L09-Accounting-for-variation.html

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<title>9&nbsp; Accounting for variation – Lessons in Statistical Thinking</title>
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  // Create main div element
  var mainDiv = document.createElement('div');
  mainDiv.id = 'qwebr-noninteractive-area-' + qwebrCounter;
  mainDiv.className = `qwebr-noninteractive-area`;
  if (qwebrOptions.classes) {
    mainDiv.className += " " + qwebrOptions.classes
  }
  
  // Add a unique cell identifier that users can customize
  if (qwebrOptions.label) {
    mainDiv.setAttribute('data-id', qwebrOptions.label);
  }
  
  // Create a status container div
  var statusContainer = createLoadingContainer(qwebrCounter);

  // Create output code area div
  var outputCodeAreaDiv = document.createElement('div');
  outputCodeAreaDiv.id = 'qwebr-output-code-area-' + qwebrCounter;
  outputCodeAreaDiv.className = 'qwebr-output-code-area';
  outputCodeAreaDiv.setAttribute('aria-live', 'assertive');

  // Create pre element inside output code area
  var preElement = document.createElement('pre');
  preElement.style.visibility = 'hidden';
  outputCodeAreaDiv.appendChild(preElement);

  // Create output graph area div
  var outputGraphAreaDiv = document.createElement('div');
  outputGraphAreaDiv.id = 'qwebr-output-graph-area-' + qwebrCounter;
  outputGraphAreaDiv.className = 'qwebr-output-graph-area';

  // Append all elements to the main div
  mainDiv.appendChild(statusContainer);
  mainDiv.appendChild(outputCodeAreaDiv);
  mainDiv.appendChild(outputGraphAreaDiv);

  return mainDiv;
};

// Function that adds a stub in the page to indicate a setup cell was used.
globalThis.qwebrCreateNonInteractiveSetupElement = function(qwebrCounter, qwebrOptions) {
  // Create main div element
  var mainDiv = document.createElement('div');
  mainDiv.id = `qwebr-noninteractive-setup-area-${qwebrCounter}`;
  mainDiv.className = `qwebr-noninteractive-setup-area`;
  if (qwebrOptions.classes) {
    mainDiv.className += " " + qwebrOptions.classes
  }


  // Add a unique cell identifier that users can customize
  if (qwebrOptions.label) {
    mainDiv.setAttribute('data-id', qwebrOptions.label);
  }

  // Create a status container div
  var statusContainer = createLoadingContainer(qwebrCounter);

  // Append status onto the main div
  mainDiv.appendChild(statusContainer);

  return mainDiv;
}


// Function to create loading container with specified ID
globalThis.createLoadingContainer = function(qwebrCounter) {

  // Create a status container
  const container = document.createElement('div');
  container.id = `qwebr-non-interactive-loading-container-${qwebrCounter}`;
  container.className = 'qwebr-non-interactive-loading-container qwebr-cell-needs-evaluation';

  // Create an R project logo to indicate its a code space
  const rProjectIcon = document.createElement('i');
  rProjectIcon.className = 'fa-brands fa-r-project fa-3x qwebr-r-project-logo';

  // Setup a loading icon from font awesome
  const spinnerIcon = document.createElement('i');
  spinnerIcon.className = 'fa-solid fa-spinner fa-spin fa-1x qwebr-icon-status-spinner';

  // Add a section for status text
  const statusText = document.createElement('p');
  statusText.id = `qwebr-status-text-${qwebrCounter}`;
  statusText.className = `qwebr-status-text qwebr-cell-needs-evaluation`;
  statusText.innerText = 'Loading webR...';

  // Incorporate an inner container
  const innerContainer = document.createElement('div');

  // Append elements to the inner container
  innerContainer.appendChild(spinnerIcon);
  innerContainer.appendChild(statusText);

  // Append elements to the main container
  container.appendChild(rProjectIcon);
  container.appendChild(innerContainer);

  return container;
}
</script>
<script type="module">
// Function to install a single package
async function qwebrInstallRPackage(packageName) {
  await mainWebR.evalRVoid(`webr::install('${packageName}');`);
}

// Function to load a single package
async function qwebrLoadRPackage(packageName) {
  await mainWebR.evalRVoid(`require('${packageName}', quietly = TRUE)`);
}

// Generic function to process R packages
async function qwebrProcessRPackagesWithStatus(packages, processType, displayStatusMessageUpdate = true) {
  // Switch between contexts
  const messagePrefix = processType === 'install' ? 'Installing' : 'Loading';

  // Modify button state
  qwebrSetInteractiveButtonState(`🟡 ${messagePrefix} package ...`, false);

  // Iterate over packages
  for (let i = 0; i < packages.length; i++) {
    const activePackage = packages[i];
    const formattedMessage = `${messagePrefix} package ${i + 1} out of ${packages.length}: ${activePackage}`;

    // Display the update in header
    if (displayStatusMessageUpdate) {
      qwebrUpdateStatusHeader(formattedMessage);
    }

    // Display the update in non-active areas
    qwebrUpdateStatusMessage(formattedMessage);

    // Run package installation
    if (processType === 'install') {
      await qwebrInstallRPackage(activePackage);
    } else {
      await qwebrLoadRPackage(activePackage);
    }
  }

  // Clean slate
  if (processType === 'load') {
    await mainWebR.flush();
  }
}

// Start a timer
const initializeWebRTimerStart = performance.now();

// Encase with a dynamic import statement
globalThis.qwebrInstance = import(qwebrCustomizedWebROptions.baseURL + "webr.mjs").then(
  async ({ WebR, ChannelType }) => {
    // Populate WebR options with defaults or new values based on `webr` meta
    globalThis.mainWebR = new WebR(qwebrCustomizedWebROptions);

    // Initialization WebR
    await mainWebR.init();

    // Setup a shelter
    globalThis.mainWebRCodeShelter = await new mainWebR.Shelter();

    // Setup a pager to allow processing help documentation 
    await mainWebR.evalRVoid('webr::pager_install()'); 

    // Override the existing install.packages() to use webr::install()
    await mainWebR.evalRVoid('webr::shim_install()'); 

    // Specify the repositories to pull from
    // Note: webR does not use the `repos` option, but instead uses `webr_pkg_repos`
    // inside of `install()`. However, other R functions still pull from `repos`.
    await mainWebR.evalRVoid(`
      options(
        webr_pkg_repos = c(${qwebrPackageRepoURLS.map(repoURL => `'${repoURL}'`).join(',')}),
        repos = c(${qwebrPackageRepoURLS.map(repoURL => `'${repoURL}'`).join(',')})
      )
    `);

