AB_testing_A_complete_guide_to_statistical_testing.ipynb

{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A/B Testing — A complete guide to statistical testing\n",
    "### https://francescocasalegno.github.io/blog/post_abtesting.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from scipy.special import binom\n",
    "from scipy.stats import chi2_contingency, mannwhitneyu, norm, t, ttest_ind\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 1. Discrete Metrics\n",
    "Let's consider first discrete metrics, e.g. click-though rate. We randomly show visitors one of two possible designs of an advertisement, and based on how many of them click on it we need to determine whether our data significantly contradict the hypothesis that the two designs are equivalently efficient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- Observations:\n",
      "  - Version A: = [1 0 0 1 1 1 1 0 0 0 1 0 0 1 1]\n",
      "  - Version B: = [0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1]\n",
      "\n",
      "- Contingency table:\n"
     ]
    },
    {
     "data": {
      "text/html": [
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       "\n",
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       "        vertical-align: top;\n",
       "    }\n",
       "\n",
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       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>A</th>\n",
       "      <th>B</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>click</th>\n",
       "      <td>7</td>\n",
       "      <td>15</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>no click</th>\n",
       "      <td>8</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          A   B\n",
       "click     7  15\n",
       "no click  8   4"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(42)\n",
    "\n",
    "x = np.random.binomial(n=1, p=0.6, size=15)\n",
    "y = np.random.binomial(n=1, p=0.4, size=19)\n",
    "\n",
    "_, (a, c) = np.unique(x, return_counts=True)\n",
    "_, (b, d) = np.unique(y, return_counts=True)\n",
    "\n",
    "df = pd.DataFrame(data=[[a, b], [c, d]], \n",
    "                 index=[\"click\", \"no click\"], \n",
    "                 columns=[\"A\", \"B\"])\n",
    "m = df.values\n",
    "\n",
    "print(\"- Observations:\")\n",
    "print(f\"  - Version A: = {x}\")\n",
    "print(f\"  - Version B: = {y}\")\n",
    "print(\"\")\n",
    "print(\"- Contingency table:\")\n",
    "display(df)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 1.1 Fisher's exact test\n",
    "Since we have a 2x2 contingency table we can use Fisher's exact test to compute an exact p-value and test our hypothesis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def hypergeom(k, K, n, N):\n",
    "    \"\"\"Probability mass funciton of the hypergeometric distribution.\"\"\"\n",
    "    return binom(K, k) * binom(N-K, n-k) / binom(N, n)\n",
    "\n",
    "\n",
    "def fisher_prob(m):\n",
    "    \"\"\"Probability of a given observed contingency table according to Fisher's exact test.\"\"\"\n",
    "    ((a, b), (c ,d)) = m\n",
    "    k = a\n",
    "    K = a+b\n",
    "    n = a+c\n",
    "    N = a+b+c+d\n",
    "    return hypergeom(k, K, n, N)\n",
    "\n",
    "def fisher_probs_histogram(m):\n",
    "    \"\"\"Computes prob mass function histogram accroding to Fisher's exact test.\"\"\"\n",
    "    neg_val = -min(m[0,0], m[1,1])\n",
    "    pos_val = min(m[1,0], m[1,0])\n",
    "    probs = []\n",
    "    for k in range(neg_val, pos_val+1):\n",
    "        m1 = m + np.array([[1, -1], [-1, 1]]) * k\n",
    "        probs.append(fisher_prob(m1))\n",
    "    return probs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- Fisher's exact test: p-val = 7.5%\n"
     ]
    },
