{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 8. Bootstrapping\n", "\n", "*This recitation was prepared by Cecelia Andrews with help from Justin Bois.*\n", "\n", "**[Law school dataset download](https://s3.amazonaws.com/bebi103.caltech.edu/data/law_school_data.csv)**\n", "\n", "**[*C. elegans* egg cross-sectional area dataset download](https://s3.amazonaws.com/bebi103.caltech.edu/data/c_elegans_egg_xa.csv)**\n", "\n", "**[Bee sperm dataset download](https://s3.amazonaws.com/bebi103.caltech.edu/data/bee_sperm.csv)**\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
\n", " \n", " Loading BokehJS ...\n", "
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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", " 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", " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", " document.head.appendChild(element);\n", " }\n", " };var element = document.getElementById(\"1001\");\n", " if (element == null) {\n", " console.error(\"Bokeh: ERROR: autoload.js configured with elementid '1001' but no matching script tag was found. \")\n", " return false;\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", " var js_urls = [\"https://cdn.pydata.org/bokeh/release/bokeh-1.3.4.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-widgets-1.3.4.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-tables-1.3.4.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-gl-1.3.4.min.js\"];\n", " var css_urls = [];\n", "\n", " var inline_js = [\n", " function(Bokeh) {\n", " Bokeh.set_log_level(\"info\");\n", " },\n", " \n", " function(Bokeh) {\n", " \n", " },\n", " function(Bokeh) {} // ensure no trailing comma for IE\n", " ];\n", "\n", " function run_inline_js() {\n", " \n", " if ((root.Bokeh !== undefined) || (force === true)) {\n", " for (var i = 0; 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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 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 console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n document.head.appendChild(element);\n }\n };var element = document.getElementById(\"1001\");\n if (element == null) {\n console.error(\"Bokeh: ERROR: autoload.js configured with elementid '1001' but no matching script tag was found. \")\n return false;\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 var js_urls = [\"https://cdn.pydata.org/bokeh/release/bokeh-1.3.4.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-widgets-1.3.4.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-tables-1.3.4.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-gl-1.3.4.min.js\"];\n var css_urls = [];\n\n var inline_js = [\n function(Bokeh) {\n Bokeh.set_log_level(\"info\");\n },\n \n function(Bokeh) {\n \n },\n function(Bokeh) {} // ensure no trailing comma for IE\n ];\n\n function run_inline_js() {\n \n if ((root.Bokeh !== undefined) || (force === true)) {\n for (var i = 0; i < inline_js.length; i++) {\n inline_js[i].call(root, root.Bokeh);\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": [ "import pandas as pd\n", "import numpy as np\n", "import numba\n", "import scipy.stats as st\n", "\n", "import bokeh_catplot\n", "\n", "import colorcet\n", "\n", "import bokeh.io\n", "import bokeh.plotting\n", "import bokeh.layouts\n", "bokeh.io.output_notebook()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "This this recitation, we will cover\n", "\n", "1. Review of bootstrapping concepts from Lesson 7.\n", "3. BCa Confidence Intervals\n", "4. Studentized Bootstrap\n", "\n", "\n", "An excellent reference, if dated, for bootstrapping calculations is Efron and Tibshirani's *An Introduction to the Bootstrap*, 1994 ([Amazon](http://www.amazon.com/Introduction-Bootstrap-Monographs-Statistics-Probability/dp/0412042312))." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1. Review\n", "\n", "### The plug-in principle\n", "\n", "This was covered in Lecture 7. \n", "\n", "**Simplified definition:** Say we have a sample of repeated measurements, and we know there is a generative distribution for these measurements. But, we don't know what that generative distribution is. We can estimate a parameter, $\\theta = T(F)$, of the generative distribution by computing the statistical functional using the empirical CDF rather than the actual CDF.\n", "\n", "\n", "**Thorough definition, from Lecture 7:** \n", "We'll use the same example covered in Lecture 7. Say we have a data set of repeated measurements of the lengths of *C. elegans* eggs. We know there is a generative distribution for these lengths, but we don't know any parameters of the distribution. How can we estimate a parameter of the distribution? \n", "\n", "Call the CDF of our generative distribution $F(x)$. We want to compute a statistical functional, $T(F)$, of the CDF. $T(F)$ could be the mean, variance, median, etc. The parameter $\\theta$ is defined by the functional, so $\\theta = T(F)$. \n", "\n", "However, we only have a sample from the generative distribution (our egg length measurements). We can compute the ECDF of our measurements, called $\\hat{F}$. The plug-in principle simply states we can estimate our parameter of interest, $\\theta$, by calculating: \n", "\n", "\\begin{align}\n", "\\hat{\\theta} &= T(\\hat{F}) \n", "\\end{align}\n", "\n", "Essentially, the plug-in principle says we can estimate the parameter $\\theta$ by computing the statistical functional on the empirical CDF rather than the actual CDF." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Confidence Intervals\n", "\n", "These were covered in Lecture 7. \n", "\n", "\"If an experiment is repeated over and over again, the estimate I compute for a parameter, $\\hat{\\theta}$ , will lie between the bounds of the 95% confidence interval for 95% of the experiments.\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Bootstrapping\n", "\n", "Also covered in Lesson 7. \n", "\n", "The basic idea of bootstrapping is to simulate many repetitions of the experiment. This allows us to get a better plug-in estimate for the statistical functional as if we did the experiment many times. To see the power of bootstrapping in action, let's consider the example given in Efron and Tibshirani's *An Introduction to the Bootstrap*. \n", "\n", "The CSV file `law_school_data.csv`, available [here](https://s3.amazonaws.com/bebi103.caltech.edu/data/law_school_data.csv), contains the population of measurements (LSAT, GPA) for N=82 American law schools for the entering classes of 1973. LSAT is the average score of the class on the LSAT (national law exam), and GPA is the average undergraduate GPA. In this example, we have a census, so we know the actual values of the summary statistics we may want to compute. Here we load in the data and view the first 10 observations." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " School LSAT GPA\n", "0 1 622 3.23\n", "1 2 542 2.83\n", "2 3 579 3.24\n", "3 4 653 3.12\n", "4 5 606 3.09\n", "5 6 576 3.39\n", "6 7 620 3.10\n", "7 8 615 3.40\n", "8 9 553 2.97\n", "9 10 607 2.91" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_law = pd.read_csv('../data/law_school_data.csv')\n", "df_law.head(10)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, let's pretend we are a researcher interested in estimating the mean LSAT and GPA for American law schools. Say we collect a random sample of 15 law schools and want to estimate the population mean from our sample. We will simulate this by drawing a random sample of size n=15:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " School LSAT GPA\n", "21 22 614 3.19\n", "51 52 580 3.07\n", "11 12 596 3.24\n", "17 18 646 3.47\n", "2 3 579 3.24\n", "10 11 558 3.11\n", "33 34 591 3.02\n", "3 4 653 3.12\n", "13 14 581 3.22\n", "36 37 615 3.37\n", "34 35 578 3.03\n", "52 53 594 2.96\n", "16 17 599 3.23\n", "70 71 570 3.01\n", "49 50 555 3.00" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rg = np.random.default_rng(seed=3252)\n", "\n", "# Draw 15 random samples and make a new dataframe\n", "indices = rg.choice(np.arange(82), replace=False, size=15)\n", "df_sample = df_law.loc[indices]\n", "df_sample" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "How does this sample compare to the population? Let's compare the population mean LSAT and GPA to the sample means:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Sample mean LSAT: 593.9333333333333\n", "Sample mean GPA: 3.1519999999999997\n", "Population mean LSAT: 597.5487804878048\n", "Population mean GPA: 3.1347560975609756\n" ] } ], "source": [ "# Convert data to numpy arrays\n", "sample_lsat = df_sample['LSAT'].values\n", "sample_gpa = df_sample['GPA'].values\n", "pop_lsat = df_law['LSAT'].values\n", "pop_gpa = df_law['GPA'].values\n", "\n", "# Print means\n", "print(f'Sample mean LSAT: {np.mean(sample_lsat)}')\n", "print(f'Sample mean GPA: {np.mean(sample_gpa)}')\n", "print(f'Population mean LSAT: {np.mean(pop_lsat)}')\n", "print(f'Population mean GPA: {np.mean(pop_gpa)}')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The accuracy of the mean LSAT and GPA of the small sample will vary based on the sample you draw, but they can be fairly far away from the mean of the population such that the standard error does not capture the actual mean.\n", "\n", "Let's compare the ECDFs as well." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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np.arange(1, len(data) + 1) / len(data)\n", "\n", "\n", "# Set up 2 plots\n", "p = bokeh.plotting.figure(\n", " width=400, height=300, x_axis_label=\"LSAT\", y_axis_label=\"ECDF\",\n", ")\n", "\n", "a = bokeh.plotting.figure(\n", " width=400, height=300, x_axis_label=\"GPA\", y_axis_label=\"ECDF\",\n", ")\n", "\n", "# Get ECDF values\n", "sample_lsat_ecdf = ecdf(sample_lsat)\n", "pop_lsat_ecdf = ecdf(pop_lsat)\n", "sample_gpa_ecdf = ecdf(sample_gpa)\n", "pop_gpa_ecdf = ecdf(pop_gpa)\n", "\n", "# Plot ECDFs\n", "p.circle(\n", " np.sort(sample_lsat),\n", " sample_lsat_ecdf,\n", " color=colorcet.b_glasbey_category10[0],\n", " alpha=2,\n", " legend=\"Sample LSAT\",\n", ")\n", "p.circle(\n", " np.sort(pop_lsat),\n", " pop_lsat_ecdf,\n", " color=colorcet.b_glasbey_category10[1],\n", " alpha=2,\n", " legend=\"Large sample LSAT\",\n", ")\n", "a.circle(\n", " np.sort(sample_gpa),\n", " sample_gpa_ecdf,\n", " color=colorcet.b_glasbey_category10[0],\n", " alpha=2,\n", " legend=\"Sample GPA\",\n", ")\n", "a.circle(\n", " np.sort(pop_gpa),\n", " pop_gpa_ecdf,\n", " color=colorcet.b_glasbey_category10[1],\n", " alpha=2,\n", " legend=\"Large sample GPA\",\n", ")\n", "\n", "a.legend.location = \"bottom_right\"\n", "p.legend.location = \"bottom_right\"\n", "\n", "bokeh.io.show(bokeh.layouts.row(p, a))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The ECDFs match up fairly well, but if we zoom in on the 50th percentile we can see the means may not overlap. Let's see if we can compute a confidence interval through bootstrapping. We'll use the function `draw_bs_sample()` to draw bootstrap samples of the same size as our sample (n=15), and compute the means of each to create 1000 replicates. Remember, our bootstrap samples are always the same size as the original sample because we want to simulate repeating the same experiment." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Sample mean LSAT: 593.9333333333333\n", "Sample mean GPA: 3.1519999999999997\n", "\n", "\n", "Mean LSAT 95% C.I.: [580.66666667 609.87166667]\n", "Mean GPA 95% C.I.: [3.08465 3.21666667]\n", "\n", "\n", "Population mean LSAT: 597.5487804878048\n", "Population mean GPA: 3.1347560975609756\n" ] } ], "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", "sample_lsat_mean_reps = np.empty(1000)\n", "sample_gpa_mean_reps = np.empty(1000)\n", "for i in range(1000):\n", " sample_lsat_mean_reps[i] = np.mean(draw_bs_sample(sample_lsat))\n", " sample_gpa_mean_reps[i] = np.mean(draw_bs_sample(sample_gpa))\n", "\n", "print(f'Sample mean LSAT: {np.mean(sample_lsat)}')\n", "print(f'Sample mean GPA: {np.mean(sample_gpa)}')\n", "print('\\n')\n", "print(f'Mean LSAT 95% C.I.: {np.percentile(sample_lsat_mean_reps, [2.5, 97.5])}')\n", "print(f'Mean GPA 95% C.I.: {np.percentile(sample_gpa_mean_reps, [2.5, 97.5])}')\n", "print('\\n')\n", "print(f'Population mean LSAT: {np.mean(pop_lsat)}')\n", "print(f'Population mean GPA: {np.mean(pop_gpa)}')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The 95% confidence interval captures the true mean for both the LSAT and GPA data. Let's compare the ECDFs again. We will create 100 bootstrap sample ECDFs and overlay them over the population ECDF." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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Application\",\"version\":\"1.3.4\"}};\n", " var render_items = [{\"docid\":\"08847847-10af-43b2-8ce9-6387c16a4a2e\",\"roots\":{\"2353\":\"7545b831-fc1b-46aa-8def-eeaead422983\"}}];\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", " embed_document(root);\n", " clearInterval(timer);\n", " }\n", " attempts++;\n", " if (attempts > 100) {\n", " console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n", " clearInterval(timer);\n", " }\n", " }, 10, root)\n", " }\n", "})(window);" ], "application/vnd.bokehjs_exec.v0+json": "" }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "id": "2353" } }, "output_type": "display_data" } ], "source": [ "p = bokeh.plotting.figure(\n", " width=400, height=300, x_axis_label=\"LSAT\", y_axis_label=\"ECDF\"\n", ")\n", "\n", "a = bokeh.plotting.figure(\n", " width=400, height=300, x_axis_label=\"GPA\", y_axis_label=\"ECDF\",\n", ")\n", "\n", "# ECDF values for plotting\n", "sample_lsat_ecdf = ecdf(sample_lsat)\n", "pop_lsat_ecdf = ecdf(pop_lsat)\n", "sample_gpa_ecdf = ecdf(sample_gpa)\n", "pop_gpa_ecdf = ecdf(pop_gpa)\n", "\n", "# Make 100 bootstrap samples and plot them\n", "for _ in range(100):\n", " lsat = draw_bs_sample(sample_lsat)\n", " gpa = draw_bs_sample(sample_gpa)\n", " # Add semitransparent ECDFs to the plot\n", " p.circle(\n", " np.sort(lsat),\n", " sample_lsat_ecdf,\n", " color=colorcet.b_glasbey_category10[0],\n", " alpha=0.02,\n", " )\n", " a.circle(\n", " np.sort(gpa),\n", " sample_gpa_ecdf,\n", " color=colorcet.b_glasbey_category10[0],\n", " alpha=0.02,\n", " )\n", " \n", "p.circle(\n", " np.sort(pop_lsat), pop_lsat_ecdf, color=colorcet.b_glasbey_category10[1], alpha=2\n", ")\n", "a.circle(\n", " np.sort(pop_gpa), pop_gpa_ecdf, color=colorcet.b_glasbey_category10[1], alpha=2\n", ")\n", "\n", "bokeh.io.show(bokeh.layouts.row(p, a))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "What do you notice about the bootstrapped ECDFs? Consider the approximate shape and mean. \n", "\n", "Now that we've shown the power of bootstrapping, let's explore some more bootstrapping methods! First we'll review pairs bootstrapping, which we learned about in class, then go into some new methods." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Pairs Bootstrap\n", "\n", "Also covered in Lesson 7. \n", "\n", "We use pairs bootstrapping when we have paired data, and we will often want to compute the correlation. The important thing to remember with pairs bootstrapping is that we draw a random sample of *pairs*, not 2 random samples of each array. For example, if we have paired data in 2 arrays: $\\mathbf{x} = [x_1, x_2,\\ldots, x_n]$ and $\\mathbf{y} = [y_1, y_2,\\ldots, y_n]$, we must draw the same indices from both arrays so we end up with paired samples: \n", "\n", "\\begin{align}\n", "&\\mathbf{\\hat{x}} = [x_i, x_j, x_k,\\ldots, x_n], \\text{ and}\\\\[1em] \n", "&\\mathbf{\\hat{y}} = [y_i, y_j, y_k,\\ldots, y_n].\n", "\\end{align}\n", "\n", "Take a look at Lesson 7 for a coding example. \n", "\n", "**Question**: Why do we have to keep pairs together to estimate correlation? What result will you likely get for the correlation if your pairs get separated?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. BCa Confidence Intervals\n", "\n", "The BCa confidence interval is the bias-corrected accelerated bootstrap confidence interval. This concept was introduced by Bradley Efron. The BCa interval is designed to automatically create confidence intervals that adjust for underlying higher order effects. In simple terms, it corrects for bias and skew in the distribution of bootstrap estimates. \n", "\n", "In this recitation, we will focus on the nonparametric case and show a coding example. If you want to dive into this concept even more, which I highly recommend, take a look at [Efron, \"Better Bootstrap Confidence Intervals.\" Journal of the American Statistical Association 82, no. 397 (1987): 171-85.](https://doi.org/10.2307/2289144). \n", "\n", "Let's consider the *C. elegans* egg data example again. We have a dataset of repeated measurements of the areas *C. elegans* eggs. We know there is a generative distribution for this data, but we don't know what it is. We want to compute a confidence interval of the mean. For the general case, we'll call the statistic we want to estimate $\\theta$. As we know, the standard confidence interval is given by: \n", "\n", "\\begin{align}\n", "\\theta \\in [\\hat{\\theta} + \\hat{\\sigma}z^{(\\alpha)}, \\hat{\\theta} + \\hat{\\sigma}z^{(1 - \\alpha)}] \\tag{2.1}\n", "\\end{align}\n", "\n", "\n", "where $\\hat{\\theta}$ is our sample estimate of $\\theta$, $\\hat{\\sigma}$ is our estimate of the standard deviation of the distribution of $\\hat{\\theta}$, and $z^{(\\alpha)}$ is the $100 \\times \\alpha$ percentile point of a standard normal variate.\n", "\n", "This standard confidence interval (Equation 2.1) is based on taking the asymptotic normal approximation \n", "\n", "\\begin{align}\n", "(\\hat{\\theta} - \\theta)/\\hat{\\sigma} \\sim N(0, 1) \\tag{2.2}\n", "\\end{align}\n", "\n", "In our case, and in many cases, we assume there is a generative distribution, $g$, for our data. Thus, we can achieve normality and constant standard error by considering the monotone transformation $\\hat{\\phi} = g(\\hat{\\theta})$ and $\\phi = g(\\theta)$ instead of $\\hat{\\theta}$ and $\\theta$. We assume this transformation is both **normalizing** and **variance stabilizing** (defined momentarily). Through this method, we can now compute the confidence interval for $\\phi$ based on: \n", "\n", "\\begin{align}\n", "(\\hat{\\phi} - \\phi)/\\tau \\sim N(-z_0, 1) \\tag{2.3}\n", "\\end{align}\n", "\n", "where $\\tau$ is the constant standard error of $\\hat{\\phi}$ and $z_0$ is the bias constant. To get the confidence interval for $\\theta$, we take the inverse transformation $\\theta = g^{-1}(\\phi)$. The great part: all of this is achieved automatically simply through bootstrapping! We do not need to know $g$ or $z_0$.\n", "\n", "How can we improve on this? That's where BCa comes in. Like we described previously, through bootstrapping alone we are assuming the transformation $g$ is both normalizing and variance stabilizing. A [variance stabilizing transformation](https://en.wikipedia.org/wiki/Variance-stabilizing_transformation) is a transformation such that the variance is no longer related to the mean. For some distributions (ex. Binomial), the variance is dependent on the mean. A variance stabilizing transformation removes this dependence. But, since we do not know the transformation, we do not know for sure that it is variance stabilizing. In the BCa confidence interval, we only need to know that $g$ is normalizing. This makes it more accurate in many cases!\n", "\n", "The BCa incorporates some bias constant, $z_0$, and some acceleration constant, $a$, such that the BCa confidence interval is based on: \n", "\n", "\\begin{align}\n", "&(\\hat{\\phi} - \\phi)/\\tau \\sim N(-z_0\\sigma_\\phi, \\sigma_\\phi^2),\\\\[1em]\n", "&\\sigma_\\phi = 1 + a\\phi. \\tag{2.4}\n", "\\end{align}\n", "\n", "We can determine $a$ and $z_0$ by resampling our data. Here, we are considering the nonparametric case. For the nonparametric case, the acceleration constant for the mean is given by\n", "\n", "\\begin{align}\n", "a = \\frac{\\sum_i (x_i - \\bar{x})^3}{6\\left(\\sum_i (x_i - \\bar{x})^2\\right)^{3/2}}.\n", "\\tag{2.5}\n", "\\end{align}\n", "\n", "We can compute $z_0$ from our bootstrap replicates. The bias-correction constant, $z_0$, is: \n", "\n", "\\begin{align}\n", "z_0 = \\phi^{-1}(\\hat{G}(\\hat{\\theta}))\n", "\\tag{2.6}\n", "\\end{align}\n", "\n", "From the plug-in principle, we know that for the nonparametric case we can approximate the CDF $\\hat{G}(\\hat{\\theta})$ as the ECDF of our bootstrap replicates. So, $\\phi^{-1}(\\hat{G}(\\hat{\\theta}))$ is given by the inverse ECDF of our bootstrap replicates. \n", "\n", "This may all be a little confusing now, so let's practice with the worm egg example! I will review how to compute each parameter as we go.\n", "\n", "Let's load in our worm egg area dataset, digitized from [Harvey and Orbidans's 2011 PLoS One paper](https://doi.org/10.1371/journal.pone.0025840)." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " food area (sq um)\n", "0 high 1683\n", "1 high 2061\n", "2 high 1792\n", "3 high 1852\n", "4 high 2091" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_worms = pd.read_csv('../data/c_elegans_egg_xa.csv', comment='#')\n", "df_worms.head()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We'll convert the data we're interested into arrays:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "fed_worms = df_worms.loc[df_worms['food'] == 'high']['area (sq um)'].values\n", "hungry_worms = df_worms.loc[df_worms['food'] == 'low']['area (sq um)'].values" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here we write some helper functions to calculate the parameters for the BCa interval. We will combine them in our main function, `compute_bca_ci()`." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [], "source": [ "st.norm.ppf?" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "def compute_jackknife_reps(data, statfunction=np.mean):\n", " '''Returns jackknife resampled replicates for the given data and statistical function'''\n", " # Set up empty array to store jackknife replicates\n", " jack_reps = np.empty(len(data))\n", " \n", " # For each observation in the dataset, compute the statistical function on the sample \n", " # with that observation removed\n", " for i in range(len(data)):\n", " jack_sample = np.delete(data, i)\n", " jack_reps[i] = statfunction(jack_sample)\n", " return jack_reps\n", "\n", "\n", "def compute_a(jack_reps):\n", " '''Returns the acceleration constant a'''\n", " mean = np.mean(jack_reps)\n", " return (1/6) * np.divide(np.sum(mean - jack_reps)**3, (np.sum(mean - jack_reps)**2)**(3/2))\n", "\n", "\n", "def bootstrap_replicates(data, n_reps=1000, statfunction=np.mean):\n", " '''Computes n_reps number of bootstrap replicates for given data and statistical function'''\n", " boot_reps = np.empty(n_reps)\n", " for i in range(n_reps):\n", " boot_reps[i] = statfunction(draw_bs_sample(data))\n", " return boot_reps\n", "\n", "\n", "def compute_z0(data, boot_reps, statfunction=np.mean):\n", " '''Computes z0 for given data and statistical function'''\n", " s = statfunction(data)\n", " return st.norm.ppf(np.sum(boot_reps < s) / len(boot_reps))\n", "\n", "\n", "def compute_bca_ci(data, alpha_level, n_reps=1000, statfunction=np.mean):\n", " '''Returns BCa confidence interval for given data at given alpha level'''\n", " # Compute bootstrap and jackknife replicates\n", " boot_reps = bootstrap_replicates(data, n_reps, statfunction)\n", " jack_reps = compute_jackknife_reps(data, statfunction)\n", " \n", " # Compute a and z0\n", " a = compute_a(jack_reps)\n", " z0 = compute_z0(data, boot_reps, statfunction)\n", " \n", " # Compute confidence interval indices\n", " alphas = np.array([alpha_level/2., 1-alpha_level/2.])\n", " zs = z0 + st.norm.ppf(alphas).reshape(alphas.shape+(1,)*z0.ndim)\n", " avals = st.norm.cdf(z0 + zs/(1-a*zs))\n", " ints = np.round((len(boot_reps)-1)*avals)\n", " ints = np.nan_to_num(ints).astype('int')\n", " \n", " # Compute confidence interval\n", " boot_reps = np.sort(boot_reps)\n", " ci_low = boot_reps[ints[0]]\n", " ci_high = boot_reps[ints[1]]\n", " return (ci_low, ci_high)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now compute the BCa confidence intervals." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Fed worms mean egg area 95% C.I.:(1791.4318181818182, 1886.8181818181818)\n", "Hungry worms mean egg area 95% C.I.:(2068.9298245614036, 2147.1052631578946)\n" ] } ], "source": [ "print(f'Fed worms mean egg area 95% C.I.:{compute_bca_ci(fed_worms, 0.05)}')\n", "print(f'Hungry worms mean egg area 95% C.I.:{compute_bca_ci(hungry_worms, 0.05)}')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 3. Studentized Bootstrap\n", "\n", "The Studentized bootstrap error aims to reduce coverage error of our resampling. It is a good idea to use Studentized bootstrapping to approximate your statistic of interest when you suspect the distribution of your sample may not be close to the actual population distribution. The Studentized bootstrapping method works because even when this is the case, the distribution of the sample statistic divided by estimated standard error could be close to the actual population statistic divided by the population standard error. This the same principle employed in the Student-t test. \n", "\n", "The Studentized bootstrap involves taking inner and outer bootstrap samples. The algorithm is as follows:\n", "\n", "1. Compute $\\hat{\\theta}$, the sample test statistic\n", "2. A large number of times (number of reps ≈1000), draw a bootstrap sample. Call this the outer sample:\n", " a. Compute the statistical functional on the sample\n", " b. Resample your bootstrap sample ≈50 to 100 times. Call this the inner sample:\n", " i. Compute the statistical function on the resample\n", " c. For your inner sample replicates, compute the standard deviation\n", "3. Compute your statistic of interest from your replicates: t = (outer sample replicates) / (standard deviation of the inner sample replicates)\n", "4. Compute `s` = standard deviation of your outer sample replicates\n", "3. Multiply `t` by `s`\n", "5. Compute the confidence interval from the $\\alpha/2$ and $(1 - \\alpha)/2$ percentiles of t*s\n", "\n", "\n", "Let's try this out on a dataset comparing bee sperm counts in the presence of a pesticide to control bees. The data come from [Straub, et al., Proc. Royal Soc. B 283(1835): 20160506](http://dx.doi.org/10.1098/rspb.2016.0506)." ] }, { "cell_type": "code", "execution_count": 14, "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
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" ], "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": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_bees = pd.read_csv('../data/bee_sperm.csv', comment='#')\n", "df_bees.head()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's take a quick look at the ECDFs." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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Application\",\"version\":\"1.3.4\"}};\n", " var render_items = [{\"docid\":\"24dca62b-aeba-4c48-ab64-d56504127740\",\"roots\":{\"5302\":\"b7ccdd89-b249-4604-8f1d-438e8f2e5d08\"}}];\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", " embed_document(root);\n", " clearInterval(timer);\n", " }\n", " attempts++;\n", " if (attempts > 100) {\n", " console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n", " clearInterval(timer);\n", " }\n", " }, 10, root)\n", " }\n", "})(window);" ], "application/vnd.bokehjs_exec.v0+json": "" }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "id": "5302" } }, "output_type": "display_data" } ], "source": [ "bokeh.io.show(\n", " bokeh_catplot.ecdf(\n", " data=df_bees,\n", " cats='Treatment',\n", " val='Alive Sperm Millions'\n", " )\n", ")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For computing the confidence intervals of the means, we'll want to convert the data of interest into arrays, and take a look at the sample mean:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Control mean alive sperm (millions): 1.8727770758620692\n", "Pesticide mean alive sperm(millions): 1.2997477111111113\n" ] } ], "source": [ "control = df_bees.loc[df_bees['Treatment'] == 'Control']['Alive Sperm Millions'].values\n", "pesticide = df_bees.loc[df_bees['Treatment'] == 'Pesticide']['Alive Sperm Millions'].values\n", "\n", "print(f'Control mean alive sperm (millions): {np.mean(control)}')\n", "print(f'Pesticide mean alive sperm(millions): {np.mean(pesticide)}')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, we'll employ the Studentized bootstrap to compute the confidence interval." ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Control Alive Sperm Millions 95% C.I.: (1.6422482149046422, 2.0943950244253298)\n", "Pesticide Alive Sperm Millions 95% C.I.: (1.1189521249528647, 1.4847851245037909)\n" ] } ], "source": [ "def studentized_bootstrap_ci(\n", " data,\n", " statfunction=np.mean,\n", " n_bs_samples=1000,\n", " n_inner_samples=100,\n", " ptiles=(2.5, 97.5),\n", "):\n", " thetas = np.empty(n_bs_samples)\n", " ts = np.zeros(n_bs_samples)\n", "\n", " # Outer sample\n", " for i in range(n_bs_samples):\n", " sample = draw_bs_sample(data)\n", " theta = statfunction(sample)\n", " thetas[i] = theta\n", "\n", " # Inner sample\n", " resample_stats = np.empty(n_inner_samples)\n", " for j in range(n_inner_samples):\n", " resample = draw_bs_sample(sample)\n", " resample_stats[j] = statfunction(resample)\n", "\n", " # Compute statistic of interest\n", " ts[i] = np.sum(resample_stats - theta) / np.std(resample_stats)\n", "\n", " s = np.std(thetas)\n", " ts = np.sort(ts)\n", "\n", " p1, p2 = np.percentile(s*ts, ptiles)\n", " ci_high = np.mean(data)-(s*p1)\n", " ci_low = np.mean(data)-(s*p2)\n", " return (ci_low, ci_high)\n", "\n", "studentized_ctrl_ci = studentized_bootstrap_ci(control)\n", "studentized_pest_ci = studentized_bootstrap_ci(pesticide)\n", "\n", "print(f\"Control Alive Sperm Millions 95% C.I.: {studentized_ctrl_ci}\")\n", "print(f\"Pesticide Alive Sperm Millions 95% C.I.: {studentized_pest_ci}\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's get a plot of these confidence intervals to compare." ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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10, root)\n", " }\n", "})(window);" ], "application/vnd.bokehjs_exec.v0+json": "" }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "id": "7109" } }, "output_type": "display_data" } ], "source": [ "def plot_with_error_bars(means, confs, names, **kwargs):\n", " \"\"\"Make a horizontal plot of means/conf ints with error bars.\"\"\"\n", " frame_height = kwargs.pop(\"frame_height\", 150)\n", " frame_width = kwargs.pop(\"frame_width\", 450)\n", "\n", " p = bokeh.plotting.figure(\n", " y_range=names, frame_height=frame_height, frame_width=frame_width, **kwargs\n", " )\n", "\n", " p.circle(x=means, y=names)\n", " for conf, name in zip(confs, names):\n", " p.line(x=conf, y=[name, name], line_width=2)\n", "\n", " return p\n", "\n", "means = [np.mean(control), np.mean(pesticide)]\n", "confs = [studentized_ctrl_ci, studentized_pest_ci]\n", "names = ['control', 'pesticide']\n", "\n", "bokeh.io.show(plot_with_error_bars(means, confs, names, x_axis_label='alive sperm (millions)'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Success!! We can see the mean alive sperm count in millions for the pesticide-exposed bees is likely lower than the control bees, but the confidence intervals do overlap so we should temper that statement.\n", "\n", "For comparison, we can compute the non-Studentized confidence intervals." ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Control Alive Sperm Millions 95% C.I.: [1.67537854 2.07238755]\n", "Pesticide Alive Sperm Millions 95% C.I.: [1.11075824 1.49738931]\n" ] } ], "source": [ "ctrl_reps = [np.mean(draw_bs_sample(control)) for _ in range(10000)]\n", "pest_reps = [np.mean(draw_bs_sample(pesticide)) for _ in range(10000)]\n", "\n", "ctrl_ci = np.percentile(ctrl_reps, [2.5, 97.5])\n", "pest_ci = np.percentile(pest_reps, [2.5, 97.5])\n", "\n", "print(f\"Control Alive Sperm Millions 95% C.I.: {ctrl_ci}\")\n", "print(f\"Pesticide Alive Sperm Millions 95% C.I.: {pest_ci}\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's plot." ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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'studentized pest', 'ctrl', 'pest']\n", "\n", "bokeh.io.show(plot_with_error_bars(means, confs, names, x_axis_label='alive sperm (millions)'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Studentized confidence intervals are fairly close, but slightly larger. In this case, our sample distribution is likely close to the population distribution, so the rigor of the studentized bootstrap may not dramatically improve our results. However, when the distributions are different, the inner sample will attempt to correct for this." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Computing environment" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "CPython 3.7.4\n", "IPython 5.8.0\n", "\n", "numpy 1.17.4\n", "scipy 1.3.1\n", "numba 0.45.1\n", "bokeh 1.3.4\n", "colorcet 1.0.0\n", "jupyterlab 1.2.0\n" ] } ], "source": [ "%load_ext watermark\n", "%watermark -v -p numpy,scipy,numba,bokeh,colorcet,jupyterlab" ] } ], "metadata": { "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": "3.7.4" } }, "nbformat": 4, "nbformat_minor": 4 }