    // Check to see if any packages need to be installed
    if (qwebrSetupRPackages) {
      // Obtain only a unique list of packages
      const uniqueRPackageList = Array.from(new Set(qwebrInstallRPackagesList));

      // Install R packages one at a time (either silently or with a status update)
      await qwebrProcessRPackagesWithStatus(uniqueRPackageList, 'install', qwebrShowStartupMessage);

      if (qwebrAutoloadRPackages) {
        // Load R packages one at a time (either silently or with a status update)
        await qwebrProcessRPackagesWithStatus(uniqueRPackageList, 'load', qwebrShowStartupMessage);
      }
    }
  }
);

// Stop timer
const initializeWebRTimerEnd = performance.now();

</script>
<script type="module">
// Function to verify a given JavaScript Object is empty
globalThis.qwebrIsObjectEmpty = function (arr) {
    return Object.keys(arr).length === 0;
}

// Global version of the Escape HTML function that converts HTML 
// characters to their HTML entities.
globalThis.qwebrEscapeHTMLCharacters = function(unsafe) {
    return unsafe
      .replace(/&/g, "&amp;")
      .replace(/</g, "&lt;")
      .replace(/>/g, "&gt;")
      .replace(/"/g, "&quot;")
      .replace(/'/g, "&#039;");
  };

// Passthrough results
globalThis.qwebrIdentity = function(x) {
    return x;
};

// Append a comment
globalThis.qwebrPrefixComment = function(x, comment) {
    return `${comment}${x}`;
};

// Function to parse the pager results
globalThis.qwebrParseTypePager = async function (msg) { 

    // Split out the event data
    const { path, title, deleteFile } = msg.data; 

    // Process the pager data by reading the information from disk
    const paged_data = await mainWebR.FS.readFile(path).then((data) => {
        // Obtain the file content
        let content = new TextDecoder().decode(data);

        // Remove excessive backspace characters until none remain
        while(content.match(/.[\b]/)){
        content = content.replace(/.[\b]/g, '');
        }

        // Returned cleaned data
        return content;
    });

    // Unlink file if needed
    if (deleteFile) { 
        await mainWebR.FS.unlink(path); 
    } 

    // Return extracted data with spaces
    return paged_data;
} 

// Function to run the code using webR and parse the output
globalThis.qwebrComputeEngine = async function(
    codeToRun, 
    elements, 
    options) {

    // Call into the R compute engine that persists within the document scope.
    // To be prepared for all scenarios, the following happens: 
    // 1. We setup a canvas device to write to by making a namespace call into the {webr} package
    // 2. We use values inside of the options array to set the figure size.
    // 3. We capture the output stream information (STDOUT and STERR)
    // 4. We disable the current device's image creation.
    // 5. Piece-wise parse the results into the different output areas

    // Create a pager variable for help/file contents
    let pager = [];

    // Handle how output is processed
    let showMarkup = options.results === "markup" && options.output !== "asis";
    let processOutput;

    if (showMarkup) {
        processOutput = qwebrEscapeHTMLCharacters;
    } else {
        processOutput = qwebrIdentity;
    }

    // ---- 
    // Convert from Inches to Pixels by using DPI (dots per inch)
    // for bitmap devices (dpi * inches = pixels)
    let fig_width = options["fig-width"] * options["dpi"]
    let fig_height = options["fig-height"] * options["dpi"]

    // Initialize webR
    await mainWebR.init();

    // Setup a webR canvas by making a namespace call into the {webr} package
    // Evaluate the R code
    // Remove the active canvas silently
    const result = await mainWebRCodeShelter.captureR(
        `webr::canvas(width=${fig_width}, height=${fig_height}, capture = TRUE)
        .webr_cvs_id <- dev.cur()
        ${codeToRun}
        invisible(dev.off(.webr_cvs_id))
        `, {
        withAutoprint: true,
        captureStreams: true,
        captureConditions: false//,
        // env: webR.objs.emptyEnv, // maintain a global environment for webR v0.2.0
    });

    // -----

    // Start attempting to parse the result data
    processResultOutput:try {
        
        // Avoid running through output processing
        if (options.results === "hide" || options.output === "false") { 
            break processResultOutput; 
        }

        // Merge output streams of STDOUT and STDErr (messages and errors are combined.)
        // Require both `warning` and `message` to be true to display `STDErr`. 
        const out = result.output
        .filter(
            evt => evt.type === "stdout" || 
            ( evt.type === "stderr" && (options.warning === "true" && options.message === "true")) 
        )
        .map((evt, index) => {
            const className = `qwebr-output-code-${evt.type}`;
            const outputResult = qwebrPrefixComment(processOutput(evt.data), options.comment);
            return `<code id="${className}-editor-${elements.id}-result-${index + 1}" class="${className}">${outputResult}</code>`;
        })
        .join("\n");


        // Clean the state
        // We're now able to process pager events.
        // As a result, we cannot maintain a true 1-to-1 output order 
        // without individually feeding each line
        const msgs = await mainWebR.flush();

        // Use `map` to process the filtered "pager" events asynchronously
        const pager = await Promise.all(
            msgs.filter(msg => msg.type === 'pager').map(
                async (msg) => {
                    return await qwebrParseTypePager(msg);
                }
            )
        );

        // Nullify the output area of content
        elements.outputCodeDiv.innerHTML = "";
        elements.outputGraphDiv.innerHTML = "";

        // Design an output object for messages
        const pre = document.createElement("pre");
        if (/\S/.test(out)) {
            // Display results as HTML elements to retain output styling
            const div = document.createElement("div");
            div.innerHTML = out;
            pre.appendChild(div);
        } else {
            // If nothing is present, hide the element.
            pre.style.visibility = "hidden";
        }

        elements.outputCodeDiv.appendChild(pre);

        // Determine if we have graphs to display
        if (result.images.length > 0) {
            // Create figure element
            const figureElement = document.createElement('figure');

            // Place each rendered graphic onto a canvas element
            result.images.forEach((img) => {
                // Construct canvas for object
                const canvas = document.createElement("canvas");

                // Set canvas size to image
                canvas.width = img.width;
                canvas.height = img.height;

                // Apply output truncations
                canvas.style.width = options["out-width"] ? options["out-width"] : `${fig_width}px`;
                if (options["out-height"]) {
                    canvas.style.height = options["out-height"];
                }

                // Apply styling
                canvas.style.display = "block";
                canvas.style.margin = "auto";