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 600x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "bars_h = np.array(fisher_probs_histogram(m))\n",
    "\n",
    "f, ax = plt.subplots(figsize=(6, 3))\n",
    "ii = np.arange(len(bars_h))\n",
    "ax.bar(ii, bars_h)\n",
    "idxs = bars_h <= fisher_prob(m)\n",
    "ax.bar(ii[idxs], bars_h[idxs], color='r')\n",
    "ax.set_ylabel(\"prob density\")\n",
    "p_val = bars_h[idxs].sum()\n",
    "neg_val = -min(m[0,0], m[1,1])\n",
    "pos_val = min(m[1,0], m[1,0])\n",
    "ax.bar(ii[-neg_val], bars_h[-neg_val], color='orange')\n",
    "\n",
    "ax.set_xticks(ii)\n",
    "ax.set_xticklabels(np.arange(neg_val, pos_val+1))\n",
    "f.tight_layout()\n",
    "print(f\"- Fisher's exact test: p-val = {100*p_val:.1f}%\")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 1.2 Pearson's chi-squared test\n",
    "Fisher's exact test has the important advantage of computing exact p-values. But if we have a large sample size, it may be computationally inefficient. In this case, we can use Pearson's chi-squared test to compute an approximate p-value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- Pearson's chi-squared t-test:\n",
      "   - χ2 value: 3.825\n",
      "   - p-value: 5.1%\n"
     ]
    }
   ],
   "source": [
    "chi2_val, p_val = chi2_contingency(m, correction=False)[:2]\n",
    "\n",
    "print(\"- Pearson's chi-squared t-test:\")\n",
    "print(f\"   - χ2 value: {chi2_val:.3f}\")\n",
    "print(f\"   - p-value: {p_val*100:.1f}%\")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2. Continuous metrics\n",
    "Let's now consider the case of a continuous metrics, e.g. average revenue per user. We randomly show visitors of our website one of two possible layouts of products for sale, and based on how much revenue each user generated in a month we want to determine whether our data significantly contradict the hypothesis that the two website layouts are equivalently efficient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- Observations:\n",
      "  - Version A: = [200 150 250 350 150 150 350 250 150 250 150 150 200   0   0 100  50]\n",
      "  - Version B: = [300 150 150 400 250 250 150 200 250 150 300 200 250 200]\n",
      "\n",
      "- Distribution plot:\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(42)\n",
    "\n",
    "n_x, n_y = 17, 14\n",
    "\n",
    "d1 = norm(loc=200, scale=100)\n",
    "d2 = norm(loc=280, scale=90)\n",
    "\n",
    "disc = 50\n",
    "x = (d1.rvs(size=n_x) / disc).astype(int) * disc\n",
    "y = (d2.rvs(size=n_y) / disc).astype(int) * disc\n",
    "\n",
    "\n",
    "print(\"- Observations:\")\n",
    "print(f\"  - Version A: = {x}\")\n",
    "print(f\"  - Version B: = {y}\")\n",
    "print(\"\")\n",
    "print(f\"- Distribution plot:\")\n",
    "f, ax = plt.subplots(figsize=(6, 3))\n",
    "for i, (x_, l_, c_) in enumerate(zip([x, y], [\"A\", \"B\"], [\"tab:blue\", \"tab:olive\"])):\n",
    "    v, c = np.unique(x_, return_counts=True)\n",
    "    ax.bar(v-5+10*i, c, width=10, label=l_, color=c_)\n",
    "\n",
    "ax.set_xlabel(\"purchase in $\")\n",
    "ax.set_ylabel(\"count\")\n",
    "ax.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_pval(distribution, t_val, xlims=(-5, 5), ylims=(0, 0.5)):\n",
    "    xxx = np.linspace(*xlims, 1000)\n",
    "    f, ax = plt.subplots(figsize=(4,3))\n",
    "    ax.plot(xxx, distribution.pdf(xxx))\n",
    "    ax.set_ylim(ylims)\n",
    "    ax.vlines(t_val, 0, distribution.pdf(t_val), color='orange')\n",
    "    ax.plot(t_val, distribution.pdf(t_val), 'o', color='orange')\n",
    "    xp = xxx <= t_val\n",
    "    ax.fill_between(xxx[xp], xxx[xp] * 0, distribution.pdf(xxx[xp]), color='r')\n",
    "    xp = xxx >= -t_val\n",
    "    ax.fill_between(xxx[xp], xxx[xp] * 0, distribution.pdf(xxx[xp]), color='r')\n",
    "    ax.set_ylabel(\"prob denisty\")\n",
    "    f.tight_layout()\n",
    "    return f, ax"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### 2.1 Z-test\n",
    "##### The Z-test can be applied under the following assumptions.\n",
    "\n",
    "* The observations are normally distributed (or the sample size is large).\n",
    "* The sampling distributions have known variance σ_X and σ_Y.\n",
    "\n",
    "Under the above assumptions, the Z-test relies on the observation that the following Z statistic has a standard normal distribution.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- Z-test:\n",
      "   - z value: -1.697\n",
      "   - p-value: 9.0%\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 400x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Known standard deviations\n",
    "s_x = 100\n",
    "s_y = 90\n",
    "\n",
    "# Z value\n",
    "z_val = (x.mean() - y.mean()) / np.sqrt(s_x**2/n_x + s_y**2/n_y)\n",
    "\n",
    "# Test statistic distribution under null hypothesis H0\n",
    "stat_distrib = norm(loc=0, scale=1)\n",
    "\n",
    "# p-value\n",
    "p_val = stat_distrib.cdf(z_val) * 2\n",
    "\n",
    "print(\"- Z-test:\")\n",
    "print(f\"   - z value: {z_val:.3f}\")\n",
    "print(f\"   - p-value: {p_val*100:.1f}%\")\n",
    "plot_pval(stat_distrib, z_val);"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2.2 Student's t-test\n",
    "In most cases, the variances of the sampling distributions are unknown, so that we need to estimate them. Student's t-test can then be applied under the following assumptions.\n",
    "\n",
    "* The observations are normally distributed (or the sample size is large).\n",
    "* The sampling distributions have \"similar\" variances σX ≈ σY.\n",
    "\n",
    "Under the above assumptions, Student's t-test relies on the observation that the following t statistic has a Student's t distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- Student's t-test:\n",
      "   - nu: 29.000\n",
      "   - t value: -1.789\n",
      "   - p-value: 8.4%\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 400x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Sample variances (w. Bessel correction)\n",
    "s_x = np.sqrt(np.var(x, ddof=1))\n",
    "s_y = np.sqrt(np.var(y, ddof=1))\n",
    "\n",
    "# Pooled variance\n",
    "s_p = np.sqrt(((n_x-1) * s_x**2 +\n",
    "               (n_y-1) * s_y**2)\n",
    "              / (n_x + n_y - 2))\n",
    "\n",
    "# Test statistic distribution under null hypothesis H0\n",
    "dofs = n_x + n_y - 2\n",
    "stat_distrib = t(df=dofs, loc=0, scale=1)\n",
    "\n",
    "# t value\n",
    "t_val = (x.mean() - y.mean()) / (s_p * np.sqrt(1/n_x + 1/n_y))\n",
    "\n",
    "# p-value\n",
    "p_val = stat_distrib.cdf(t_val) * 2\n",
    "\n",
    "print(\"- Student's t-test:\")\n",
    "print(f\"   - nu: {dofs:.3f}\")\n",
    "print(f\"   - t value: {t_val:.3f}\")\n",
    "print(f\"   - p-value: {p_val*100:.1f}%\")\n",
    "plot_pval(stat_distrib, z_val);"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2.3 Welch's t-test\n",
    "In most cases Student's t test can be effectively applied with good results. However, it may rarely happen that its second assumption (similar variance of the sampling distributions) is violated. In that case, we cannot compute a pooled variance and rather than Student's t test we should use Welch's t-test.\n",
    "\n",
    "This test operates under the same assumptions of Student's t-test but removes the requirement on the similar variances. Then, we can use a slightly different t statistic, which also has a Student's t distribution, but with a different number of degrees of freedom ν."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- Welch' t-test:\n",
      "   - nu: 28.505\n",
      "   - t value: -1.848\n",
