|:-----------:|:------------:|:----:|\n",
        "| Tea first |  a |  b  | a + b |\n",
        "| Milk first | c | d  | c + d |\n",
        "|     | a + c | b + d | \n",
        "\n",
        "We would like to test whether the true positive prediction of Tea happens by chance. This test can be done by using Fisher's Exact Test. The test is based on the foundation that the true positive prediction variable $X$ is hypergeometrically distributed. Let $r=a+c$, $m=a+b$, and $N=a+b+c+d$. We can show that there are:\n",
        "- $r \\choose a$ possibilities to have **correct** predictions when predicting tea (True Positive);\n",
        "- $N - r \\choose m - a$ possibilities to have **wrong** predictions when predicting tea (False Positive);\n",
        "- $N \\choose m$ possibiles for any predictions.\n",
        "\n",
        "Therefore, the probability that we observe $T$ is \n",
        "\n",
        "$$\n",
        "P( T ) = P(X=a)  = \\frac{ {r \\choose a} {N - r \\choose m- a}}{N \\choose m}, \n",
        "$$\n",
        "where $a$ is the number of observed successes and $N, r, m$ are the parameters of the hypergeometric distribution:\n",
        "- $N$ is the populatino size;\n",
        "- $r$ is the number of success states in the population;\n",
        "- $m$ is the number of draws.\n",
        "\n",
        "To get the significant value of the test, we need to look at the probablity that we observe the table and other equally extreme tables by consider all the tables that have the same marginal distributions, which are $a+b$, $c+d$, $a+c$, and $c+d$ respectively. For two-sided test, the significant value is the sum of probablities of all configurations that are equal or below the one of the observed data."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "fQhm9Pj9dYO9",
        "colab_type": "code",
        "outputId": "0421bd6a-b63a-47eb-fa03-46a7985f388a",
        "cellView": "form",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 706
        }
      },
      "source": [
        "#@title Contingency Table { run: \"auto\" }\n",
        "\n",
        "a = 7 #@param {type: \"slider\", min: 1, max: 20}\n",
        "b = 10 #@param {type: \"slider\", min: 1, max: 20}\n",
        "c = 12 #@param {type: \"slider\", min: 1, max: 20}\n",
        "d = 14 #@param {type: \"slider\", min: 1, max: 20}\n",
        "\n",
        "tab = [\n",
        "  [18, 2],\n",
        "  [11, 9]\n",
        "]\n",
        "def fisher_exact_test_two_sided(tab):\n",
        "  a = tab[0][0]\n",
        "  b = tab[0][1]\n",
        "  c = tab[1][0]\n",
        "  d = tab[1][1]\n",
        "\n",
        "  print(\"| %2d | %2d |\" % (a, b))\n",
        "  print(\"-----+-----\")\n",
        "  print(\"| %2d | %2d |\" % (c, d))\n",
        "\n",
        "  N = a + b + c + d\n",
        "  r = a + c\n",
        "  m = a + b\n",
        "\n",
        "  print('r: ', r)\n",
        "  print('m: ', m)\n",
        "  print('N: ', N)\n",
        "\n",
        "  rv = hypergeom(N, r, m)\n",
        "  threshold = rv.pmf(a)\n",
        "\n",
        "  p_value = 0\n",
        "  print(\"Equally or more extreme configurations\")\n",
        "  print(\"a, b, c, d \\t Probability\")\n",
        "  for i in range(m):\n",
        "    aa = m - i\n",
        "    cc = r - aa\n",
        "    dd = N - (aa + i + cc)\n",
        "\n",
        "    tt = rv.pmf(aa)\n",
        "    if tt <= threshold:\n",
        "      print(aa, i, cc, dd, \"\\t\", tt)\n",
        "      p_value += tt\n",
        "\n",
        "    if dd == 0:\n",
        "      break\n",
        "\n",
        "  print(\"Two-sided p-value: %.4f\" % p_value)\n",
        "  return rv, (N, r, m, a, p_value)\n",
        "\n",