                // Draw image onto Canvas
                const ctx = canvas.getContext("2d");
                ctx.drawImage(img, 0, 0, img.width, img.height);
          
                // Append canvas to figure output area
                figureElement.appendChild(canvas);
            });

            if (options['fig-cap']) {
                // Create figcaption element
                const figcaptionElement = document.createElement('figcaption');
                figcaptionElement.innerText = options['fig-cap'];
                // Append figcaption to figure
                figureElement.appendChild(figcaptionElement);    
            }

            elements.outputGraphDiv.appendChild(figureElement);
        }

        // Display the pager data
        if (pager) {
        // Use the `pre` element to preserve whitespace.
        pager.forEach((paged_data, index) => {
            let pre_pager = document.createElement("pre");
            pre_pager.innerText = paged_data;
            pre_pager.classList.add("qwebr-output-code-pager");
            pre_pager.setAttribute("id", `qwebr-output-code-pager-editor-${elements.id}-result-${index + 1}`);
            elements.outputCodeDiv.appendChild(pre_pager);
        });
        }
    } finally {
        // Clean up the remaining code
        mainWebRCodeShelter.purge();
    }
}

// Function to execute the code (accepts code as an argument)
globalThis.qwebrExecuteCode = async function (
    codeToRun,
    id,
    options = {}) {

    // If options are not passed, we fall back on the bare minimum to handle the computation
    if (qwebrIsObjectEmpty(options)) {
        options = { 
            "context": "interactive", 
            "fig-width": 7, "fig-height": 5, 
            "out-width": "700px", "out-height": "", 
            "dpi": 72,
            "results": "markup", 
            "warning": "true", "message": "true",
        };
    }

    // Next, we access the compute areas values
    const elements = {
        runButton: document.getElementById(`qwebr-button-run-${id}`),
        outputCodeDiv: document.getElementById(`qwebr-output-code-area-${id}`),
        outputGraphDiv: document.getElementById(`qwebr-output-graph-area-${id}`),
        id: id,
    }

    // Disallowing execution of other code cells
    document.querySelectorAll(".qwebr-button-run").forEach((btn) => {
        btn.disabled = true;
    });

    if (options.context == EvalTypes.Interactive) {
        // Emphasize the active code cell
        elements.runButton.innerHTML = '<i class="fa-solid fa-spinner fa-spin qwebr-icon-status-spinner"></i> <span>Run Code</span>';
    }

    // Evaluate the code and parse the output into the document
    await qwebrComputeEngine(codeToRun, elements, options);

    // Switch to allowing execution of code
    document.querySelectorAll(".qwebr-button-run").forEach((btn) => {
        btn.disabled = false;
    });

    if (options.context == EvalTypes.Interactive) {
        // Revert to the initial code cell state
        elements.runButton.innerHTML = '<i class="fa-solid fa-play qwebr-icon-run-code"></i> <span>Run Code</span>';
    }
}

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<h1 class="title"><span id="sec-accounting-for-variation" class="quarto-section-identifier"><span class="chapter-number">9</span>&nbsp; <span class="chapter-title">Accounting for variation</span></span></h1>
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<p>Lesson <a href="#sec-variation" class="quarto-xref"><span class="quarto-unresolved-ref ref-noprefix">sec-variation</span></a> listed three foundations for statistical thinking and provided instruction on the first one:</p>
<ol type="1">
<li>How to measure the amount of variation in a variable</li>
</ol>
<p>In this Lesson, we will take on two and three:</p>
<ol start="2" type="1">
<li>Accounting for variation in a response variable.</li>
<li>Measuring what remains unaccounted for.</li>
</ol>
<p>In business, accounting keeps track of money and value: assets, receipts, expenditures, etc. Likewise, in statistics, accounting for variation is the process of keeping track of variation. At its most basic, accounting for variation splits a variable into components: those parts which can be attributed to one or more other variables and the remaining (or residual<a href="Glossary.html#def-residual" class="quarto-xref" target="popup" onclick="window.open('Glossary.html#def-residual', 'popup', 'width=500, height=300'); return false;"><sup><i class="bi bi-card-text"></i></sup></a> ) which cannot be so attributed.</p>
<p>The variable being split up is called the “<strong>response variable</strong>.” The human modeler chooses a response variable, depending on her purpose for undertaking the work. The modeler also chooses <em>explanatory variables</em> that may account for some of the variability in the response variable. There is almost always something left over, variation that cannot be attributed to the explanatory variables. This left-over part is called the “<strong>residual</strong>.”</p>
<p>To illustrate, we return to a historical statistical landmark, the data collected in the 1880s by Francis Galton on the heights of parents and their full-grown children. In that era, there was much excitement, uncertainty, and controversy about Charles Darwin’s <em>theory of evolution</em>.</p>
<!-- [Remarkably, Galton was Darwin's cousin.]{.aside} Today, we understand the role of DNA in encoding genetic information. But in Galton's day, the concept of "gene" was unknown. -->
<p>Everyone knows that height varies from person to person. How much variation in height is there? We quantify variation using the <em>variance</em>. <code>var()</code> will do the calculation. The units of the output will be in <em>square-inches</em>, since <code>height</code> itself is measured in inches.</p>
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<li>Explain why <code>var(height)</code> has units of “square-inches” when the variable <code>height</code> is measured in inches.</li>
<li>Suppose the variance of a variable <code>price</code> is 144 square inches. What is the standard deviation of <code>price</code>? (Make sure to include the units!)</li>
<li>A highway traffic monitor records the speed of vehicles passing over a spot in the road. The units of speed are miles-per-hour. But the vehicles differ in speed somewhat. What will be the units of the variance of speed?</li>
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<li>When calculating the variance of a variable, we take many pairs of values, find their difference, and square them. For instance, suppose <code>height</code> includes the two values 57 inches and 54 inches. The difference between them is 3 inches. The square of this difference is 9 square-inches. The variance is the average of all the different pairs of values.</li>
<li>The standard deviation is simply the square root of the variance. <span class="math inline">\(\sqrt{\text{144 square inches}} = \text{12 inches}\)</span></li>
<li>Since speed is measured in miles/hour, the variance of speed will have units (miles/hour)<sup>2</sup>, which might equally well be written “square-miles per square hour.” Admittedly, “square hour” is a strange sounding unit, but speed and the variance of speed are different kinds of things.</li>
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[Each specimen is a full-grown child in the `Galton` data frame. The variables recorded are the `height` of the child in inches, the heights of the `mother` and `father`, and the `sex` of the child according to the conventions of Victorian England, where the data were collected.]{.aside}

