{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Introduction to the bootstrap\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", "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/javascript": [ "\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", " var JS_MIME_TYPE = 'application/javascript';\n", " var HTML_MIME_TYPE = 'text/html';\n", " var EXEC_MIME_TYPE = 'application/vnd.bokehjs_exec.v0+json';\n", " var CLASS_NAME = 'output_bokeh rendered_html';\n", "\n", " /**\n", " * Render data to the DOM node\n", " */\n", " function render(props, node) {\n", " var script = document.createElement(\"script\");\n", " node.appendChild(script);\n", " }\n", "\n", " /**\n", " * Handle when an output is cleared or removed\n", " */\n", " function handleClearOutput(event, handle) {\n", " var cell = handle.cell;\n", "\n", " var id = cell.output_area._bokeh_element_id;\n", " var server_id = cell.output_area._bokeh_server_id;\n", " // Clean up Bokeh references\n", " if (id != null && id in Bokeh.index) {\n", " Bokeh.index[id].model.document.clear();\n", " delete Bokeh.index[id];\n", " }\n", "\n", " if (server_id !== undefined) {\n", " // Clean up Bokeh references\n", " var cmd = \"from bokeh.io.state import curstate; print(curstate().uuid_to_server['\" + server_id + \"'].get_sessions()[0].document.roots[0]._id)\";\n", " cell.notebook.kernel.execute(cmd, {\n", " iopub: {\n", " output: function(msg) {\n", " var id = msg.content.text.trim();\n", " if (id in Bokeh.index) {\n", " Bokeh.index[id].model.document.clear();\n", " delete Bokeh.index[id];\n", " }\n", " }\n", " }\n", " });\n", " // Destroy server and session\n", " var cmd = \"import bokeh.io.notebook as ion; ion.destroy_server('\" + server_id + \"')\";\n", " cell.notebook.kernel.execute(cmd);\n", " }\n", " }\n", "\n", " /**\n", " * Handle when a new output is added\n", " */\n", " function handleAddOutput(event, handle) {\n", " var output_area = handle.output_area;\n", " var output = handle.output;\n", "\n", " // limit handleAddOutput to display_data with EXEC_MIME_TYPE content only\n", " if ((output.output_type != \"display_data\") || (!output.data.hasOwnProperty(EXEC_MIME_TYPE))) {\n", " return\n", " }\n", "\n", " var toinsert = output_area.element.find(\".\" + CLASS_NAME.split(' ')[0]);\n", "\n", " if (output.metadata[EXEC_MIME_TYPE][\"id\"] !== undefined) {\n", " toinsert[toinsert.length - 1].firstChild.textContent = output.data[JS_MIME_TYPE];\n", " // store reference to embed id on output_area\n", " output_area._bokeh_element_id = output.metadata[EXEC_MIME_TYPE][\"id\"];\n", " }\n", " if (output.metadata[EXEC_MIME_TYPE][\"server_id\"] !== undefined) {\n", " var bk_div = document.createElement(\"div\");\n", " bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n", " var script_attrs = bk_div.children[0].attributes;\n", " for (var i = 0; i < script_attrs.length; i++) {\n", " toinsert[toinsert.length - 1].firstChild.setAttribute(script_attrs[i].name, script_attrs[i].value);\n", " toinsert[toinsert.length - 1].firstChild.textContent = bk_div.children[0].textContent\n", " }\n", " // store reference to server id on output_area\n", " output_area._bokeh_server_id = output.metadata[EXEC_MIME_TYPE][\"server_id\"];\n", " }\n", " }\n", "\n", " function register_renderer(events, OutputArea) {\n", "\n", " function append_mime(data, metadata, element) {\n", " // create a DOM node to render to\n", " var toinsert = this.create_output_subarea(\n", " metadata,\n", " CLASS_NAME,\n", " EXEC_MIME_TYPE\n", " );\n", " this.keyboard_manager.register_events(toinsert);\n", " // Render to node\n", " var props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n", " render(props, toinsert[toinsert.length - 1]);\n", " element.append(toinsert);\n", " return toinsert\n", " }\n", "\n", " /* Handle when an output is cleared or removed */\n", " events.on('clear_output.CodeCell', handleClearOutput);\n", " events.on('delete.Cell', handleClearOutput);\n", "\n", " /* Handle when a new output is added */\n", " events.on('output_added.OutputArea', handleAddOutput);\n", "\n", " /**\n", " * Register the mime type and append_mime function with output_area\n", " */\n", " OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n", " /* Is output safe? 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\\n\"+\n", " \"

