low_power_experiment.ipynb

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  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "910d7cd8-7ddc-4359-857a-ccf61dc62fd2",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import seaborn as sns\n",
    "from matplotlib_inline.backend_inline import set_matplotlib_formats\n",
    "from scipy.stats import ttest_ind, ttest_ind_from_stats\n",
    "from statsmodels.stats.power import TTestIndPower\n",
    "\n",
    "# from IPython.display import set_matplotlib_formats\n",
    "# set_matplotlib_formats(\"svg\")\n",
    "\n",
    "set_matplotlib_formats(\"svg\")\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "e1b30e22-4137-4566-9479-f6bf9ed0bd77",
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   "outputs": [
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     "execution_count": 61,
     "metadata": {},
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   ],
   "source": [
    "# no effect -> low power\n",
    "TTestIndPower().power(\n",
    "    effect_size=0,\n",
    "    nobs1=1_000_000,\n",
    "    alpha=0.05,\n",
    "    ratio=0.5,\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "db9f4a81-e540-4927-9ecb-f9c7437de441",
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   "outputs": [],
   "source": [
    "def generate_experiment(size: int, p_a: float, p_b: float) -> dict:\n",
    "    # bernoulli distribution = binomial n=1\n",
    "    sample_a = np.random.binomial(n=1, p=p_a, size=size)\n",
    "    sample_b = np.random.binomial(n=1, p=p_b, size=size)\n",
    "\n",
    "    stat, p_val = ttest_ind(sample_a, sample_b, equal_var=False)\n",
    "\n",
    "    with np.errstate(divide=\"ignore\", invalid=\"ignore\"):\n",
    "\n",
    "        mean_a, mean_b = sample_a.mean(), sample_b.mean()\n",
    "        rel_effect_size = (mean_b - mean_a) / np.abs(mean_a)\n",
    "\n",
    "        return {\n",
    "            \"mean_a\": mean_a,\n",
    "            \"mean_b\": mean_b,\n",
    "            \"std_a\": sample_a.std(),\n",
    "            \"std_b\": sample_b.std(),\n",
    "            \"rel_effect_size\": rel_effect_size,\n",
    "            \"stat\": stat,\n",
    "            \"p_val\": p_val,\n",
    "        }"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "d5b06694-a564-41f0-b3ef-98ddc28dc73a",
   "metadata": {},
   "outputs": [],
   "source": [
    "def simulate_two_sizes(\n",
    "    size_1: int,\n",
    "    size_2: int,\n",
    "    p_a: float,\n",
    "    p_b: float,\n",
    "    repeats: int = 1000,\n",
    ") -> pd.DataFrame:\n",
    "    def generate_experiments(size: int) -> pd.DataFrame:\n",
    "        experiments = pd.DataFrame(\n",
    "            generate_experiment(size, p_a, p_b) for _ in range(repeats)\n",
    "        )\n",
    "        experiments[\"size\"] = size\n",
    "        experiments[\"effect_exists\"] = p_a != p_b\n",
    "        return experiments\n",
    "\n",
    "    return pd.concat(\n",
    "        objs=[\n",
    "            generate_experiments(size_1),\n",
    "            generate_experiments(size_2),\n",
    "        ],\n",
    "        ignore_index=True,\n",
    "    )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "453e497c-41f2-4e09-b4d7-91c6c30d992e",
   "metadata": {},
   "outputs": [],
   "source": [
    "small = 100\n",
    "large = 50_000\n",
    "\n",
    "results_effect_exists = simulate_two_sizes(\n",
    "    size_1=small,\n",
    "    size_2=large,\n",
    "    p_a=0.03,\n",
    "    p_b=0.032,\n",
    ")\n",
    "\n",
    "results_no_effect = simulate_two_sizes(\n",
    "    size_1=small,\n",
    "    size_2=large,\n",
    "    p_a=0.03,\n",
    "    p_b=0.03,\n",
    ")\n",
    "\n",
    "results = pd.concat(objs=[results_effect_exists, results_no_effect], ignore_index=True)"
   ]
  },
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   "execution_count": 71,
   "id": "ff145213-acd3-4c96-afae-f65b8d4ee9f6",
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   ],
   "source": [
    "# low power -> noisy p-value\n",
    "# no effect -> always low power\n",
    "g = sns.FacetGrid(results, col=\"effect_exists\", sharey=False, sharex=False, height=4)\n",
    "g.map_dataframe(sns.kdeplot, x=\"p_val\", hue=\"size\", legend=True)\n",
    "plt.legend(title=\"Size\", loc=\"upper right\", labels=[large, small])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "id": "73153455-7418-4319-b797-56e893d59bfa",
   "metadata": {},
   "outputs": [
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   "source": [
    "sns.violinplot(\n",
    "    data=results,\n",
    "    x=\"size\",\n",
    "    y=\"p_val\",\n",
    "    hue=\"effect_exists\",\n",
    "    split=True,\n",
    "    inner=\"quart\",\n",
    "    linewidth=1,\n",
    "    palette={True: \"b\", False: \".85\"},\n",
    ")\n",
    "sns.despine(left=True)\n",
    "plt.title(\"Low power generates noisy $p$-values\")\n",
    "plt.show()"
   ]
  }
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