AND MORE STILL
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<div class="page-columns page-full"><p>Galton’s overall project was to quantify the heritability of traits (such as height) from parents to offspring. Darwin had famously measured the beaks of finches on the Galapagos Islands. Galton, in London, worked with the most readily available local species: humans. Height is easy and socially acceptable to measure, a good candidate for data collection. </p><div class="no-row-height column-margin column-container"><span class="margin-aside">Galton <em>invented</em> the methods we are going to use in this example.</span></div></div>
<p>Galton sought to divide the height variation in the children into two parts: that which could be attributed to the parents and that which remained unaccounted for, which we call the “<strong>residual</strong>.” We will start with a mathematically more straightforward project: dividing the variation in the children’s heights into that part associated with the child’s sex and the residual.</p>
<p><a href="#lst-galton-mean" class="quarto-xref">Active R chunk&nbsp;<span class="quarto-unresolved-ref">lst-galton-mean</span></a> shows how to calculate the average <code>height</code> for each <code>sex</code> using the techniques from Lesson <a href="#sec-wrangling" class="quarto-xref"><span class="quarto-unresolved-ref ref-noprefix">sec-wrangling</span></a>. The line numbers in the code listing explain each of the components of the command.</p>
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<div class="sourceCode cell-code" id="annotated-cell-3"><pre class="sourceCode r code-annotation-code code-with-copy code-annotated"><code class="sourceCode r"><span id="annotated-cell-3-1"><a href="#annotated-cell-3-1" aria-hidden="true" tabindex="-1"></a>Galton <span class="sc">|&gt;</span></span>
<a class="code-annotation-anchor" data-target-cell="annotated-cell-3" data-target-annotation="1" onclick="event.preventDefault();">1</a><span id="annotated-cell-3-2" class="code-annotation-target"><a href="#annotated-cell-3-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">summarize</span>(</span>
<a class="code-annotation-anchor" data-target-cell="annotated-cell-3" data-target-annotation="2" onclick="event.preventDefault();">2</a><span id="annotated-cell-3-3" class="code-annotation-target"><a href="#annotated-cell-3-3" aria-hidden="true" tabindex="-1"></a>    <span class="at">modval =</span> <span class="fu">mean</span>(height),</span>
<a class="code-annotation-anchor" data-target-cell="annotated-cell-3" data-target-annotation="3" onclick="event.preventDefault();">3</a><span id="annotated-cell-3-4" class="code-annotation-target"><a href="#annotated-cell-3-4" aria-hidden="true" tabindex="-1"></a>   <span class="at">.by =</span> sex</span>
<span id="annotated-cell-3-5"><a href="#annotated-cell-3-5" aria-hidden="true" tabindex="-1"></a>   )</span><div class="code-annotation-gutter-bg"></div><div class="code-annotation-gutter"></div></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<dt data-target-cell="annotated-cell-3" data-target-annotation="1">1</dt>
<dd>
<span data-code-cell="annotated-cell-3" data-code-lines="2" data-code-annotation="1"><code>summarize()</code> reduces many rows to a single row.</span>
</dd>
<dt data-target-cell="annotated-cell-3" data-target-annotation="2">2</dt>
<dd>
<span data-code-cell="annotated-cell-3" data-code-lines="3" data-code-annotation="2">Calculate the mean of <code>height</code> and name the corresponding variable in the output <code>modval</code>.</span>
</dd>
<dt data-target-cell="annotated-cell-3" data-target-annotation="3">3</dt>
<dd>
<span data-code-cell="annotated-cell-3" data-code-lines="4" data-code-annotation="3">Break down the calculation in (2) group by group.<br>
</span>
</dd>
</dl>
</div>
<div class="cell-output-display">
<div style="font-family: Monospace;">
<table class="lightable-minimal caption-top table table-sm table-striped small" data-quarto-postprocess="true">
<thead>
<tr class="header">
<th style="text-align: left;" data-quarto-table-cell-role="th">sex</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">modval</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">M</td>
<td style="text-align: right;">69.23</td>
</tr>
<tr class="even">
<td style="text-align: left;">F</td>
<td style="text-align: right;">64.11</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<p>To see the results of the command, run it!</p>
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<p>Note that the output has one row for each <code>sex</code>. As always, the output from summarize is set by the arguments to the function. Here, there will be two variables in the output: <code>sex</code> and the mean height for each sex, which we’re calling <code>modval</code>, short for <strong>model value</strong>.</p>
<p>This simple report indicates that the sexes differ, on average, by about 5 inches in height. But this report offers no insight into how much of the person-to-person variation in height is attributable to <code>sex</code>. For that, we need to look at the group averages <em>in the context of the variation of individuals</em>. That is, instead of computing the model value for each <code>sex</code>, we will compute it for each specimen using <code>mutate()</code>. This is simple, since all <code>F</code> specimens will have a model value of 64.1 inches, while the <code>M</code> specimens all have a model value of 69.2 inches. However, anticipating what will come later in this Lesson, we will do the calculation using <code>mutate()</code>.</p>
<div id="lst-galton-mutate" class="listing quarto-float quarto-figure quarto-figure-left anchored">
<figure class="quarto-float quarto-float-lst figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-lst quarto-uncaptioned" id="lst-galton-mutate-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Active R chunk&nbsp;9.1
</figcaption>
<div aria-describedby="lst-galton-mutate-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
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</figure>
</div>
<p>Note: To save space on the screen, <a href="#lst-galton-mutate" class="quarto-xref">Active R chunk&nbsp;<span class="quarto-unresolved-ref">lst-galton-mutate</span></a> is written to show only a handful of the 898 rows that would otherwise be in the output.</p>
<p><a href="#lst-galton-mutate2" class="quarto-xref">Active R chunk&nbsp;<span class="quarto-unresolved-ref">lst-galton-mutate2</span></a> repeats the above calculation, but summarizes to show the variance of <code>height</code> and the variance of the model values.</p>
<div id="lst-galton-mutate2" class="listing quarto-float quarto-figure quarto-figure-left anchored">
