cuped.ipynb

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   "id": "0a835b1a-80fe-4ca4-a78c-a6381029fa4d",
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   "source": [
    "import seaborn as sns\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import ttest_ind\n",
    "\n",
    "from matplotlib_inline.backend_inline import set_matplotlib_formats\n",
    "set_matplotlib_formats(\"svg\")\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "da222694-c972-4d7c-b010-2cc31b1247f5",
   "metadata": {},
   "source": [
    "### Controlled-Experiment using Pre-Experiment Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "id": "6b5b0138-e78b-49a1-b9a7-d1ceb7c20dbd",
   "metadata": {},
   "outputs": [],
   "source": [
    "def normal(mean: float, sd: float, size: int) -> np.array:\n",
    "    return np.random.randn(size) * sd + mean\n",
    "\n",
    "\n",
    "def ate(Y_A: np.array, Y_B: np.array) -> float:\n",
    "    return Y_B.mean() - Y_A.mean()\n",
    "\n",
    "\n",
    "def generate_experiment() -> pd.DataFrame:    \n",
    "    n_samples = 1000\n",
    "    \n",
    "    sd = 5\n",
    "    mean = 100\n",
    "    effect = 2\n",
    "    \n",
    "    Y_A_pre = normal(mean, sd, n_samples)\n",
    "    Y_B_pre = normal(mean, sd, n_samples)\n",
    "\n",
    "    Y_A_post = Y_A_pre + 0 + normal(0, 2, n_samples)\n",
    "    Y_B_post = Y_B_pre + effect + normal(0, 2, n_samples)\n",
    "    \n",
    "    Y_pre = np.concatenate([Y_A_pre, Y_B_pre])\n",
    "    Y_post = np.concatenate([Y_A_post, Y_B_post])\n",
    "    \n",
    "    cov_matrix = np.cov(Y_pre, Y_post)\n",
    "    theta = cov_matrix[0, 1] / cov_matrix[0, 0]\n",
    "    \n",
    "    Y_A_adj = Y_A_post - (Y_A_pre - Y_pre.mean()) * theta\n",
    "    Y_B_adj = Y_B_post - (Y_B_pre - Y_pre.mean()) * theta\n",
    "\n",
    "    return pd.DataFrame(\n",
    "        {\n",
    "            \"Y_A_post\": Y_A_post,\n",
    "            \"Y_B_post\": Y_B_post,\n",
    "            \"Y_A_adj\": Y_A_adj,\n",
    "            \"Y_B_adj\": Y_B_adj,\n",
    "        }\n",
    "    )\n",
    "\n",
    "\n",
    "def evaluate(exp: pd.DataFrame) -> dict:\n",
    "\n",
    "    test = ttest_ind(exp[\"Y_A_post\"], exp[\"Y_B_post\"])\n",
    "    test_adj = ttest_ind(exp[\"Y_A_adj\"], exp[\"Y_B_adj\"])\n",
    "\n",
    "    return {\n",
    "        \"effect\": ate(exp[\"Y_A_post\"], exp[\"Y_B_post\"]),\n",
    "        \"effect_adj\": ate(exp[\"Y_A_adj\"], exp[\"Y_B_adj\"]),\n",
    "        \"stat\": test.statistic,\n",
    "        \"stat_adj\": test_adj.statistic,\n",
    "        \"pval\": test.pvalue,\n",
    "        \"pval_adj\": test_adj.pvalue,\n",
    "    }"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "id": "26f5f190-8a62-4152-83e9-b15afe835fea",
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(0)\n",
    "experiments = pd.DataFrame(evaluate(generate_experiment()) for _ in range(1000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "id": "888249c2-9a38-415b-9a90-9b37374a7edd",
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    {
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       "    }\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>effect</th>\n",
       "      <th>effect_adj</th>\n",
       "      <th>stat</th>\n",
       "      <th>stat_adj</th>\n",
       "      <th>pval</th>\n",
       "      <th>pval_adj</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>1000.000000</td>\n",
       "      <td>1000.000000</td>\n",
       "      <td>1000.000000</td>\n",
       "      <td>1000.000000</td>\n",
       "      <td>1.000000e+03</td>\n",
       "      <td>1.000000e+03</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>2.005252</td>\n",
       "      <td>1.998557</td>\n",
       "      <td>-8.333405</td>\n",
       "      <td>-22.362584</td>\n",
       "      <td>1.919109e-09</td>\n",
       "      <td>2.255989e-75</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>0.236180</td>\n",
       "      <td>0.088344</td>\n",
       "      <td>0.989225</td>\n",
       "      <td>1.050246</td>\n",
       "      <td>3.630413e-08</td>\n",
       "      <td>7.133961e-74</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>1.180798</td>\n",
       "      <td>1.718413</td>\n",
       "      <td>-11.861885</td>\n",
       "      <td>-26.340592</td>\n",
       "      <td>2.059561e-31</td>\n",
       "      <td>1.671825e-131</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>1.854896</td>\n",
