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    <h2 id="toc-title">Table of contents</h2>

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  <li><a href="#task" id="toc-task" class="nav-link active" data-scroll-target="#task">Task</a>
  <ul class="collapse">
  <li><a href="#research-question" id="toc-research-question" class="nav-link" data-scroll-target="#research-question">Research Question</a></li>
  <li><a href="#hypotheses" id="toc-hypotheses" class="nav-link" data-scroll-target="#hypotheses">Hypotheses</a></li>
  </ul></li>
  <li><a href="#data-overview" id="toc-data-overview" class="nav-link" data-scroll-target="#data-overview">Data Overview</a></li>
  <li><a href="#visualising-the-difference" id="toc-visualising-the-difference" class="nav-link" data-scroll-target="#visualising-the-difference">Visualising the difference</a>
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  <li><a href="#boxplot-distribution-of-mortality-rates" id="toc-boxplot-distribution-of-mortality-rates" class="nav-link" data-scroll-target="#boxplot-distribution-of-mortality-rates">Boxplot: Distribution of Mortality Rates</a></li>
  <li><a href="#time-series-north-vs-south-over-time" id="toc-time-series-north-vs-south-over-time" class="nav-link" data-scroll-target="#time-series-north-vs-south-over-time">Time Series: North vs South Over Time</a></li>
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  <li><a href="#statistical-testing" id="toc-statistical-testing" class="nav-link" data-scroll-target="#statistical-testing">Statistical Testing</a>
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  <li><a href="#assumption-checks" id="toc-assumption-checks" class="nav-link" data-scroll-target="#assumption-checks">Assumption Checks</a></li>
  <li><a href="#understanding-the-variance-ratio" id="toc-understanding-the-variance-ratio" class="nav-link" data-scroll-target="#understanding-the-variance-ratio">Understanding the Variance Ratio</a></li>
  <li><a href="#why-use-welchs-ttest" id="toc-why-use-welchs-ttest" class="nav-link" data-scroll-target="#why-use-welchs-ttest">Why Use Welch’s t‑test?</a></li>
  <li><a href="#twosample-ttest-welch" id="toc-twosample-ttest-welch" class="nav-link" data-scroll-target="#twosample-ttest-welch">Two‑Sample t‑test (Welch)</a></li>
  <li><a href="#understanding-p-value" id="toc-understanding-p-value" class="nav-link" data-scroll-target="#understanding-p-value">Understanding p-value</a></li>
  <li><a href="#effect-size-cohens-d" id="toc-effect-size-cohens-d" class="nav-link" data-scroll-target="#effect-size-cohens-d">Effect Size (Cohen’s d)</a></li>
  <li><a href="#what-the-effect-size-tells-us" id="toc-what-the-effect-size-tells-us" class="nav-link" data-scroll-target="#what-the-effect-size-tells-us">What the Effect Size Tells Us</a></li>
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  <li><a href="#note-on-covid19-impact" id="toc-note-on-covid19-impact" class="nav-link" data-scroll-target="#note-on-covid19-impact">Note on COVID‑19 Impact</a></li>
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  <li><a href="#limitations" id="toc-limitations" class="nav-link" data-scroll-target="#limitations">Limitations</a></li>
  <li><a href="#conclusion" id="toc-conclusion" class="nav-link" data-scroll-target="#conclusion">Conclusion</a></li>
  <li><a href="#appendix" id="toc-appendix" class="nav-link" data-scroll-target="#appendix">Appendix</a></li>
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<main class="content" id="quarto-document-content">

<header id="title-block-header" class="quarto-title-block default">
<div class="quarto-title">
<h1 class="title">Statistical Comparison - North vs South Respiratory Mortality</h1>
</div>


<div class="quarto-title-meta">

    <div>
    <div class="quarto-title-meta-heading">Author</div>
    <div class="quarto-title-meta-contents">
             <p>Marco Sorbona, PhD </p>
          </div>
  </div>

    <div>
    <div class="quarto-title-meta-heading">Published</div>
    <div class="quarto-title-meta-contents">
      <p class="date">March 1, 2026</p>
    </div>
  </div>