<figure class="quarto-float quarto-float-lst figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-lst quarto-uncaptioned" id="lst-galton-mutate2-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Active R chunk&nbsp;9.2
</figcaption>
<div aria-describedby="lst-galton-mutate2-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
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</figure>
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<p>The output shows that the variability in the model values of height, is about half the variability in the height itself. This is the the answer to (2) from the introduction to this Lesson; <code>sex</code> accounts for about half of the variation in height.</p>
<p>Item (3) in the introduction was about measuring what remains unexplained. This “unexplained” part of height is called the <strong>residual</strong> from the model values. The residual is easy to calculate. <a href="#lst-galton-resid-var" class="quarto-xref">Active R chunk&nbsp;<span class="quarto-unresolved-ref">lst-galton-resid-var</span></a> calculates the residuals from the simple model that accounts for <code>height</code> by <code>sex</code>.</p>
<div id="lst-galton-resid-var" class="listing quarto-float quarto-figure quarto-figure-left anchored">
<figure class="quarto-float quarto-float-lst figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-lst quarto-uncaptioned" id="lst-galton-resid-var-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Active R chunk&nbsp;9.3
</figcaption>
<div aria-describedby="lst-galton-resid-var-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
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</figure>
</div>
<p>It’s essential to note from the output of <a href="#lst-galton-resid-var" class="quarto-xref">Active R chunk&nbsp;<span class="quarto-unresolved-ref">lst-galton-resid-var</span></a> that the variance in the response variable (<code>height</code>) is exactly the sum of the the variance of the model values and the variance of the residuals. In other words, the variance in the response variable is <strong>perfectly split into two parts</strong>:</p>
<ol type="1">
<li>the part accounted for by the explanatory variable and</li>
<li>the part that remains unaccounted for.</li>
</ol>
<div id="lck-9-02." class="callout callout-style-default callout-tip callout-titled">
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<div class="callout-icon-container">
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<div class="callout-title-container flex-fill">
Tip&nbsp;9.2
</div>
<div class="callout-btn-toggle d-inline-block border-0 py-1 ps-1 pe-0 float-end"><i class="callout-toggle"></i></div>
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<div id="callout-2" class="callout-2-contents callout-collapse collapse">
<div class="callout-body-container callout-body">
<p><a href="#lst-galton-resid-var" class="quarto-xref">Active R chunk&nbsp;<span class="quarto-unresolved-ref">lst-galton-resid-var</span></a> shows how to calculate the residual for each specimen: subtract the model value from the value of the response variable. Let’s look at the values of the residuals from the model of height versus sex.</p>
<div id="lst-resid-point-plot" class="listing quarto-float quarto-figure quarto-figure-left anchored">
<figure class="quarto-float quarto-float-lst figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-lst quarto-uncaptioned" id="lst-resid-point-plot-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Active R chunk&nbsp;9.4
</figcaption>
<div aria-describedby="lst-resid-point-plot-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<div class="cell">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a>Galton <span class="sc">|&gt;</span> </span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">modval =</span> <span class="fu">mean</span>(height), <span class="at">.by =</span> sex) <span class="sc">|&gt;</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">resid =</span> height <span class="sc">-</span> modval) <span class="sc">|&gt;</span></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">point_plot</span>(resid <span class="sc">~</span> <span class="dv">1</span>, <span class="at">annot =</span> <span class="st">"violin"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div>
<figure class="figure">
<p><a href="L09-Accounting-for-variation_files/figure-html/unnamed-chunk-3-1.png" class="lightbox" data-gallery="quarto-lightbox-gallery-1" title="Active R chunk&nbsp;9.4: "><img src="L09-Accounting-for-variation_files/figure-html/unnamed-chunk-3-1.png" class="img-fluid figure-img" width="672"></a></p>
</figure>
</div>
</div>
</div>
</div>
</figure>
</div>
<ol type="a">
<li><p>Which line of <a href="#lst-resid-point-plot" class="quarto-xref">Active R chunk&nbsp;<span class="quarto-unresolved-ref">lst-resid-point-plot</span></a> finds the residual for each of the specimens?</p></li>
<li><p>Some residuals are positive and some negative. Explain how a residual can be negative.</p></li>
</ol>
<button class="btn btn-light" data-bs-toggle="collapse" data-bs-target="#hintieFIaK">
Answer
</button>
<div id="hintieFIaK" class="collapse card card-body">
<p>If the model value is larger than the response variable, the residual will be negative. Similarly, if the model value is less than the response variable, the residual is positive.</p>
</div>
</div>
</div>
</div>
<div id="lck-9.03" class="callout callout-style-default callout-tip callout-titled">
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<div class="callout-icon-container">
<i class="callout-icon"></i>
</div>
<div class="callout-title-container flex-fill">
Tip&nbsp;9.3
</div>
<div class="callout-btn-toggle d-inline-block border-0 py-1 ps-1 pe-0 float-end"><i class="callout-toggle"></i></div>
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<div id="callout-3" class="callout-3-contents callout-collapse collapse">
<div class="callout-body-container callout-body">
<p>Since Lesson <span class="citation" data-cites="sec">[-@sec--variation-and-distribution]</span> we have used the <strong>variance</strong> as the prefered way to measure the amount of <em>variation</em>. The advantage of the variance is that it perfectly partitions variation in the response variable into two parts: (i) variation in the model values and (ii) variation in the residuals.</p>
<p>The following chunk demonstrates the perfect partitioning: Run it!</p>
<div id="lst-resid-point-plot" class="listing quarto-float quarto-figure quarto-figure-left anchored">
<figure class="quarto-float quarto-float-lst figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-lst quarto-uncaptioned" id="lst-resid-point-plot-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Active R chunk&nbsp;9.5
</figcaption>
