{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Hacker's approach to NHST\n", "\n", "**[Dataset download](https://s3.amazonaws.com/bebi103.caltech.edu/data/bee_sperm.csv)**\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
\n", " \n", " Loading BokehJS ...\n", "
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\\n\"+\n", " \"\\n\"+\n", " \"\\n\"+\n", " \"from bokeh.resources import INLINE\\n\"+\n", " \"output_notebook(resources=INLINE)\\n\"+\n", " \"\\n\"+\n", " \"
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i < inline_js.length; i++) {\n", " inline_js[i].call(root, root.Bokeh);\n", " }\n", " if (force === true) {\n", " display_loaded();\n", " }} else if (Date.now() < root._bokeh_timeout) {\n", " setTimeout(run_inline_js, 100);\n", " } else if (!root._bokeh_failed_load) {\n", " console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n", " root._bokeh_failed_load = true;\n", " } else if (force !== true) {\n", " var cell = $(document.getElementById(\"1001\")).parents('.cell').data().cell;\n", " cell.output_area.append_execute_result(NB_LOAD_WARNING)\n", " }\n", "\n", " }\n", "\n", " if (root._bokeh_is_loading === 0) {\n", " console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n", " run_inline_js();\n", " } else {\n", " load_libs(css_urls, js_urls, function() {\n", " console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n", " run_inline_js();\n", " });\n", " }\n", "}(window));" ], "application/vnd.bokehjs_load.v0+json": "\n(function(root) {\n function now() {\n return new Date();\n }\n\n var force = true;\n\n if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n root._bokeh_onload_callbacks = [];\n root._bokeh_is_loading = undefined;\n }\n\n \n\n \n if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n root._bokeh_timeout = Date.now() + 5000;\n root._bokeh_failed_load = false;\n }\n\n var NB_LOAD_WARNING = {'data': {'text/html':\n \"
\\n\"+\n \"

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\\n\"+\n \"\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"
\"}};\n\n function display_loaded() {\n var el = document.getElementById(\"1001\");\n if (el != null) {\n el.textContent = \"BokehJS is loading...\";\n }\n if (root.Bokeh !== undefined) {\n if (el != null) {\n el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n }\n } else if (Date.now() < root._bokeh_timeout) {\n setTimeout(display_loaded, 100)\n }\n }\n\n\n function run_callbacks() {\n try {\n root._bokeh_onload_callbacks.forEach(function(callback) {\n if (callback != null)\n callback();\n });\n } finally {\n delete root._bokeh_onload_callbacks\n }\n console.debug(\"Bokeh: all callbacks have finished\");\n }\n\n function load_libs(css_urls, js_urls, callback) {\n if (css_urls == null) css_urls = [];\n if (js_urls == null) js_urls = [];\n\n root._bokeh_onload_callbacks.push(callback);\n if (root._bokeh_is_loading > 0) {\n console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n return null;\n }\n if (js_urls == null || js_urls.length === 0) {\n run_callbacks();\n return null;\n }\n console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n function on_load() {\n root._bokeh_is_loading--;\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n run_callbacks()\n }\n }\n\n function on_error() {\n console.error(\"failed to load \" + url);\n }\n\n for (var i = 0; i < css_urls.length; i++) {\n var url = css_urls[i];\n const element = document.createElement(\"link\");\n element.onload = on_load;\n element.onerror = on_error;\n element.rel = \"stylesheet\";\n element.type = \"text/css\";\n element.href = url;\n console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n document.body.appendChild(element);\n }\n\n const hashes = {\"https://cdn.bokeh.org/bokeh/release/bokeh-2.2.3.min.js\": \"T2yuo9Oe71Cz/I4X9Ac5+gpEa5a8PpJCDlqKYO0CfAuEszu1JrXLl8YugMqYe3sM\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.2.3.min.js\": \"98GDGJ0kOMCUMUePhksaQ/GYgB3+NH9h996V88sh3aOiUNX3N+fLXAtry6xctSZ6\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.2.3.min.js\": \"89bArO+nlbP3sgakeHjCo1JYxYR5wufVgA3IbUvDY+K7w4zyxJqssu7wVnfeKCq8\"};\n\n for (var i = 0; i < js_urls.length; i++) {\n var url = js_urls[i];\n var element = document.createElement('script');\n element.onload = on_load;\n element.onerror = on_error;\n element.async = false;\n element.src = url;\n if (url in hashes) {\n element.crossOrigin = \"anonymous\";\n element.integrity = \"sha384-\" + hashes[url];\n }\n console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n document.head.appendChild(element);\n }\n };\n\n function inject_raw_css(css) {\n const element = document.createElement(\"style\");\n element.appendChild(document.createTextNode(css));\n document.body.appendChild(element);\n }\n\n \n var js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-2.2.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.2.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.2.3.min.js\"];\n var css_urls = [];\n \n\n var inline_js = [\n function(Bokeh) {\n Bokeh.set_log_level(\"info\");\n },\n function(Bokeh) {\n \n \n }\n ];\n\n function run_inline_js() {\n \n if (root.Bokeh !== undefined || force === true) {\n \n for (var i = 0; i < inline_js.length; i++) {\n inline_js[i].call(root, root.Bokeh);\n }\n if (force === true) {\n display_loaded();\n }} else if (Date.now() < root._bokeh_timeout) {\n setTimeout(run_inline_js, 100);\n } else if (!root._bokeh_failed_load) {\n console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n root._bokeh_failed_load = true;\n } else if (force !== true) {\n var cell = $(document.getElementById(\"1001\")).parents('.cell').data().cell;\n cell.output_area.append_execute_result(NB_LOAD_WARNING)\n }\n\n }\n\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n run_inline_js();\n } else {\n load_libs(css_urls, js_urls, function() {\n console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n run_inline_js();\n });\n }\n}(window));" }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Colab setup ------------------\n", "import os, sys, subprocess\n", "if \"google.colab\" in sys.modules:\n", " cmd = \"pip install --upgrade iqplot colorcet bebi103 watermark\"\n", " process = subprocess.Popen(cmd.split(), stdout=subprocess.PIPE, stderr=subprocess.PIPE)\n", " stdout, stderr = process.communicate()\n", " data_path = \"https://s3.amazonaws.com/bebi103.caltech.edu/data/\"\n", "else:\n", " data_path = \"../data/\"\n", "# ------------------------------\n", "\n", "\n", "import numpy as np\n", "import pandas as pd\n", "\n", "import numba\n", "\n", "import iqplot\n", "\n", "import colorcet\n", "\n", "import bokeh.io\n", "bokeh.io.output_notebook()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this part of the lesson, we will perform null hypothesis significance tests (NHST) using the [drone sperm quality data set](https://s3.amazonaws.com/bebi103.caltech.edu/data/bee_sperm.csv). You should be sure to carefully read the lecture material on NHST so that you are very clear on what NHST is and what is entails.\n", "\n", "Of course, we need to load in the data set first." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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SpecimenTreatmentEnvironmentTreatmentNCSSSample IDColonyCageSampleSperm Volume per 500 ulQuantityViabilityRaw (%)QualityAge (d)InfertilAliveSpermQuantity MillionsAlive Sperm MillionsDead Sperm Millions
0227ControlCage1C2-1-12112150000215000096.726381461675696.72638114020796172.15002.0796170.070383
1228ControlCage1C2-1-22122287500228750096.349807976059596.34980814022040012.28752.2040010.083499
2229ControlCage1C2-1-3213875008750098.7598.750000140864060.08750.0864060.001094
3230ControlCage1C2-1-42141875000187500093.287420833694193.28742114017491391.87501.7491390.125861
4231ControlCage1C2-1-52151587500158750097.792506105006197.79250614015524561.58751.5524560.035044
\n", "