\\n\"+\n", " \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n", " \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n", " \"

\\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));" ], "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 \"

\\n\"+\n \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n \"

\\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 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", "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": [ "We have discussed the **plug-in principle**, in which we approximate the distribution function of the generative model, $F$, with the **empirical distribution function**, $\\hat{F}$. We then approximate all statistical functionals of $F$ using $\\hat{F}$; $T(F)\\approx T(\\hat{F})$. Using the frequentist perspective, we talked about using the plug-in principle and **bootstrapping** to compute confidence intervals and do hypothesis testing. In this lesson, we will go over how we implement this in practice. We will work with some real data on the reproductive health of male bees that have been treated with pesticide.\n", "\n", "It is important as you go through this lesson to remember that you do not know what the generative distribution is. We are only approximating it using the plug-in principle. Since we do not know what $F$ is, we do not know what its parameters are, so we therefore cannot estimate them. Rather, we can study **summary statistics**, which are plug-in estimates for statistical functionals of the generative distribution, such as means, variances, or even the ECDF itself, and how they may vary from experiment to experiment. In this sense, we are doing **nonparametric inference**. We do not have a parametrized model in mind, but are investigating how experimental results might vary upon repetition of the experiment." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The data set\n", "\n", "Neonicotinoid pesticides are thought to have inadvertent effects on service-providing insects such as bees. A study of this was [featured in the New York Times](http://www.nytimes.com/2016/07/29/science/neonicotinoid-insecticide-bee-sperm.html). The original paper by Straub and coworkers may be found [here](http://dx.doi.org/10.1098/rspb.2016.0506). The authors put their [data in the Dryad repository](http://dx.doi.org/10.5061/dryad.bs515), which means that it is free to all to work with!\n", "\n", "(Do you see a trend here? If you want people to think deeply about your results, explore them, learn from them, in general further science with them, *make your data publicly available.* Strongly encourage the members of your lab to do the same.)\n", "\n", "We will look at the sperm quality of drone bees using [this data set](https://s3.amazonaws.com/bebi103.caltech.edu/data/bee_sperm.csv). First, let's load in the data set and check it out." ] }, { "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": [ "We are interested in the number of alive sperm in the samples. Let's first explore the data by making ECDFs for the two groups we will compare, those treated with pesticide (\"Pesticide\") and those that are not (\"Control\")." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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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": [ "p = iqplot.ecdf(df, q=\"Alive Sperm Millions\", cats=\"Treatment\")\n", "\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The visual inspection of the ECDFs suggests that indeed the control drones have more alive sperm than those treated with pesticide. But how variable would these ECDFs be if we repeated the experiment?