<div aria-describedby="lst-resid-point-plot-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<div id="qwebr-insertion-location-6"></div>
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</div>
</figure>
</div>
<ol type="a">
<li>Modify the above chunk to model the age of a mother (<code>mage</code>) from the <code>Births2022</code> data frame by the mother’s education level (<code>meduc</code>). Is the variation in <code>mage</code> perfectly partitioned between the model values and the residuals?</li>
</ol>
<button class="btn btn-light" data-bs-toggle="collapse" data-bs-target="#hintcXshjv">
Hint
</button>
<div id="hintcXshjv" class="collapse card card-body">
<p>In addition to replacing <code>Galton</code> with <code>Births2022</code>, you will need to replace the explanatory variable <code>sex</code> with <code>meduc</code>. Similarly, replace <code>height</code> with <code>mage</code>.</p>
</div>
<ol start="2" type="a">
<li>Some people prefer to use the standard deviation (<code>sd()</code>) to measure variation. Modify the above chunk, replacing <code>var()</code> with <code>sd()</code>. Looking at the results, does the standard deviation perfectly partition the response variable into model values and residuals?</li>
</ol>
<button class="btn btn-light" data-bs-toggle="collapse" data-bs-target="#hintSaoDP2">
Answers
</button>
<div id="hintSaoDP2" class="collapse card card-body">
<ol type="a">
<li>The variance of any response variable will always be split perfectly between the variance of the model values and the variance of the residuals. This is a mathematical fact that stems from the same phenomenon as in the Pythagorean Theorem.</li>
<li>The standard deviation of the response is not split perfectly by the standard deviation of the model values and the standard deviation of the residuals.</li>
</ol>
</div>
</div>
</div>
</div>
<section id="numerical-explanatory-variables" class="level2">
<h2 class="anchored" data-anchor-id="numerical-explanatory-variables">Numerical explanatory variables</h2>
<p>The previous section used the categorical variable <code>sex</code> as the explanatory variable. Galton’s interest, however, was in the relationship between children’s and parents’ height.</p>
<p><code>sex</code> is a <em>categorical</em> explanatory variable. Using <code>mutate(..., .by = sex)</code> is a good method of handling categorical explanatory variables. However, this does not work for <em>quantitative</em> explanatory variables. The reason is that <code>.by</code> translates a quantitative variable into a categorical one, with a result for each unique quantitative value. It’s trivial to substitute the height of the <code>mother</code> or <code>father</code> in place of <code>sex</code> in the method introduced in the previous section. However, as we shall see, the results are not satisfactory. Galton’s key discovery was the proper method for relating two <em>quantitative</em> variables such as <code>height</code> and <code>mother</code>.</p>
<p>First, let’s try simply substituting in <code>mother</code> as the explanatory variable and using <code>mean()</code> to create the model values. Then, use <code>point_plot()</code> to show the model values as a function of <code>mother</code>.</p>
<div id="lst-galton-by-mother" class="listing quarto-float quarto-figure quarto-figure-left anchored">
<figure class="quarto-float quarto-float-lst figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-lst quarto-uncaptioned" id="lst-galton-by-mother-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Active R chunk&nbsp;9.6
</figcaption>
<div aria-describedby="lst-galton-by-mother-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
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</div>
</figure>
</div>
<p>The last line of <a href="#lst-galton-by-mother" class="quarto-xref">Active R chunk&nbsp;<span class="quarto-unresolved-ref">lst-galton-by-mother</span></a> connects adjacent points with lines. It’s hard to see any pattern in the model values.</p>
<p>It is common sense that the model linking mothers’ heights to children’s height should be <strong>smooth</strong>, not the jagged pattern seen in <a href="#lst-galton-by-mother" class="quarto-xref">Active R chunk&nbsp;<span class="quarto-unresolved-ref">lst-galton-by-mother</span></a>. The source of the jaggedness is the use of <code>.by = mother</code> in <code>mutate(modval = mean(height), .by = mother)</code>. There is nothing in <code>.by = mother</code> to enforce the idea that the model value for mid-height mothers should be in-between the model values for short and for tall mothers.</p>
<p>The solution to the problem of jagged model values is to avoid the absolute splitting into non-overlapping groups by mother’s height. Instead, we want to find a smooth relationship. Galton invented the method for accomplishing this. A modern form of his method is provided by the <code>model_values()</code> function., which we shall use to construct <code>Model3</code>.</p>
<div id="lst-mother-model3" class="listing quarto-float quarto-figure quarto-figure-left anchored">
<figure class="quarto-float quarto-float-lst figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-lst quarto-uncaptioned" id="lst-mother-model3-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Active R chunk&nbsp;9.7
</figcaption>
<div aria-describedby="lst-mother-model3-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
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</figure>
</div>
<p>Notice that the <code>.by = mother</code> step has been entirely removed. Notice also that <code>model_values()</code> uses the same kind of <strong>tilde expression</strong> as we have employed when plotting. The response variable is listed on the left of the <img src="www/tilde.png" class="img-fluid">, the explanatory variable on the right side. In other words, we are modeling the child’s <code>height</code> as a function of <code>mother</code>’s height.</p>
<p>As always, modeling splits the variance of the response variable into two parts, one associated with the explanatory variable and the other holding what’s left over: the residual. Here’s the split for <code>Model3</code> which uses <code>mother</code> as an explanatory variable:</p>
<div id="lst-mother-model4" class="listing quarto-float quarto-figure quarto-figure-left anchored">
<figure class="quarto-float quarto-float-lst figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-lst quarto-uncaptioned" id="lst-mother-model4-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Active R chunk&nbsp;9.8
</figcaption>
<div aria-describedby="lst-mother-model4-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
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</figure>
</div>