        "rv, (N, r, m, a, p_value) = fisher_exact_test_two_sided([\n",
        "  [a, b],\n",
        "  [c, d]                                                \n",
        "]);\n",
        "\n",
        "probs = []\n",
        "x = np.array(list(range(min(m, r)+1)))\n",
        "for i in x:\n",
        "  probs.append(rv.pmf(i))\n",
        "probs = np.array(probs)\n",
        "abv_ix = np.argwhere(probs > rv.pmf(a))\n",
        "blw_ix = np.argwhere(probs <= rv.pmf(a))\n",
        "\n",
        "if len(abv_ix) > 0:\n",
        "  plt.bar(x[abv_ix].reshape(-1), probs[abv_ix].reshape(-1), label=\"Non-extreme settings\")\n",
        "\n",
        "if len(blw_ix) > 0:\n",
        "  plt.bar(x[blw_ix].reshape(-1), probs[blw_ix].reshape(-1), color=\"red\", label=\"Equally ore more extreme settings\")\n",
        "plt.vlines(a, 0, max(probs), color=\"red\", label=\"Observed data\", linestyles=\"dotted\")\n",
        "plt.title(\"Fisher's 2-sided Exact Test: p=%.4f\" % p_value)\n",
        "plt.legend();"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "|  7 | 10 |\n",
            "-----+-----\n",
            "| 12 | 14 |\n",
            "r:  19\n",
            "m:  17\n",
            "N:  43\n",
            "Equally or more extreme configurations\n",
            "a, b, c, d \t Probability\n",
            "17 0 2 24 \t 4.060102347920716e-10\n",
            "16 1 3 23 \t 5.521739193172201e-08\n",
            "15 2 4 22 \t 2.5400000288592035e-06\n",
            "14 3 5 21 \t 5.5880000634902585e-05\n",
            "13 4 6 20 \t 0.0006845300077775567\n",
            "12 5 7 19 \t 0.005085080057776125\n",
            "11 6 8 18 \t 0.02415413027443666\n",
            "10 7 9 17 \t 0.07591298086251508\n",
            "9 8 10 16 \t 0.16131508433284467\n",
            "7 10 12 14 \t 0.23464012266595552\n",
            "6 11 13 13 \t 0.1608023218270186\n",
            "5 12 14 12 \t 0.0746582208482587\n",
            "4 13 15 11 \t 0.02297176026100261\n",
            "3 14 16 10 \t 0.004512310051268394\n",
            "2 15 17 9 \t 0.0005308600060315754\n",
            "1 16 18 8 \t 3.317875037697345e-05\n",
            "Two-sided p-value: 0.7654\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXoAAAEICAYAAABRSj9aAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "vNYoS_1W3thv",
        "colab_type": "text"
      },
      "source": [
        "## One-sample T-Test and Bayesian T-test\n",
        "\n",
        "Using data from : https://drive.google.com/drive/folders/111KtiKFTMPOy1_kQOWx_OSXQtmPMLmB4. \n",
        "\n",
        "**Question**:\n",
        "\n",
        "Use the one-sample t-test data set from the JASP library (also available here: ) to perform a Frequentist and Bayesian t-test (two-tailed). H_0 (Null hypothesis): The sample mean is not significantly different from the population mean of 178cm. What is the p-value and the Bayes factor? In the Bayesian approach use the default value for the scale parameter (= 0.707). To change this value collapse the 'Prior' field. What happens with the Bayes factor if you increase/decrease this value? Have a look at the respective plots. In our lectures we have not yet discussed the role of priors we will do so in the Monday session.\n",
        "\n",
        "**T-Test**\n",
        "![](https://i.imgur.com/vyYNYuW.png)\n",
        "\n",
        "**Bayes Factor**\n",
        "![](https://i.imgur.com/nsXq739.png)\n",
        "\n",
        "**Influence of Cauchy Prior's scale**\n",
        "![](https://i.imgur.com/l3QuVzl.png)"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "vVIfPQ9M5D_9",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        ""
      ],
      "execution_count": 0,
      "outputs": []
    }
  ]
}