" ], "text/plain": [ " Specimen Treatment Environment TreatmentNCSS Sample ID Colony Cage \\\n", "0 227 Control Cage 1 C2-1-1 2 1 \n", "1 228 Control Cage 1 C2-1-2 2 1 \n", "2 229 Control Cage 1 C2-1-3 2 1 \n", "3 230 Control Cage 1 C2-1-4 2 1 \n", "4 231 Control Cage 1 C2-1-5 2 1 \n", "\n", " Sample Sperm Volume per 500 ul Quantity ViabilityRaw (%) Quality \\\n", "0 1 2150000 2150000 96.7263814616756 96.726381 \n", "1 2 2287500 2287500 96.3498079760595 96.349808 \n", "2 3 87500 87500 98.75 98.750000 \n", "3 4 1875000 1875000 93.2874208336941 93.287421 \n", "4 5 1587500 1587500 97.7925061050061 97.792506 \n", "\n", " Age (d) Infertil AliveSperm Quantity Millions Alive Sperm Millions \\\n", "0 14 0 2079617 2.1500 2.079617 \n", "1 14 0 2204001 2.2875 2.204001 \n", "2 14 0 86406 0.0875 0.086406 \n", "3 14 0 1749139 1.8750 1.749139 \n", "4 14 0 1552456 1.5875 1.552456 \n", "\n", " Dead Sperm Millions \n", "0 0.070383 \n", "1 0.083499 \n", "2 0.001094 \n", "3 0.125861 \n", "4 0.035044 " ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df = pd.read_csv(os.path.join(data_path, \"bee_sperm.csv\"), comment='#')\n", "df.head()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Because we will be using Numba'd functions to rapidly do our sampling, we will go ahead and slice out the subsets of the data frame we will use for inference as Numpy arrays. We will use the specimens that have nonzero sperm count of alive and dead sperm." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "# Extract NumPy arrays (only use values greater than zero for logs)\n", "inds = (df[\"Alive Sperm Millions\"] > 0) & (df[\"Dead Sperm Millions\"] > 0)\n", "df = df.loc[inds, :]\n", "\n", "alive_ctrl = df.loc[df[\"Treatment\"] == \"Control\", \"Alive Sperm Millions\"].values\n", "\n", "alive_pest = df.loc[df[\"Treatment\"] == \"Pesticide\", \"Alive Sperm Millions\"].values\n", "\n", "dead_ctrl = df.loc[df[\"Treatment\"] == \"Control\", \"Dead Sperm Millions\"].values\n", "\n", "dead_pest = df.loc[df[\"Treatment\"] == \"Pesticide\", \"Dead Sperm Millions\"].values" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, so we have them available, we will define some of the utility functions we defined earlier in the lesson." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "@numba.njit\n", "def draw_bs_sample(data):\n", " \"\"\"Draw a bootstrap sample from a 1D data set.\"\"\"\n", " return np.random.choice(data, size=len(data))\n", "\n", "\n", "@numba.njit\n", "def bivariate_r(x, y):\n", " \"\"\"\n", " Compute plug-in estimate for the bivariate correlation coefficient.\n", " \"\"\"\n", " return (\n", " np.sum((x - np.mean(x)) * (y - np.mean(y)))\n", " / np.std(x)\n", " / np.std(y)\n", " / np.sqrt(len(x))\n", " / np.sqrt(len(y))\n", " )" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Permutation hypothesis tests\n", "\n", "Recall the steps for performing a NHST.\n", "\n", "1. Clearly state the null hypothesis.\n", "2. Define a **test statistic**, a scalar value that you can compute from data. Compute it directly from your measured data.\n", "3. *Simulate* data acquisition for the scenario where the null hypothesis is true. Do this many times, computing and storing the value of the test statistic each time.\n", "4. The fraction of simulations for which the test statistic is at least as extreme as the test statistic computed from the measured data is called the **p-value**, which is what you report.\n", "\n", "For one special type of hypothesis, there is a very straight-forward way of simulating it. Here is our hypothesis: *the control and pesticide-treated samples have exactly the same distribution.* To simulate this, we take the following steps for two data sets, a control with $n$ measurements and a test the other with $m$.\n", "\n", "1. Concatenate the two data sets into one.\n", "2. Randomly scramble the order of the combined data set.\n", "3. Designate the first $n$ entries in this scrambled array to be \"control\" and the remaining to be \"test.\"\n", "\n", "This simulation is exact; it is as if the label of the data set has no meaning; hence the distributions of the two data sets are entirely equal. We test such a null hypothesis with a **permutation test**. A **permutation sample** is akin to a bootstrap sample; it is a new pair of data sets generated after scrambling the concatenated data set. A **permutation replicate** is a value of the test statistic computed from a permutation sample. For this test, we will use the difference of means as the test statistic. \n", "\n", "Let's code this up. We can write a function to draw a permutation sample, and then a function to draw permutation replicates for an arbitrary user-supplied summary statistic." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "@numba.njit\n", "def draw_perm_sample(x, y):\n", " \"\"\"Generate a permutation sample.\"\"\"\n", " concat_data = np.concatenate((x, y))\n", " np.random.shuffle(concat_data)\n", " \n", " return concat_data[:len(x)], concat_data[len(x):]\n", "\n", "\n", "def draw_perm_reps(x, y, stat_fun, size=1):\n", " \"\"\"Generate array of permuation replicates.\"\"\"\n", " return np.array([stat_fun(*draw_perm_sample(x, y)) for _ in range(size)])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since we will look at the difference of means specifically, we will also code up a custom function to draw replicates of the difference of means." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "@numba.njit\n", "def draw_perm_reps_diff_mean(x, y, size=1):\n", " \"\"\"Generate array of permuation replicates.\"\"\"\n", " out = np.empty(size)\n", " for i in range(size):\n", " x_perm, y_perm = draw_perm_sample(x, y)\n", " out[i] = np.mean(x_perm) - np.mean(y_perm)\n", "\n", " return out" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To perform the hypothesis test, then, with the difference of means as our test statistic, we have only to draw many replicates and then tally up how many of them are more extreme than the observed difference of means." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "p-value = 0.0\n" ] } ], "source": [ "# Compute test statistic for original data set\n", "diff_mean = np.mean(alive_ctrl) - np.mean(alive_pest)\n", "\n", "# Draw replicates\n", "perm_reps = draw_perm_reps_diff_mean(alive_ctrl, alive_pest, size=1)\n", "\n", "# Compute p-value\n", "p_val = np.sum(perm_reps >= diff_mean) / len(perm_reps)\n", "\n", "print('p-value =', p_val)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Whoa! Wait a minute. The p-value is *zero*? This just means that in all of the 10,000 replicates we took, not one had a test statistic as extreme as was observed. We cannot resolve p-values much less than 0.001 with 10,000 permutation replicates. Let's try taking ten million replicates as see how we do. This will take a few seconds to run (which again highlights the utility of using Numpy arrays with Numba)." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "p-value = 3.1e-05\n" ] } ], "source": [ "# Compute test statistic for original data set\n", "diff_mean = np.mean(alive_ctrl) - np.mean(alive_pest)\n", "\n", "# Draw replicates\n", "perm_reps = draw_perm_reps_diff_mean(alive_ctrl, alive_pest, size=10000000)\n", "\n", "# Compute p-value\n", "p_val = np.sum(perm_reps >= diff_mean) / len(perm_reps)\n", "\n", "print('p-value =', p_val)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So, our p-value is quite small, less than $10^{-4}$. This means that the probability of getting a difference of means as extreme as was observed under the null hypothesis that the control and test samples were drawn from identical distribution is quite small." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Permutation hypothesis test for correlation\n", "\n", "As we continue to explore the data set, we might want to consider a hypothesis: dead and alive sperm counts are uncorrelated. It is possible that under this hypothesis, we could see correlation at the same level observed in the real data. We can test this hypothesis by permutation. In this case, we scramble the labels \"dead\" and \"alive\" and see what correlation we get. We will do 10 million permutation replicates. This will take a while to compute." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "p-value control: 0.000e+00\n", "p-value pesticide: 1.400e-06\n", "\n" ] } ], "source": [ "# Compute bivariate r for actual data\n", "r_ctrl = bivariate_r(np.log(alive_ctrl), np.log(dead_ctrl))\n", "r_pest = bivariate_r(np.log(alive_pest), np.log(dead_pest))\n", "\n", "# Get permutation replicates\n", "perm_reps_ctrl = draw_perm_reps(\n", " np.log(alive_ctrl), np.log(dead_ctrl), bivariate_r, size=10000000\n", ")\n", "perm_reps_pest = draw_perm_reps(\n", " np.log(alive_pest), np.log(dead_pest), bivariate_r, size=10000000\n", ")\n", "\n", "# Compute p-value\n", "p_ctrl = np.sum(np.abs(perm_reps_ctrl) > np.abs(r_ctrl)) / len(perm_reps_ctrl)\n", "p_pest = np.sum(np.abs(perm_reps_pest) > np.abs(r_pest)) / len(perm_reps_pest)\n", "\n", "print(\n", " \"\"\"\n", "p-value control: {0:.3e}\n", "p-value pesticide: {1:.3e}\n", "\"\"\".format(\n", " p_ctrl, p_pest\n", " )\n", ")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that here we used the absolute value of the permutation replicates. We did this because we are interested to see if we get correlation at all, including anticorrelation. This is another essential piece of a hypothesis test: you need to define what it means to be more extreme. \n", "\n", "The p-value for both is tiny; not one in 10 million permutations for the control sample gave a correlation as high as was observed, and only 10 or so of the pesticide treated group. So, we would report something like $p < 10^{-6}$ for control and $p < 10^{-5}$ for pesticide." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Bootstrap hypothesis tests\n", "\n", "Permutation tests are great: they exactly simulate the null hypothesis that the two samples are identically distributed. But they are limited to this specific null hypothesis. What if we had a different hypothesis, say only that the means of the two distributions we are comparing are equal, but other properties of the distributions need not be identical? In this case, we cannot use the permutations to simulate the null hypothesis.\n", "\n", "Instead, we can simulate the null hypothesis by *shifting* the means of the control and test distributions so that they are equal. We then take a bootstrap sample out of each of the shifted data sets. We compute our test statistic from these two bootstrap samples to get a bootstrap replicate. Then, the number of bootstrap replicates that are at least as extreme as the test statistic from the original data is used to compute the p-value. Let's see this in action. First, we'll see how to shift the data sets and what the resulting ECDFs look like." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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Application\",\"version\":\"2.2.3\"}};\n", " var render_items = [{\"docid\":\"554caae8-40dc-43d4-b5f7-6bcab510e90d\",\"root_ids\":[\"1002\"],\"roots\":{\"1002\":\"fa18ad80-0cef-47bb-ab9a-c5f33c2354df\"}}];\n", " root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n", "\n", " }\n", " if (root.Bokeh !== undefined) {\n", " embed_document(root);\n", " } else {\n", " var attempts = 0;\n", " var timer = setInterval(function(root) {\n", " if (root.Bokeh !== undefined) {\n", " clearInterval(timer);\n", " embed_document(root);\n", " } else {\n", " attempts++;\n", " if (attempts > 100) {\n", " clearInterval(timer);\n", " console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n", " }\n", " }\n", " }, 10, root)\n", " }\n", "})(window);" ], "application/vnd.bokehjs_exec.v0+json": "" }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "id": "1002" } }, "output_type": "display_data" } ], "source": [ "# Shift data sets\n", "total_mean = np.mean(np.concatenate((alive_ctrl, alive_pest)))\n", "alive_ctrl_shift = alive_ctrl - np.mean(alive_ctrl) + total_mean\n", "alive_pest_shift = alive_pest - np.mean(alive_pest) + total_mean\n", "\n", "# Plot the ECDFs\n", "df_shift = pd.DataFrame(\n", " data={\n", " \"treatment\": [\"control\"] * len(alive_ctrl_shift)\n", " + [\"pesticide\"] * len(alive_pest_shift),\n", " \"alive sperm (millions)\": np.concatenate((alive_ctrl_shift, alive_pest_shift)),\n", " }\n", ")\n", "p = iqplot.ecdf(df_shift, q=\"alive sperm (millions)\", cats=\"treatment\")\n", "\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The distributions now have the same mean, but nothing else about them has changed. They still have the same shape as before. Now, let's draw bootstrap samples out of these shifted distributions and see how they compare to the original ECDFs." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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[{\"docid\":\"8352bea2-29b1-4874-a6da-8a037f86aae9\",\"root_ids\":[\"1134\"],\"roots\":{\"1134\":\"b2c104ee-3983-4aef-8816-cf8be066957a\"}}];\n", " root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n", "\n", " }\n", " if (root.Bokeh !== undefined) {\n", " embed_document(root);\n", " } else {\n", " var attempts = 0;\n", " var timer = setInterval(function(root) {\n", " if (root.Bokeh !== undefined) {\n", " clearInterval(timer);\n", " embed_document(root);\n", " } else {\n", " attempts++;\n", " if (attempts > 100) {\n", " clearInterval(timer);\n", " console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n", " }\n", " }\n", " }, 10, root)\n", " }\n", "})(window);" ], "application/vnd.bokehjs_exec.v0+json": "" }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "id": "1134" } }, "output_type": "display_data" } ], "source": [ "p = iqplot.ecdf(\n", " df,\n", " q=\"Alive Sperm Millions\",\n", " cats=\"Treatment\",\n", " x_axis_label=\"alive sperm (millions)\",\n", ")\n", "\n", "for _ in range(100):\n", " ctrl_rep = draw_bs_sample(alive_ctrl_shift)\n", " pest_rep = draw_bs_sample(alive_pest_shift)\n", " df_rep = pd.DataFrame(\n", " data={\n", " \"Treatment\": [\"Control\"] * len(ctrl_rep) + [\"Pesticide\"] * len(pest_rep),\n", " \"Alive Sperm Millions\": np.concatenate((ctrl_rep, pest_rep)),\n", " }\n", " )\n", "\n", " p = iqplot.ecdf(\n", " df_rep,\n", " q=\"Alive Sperm Millions\",\n", " cats=\"Treatment\",\n", " p=p,\n", " marker_kwargs=dict(alpha=0.02),\n", " )\n", "\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The blue-orange haze in between the ECDFs of the original data are the bootstrap samples under the null hypothesis. Only on rare occurrence does that haze reach the original ECDFs, so we might suspect that the observed data might not be all that probable under the null hypothesis. Let's perform the hypothesis test." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "p-value = 5.5e-06\n" ] } ], "source": [ "@numba.njit\n", "def draw_bs_reps_diff_mean(x, y, size=1):\n", " \"\"\"\n", " Generate bootstrap replicates with difference of means\n", " as the test statistic.\n", " \"\"\"\n", " out = np.empty(size)\n", " for i in range(size):\n", " out[i] = np.mean(draw_bs_sample(x)) - np.mean(draw_bs_sample(y))\n", "\n", " return out\n", "\n", "\n", "# Generate samples (10 million again)\n", "bs_reps = draw_bs_reps_diff_mean(alive_ctrl_shift, alive_pest_shift, size=10000000)\n", "\n", "# Compute p-value\n", "p_val = np.sum(bs_reps >= diff_mean) / len(bs_reps)\n", "\n", "print(\"p-value =\", p_val)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This p-value is of similar magnitude as what we got from the permutation test." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Simulating a hypothesis: challenging and rewarding\n", "\n", "As you can see from the above examples with permutation and bootstrap hypothesis tests, the part of the procedure that requires the most thought and care is simulating the null hypothesis. For a test where we are comparing means, we needed to shift the distributions so that all other aspects of the distribution remain the same, and then resample from the shifted distribution. \n", "\n", "The strength of doing these kinds of hypothesis tests is that they are entirely nonparametric. There is not need to specify a probability distribution for the null hypothesis; you can just make statements about how the two distributions you are comparing are related." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Computing environment" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "CPython 3.8.5\n", "IPython 7.18.1\n", "\n", "numpy 1.19.1\n", "pandas 1.1.3\n", "numba 0.51.2\n", "altair not installed\n", "bokeh 2.2.3\n", "colorcet 2.0.2\n", "iqplot 0.1.6\n", "jupyterlab 2.2.6\n" ] } ], "source": [ "%load_ext watermark\n", "\n", "%watermark -v -p numpy,pandas,numba,altair,bokeh,colorcet,iqplot,jupyterlab" ] } ], "metadata": { "anaconda-cloud": {}, "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": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }