{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Pairs bootstrap and correlation\n", "\n", "[Dataset download](https://s3.amazonaws.com/bebi103.caltech.edu/data/bee_sperm.csv)\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "tags": [] }, "outputs": [ { "data": { "text/html": [ "\n", "
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\\n\"+\n \"

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\\n\"+\n \"\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"
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" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We continue our analysis of the [drone sperm quality data set](https://s3.amazonaws.com/bebi103.caltech.edu/data/bee_sperm.csv). Let's load it and remind ourselves of the content." ] }, { "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(\"../data/bee_sperm.csv\", comment='#')\n", "df.head()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Correlation\n", "\n", "We might wish to investigate how two measured quantities are correlated. For example, if the number of dead sperm and the number of alive sperm are closely correlated, this would mean that a given drone produces some quantity of sperm and some fraction tend to be dead. Let's take a look at this." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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" const render_items = [{\"docid\":\"6c1cdc71-8f94-4f57-987c-ea47f3d3bc72\",\"roots\":{\"p1002\":\"d663b412-4b41-457e-9013-c5ef61145072\"},\"root_ids\":[\"p1002\"]}];\n", " 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": "p1002" } }, "output_type": "display_data" } ], "source": [ "# Only use values greater than zero for log scale\n", "inds = (df[\"Alive Sperm Millions\"] > 0) & (df[\"Dead Sperm Millions\"] > 0)\n", "\n", "p = bokeh.plotting.figure(\n", " frame_height=300,\n", " frame_width=300,\n", " x_axis_label=\"alive sperm (millions)\",\n", " y_axis_label=\"dead sperm (millions)\",\n", " x_axis_type=\"log\",\n", " y_axis_type=\"log\",\n", ")\n", "\n", "p.circle(\n", " source=df.loc[inds & (df[\"Treatment\"] == \"Control\"), :],\n", " x=\"Alive Sperm Millions\",\n", " y=\"Dead Sperm Millions\",\n", " legend_label=\"control\",\n", ")\n", "p.circle(\n", " source=df.loc[inds & (df[\"Treatment\"] == \"Pesticide\"), :],\n", " x=\"Alive Sperm Millions\",\n", " y=\"Dead Sperm Millions\",\n", " color='orange',\n", " legend_label=\"pesticide\",\n", ")\n", "\n", "p.legend.location = 'top_left'\n", "\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "There seems to be some correlation (on a log scale), but it is difficult to tell. We can compute the correlation with the **bivariate correlation coefficient**, also known as the Pearson correlation. It is the plug-in estimate of the correlation between variables (in this case alive and dead sperm). The **correlation** is the covariance divided by the geometric mean of the individual variances\n", "\n", "The bivariate correlation coefficient is implemented with `np.corrcoef()`, but we will code our own. We will JIT it for speed." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "@numba.njit\n", "def bivariate_r(x, y):\n", " \"\"\"\n", " Compute plug-in estimate for the bivariate correlation coefficient.\n", " \"\"\"\n", " return (\n", " np.sum((x - np.mean(x)) * (y - np.mean(y)))\n", " / np.std(x)\n", " / np.std(y)\n", " / np.sqrt(len(x))\n", " / np.sqrt(len(y))\n", " )" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can use it to compute the bivariate correlation coefficient for the logarithm of alive and dead sperm." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.1785839325544448" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "bivariate_r(\n", " df.loc[inds, \"Alive Sperm Millions\"].values,\n", " df.loc[inds, \"Dead Sperm Millions\"].values,\n", ")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Pairs bootstrap confidence intervals\n", "\n", "How can we get a confidence interval on a correlation coefficient? We can again apply the bootstrap, but this time, the replicate is a *pair* of data, in this case a dead sperm count/alive sperm count pair. The process of drawing pairs of data points from an experiment and then computing bootstrap replicates from them is called **pairs bootstrap**. Let's code it up for this example with the bivariate correlation.\n", "\n", "Our strategy in coding up the pairs bootstrap is to draw bootstrap samples of the *indices* of measurement and use those indices to select the pairs." ] }, { "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))\n", "\n", "\n", "@numba.njit\n", "def draw_bs_pairs(x, y):\n", " \"\"\"Draw a pairs bootstrap sample.\"\"\"\n", " inds = np.arange(len(x))\n", " bs_inds = draw_bs_sample(inds)\n", " \n", " return x[bs_inds], y[bs_inds]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "With our pairs sampling function in place, we can write a function to compute replicates." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "@numba.njit\n", "def draw_bs_pairs_reps_bivariate(x, y, size=1):\n", " \"\"\"\n", " Draw bootstrap pairs replicates.\n", " \"\"\"\n", " out = np.empty(size)\n", "\n", " for i in range(size):\n", " out[i] = bivariate_r(*draw_bs_pairs(x, y))\n", "\n", " return out" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, we can put it all together to compute confidence intervals on the correlation. To start, we extract all of the relevant measurements as Numpy arrays to allow for faster resampling (and that's what our Numba'd functions require)." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "# Extract NumPy arrays (only use values greater than zero for logs)\n", "inds = (df[\"Alive Sperm Millions\"] > 0) & (df[\"Dead Sperm Millions\"] > 0)\n", "\n", "alive_ctrl = df.loc[\n", " (inds) & (df[\"Treatment\"] == \"Control\"), \"Alive Sperm Millions\"\n", "].values\n", "\n", "alive_pest = df.loc[\n", " (inds) & (df[\"Treatment\"] == \"Pesticide\"), \"Alive Sperm Millions\"\n", "].values\n", "\n", "dead_ctrl = df.loc[\n", " (inds) & (df[\"Treatment\"] == \"Control\"), \"Dead Sperm Millions\"\n", "].values\n", "\n", "dead_pest = df.loc[\n", " (inds) & (df[\"Treatment\"] == \"Pesticide\"), \"Dead Sperm Millions\"\n", "].values" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can compute the bootstrap replicates using our `draw_bs_pairs_reps_bivariate()` function." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "# Get reps\n", "bs_reps_ctrl = draw_bs_pairs_reps_bivariate(\n", " np.log(alive_ctrl), np.log(dead_ctrl), size=10000\n", ")\n", "\n", "bs_reps_pest = draw_bs_pairs_reps_bivariate(\n", " np.log(alive_pest), np.log(dead_pest), size=10000\n", ")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And from the replicates, we can compute and plot the 95% confidence interval." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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" const render_items = [{\"docid\":\"2734087b-5a2e-44d4-aa3e-ac277ecce022\",\"roots\":{\"p1087\":\"c89e6246-8c02-4eec-aef1-4753cd6f4933\"},\"root_ids\":[\"p1087\"]}];\n", " 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": "p1087" } }, "output_type": "display_data" } ], "source": [ "# Get the confidence intervals\n", "conf_int_ctrl = np.percentile(bs_reps_ctrl, [2.5, 97.5])\n", "conf_int_pest = np.percentile(bs_reps_pest, [2.5, 97.5])\n", "\n", "# Plot confidence intervals\n", "summaries = [\n", " dict(\n", " label=\"control\",\n", " estimate=bivariate_r(np.log(alive_ctrl), np.log(dead_ctrl)),\n", " conf_int=conf_int_ctrl,\n", " ),\n", " dict(\n", " label=\"treatment\",\n", " estimate=bivariate_r(np.log(alive_pest), np.log(dead_pest)),\n", " conf_int=conf_int_pest,\n", " ),\n", "]\n", "p = bebi103.viz.confints(summaries, x_axis_label=\"bivariate correlation of logs\")\n", "\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see a clear correlation in both samples, with a wide, but positive, confidence interval. Note that we did this analysis on a log scale, since the data span several orders of magnitude." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Computing environment" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Python implementation: CPython\n", "Python version : 3.11.4\n", "IPython version : 8.12.2\n", "\n", "numpy : 1.24.3\n", "pandas : 2.0.3\n", "numba : 0.57.0\n", "bokeh : 3.2.1\n", "bebi103 : 0.1.15\n", "jupyterlab: 4.0.5\n", "\n" ] } ], "source": [ "%load_ext watermark\n", "%watermark -v -p numpy,pandas,numba,bokeh,bebi103,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": "3.11.4" } }, "nbformat": 4, "nbformat_minor": 4 }