      "   - p-value: 7.5%\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 400x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Sample variances (w. Bessel correction)\n",
    "s_x = np.sqrt(np.var(x, ddof=1))\n",
    "s_y = np.sqrt(np.var(y, ddof=1))\n",
    "\n",
    "# Denominator (not a pooled variance!)\n",
    "s_d = np.sqrt(s_x**2/n_x + s_y**2/n_y)\n",
    "    \n",
    "    \n",
    "# Test statistic distribution under null hypothesis H0\n",
    "dofs = s_d**4 / ((s_x**2/n_x)**2/(n_x-1) +\n",
    "                 (s_y**2/n_y)**2/(n_y-1))\n",
    "stat_distrib = t(df=dofs, loc=0, scale=1)\n",
    "\n",
    "# t value\n",
    "t_val = (x.mean() - y.mean()) / s_d\n",
    "\n",
    "# p-value\n",
    "p_val = stat_distrib.cdf(t_val) * 2\n",
    "\n",
    "print(\"- Welch' t-test:\")\n",
    "print(f\"   - nu: {dofs:.3f}\")\n",
    "print(f\"   - t value: {t_val:.3f}\")\n",
    "print(f\"   - p-value: {p_val*100:.1f}%\")\n",
    "plot_pval(stat_distrib, z_val);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2.4 Mann-Whitney U test\n",
    "This test makes the least assumptions about the nature of our observations, as it is fully nonparametric. The idea of Mann-Whitney U test is to compute the U statistic.\n",
    "\n",
    "The values of this test statistic are tabulated, as the distribution can be computed under the null hypothesis that, for random samples X and Y from the two populations, the probability P(X < Y) is the same as P(X > Y)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- Mann-Whitney U test:\n",
      "   - U value: 76.000\n",
      "   - p-value: 8.0%\n"
     ]
    }
   ],
   "source": [
    "mwu = mannwhitneyu(x, y, use_continuity=False, alternative=\"two-sided\")\n",
    "\n",
    "print(\"- Mann-Whitney U test:\")\n",
    "print(f\"   - U value: {mwu.statistic:.3f}\")\n",
    "print(f\"   - p-value: {mwu.pvalue*100:.1f}%\")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3. Data distribution VS Stats distribution\n",
    "Tests that rely on the assumption of normally distributed test statistics can also be applied if the original sampling distribution is highly non-normal!\n",
    "\n",
    "Indeed, thanks to the Central Limit Theorem, the distribution of the test statistic is asymptotically normal as the sample size increases.\n",
    "\n",
    "This is very useful in the common case of A/B tests that produce observations that are zero-inflated and/or multimodal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = np.array([245, 0, 0, 0, 0, 0, 0, 0, 0,  1, 3, 1, 3, 2, 9, 18, 22, 10, 6, 2, 2, 1, 2, 1, 1])\n",
    "a = np.arange(len(p))\n",
    "p = p / p.sum()\n",
    "\n",
    "f, ax = plt.subplots(figsize=(6, 3))\n",
    "ax.bar(a, p)\n",
    "ax.set_ylabel(\"prob density\")\n",
    "ax.set_xlabel(\"x\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 700x460 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "loc = (p*np.arange(len(p))).sum()\n",
    "\n",
    "nn = [1, 10, 20, 40]\n",
    "f, axx = plt.subplots(2, 2, figsize=(3.5*2, 2.3*2))    \n",
    "\n",
    "for i, n in enumerate(nn):\n",
    "    r, c = divmod(i, 2)        \n",
    "    x = p\n",
    "    ax = axx[r, c]\n",
    "    for _ in range(n):\n",
    "        x = np.convolve(x, p)\n",
    "\n",
    "    ax.bar(np.arange(len(x))/n, x*n, width=1/n)\n",
    "    ax.set_ylim([0, 0.51])\n",
    "    ax.vlines(loc, 0, 0.51, 'r')\n",
    "    ax.set_xlabel(\"test statistic\")\n",
    "    ax.set_ylabel(\"prob density\")\n",
    "    ax.set_title(f\"sample size = {n}\")\n",
    "\n",
    "f.tight_layout()"
   ]
  }
 ],
 "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": "3.11.1"
  },
  "orig_nbformat": 4,
  "vscode": {
   "interpreter": {
    "hash": "5b88b63acc15a622adab9e20a362eef09d5c387cb76f0f81f96733f0cb5c1aa9"
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}