{ "cells": [ { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "# Performing bootstrap calculations {#sec-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": { "nbsphinx": "hidden", "tags": [] }, "outputs": [], "source": [ "#| code-fold: true\n", "\n", "# Colab setup ------------------\n", "import os, sys, subprocess\n", "if \"google.colab\" in sys.modules:\n", " cmd = \"pip install --upgrade polars iqplot 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", "# ------------------------------" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "tags": [] }, "outputs": [ { "data": { "text/html": [ " \n", "
\n", " \n", " Loading BokehJS ...\n", "
\n" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/javascript": [ "'use strict';\n", "(function(root) {\n", " function now() {\n", " return new Date();\n", " }\n", "\n", " const 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", "const JS_MIME_TYPE = 'application/javascript';\n", " const HTML_MIME_TYPE = 'text/html';\n", " const EXEC_MIME_TYPE = 'application/vnd.bokehjs_exec.v0+json';\n", " const CLASS_NAME = 'output_bokeh rendered_html';\n", "\n", " /**\n", " * Render data to the DOM node\n", " */\n", " function render(props, node) {\n", " const 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", " function drop(id) {\n", " const view = Bokeh.index.get_by_id(id)\n", " if (view != null) {\n", " view.model.document.clear()\n", " Bokeh.index.delete(view)\n", " }\n", " }\n", "\n", " const cell = handle.cell;\n", "\n", " const id = cell.output_area._bokeh_element_id;\n", " const server_id = cell.output_area._bokeh_server_id;\n", "\n", " // Clean up Bokeh references\n", " if (id != null) {\n", " drop(id)\n", " }\n", "\n", " if (server_id !== undefined) {\n", " // Clean up Bokeh references\n", " const cmd_clean = \"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_clean, {\n", " iopub: {\n", " output: function(msg) {\n", " const id = msg.content.text.trim()\n", " drop(id)\n", " }\n", " }\n", " });\n", " // Destroy server and session\n", " const cmd_destroy = \"import bokeh.io.notebook as ion; ion.destroy_server('\" + server_id + \"')\";\n", " cell.notebook.kernel.execute(cmd_destroy);\n", " }\n", " }\n", "\n", " /**\n", " * Handle when a new output is added\n", " */\n", " function handleAddOutput(event, handle) {\n", " const output_area = handle.output_area;\n", " const output = handle.output;\n", "\n", " // limit handleAddOutput to display_data with EXEC_MIME_TYPE content only\n", " if ((output.output_type != \"display_data\") || (!Object.prototype.hasOwnProperty.call(output.data, EXEC_MIME_TYPE))) {\n", " return\n", " }\n", "\n", " const 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", " const bk_div = document.createElement(\"div\");\n", " bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n", " const script_attrs = bk_div.children[0].attributes;\n", " for (let 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", " const 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", " const 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(error = null) {\n", " const el = document.getElementById(\"e0151286-af70-41b1-9e44-50dc6caefcf0\");\n", " if (el != null) {\n", " const html = (() => {\n", " if (typeof root.Bokeh === \"undefined\") {\n", " if (error == null) {\n", " return \"BokehJS is loading ...\";\n", " } else {\n", " return \"BokehJS failed to load.\";\n", " }\n", " } else {\n", " const prefix = `BokehJS ${root.Bokeh.version}`;\n", " if (error == null) {\n", " return `${prefix} successfully loaded.`;\n", " } else {\n", " return `${prefix} encountered errors while loading and may not function as expected.`;\n", " }\n", " }\n", " })();\n", " el.innerHTML = html;\n", "\n", " if (error != null) {\n", " const wrapper = document.createElement(\"div\");\n", " wrapper.style.overflow = \"auto\";\n", " wrapper.style.height = \"5em\";\n", " wrapper.style.resize = \"vertical\";\n", " const content = document.createElement(\"div\");\n", " content.style.fontFamily = \"monospace\";\n", " content.style.whiteSpace = \"pre-wrap\";\n", " content.style.backgroundColor = \"rgb(255, 221, 221)\";\n", " content.textContent = error.stack ?? error.toString();\n", " wrapper.append(content);\n", " el.append(wrapper);\n", " }\n", " } else if (Date.now() < root._bokeh_timeout) {\n", " setTimeout(() => display_loaded(error), 100);\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(url) {\n", " console.error(\"failed to load \" + url);\n", " }\n", "\n", " for (let i = 0; i < css_urls.length; i++) {\n", " const url = css_urls[i];\n", " const element = document.createElement(\"link\");\n", " element.onload = on_load;\n", " element.onerror = on_error.bind(null, url);\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", " for (let i = 0; i < js_urls.length; i++) {\n", " const url = js_urls[i];\n", " const element = document.createElement('script');\n", " element.onload = on_load;\n", " element.onerror = on_error.bind(null, url);\n", " element.async = false;\n", " element.src = url;\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", " const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.7.3.min.js\"];\n", " const css_urls = [];\n", "\n", " const inline_js = [ function(Bokeh) {\n", " Bokeh.set_log_level(\"info\");\n", " },\n", "function(Bokeh) {\n", " }\n", " ];\n", "\n", " function run_inline_js() {\n", " if (root.Bokeh !== undefined || force === true) {\n", " try {\n", " for (let i = 0; i < inline_js.length; i++) {\n", " inline_js[i].call(root, root.Bokeh);\n", " }\n", "\n", " } catch (error) {display_loaded(error);throw error;\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", " const cell = $(document.getElementById(\"e0151286-af70-41b1-9e44-50dc6caefcf0\")).parents('.cell').data().cell;\n", " cell.output_area.append_execute_result(NB_LOAD_WARNING)\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": "'use strict';\n(function(root) {\n function now() {\n return new Date();\n }\n\n const 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 if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n root._bokeh_timeout = Date.now() + 5000;\n root._bokeh_failed_load = false;\n }\n\n const 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(error = null) {\n const el = document.getElementById(\"e0151286-af70-41b1-9e44-50dc6caefcf0\");\n if (el != null) {\n const html = (() => {\n if (typeof root.Bokeh === \"undefined\") {\n if (error == null) {\n return \"BokehJS is loading ...\";\n } else {\n return \"BokehJS failed to load.\";\n }\n } else {\n const prefix = `BokehJS ${root.Bokeh.version}`;\n if (error == null) {\n return `${prefix} successfully loaded.`;\n } else {\n return `${prefix} encountered errors while loading and may not function as expected.`;\n }\n }\n })();\n el.innerHTML = html;\n\n if (error != null) {\n const wrapper = document.createElement(\"div\");\n wrapper.style.overflow = \"auto\";\n wrapper.style.height = \"5em\";\n wrapper.style.resize = \"vertical\";\n const content = document.createElement(\"div\");\n content.style.fontFamily = \"monospace\";\n content.style.whiteSpace = \"pre-wrap\";\n content.style.backgroundColor = \"rgb(255, 221, 221)\";\n content.textContent = error.stack ?? error.toString();\n wrapper.append(content);\n el.append(wrapper);\n }\n } else if (Date.now() < root._bokeh_timeout) {\n setTimeout(() => display_loaded(error), 100);\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(url) {\n console.error(\"failed to load \" + url);\n }\n\n for (let i = 0; i < css_urls.length; i++) {\n const url = css_urls[i];\n const element = document.createElement(\"link\");\n element.onload = on_load;\n element.onerror = on_error.bind(null, url);\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 for (let i = 0; i < js_urls.length; i++) {\n const url = js_urls[i];\n const element = document.createElement('script');\n element.onload = on_load;\n element.onerror = on_error.bind(null, url);\n element.async = false;\n element.src = url;\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 const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.7.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.7.3.min.js\"];\n const css_urls = [];\n\n const inline_js = [ function(Bokeh) {\n Bokeh.set_log_level(\"info\");\n },\nfunction(Bokeh) {\n }\n ];\n\n function run_inline_js() {\n if (root.Bokeh !== undefined || force === true) {\n try {\n for (let i = 0; i < inline_js.length; i++) {\n inline_js[i].call(root, root.Bokeh);\n }\n\n } catch (error) {display_loaded(error);throw error;\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 const cell = $(document.getElementById(\"e0151286-af70-41b1-9e44-50dc6caefcf0\")).parents('.cell').data().cell;\n cell.output_area.append_execute_result(NB_LOAD_WARNING)\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": [ "import numpy as np\n", "import polars as pl\n", "\n", "import numba\n", "\n", "import iqplot\n", "\n", "import bebi103\n", "\n", "import bokeh.io\n", "bokeh.io.output_notebook()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "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**." ] }, { "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": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "shape: (5, 18)
