p-values_simulations.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "from scipy import stats\n",
    "import matplotlib.pyplot as plt "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following code snippets are implemented to reproduce the results of the blog [UNDERSTANDING P-VALUES VIA SIMULATIONS](https://researchutopia.wordpress.com/2013/11/10/understanding-p-values-via-simulations/). First, we simulate p-values distribution under the assumption that the null is true."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Set the number of simulated experiments\n",
    "num_sims = 100000\n",
    "# Set the zero array to store computed p-values\n",
    "p_array = np.zeros(num_sims)\n",
    "mu = 100\n",
    "sigma = 20\n",
    "size = 100\n",
    "for i in range(num_sims):\n",
    "    x = np.random.normal(loc=mu, scale=sigma, size=size)\n",
    "    y = np.random.normal(loc=mu, scale=sigma, size=size)\n",
    "    _, p_array[i] = stats.ttest_ind(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(p_array, bins=100)\n",
    "plt.title(\"Histogram of p-values under the null\")\n",
    "plt.xlabel(\"Observed p-value\")\n",
    "plt.ylabel(\"Frequency\")\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Set the number of simulated experiments\n",
    "num_sims = 100000\n",
    "# Set the zero array to store computed p-values\n",
    "p_array = np.zeros(num_sims)\n",
    "mu = 100\n",
    "sigma = 20\n",
    "size = 500\n",
    "for i in range(num_sims):\n",
    "    x = np.random.normal(loc=mu, scale=sigma, size=size)\n",
    "    y = np.random.normal(loc=mu, scale=sigma, size=size)\n",
    "    _, p_array[i] = stats.ttest_ind(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(p_array, bins=100)\n",
    "plt.title(\"Histogram of p-values under the null\")\n",
    "plt.xlabel(\"Observed p-value\")\n",
    "plt.ylabel(\"Frequency\")\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Second, we simulate p-values distribution under the assumption that the null is false."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Set the number of simulated experiments\n",
    "num_sims = 100000\n",
    "# Set the zero array to store computed p-values\n",
    "p_array = np.zeros(num_sims)\n",
    "mu = 100\n",
    "sigma = 20\n",
    "size = 100\n",
    "for i in range(num_sims):\n",
    "    x = np.random.normal(loc=mu+3, scale=sigma, size=size)\n",
    "    y = np.random.normal(loc=mu, scale=sigma, size=size)\n",
    "    _, p_array[i] = stats.ttest_ind(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(p_array, bins=50)\n",
    "plt.title(\"Histogram of p-values (true group difference)\")\n",
    "plt.xlabel(\"Observed p-value\")\n",
    "plt.ylabel(\"Frequency\")\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Set the number of simulated experiments\n",
    "num_sims = 100000\n",
    "# Set the zero array to store computed p-values\n",
    "p_array = np.zeros(num_sims)\n",
    "mu = 100\n",
    "sigma = 20\n",
    "size = 500\n",
    "for i in range(num_sims):\n",
    "    x = np.random.normal(loc=mu+3, scale=sigma, size=size)\n",
    "    y = np.random.normal(loc=mu, scale=sigma, size=size)\n",
    "    _, p_array[i] = stats.ttest_ind(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(p_array, bins=50)\n",
    "plt.title(\"Histogram of p-values (true group difference)\")\n",
    "plt.xlabel(\"Observed p-value\")\n",
    "plt.ylabel(\"Frequency\")\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}