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Bootstrap samples and ECDFs\n", "\n", "To address this question, we can generate bootstrap samples from the experimental data and make lots of ECDFs. We can then plot them all to see how the ECDF might vary. Recall that a bootstrap sample from a data set of $n$ repeated measurements is generated by drawing $n$ data points out of the original data set with replacement. The `rg.choice()` function allows for randomly drawing elements out of an array. Let's generate 100 bootstrap samples and plot their ECDFs to visualize how our data set might change as we repeat the experiment." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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Set up Numpy arrays for convenience (also much better performance)\n", "alive_ctrl = df.loc[df[\"Treatment\"] == \"Control\", \"Alive Sperm Millions\"].values\n", "alive_pest = df.loc[df[\"Treatment\"] == \"Pesticide\", \"Alive Sperm Millions\"].values\n", "\n", "# ECDF values for plotting\n", "ctrl_ecdf = np.arange(1, len(alive_ctrl)+1) / len(alive_ctrl)\n", "pest_ecdf = np.arange(1, len(alive_pest)+1) / len(alive_pest)\n", "\n", "# Make 100 bootstrap samples and plot them\n", "for _ in range(100):\n", " bs_ctrl = rg.choice(alive_ctrl, size=len(alive_ctrl))\n", " bs_pest = rg.choice(alive_pest, size=len(alive_pest))\n", "\n", " # Add semitransparent ECDFs to the plot\n", " p.circle(np.sort(bs_ctrl), ctrl_ecdf, color=colorcet.b_glasbey_category10[0], alpha=0.02)\n", " p.circle(np.sort(bs_pest), pest_ecdf, color=colorcet.b_glasbey_category10[1], alpha=0.02)\n", "\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From this graphical display, we can already see that the ECDFs do not strongly overlap in 100 bootstrap samples, so there is likely a real difference between the two treatments." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Speeding up sampling with Numba\n", "\n", "We can make sampling faster (though note that the slowness in generating the above plot is not in resampling, but in adding glyphs to the plot) by employing **just-in-time compilation**, wherein the Python code is compiled at runtime into machine code. This results in a substantial speed boost. [Numba](http://numba.pydata.org) is a powerful tool for this purpose, and we will use it. Bear in mind, though, that not all Python code is Numba-able. Specifically, only the Mersenne Twister bit generator is supported, so we need to use `np.random.choice()` in Numba's code to do our resampling. \n", "\n", "In order to jit-in-time compile a function, we need to decorate its definition with the `@numba.njit` decorator, which tells Python to use Numba to just-in-time compile the function. Note that Numba does not currently work with Pandas data frames; it typically requires Numpy arrays to do any operations on arrays." ] }, { "cell_type": "code", "execution_count": 5, "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))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's quickly quantify the speed boost. Before we do the timing, we will run the Numba'd version once to give it a chance to do the JIT compilation." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Using the Numpy default RNG (PCG64):\n", "18.3 µs ± 380 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n", "\n", "Using the Numpy Mersenne Twister:\n", "18.5 µs ± 146 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n", "\n", "Using a Numba'd Mersenne Twister:\n", "2.86 µs ± 447 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n" ] } ], "source": [ "# Run once to JIT\n", "draw_bs_sample(alive_ctrl)\n", "\n", "print('Using the Numpy default RNG (PCG64):')\n", "%timeit rg.choice(alive_ctrl, size=len(alive_ctrl))\n", "\n", "print('\\nUsing the Numpy Mersenne Twister:')\n", "%timeit np.random.choice(alive_ctrl, size=len(alive_ctrl))\n", "\n", "print(\"\\nUsing a Numba'd Mersenne Twister:\")\n", "%timeit draw_bs_sample(alive_ctrl)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "That's a factor of 7 or so increase. That can make a big difference when generating lots of bootstrap samples." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Bootstrap replicates and confidence intervals\n", "\n", "We have plotted the ECDF of the data, which is instructive, but we would also like to get plug-in estimates for the generative distribution. Remember, when doing nonparametric plug-in estimates, we plug in the ECDF for the CDF. We do not need to specify what the distribution (described mathematically by the CDF, or equivalently by the PDF) is, just that we approximate it by the empirical distribution.\n", "\n", "In lecture, we laid out the procedure to compute a confidence interval.\n", "\n", "1. Generate $B$ independent bootstrap samples.\n", "2. Compute the plug-in estimate of the statistical functional of interest for each bootstrap sample to get the **bootstrap replicates**.\n", "3. The $100(1-\\alpha)$ percent confidence interval consists of the percentiles $100\\alpha/2$ and $100(1 - \\alpha/2)$ of the bootstrap replicates.\n", "\n", "A key step here is computing the bootstrap replicate. We will write a couple functions for this. The first is generic; it takes as an argument the function to be used to compute the statistic of interest (e.g., `np.mean` or `np.median`). We will also write a few functions for commonly computed statistics, which enables us to use Numba to greatly speed up the process of generating bootstrap replicates." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "def draw_bs_reps(data, stat_fun, size=1):\n", " \"\"\"Draw boostrap replicates computed with stat_fun from 1D data set.\"\"\"\n", " return np.array([stat_fun(draw_bs_sample(data)) for _ in range(size)])\n", "\n", "\n", "@numba.njit\n", "def draw_bs_reps_mean(data, size=1):\n", " \"\"\"Draw boostrap replicates of the mean from 1D data set.\"\"\"\n", " out = np.empty(size)\n", " for i in range(size):\n", " out[i] = np.mean(draw_bs_sample(data))\n", " return out\n", "\n", "\n", "@numba.njit\n", "def draw_bs_reps_median(data, size=1):\n", " \"\"\"Draw boostrap replicates of the median from 1D data set.\"\"\"\n", " out = np.empty(size)\n", " for i in range(size):\n", " out[i] = np.median(draw_bs_sample(data))\n", " return out\n", "\n", "\n", "@numba.njit\n", "def draw_bs_reps_std(data, size=1):\n", " \"\"\"Draw boostrap replicates of the standard deviation from 1D data set.\"\"\"\n", " out = np.empty(size)\n", " for i in range(size):\n", " out[i] = np.std(draw_bs_sample(data))\n", " return out" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, let's get bootstrap replicates for the mean of each of the two treatments." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "bs_reps_mean_ctrl = draw_bs_reps_mean(alive_ctrl, size=10000)\n", "bs_reps_mean_pest = draw_bs_reps_mean(alive_pest, size=10000)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now compute the confidence intervals by computing the percentiles using the `np.percentile()` function." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Mean alive sperm count 95% conf int control (millions): [1.68, 2.07]\n", "Mean alive sperm count 95% conf int treatment (millions): [1.11, 1.49]\n", "\n" ] } ], "source": [ "# 95% confidence intervals\n", "mean_ctrl_conf_int = np.percentile(bs_reps_mean_ctrl, [2.5, 97.5])\n", "mean_pest_conf_int = np.percentile(bs_reps_mean_pest, [2.5, 97.5])\n", "\n", "print(\"\"\"\n", "Mean alive sperm count 95% conf int control (millions): [{0:.2f}, {1:.2f}]\n", "Mean alive sperm count 95% conf int treatment (millions): [{2:.2f}, {3:.2f}]\n", "\"\"\".format(*(tuple(mean_ctrl_conf_int) + tuple(mean_pest_conf_int))))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can also use the bootstrap replicates to plot the probability distribution of mean alive sperm count. *Remember*: This is not a confidence interval on a parameter value. That does not make sense in frequentist statistics. It is the confidence interval describing what we would get as a plug-in estimate for the mean if we did the experiment over and over again.\n", "\n", "When we plot the ECDF of the bootstrap replicates, we thin them so as not to choke the browser with too many points. Since we will do this again, we write a quick function to do it." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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data={\n", " \"treatment\": [\"control\"] * len(x_ctrl) + [\"pesticide\"] * len(x_pest),\n", " q: np.concatenate((x_ctrl, x_pest)),\n", " }\n", " )\n", "\n", " p = iqplot.ecdf(\n", " df, q=q, cats=\"treatment\", frame_height=200, frame_width=350, **kwargs\n", " )\n", "\n", " return p\n", "\n", "\n", "p = plot_bs_reps_ecdf(\n", " bs_reps_mean_ctrl, bs_reps_mean_pest, \"mean alive sperm (millions)\", thin=50\n", ")\n", "p.legend.location = \"top_left\"\n", "\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "These are both nice, Normal distributions with almost no overlap in the tails. We also saw in the computation of the 95% confidence intervals above that there is no overlap. This is expected, the mean alive sperm count should be Normally distributed as per the central limit theorem.\n", "\n", "We can do the same procedure for other statistical quantities that do not follow the central limit theorem. The procedure is exactly the same. We will do it for the median." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Median alive sperm count 95% conf int control (millions): [1.63, 2.11]\n", "Median alive sperm count 95% conf int treatment (millions): [0.95, 1.47]\n", "\n" ] }, { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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alive sperm (millions)\"},\"y\":{\"field\":\"__ECDF\"}},\"id\":\"47182\",\"type\":\"Circle\"},{\"attributes\":{\"text\":\"\"},\"id\":\"47169\",\"type\":\"Title\"},{\"attributes\":{},\"id\":\"47139\",\"type\":\"LinearScale\"},{\"attributes\":{\"overlay\":{\"id\":\"47155\"}},\"id\":\"47151\",\"type\":\"BoxZoomTool\"},{\"attributes\":{\"fill_alpha\":{\"value\":0.1},\"fill_color\":{\"value\":\"#1f77b3\"},\"line_alpha\":{\"value\":0.1},\"line_color\":{\"value\":\"#1f77b3\"},\"x\":{\"field\":\"median alive sperm (millions)\"},\"y\":{\"field\":\"__ECDF\"}},\"id\":\"47166\",\"type\":\"Circle\"},{\"attributes\":{\"fill_color\":{\"value\":\"#1f77b3\"},\"line_color\":{\"value\":\"#1f77b3\"},\"x\":{\"field\":\"median alive sperm (millions)\"},\"y\":{\"field\":\"__ECDF\"}},\"id\":\"47165\",\"type\":\"Circle\"},{\"attributes\":{},\"id\":\"47135\",\"type\":\"DataRange1d\"}],\"root_ids\":[\"47132\"]},\"title\":\"Bokeh Application\",\"version\":\"2.2.3\"}};\n", " var render_items = [{\"docid\":\"49d080e5-ddba-4e02-90b9-2f44101246c0\",\"root_ids\":[\"47132\"],\"roots\":{\"47132\":\"c7113d6d-d0b5-4627-996f-3cabf2a70a50\"}}];\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": "47132" } }, "output_type": "display_data" } ], "source": [ "# Get the bootstrap replicates\n", "bs_reps_median_ctrl = draw_bs_reps_median(alive_ctrl, size=10000)\n", "bs_reps_median_pest = draw_bs_reps_median(alive_pest, size=10000)\n", "\n", "# 95% confidence intervals\n", "median_ctrl_conf_int = np.percentile(bs_reps_median_ctrl, [2.5, 97.5])\n", "median_pest_conf_int = np.percentile(bs_reps_median_pest, [2.5, 97.5])\n", "\n", "print(\n", " \"\"\"\n", "Median alive sperm count 95% conf int control (millions): [{0:.2f}, {1:.2f}]\n", "Median alive sperm count 95% conf int treatment (millions): [{2:.2f}, {3:.2f}]\n", "\"\"\".format(\n", " *(tuple(median_ctrl_conf_int) + tuple(median_pest_conf_int))\n", " )\n", ")\n", "\n", "# Plot ECDFs of bootstrap replicates\n", "p = plot_bs_reps_ecdf(\n", " bs_reps_median_ctrl, bs_reps_median_pest, \"median alive sperm (millions)\", thin=50\n", ")\n", "p.legend.location = \"top_left\"\n", "\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The results are similar, and we clearly see clear non-normality in the ECDFs, since the central limit theorem does not hold here." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Computing environment" ] }, { "cell_type": "code", "execution_count": 12, "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", "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", "%watermark -v -p numpy,pandas,numba,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 }