SpecimenTreatmentEnvironmentTreatmentNCSSSample IDColonyCageSampleSperm Volume per 500 ulQuantityViabilityRaw (%)QualityAge (d)InfertilAliveSpermQuantity MillionsAlive Sperm MillionsDead Sperm Millions
i64strstri64stri64i64i64i64i64f64f64i64i64i64f64f64f64
227"Control""Cage"1"C2-1-1"2112150000215000096.72638196.72638114020796172.152.0796170.070383
228"Control""Cage"1"C2-1-2"2122287500228750096.34980896.34980814022040012.28752.2040010.083499
229"Control""Cage"1"C2-1-3"213875008750098.7598.75140864060.08750.0864060.001094
230"Control""Cage"1"C2-1-4"2141875000187500093.28742193.28742114017491391.8751.7491390.125861
231"Control""Cage"1"C2-1-5"2151587500158750097.79250697.79250614015524561.58751.5524560.035044
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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": "p1003" } }, "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 `rng.choice()` function enables random draws of 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": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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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": "p1003" } }, "output_type": "display_data" } ], "source": [ "rng = np.random.default_rng()\n", "\n", "# Set up Numpy arrays for convenience (also much better performance)\n", "dfs = df.partition_by('Treatment', as_dict=True)\n", "\n", "alive_ctrl = dfs[('Control',)].get_column(\"Alive Sperm Millions\").to_numpy()\n", "alive_pest = dfs[('Pesticide',)].get_column(\"Alive Sperm Millions\").to_numpy()\n", "\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 = rng.choice(alive_ctrl, size=len(alive_ctrl))\n", " bs_pest = rng.choice(alive_pest, size=len(alive_pest))\n", "\n", " # Add semitransparent ECDFs to the plot\n", " p.scatter(np.sort(bs_ctrl), ctrl_ecdf, color=bokeh.palettes.Category10_3[0], alpha=0.02)\n", " p.scatter(np.sort(bs_pest), pest_ecdf, color=bokeh.palettes.Category10_3[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. In order to just-in-time compile a function, we need to decorate its definition with the `@numba.njit` decorator, which tells the Python interpreter to use Numba to just-in-time compile the function. Note that Numba works on basic Python objects like tuples and Numpy arrays, so be sure to pass Numpy arrays into the respective functions.\n", "\n", "Random number generation is supported, and instances of Numpy's generators can be passed into Numba'd functions. However, to avoid always having to pass in an instance of a Numpy RNG into Numba functions, we will use Numpy's old interface to the the Mersenne Twister bit generator, which uses a syntax like `np.random.normal()`." ] }, { "cell_type": "code", "execution_count": 6, "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": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Using the Numpy default RNG (PCG64):\n", "4.33 μs ± 10.6 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n", "\n", "Using the Numpy Mersenne Twister:\n", "4.87 μs ± 16.9 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n", "\n", "Using a Numba'd Mersenne Twister:\n", "1.73 μs ± 1.81 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 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 rng.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 significant increase and 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", "We have 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 – α) percent confidence interval consists of the percentiles 100 α/2 and 100(1 – α/2) of the bootstrap replicates.\n", "\n", "A key step here is computing the bootstrap replicate. We will write a function to draw bootstrap replicates of a statistic of interest. To do so, we pass in a function, which we will call `stat_fun`. Because we do not know what function we will pass in, we cannot just-in-time compile this functions." ] }, { "cell_type": "code", "execution_count": 8, "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)])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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. Note that in our decorator, we use the `parallel=True` keyword argument. Within the `for` loop to compute the bootstrap replicates, as use `numba.prange()` as a drop-in replacement for `range()`. When we do it this way, Numba automatically parallelizes the calculation across available CPUs." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "@numba.njit(parallel=True)\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 numba.prange(size):\n", " out[i] = np.mean(draw_bs_sample(data))\n", " return out\n", "\n", "\n", "@numba.njit(parallel=True)\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 numba.prange(size):\n", " out[i] = np.median(draw_bs_sample(data))\n", " return out\n", "\n", "\n", "@numba.njit(parallel=True)\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 numba.prange(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": 10, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "OMP: Info #276: omp_set_nested routine deprecated, please use omp_set_max_active_levels instead.\n" ] } ], "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": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Mean alive sperm count 95% conf int control (millions): [1.67, 2.07]\n", "Mean alive sperm count 95% conf int treatment (millions): [1.11, 1.50]\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": [ "### Display of confidence intervals\n", "\n", "While the textual confidence intervals shown above are useful, it is often desired to display confidence intervals graphically. The [bebi103 package](https://bebi103.github.io/) has a utility to do this. It requires as input a list of dictionaries where each dictionary contains an estimate, a confidence interval, and a label. " ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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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": "p3302" } }, "output_type": "display_data" } ], "source": [ "summaries = [\n", " dict(label=\"control\", estimate=np.mean(alive_ctrl), conf_int=mean_ctrl_conf_int),\n", " dict(label=\"treated\", estimate=np.mean(alive_pest), conf_int=mean_pest_conf_int),\n", "]\n", "\n", "bokeh.io.show(bebi103.viz.confints(summaries, x_axis_label=\"alive sperm (millions)\"))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### ECDF of bootstrap replicates\n", "\n", "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, and furthermore we have not assumed any parametric model. 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 will 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": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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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](https://en.wikipedia.org/wiki/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": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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alive sperm (millions)\",\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p3389\"}}}],\"center\":[{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p3390\",\"attributes\":{\"axis\":{\"id\":\"p3386\"}}},{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p3395\",\"attributes\":{\"dimension\":1,\"axis\":{\"id\":\"p3391\"}}}],\"frame_width\":350,\"frame_height\":200}}]}}]}};\n", " const render_items = [{\"docid\":\"cad95c4b-98ac-4b59-81e3-9d9bc8475401\",\"roots\":{\"p3633\":\"f966ba2e-a1df-4233-81fc-4ff36594ef1f\"},\"root_ids\":[\"p3633\"]}];\n", " void root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n", " }\n", " if (root.Bokeh !== undefined) {\n", " embed_document(root);\n", " } else {\n", " let attempts = 0;\n", " const 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": "p3633" } }, "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", "# Plot of confidence interval\n", "summaries = [\n", " dict(label=\"control\", estimate=np.median(alive_ctrl), conf_int=median_ctrl_conf_int),\n", " dict(label=\"treated\", estimate=np.median(alive_pest), conf_int=median_pest_conf_int),\n", "]\n", "\n", "p_conf_int = bebi103.viz.confints(summaries, x_axis_label=\"alive sperm (millions)\")\n", "\n", "# Plot ECDFs of bootstrap replicates\n", "p_ecdf = plot_bs_reps_ecdf(\n", " bs_reps_median_ctrl, bs_reps_median_pest, \"median alive sperm (millions)\", thin=50\n", ")\n", "\n", "# Show both plots\n", "bokeh.io.show(bokeh.layouts.column([p_conf_int, bokeh.models.Spacer(height=50), p]))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The results are similar, and we clearly see clear non-normality in the ECDFs.\n", "\n", "Note that plotting ECDFs of bootstrap replicates is not a normal reporting mechanism for bootstrap confidence intervals. We show them here to illustrate how bootstrapping works." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Computing environment" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Python implementation: CPython\n", "Python version : 3.13.5\n", "IPython version : 9.4.0\n", "\n", "numpy : 2.2.6\n", "polars : 1.31.0\n", "numba : 0.61.2\n", "bokeh : 3.7.3\n", "iqplot : 0.3.7\n", "jupyterlab: 4.4.5\n", "\n" ] } ], "source": [ "%load_ext watermark\n", "%watermark -v -p numpy,polars,numba,bokeh,iqplot,jupyterlab" ] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.13.5" } }, "nbformat": 4, "nbformat_minor": 4 }