       "      <td>1.940064</td>\n",
       "      <td>-8.996329</td>\n",
       "      <td>-23.084122</td>\n",
       "      <td>5.288188e-19</td>\n",
       "      <td>1.038190e-104</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>2.002009</td>\n",
       "      <td>1.997867</td>\n",
       "      <td>-8.308386</td>\n",
       "      <td>-22.352968</td>\n",
       "      <td>1.767721e-16</td>\n",
       "      <td>5.756030e-99</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>2.163745</td>\n",
       "      <td>2.059455</td>\n",
       "      <td>-7.690873</td>\n",
       "      <td>-21.632170</td>\n",
       "      <td>2.278208e-14</td>\n",
       "      <td>2.060756e-93</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>2.822389</td>\n",
       "      <td>2.269974</td>\n",
       "      <td>-4.920953</td>\n",
       "      <td>-18.748329</td>\n",
       "      <td>9.317697e-07</td>\n",
       "      <td>2.255956e-72</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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       "            effect   effect_adj         stat     stat_adj          pval  \\\n",
       "count  1000.000000  1000.000000  1000.000000  1000.000000  1.000000e+03   \n",
       "mean      2.005252     1.998557    -8.333405   -22.362584  1.919109e-09   \n",
       "std       0.236180     0.088344     0.989225     1.050246  3.630413e-08   \n",
       "min       1.180798     1.718413   -11.861885   -26.340592  2.059561e-31   \n",
       "25%       1.854896     1.940064    -8.996329   -23.084122  5.288188e-19   \n",
       "50%       2.002009     1.997867    -8.308386   -22.352968  1.767721e-16   \n",
       "75%       2.163745     2.059455    -7.690873   -21.632170  2.278208e-14   \n",
       "max       2.822389     2.269974    -4.920953   -18.748329  9.317697e-07   \n",
       "\n",
       "            pval_adj  \n",
       "count   1.000000e+03  \n",
       "mean    2.255989e-75  \n",
       "std     7.133961e-74  \n",
       "min    1.671825e-131  \n",
       "25%    1.038190e-104  \n",
       "50%     5.756030e-99  \n",
       "75%     2.060756e-93  \n",
       "max     2.255956e-72  "
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     "execution_count": 158,
     "metadata": {},
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   ],
   "source": [
    "experiments.describe()"
   ]
  },
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   "cell_type": "code",
   "execution_count": 136,
   "id": "e8cd081e-2c32-4eae-be7b-0ad25c86f84a",
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    {
     "data": {
      "text/plain": [
       "Y_A_post    5.208261\n",
       "Y_B_post    5.350237\n",
       "Y_A_adj     2.015308\n",
       "Y_B_adj     2.015315\n",
       "dtype: float64"
      ]
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "experiment = generate_experiment()\n",
    "experiment.std()"
   ]
  },
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   "execution_count": 156,
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   "source": [
    "experiment_long = experiment.melt(\n",
    "    value_vars=[\"Y_A_post\", \"Y_B_post\", \"Y_A_adj\", \"Y_B_adj\"],\n",
    "    var_name=\"group\",\n",
    "    value_name=\"response\"\n",
    ")\n",
    "\n",
    "sns.histplot(experiment_long, x=\"response\", hue=\"group\", kde=True)\n",
    "plt.title(\"One Experiment: CUPED and Non-adjusted\")\n",
    "\n",
    "plt.show()"
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  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "id": "f1b4bac6-5340-4525-9cba-2f44c620c45e",
   "metadata": {},
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   "source": [
    "experiments_long = (\n",
    "    experiments[[\"effect\", \"effect_adj\"]]\n",
    "    .rename(columns={\"effect\": \"non-adjusted\", \"effect_adj\": \"cuped\"})\n",
    "    .melt(\n",
    "        value_vars=[\"non-adjusted\", \"cuped\"],\n",
    "        var_name=\"method\",\n",
    "        value_name=\"effect\"\n",
    "    )\n",
    ")"
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  },
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   "cell_type": "code",
   "execution_count": 166,
   "id": "77213250-cfc2-4282-b29a-a16dadd09d96",
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    "plt.title(\"1000 Experiments: CUPED and Non-adjusted\")\n",
    "plt.show()"
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