<p>The mother’s height doesn’t account for much of the variation in the children’s heights.</p>
<div id="lck-9-04" class="callout callout-style-default callout-tip callout-titled">
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<div class="callout-title-container flex-fill">
Tip&nbsp;9.4
</div>
<div class="callout-btn-toggle d-inline-block border-0 py-1 ps-1 pe-0 float-end"><i class="callout-toggle"></i></div>
</div>
<div id="callout-4" class="callout-4-contents callout-collapse collapse">
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<p>The <code>model_values()</code> approach to modeling works for both quantitative and categorical explanatory variables. For convenience, you can annotate <code>point_plot()</code> to show the model along with the raw data. Try it:</p>
<div id="lst-resid-point-plot" class="listing quarto-float quarto-figure quarto-figure-left anchored">
<figure class="quarto-float quarto-float-lst figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-lst quarto-uncaptioned" id="lst-resid-point-plot-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Active R chunk&nbsp;9.9
</figcaption>
<div aria-describedby="lst-resid-point-plot-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
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</div>
</figure>
</div>
<ol type="a">
<li>From the graph, estimate the largest positive residual among the females.</li>
</ol>
<button class="btn btn-light" data-bs-toggle="collapse" data-bs-target="#hinthxDevn">
Answer
</button>
<div id="hinthxDevn" class="collapse card card-body">
<p>The model value for females is about 64 inches. The tallest female is 71 inches. This makes the largest residual 71 - 64 = 5 inches.</p>
</div>
<ol start="2" type="a">
<li>Change the code in the chunk to show the model <code>height ~ mother</code>. Find a point that has a large negative residual from this model.</li>
</ol>
</div>
</div>
</div>
</section>
<section id="sec-multiple-vars1" class="level2 page-columns page-full">
<h2 class="anchored" data-anchor-id="sec-multiple-vars1">Multiple explanatory variables</h2>
<div class="page-columns page-full"><p>The models we work with in these Lessons <em>always</em> have exactly one response variable.  But models can have any number of <em>explanatory</em> variables.</p><div class="no-row-height column-margin column-container"><span class="margin-aside">Note that the idea of “response” and “explanatory” variables refers to a model and are not at all intrinsic to a bare data frame. A data frame can contain many variables, any of which can be used as explanatory variables. The choice of response variable depends on the modeler’s goals.</span></div></div>
<div class="page-columns page-full"><p>Whatever the number of explanatory variables and however many levels a categorical explanatory variable has the model splits the variance of the response into two complementary pieces: the variance accounted for by the explanatory variables and the part not accounted for, that is, the <em>residual</em> variance. </p><div class="no-row-height column-margin column-container"><span class="margin-aside">Many statistical terms mean something different in statistical than in everyday use. “Residual” is a pleasant exception: the statistical meaning is closely matched by its everyday dictionary definition.</span></div></div>
<p>To illustrate, here is a sequence of models of <code>height</code> with different numbers of explanatory variables.</p>
<table class="caption-top table">
<thead>
<tr class="header">
<th>Number of explanatory variables</th>
<th>tilde expression</th>
<th><code>var(resid)</code></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>0</td>
<td><code>height ~ 1</code></td>
<td></td>
</tr>
<tr class="even">
<td>1</td>
<td><code>height ~ sex</code></td>
<td></td>
</tr>
<tr class="odd">
<td>2</td>
<td><code>height ~ sex + mother</code></td>
<td></td>
</tr>
<tr class="even">
<td>3</td>
<td><code>height ~ sex + mother + father</code></td>
<td></td>
</tr>
</tbody>
</table>
<p>Calculating the residuals and their variance for any of these four models can be accomplished with</p>
<div class="cell">
<div class="sourceCode cell-code" id="annotated-cell-1"><pre class="sourceCode r code-annotation-code code-with-copy"><code class="sourceCode r"><span id="annotated-cell-1-1"><a href="#annotated-cell-1-1" aria-hidden="true" tabindex="-1"></a>Galton <span class="sc">|&gt;</span></span>
<span id="annotated-cell-1-2"><a href="#annotated-cell-1-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">modval =</span></span>
<span id="annotated-cell-1-3"><a href="#annotated-cell-1-3" aria-hidden="true" tabindex="-1"></a>           <span class="fu">model_values</span>(</span>
<span id="annotated-cell-1-4"><a href="#annotated-cell-1-4" aria-hidden="true" tabindex="-1"></a>             ..tilde.expression.here.</span>
<span id="annotated-cell-1-5"><a href="#annotated-cell-1-5" aria-hidden="true" tabindex="-1"></a>             ),</span>
<span id="annotated-cell-1-6"><a href="#annotated-cell-1-6" aria-hidden="true" tabindex="-1"></a>         <span class="at">resid =</span> height <span class="sc">-</span> modval) <span class="sc">|&gt;</span></span>
<span id="annotated-cell-1-7"><a href="#annotated-cell-1-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">summarize</span>(</span>
<span id="annotated-cell-1-8"><a href="#annotated-cell-1-8" aria-hidden="true" tabindex="-1"></a>    <span class="fu">var</span>(height), <span class="fu">var</span>(modval), <span class="fu">var</span>(resid)</span>
<span id="annotated-cell-1-9"><a href="#annotated-cell-1-9" aria-hidden="true" tabindex="-1"></a>    )</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p><strong>Three explanatory variables</strong></p>
<div id="qwebr-insertion-location-11"></div>
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<p><strong>Two explanatory variables</strong></p>
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<p><strong>One explanatory variable</strong></p>
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<p><strong>Zero explanatory variables</strong></p>
<div class="cell page-columns page-full">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>Galton <span class="sc">|&gt;</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">modval =</span> <span class="fu">model_values</span>(height <span class="sc">~</span> <span class="dv">1</span>),</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a>         <span class="at">resid =</span> height <span class="sc">-</span> modval) <span class="sc">|&gt;</span></span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">summarize</span>(<span class="fu">var</span>(height), <span class="fu">var</span>(modval), <span class="fu">var</span>(resid))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>

