simulation.ipynb

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   "source": [
    "$$P(A)=1/2$$\n",
    "$P(B)=p$ unknown\n",
    "$A$ and $B$ are independent"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "from scipy.stats import chi2_contingency\n",
    "from tqdm.auto import tqdm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
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   "source": [
    "def p_value(df, correction):\n",
    "    chi2, p, dof, expected = chi2_contingency(df, correction=correction)\n",
    "    return p"
   ]
  },
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   "source": [
    "# A and B are independent\n",
    "# A is chosen exactly q * samples_size times\n",
    "# B is chosen randomly with probability p\n",
    "\n",
    "sample_size = 100\n",
    "q = 0.5\n",
    "n_iterations = 1000\n",
    "output = []\n",
    "for correction in tqdm([True, False]):\n",
    "    for p in tqdm([0.3, 0.5, 0.9]):\n",
    "        for it in tqdm(range(n_iterations)):\n",
    "            a = np.array([0] * int(sample_size * q) + [1] * int(sample_size * (1 - q)))\n",
    "            b = np.random.choice([0, 1], p=[1 - p, p], size=sample_size)\n",
    "            null_df = (\n",
    "                pd.DataFrame({\"a\": a, \"b\": b}).groupby([\"a\", \"b\"]).size().unstack()\n",
    "            )\n",
    "            output.append(\n",
    "                {\n",
    "                    \"p\": p,\n",
    "                    \"correction\": correction,\n",
    "                    \"p_value\": p_value(null_df, correction),\n",
    "                }\n",
    "            )\n",
    "result = pd.DataFrame(output)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>correction</th>\n",
       "      <th>False</th>\n",
       "      <th>True</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>p</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0.3</th>\n",
       "      <td>0.056</td>\n",
       "      <td>0.028</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.5</th>\n",
       "      <td>0.051</td>\n",
       "      <td>0.034</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>0.9</th>\n",
       "      <td>0.034</td>\n",
       "      <td>0.012</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      ],
      "text/plain": [
       "correction  False  True \n",
       "p                       \n",
       "0.3         0.056  0.028\n",
       "0.5         0.051  0.034\n",
       "0.9         0.034  0.012"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "false_positive_rates = (\n",
    "    result.assign(is_significant=lambda df: df[\"p_value\"] < 0.05)\n",
    "    .groupby([\"p\", \"correction\"])[\"is_significant\"]\n",
    "    .mean()\n",
    ")\n",
    "false_positive_rates.unstack()\n"
   ]
  }
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