<div class="no-row-height column-margin column-container"><div class="cell-output-display">
<div style="font-family: Monospace;">
<table class="lightable-minimal caption-top table table-sm table-striped small" data-quarto-postprocess="true">
<thead>
<tr class="header">
<th style="text-align: right;" data-quarto-table-cell-role="th">var(height)</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">var(modval)</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">var(resid)</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">12.8373</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">12.8373</td>
</tr>
</tbody>
</table>
</div>
</div></div></div></section>
<section id="comparing-models-with-r2" class="level2 page-columns page-full">
<h2 class="anchored" data-anchor-id="comparing-models-with-r2">Comparing models with R<sup>2</sup></h2>
<p>When selecting explanatory variables, comparing two or more different models sharing the same response variable is often helpful: a simple model and a model that adds one or more explanatory variables to the simple model. The model with <em>no explanatory variables</em>, is always the simplest possible model. For example, in <a href="#sec-multiple-vars1" class="quarto-xref"><span class="quarto-unresolved-ref">sec-multiple-vars1</span></a>, the model <code>height ~ 1</code> is the simplest. Compared to the simplest model, the model <code>height ~ sex</code> has one additional explanatory variable, <code>sex</code>. Similarly, <code>height ~ sex + mother</code> has one additional explanatory variable compared to <code>height ~ sex</code>, and <code>height ~ sex + mother + father</code> adds in still another explanatory variable.</p>
<p>The simpler model is said to be “<strong>nested in</strong>” the more extensive model, analogous to a series of <a href="https://en.wikipedia.org/wiki/Matryoshka_doll">Matroshka dolls</a>. A simple measure of how much of the response variance is accounted for by the explanatory variables is the ratio of the variance of the model values divided by the variance of the response variable itself. This ratio is called “R<sup>2</sup>”, pronounced “R-squared.”</p>
<div id="fig-matroshka-dolls" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-matroshka-dolls-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<a href="www/Russian-Matroshka_no_bg.jpeg" class="lightbox" data-gallery="quarto-lightbox-gallery-2" title="Figure&nbsp;9.1: A sequence of five nested Matroshka dolls. Each smaller doll fits inside a larger one."><img src="www/Russian-Matroshka_no_bg.jpeg" class="img-fluid figure-img"></a>
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-matroshka-dolls-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure&nbsp;9.1: A sequence of five nested <em>Matroshka</em> dolls. Each smaller doll fits inside a larger one.
</figcaption>
</figure>
</div>
<div class="page-columns page-full"><p>For instance, R^2 for the model <code>height ~ sex + mother</code> is <span class="math display">\[\text{R}^2 = \frac{7.21}{12.84} = 0.56\]</span>  By comparison, R^2 for the simpler model, <code>height ~ sex,</code> is slightly smaller:</p><div class="no-row-height column-margin column-container"><span class="margin-aside">R<sup>2</sup> is also known as the “coefficient of determination,” a little-used term we shall avoid. Still, it’s worth noting the attitude behind the term; it quantifies the extent to which the response variable is “determined” by the explanatory variables.</span></div></div>
<div class="page-columns page-full"><p><span class="math display">\[\text{R}^2 = \frac{6.55}{12.84} = 0.51\]</span> For all models, <span class="math inline">\(0 \leq\)</span> R<sup>2</sup> <span class="math inline">\(\leq 1\)</span>.  It is tempting to believe that the “best” model in a set of nested models is the one with the highest R<sup>2</sup>, but statistical thinkers understand that “best” ought to depend on the purpose for which the model is being built. This matter will be a major theme in the remaining Lessons.</p><div class="no-row-height column-margin column-container"><span class="margin-aside">Instructor Note: Strictly speaking, “all” should be qualified to mean “linear least-squares models with an intercept term.”</span></div></div>
</section>
<section id="exercises" class="level2">
<h2 class="anchored" data-anchor-id="exercises">Exercises</h2>
<!-- Do NOT EDIT this file. It is produced automatically
by emit_exercise_markup().-->
<div id="exr-Q09-101" class="theorem exercise">
<p><span class="theorem-title"><strong>Activity 9.1</strong></span> The data frame <code>LSTbook::Gilbert</code> records 1641 shifts at a VA hospital. (For the story behind the data, give the command <code>?Gilbert</code>.) Whether or not a patient died during a shift is indicated by the variable <code>death</code>. The variable <code>gilbert</code> marks whether or not nurse Kristen Gilbert was on duty during the shift.</p>
<p>A. Are the variables numerical or categorical?</p>
<p>B. The data were used as evidence during Gilbert’s trial for murder. The data can be reduced to a compact form by data wrangling:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a>LSTbook<span class="sc">::</span>Gilbert <span class="sc">|&gt;</span> </span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">summarize</span>(<span class="at">count =</span> <span class="fu">n</span>(), <span class="at">.by =</span> <span class="fu">c</span>(death, gilbert))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div style="font-family: Monospace;">
<table class="lightable-minimal caption-top table table-sm table-striped small" data-quarto-postprocess="true">
<thead>
<tr class="header">
<th style="text-align: left;" data-quarto-table-cell-role="th">death</th>
<th style="text-align: left;" data-quarto-table-cell-role="th">gilbert</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">count</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">no</td>
<td style="text-align: left;">not</td>
<td style="text-align: right;">1027</td>
</tr>
<tr class="even">
<td style="text-align: left;">yes</td>
<td style="text-align: left;">not</td>
<td style="text-align: right;">357</td>
</tr>
<tr class="odd">
<td style="text-align: left;">no</td>
<td style="text-align: left;">on_duty</td>
<td style="text-align: right;">157</td>
</tr>
<tr class="even">
<td style="text-align: left;">yes</td>
<td style="text-align: left;">on_duty</td>
<td style="text-align: right;">100</td>
</tr>
</tbody>
</table>
</div>
</div>
<p>Calculate the proportion of times a death occurred when Gilbert was on duty, and the proportion of time when Gilbert was not on duty. Does the shift evidence support the claim that Gilbert was a contributing factor to the deaths.</p>
<p>C. In the Lessons, we will mainly use <em>regression modeling</em> rather than wrangling as a data-reduction technique. In the trial, the relevant question was whether Gilbert was a factor in the deaths. Express this question as a tilde expression involving the variables <code>death</code> and <code>gilbert</code>.</p>
<p>D. The response variable in your tilde expression from (C) is categorical. Regression modeling requires a <em>numerical</em> response variable. Use <code>mutate()</code> and <code>zero_one()</code> to transform the variable <code>death</code> from categorical form to a numerical form that has a value 1 when Gilbert was on duty and 0 otherwise. This will become the response variable for the regression: <code>death ~ gilbert</code>.</p>
<p>E. Carry out the regression using <code>model_train()</code>. Look at the coefficients from the fitted model with <code>conf_interval()</code>; these are in <code>.coef</code> column. Explain how the coefficients correspond to the proportions you calculated in (B).</p>
<p><a name="Q09-101"><i class="bi bi-card-text"></i>id=Q09-101</a></p>
</div>
<hr>
<div id="exr-Q09-102" class="theorem exercise">
<p><span class="theorem-title"><strong>Activity 9.2</strong></span> &nbsp;</p>
<!-- Q09-102 -->
<p>DRAFT: Which variables could be used as response variables?</p>
<p><a name="Q09-102"><i class="bi bi-card-text"></i>id=Q09-102</a></p>
</div>
<hr>
<div id="exr-Q09-103" class="theorem exercise">
<p><span class="theorem-title"><strong>Activity 9.3</strong></span> &nbsp;</p>
<!-- Q09-103 -->
<p>DRAFT: Do a nested model approach with <code>Anthro_F</code> and R<sup>2</sup></p>
<p><a name="Q09-103"><i class="bi bi-card-text"></i>id=Q09-103</a></p>
</div>
<hr>
<div id="exr-Q09-104" class="theorem exercise">
<p><span class="theorem-title"><strong>Activity 9.4</strong></span> &nbsp;</p>
<!-- Q09-104 -->
<p>DRAFT: Show that the sum of squares and mean square for modeling using the median is larger than for the mean.</p>
<p><a name="Q09-104"><i class="bi bi-card-text"></i>id=Q09-104</a></p>
</div>
<hr>
<div id="exr-Q09-105" class="theorem exercise">
<p><span class="theorem-title"><strong>Activity 9.5</strong></span> &nbsp;</p>
<!-- Q09-105 -->
<p>DRAFT: Simple calculation of <span class="math inline">\(R^2\)</span>.</p>
<p><a name="Q09-105"><i class="bi bi-card-text"></i>id=Q09-105</a></p>
</div>
<hr>
<div id="exr-Q09-106" class="theorem exercise">
<p><span class="theorem-title"><strong>Activity 9.6</strong></span> &nbsp;</p>
<!-- Q09-106 -->
<p><a name="Q09-106"><i class="bi bi-card-text"></i>id=Q09-106</a></p>
</div>
<hr>


</div></section>

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