  </div>


</header>


<section id="task" class="level2">
<h2 class="anchored" data-anchor-id="task">Task</h2>
<p>The North‑South divide in health outcomes is well documented, but this analysis tests whether the difference in respiratory mortality is statistically significant. The question is whether the observed inequality could have occurred by chance.</p>
<p><strong>Rationale</strong>: Statistical tests quantify the probability that an observed difference is due to random variation. This moves beyond descriptive observation to inferential evidence.</p>
<section id="research-question" class="level3">
<h3 class="anchored" data-anchor-id="research-question">Research Question</h3>
<blockquote class="blockquote">
<p><em>“Is there a statistically significant difference in respiratory mortality rates between northern and southern regions of England?”</em></p>
</blockquote>
</section>
<section id="hypotheses" class="level3">
<h3 class="anchored" data-anchor-id="hypotheses">Hypotheses</h3>
<ul>
<li><strong>Null hypothesis (H₀):</strong> There is no difference in mean mortality rates between North and South</li>
<li><strong>Alternative hypothesis (H₁):</strong> There is a difference in mean mortality rates between North and South</li>
</ul>
<p><strong>Rationale</strong>: Formal hypotheses provide a framework for statistical testing. The null hypothesis assumes no effect; the alternative proposes an effect. The test determines which is supported by the data.</p>
<hr>
</section>
</section>
<section id="data-overview" class="level2">
<h2 class="anchored" data-anchor-id="data-overview">Data Overview</h2>
<p>The summary table shows the number of regions, observations, mean rates, standard deviations, and minimum and maximum values for each group.</p>
<p><strong>Rationale</strong>: Understanding the basic properties of each group is essential before conducting statistical tests. This table confirms the North has a higher mean rate and greater variability.</p>
<div class="cell" data-tbl-cap="Summary statistics by region group">
<div class="code-copy-outer-scaffold"><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>north_south_data <span class="sc">|&gt;</span> </span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">group_by</span>(region_group) <span class="sc">|&gt;</span> </span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">summarise</span>(</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a>    <span class="at">n_regions =</span> <span class="fu">n_distinct</span>(region),</span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>    <span class="at">n_observations =</span> <span class="fu">n</span>(),</span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>    <span class="at">mean_rate =</span> <span class="fu">mean</span>(mortality_rate, <span class="at">na.rm =</span> <span class="cn">TRUE</span>),</span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a>    <span class="at">sd_rate =</span> <span class="fu">sd</span>(mortality_rate, <span class="at">na.rm =</span> <span class="cn">TRUE</span>),</span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a>    <span class="at">min_rate =</span> <span class="fu">min</span>(mortality_rate, <span class="at">na.rm =</span> <span class="cn">TRUE</span>),</span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a>    <span class="at">max_rate =</span> <span class="fu">max</span>(mortality_rate, <span class="at">na.rm =</span> <span class="cn">TRUE</span>)</span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a>  ) <span class="sc">|&gt;</span> </span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="fu">across</span>(<span class="fu">where</span>(is.numeric), <span class="sc">~</span><span class="fu">round</span>(., <span class="dv">1</span>))) <span class="sc">|&gt;</span> </span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a>  <span class="fu">kable</span>(<span class="at">caption =</span> <span class="st">"Summary statitics by region group"</span>)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<table class="caption-top table table-sm table-striped small">
<caption>Summary statistics by region group</caption>
<colgroup>
<col style="width: 17%">
<col style="width: 13%">
<col style="width: 20%">
<col style="width: 13%">
<col style="width: 10%">
<col style="width: 12%">
<col style="width: 12%">
</colgroup>
<thead>
<tr class="header">
<th style="text-align: left;">region_group</th>
<th style="text-align: right;">n_regions</th>
<th style="text-align: right;">n_observations</th>
<th style="text-align: right;">mean_rate</th>
<th style="text-align: right;">sd_rate</th>
<th style="text-align: right;">min_rate</th>
<th style="text-align: right;">max_rate</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">North</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">414</td>
<td style="text-align: right;">43.1</td>
<td style="text-align: right;">6.0</td>
<td style="text-align: right;">27.8</td>
<td style="text-align: right;">61.4</td>
</tr>
<tr class="even">
<td style="text-align: left;">South</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">552</td>
<td style="text-align: right;">29.0</td>
<td style="text-align: right;">6.2</td>
<td style="text-align: right;">16.8</td>
<td style="text-align: right;">56.0</td>
</tr>
</tbody>
</table>
</div>
</div>
<hr>
</section>
<section id="visualising-the-difference" class="level2">
<h2 class="anchored" data-anchor-id="visualising-the-difference">Visualising the difference</h2>
<section id="boxplot-distribution-of-mortality-rates" class="level3">
<h3 class="anchored" data-anchor-id="boxplot-distribution-of-mortality-rates">Boxplot: Distribution of Mortality Rates</h3>
<p>The boxplot displays the distribution of mortality rates for each group, showing median, quartiles, and outliers.</p>
<p><strong>Rationale</strong>: A boxplot provides an immediate visual comparison of central tendency and spread. It confirms the North has higher rates and suggests the difference is substantial.</p>
<div class="cell">
<div class="code-copy-outer-scaffold"><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>p_box <span class="ot">&lt;-</span> <span class="fu">ggplot</span>(</span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>  north_south_data,</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">aes</span>(<span class="at">x =</span> region_group, <span class="at">y =</span> mortality_rate, <span class="at">fill =</span> region_group)</span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a>) <span class="sc">+</span></span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_boxplot</span>(<span class="at">alpha =</span> <span class="fl">0.7</span>) <span class="sc">+</span></span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_fill_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="st">"North"</span> <span class="ot">=</span> <span class="st">"#e74c3c"</span>, <span class="st">"South"</span> <span class="ot">=</span> <span class="st">"#3498db"</span>)) <span class="sc">+</span></span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(</span>
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a>    <span class="at">title =</span> <span class="st">"Respiratory mortality rates by region group"</span>,</span>
<span id="cb2-9"><a href="#cb2-9" aria-hidden="true" tabindex="-1"></a>    <span class="at">subtitle =</span> <span class="st">"North shows consistently higher rates and greater variability"</span>,</span>
<span id="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a>    <span class="at">x =</span> <span class="st">""</span>,</span>
<span id="cb2-11"><a href="#cb2-11" aria-hidden="true" tabindex="-1"></a>    <span class="at">y =</span> <span class="st">"Rate per 100,000"</span>,</span>
<span id="cb2-12"><a href="#cb2-12" aria-hidden="true" tabindex="-1"></a>    <span class="at">fill =</span> <span class="st">""</span></span>
<span id="cb2-13"><a href="#cb2-13" aria-hidden="true" tabindex="-1"></a>  ) <span class="sc">+</span></span>
<span id="cb2-14"><a href="#cb2-14" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme_minimal</span>() <span class="sc">+</span></span>
<span id="cb2-15"><a href="#cb2-15" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"none"</span>)</span>
<span id="cb2-16"><a href="#cb2-16" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-17"><a href="#cb2-17" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplotly</span>(p_box, <span class="at">tooltip =</span> <span class="fu">c</span>(<span class="st">"region_group"</span>, <span class="st">"mortality_rate"</span>))</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<div class="plotly html-widget html-fill-item" id="htmlwidget-b4e8dbeeeb8169d8ad39" style="width:100%;height:464px;"></div>
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<p>Distribution of respiratory mortality rates: North vs South</p>
</div>
</div>
</section>
<section id="time-series-north-vs-south-over-time" class="level3">
<h3 class="anchored" data-anchor-id="time-series-north-vs-south-over-time">Time Series: North vs South Over Time</h3>
<p>The time series plot shows annual mean rates for each group with 95% confidence intervals represented by shaded bands.</p>
<p><strong>Rationale</strong>: This plot reveals whether the gap is consistent over time or driven by specific years. Confidence bands show the precision of each annual estimate. Non‑overlapping bands across most years support the hypothesis of a sustained difference.</p>
<div class="cell">
<div class="code-copy-outer-scaffold"><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><span class="co"># Calculate annual means and confidence intervals by group</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>annual_ci <span class="ot">&lt;-</span> north_south_data <span class="sc">|&gt;</span></span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">group_by</span>(region_group, year) <span class="sc">|&gt;</span></span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">summarise</span>(</span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a>    <span class="at">mean_rate =</span> <span class="fu">mean</span>(mortality_rate, <span class="at">na.rm =</span> <span class="cn">TRUE</span>),</span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a>    <span class="at">se =</span> <span class="fu">sd</span>(mortality_rate, <span class="at">na.rm =</span> <span class="cn">TRUE</span>) <span class="sc">/</span> <span class="fu">sqrt</span>(<span class="fu">n</span>()),</span>
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a>    <span class="at">ci_lower =</span> mean_rate <span class="sc">-</span> <span class="fl">1.96</span> <span class="sc">*</span> se,</span>
<span id="cb3-8"><a href="#cb3-8" aria-hidden="true" tabindex="-1"></a>    <span class="at">ci_upper =</span> mean_rate <span class="sc">+</span> <span class="fl">1.96</span> <span class="sc">*</span> se,</span>
<span id="cb3-9"><a href="#cb3-9" aria-hidden="true" tabindex="-1"></a>    <span class="at">.groups =</span> <span class="st">"drop"</span></span>
<span id="cb3-10"><a href="#cb3-10" aria-hidden="true" tabindex="-1"></a>  )</span>
<span id="cb3-11"><a href="#cb3-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb3-12"><a href="#cb3-12" aria-hidden="true" tabindex="-1"></a><span class="co"># Create plot with confidence ribbons</span></span>
<span id="cb3-13"><a href="#cb3-13" aria-hidden="true" tabindex="-1"></a>p_ts_ci <span class="ot">&lt;-</span> <span class="fu">ggplot</span>(annual_ci, <span class="fu">aes</span>(<span class="at">x =</span> year, <span class="at">y =</span> mean_rate, <span class="at">color =</span> region_group, <span class="at">fill =</span> region_group)) <span class="sc">+</span></span>
<span id="cb3-14"><a href="#cb3-14" aria-hidden="true" tabindex="-1"></a>  <span class="co"># Confidence interval ribbons</span></span>
<span id="cb3-15"><a href="#cb3-15" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_ribbon</span>(<span class="fu">aes</span>(<span class="at">ymin =</span> ci_lower, <span class="at">ymax =</span> ci_upper), <span class="at">alpha =</span> <span class="fl">0.2</span>, <span class="at">color =</span> <span class="cn">NA</span>) <span class="sc">+</span></span>
<span id="cb3-16"><a href="#cb3-16" aria-hidden="true" tabindex="-1"></a>  <span class="co"># Mean lines</span></span>
<span id="cb3-17"><a href="#cb3-17" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">linewidth =</span> <span class="dv">1</span>) <span class="sc">+</span></span>
<span id="cb3-18"><a href="#cb3-18" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">size =</span> <span class="dv">2</span>) <span class="sc">+</span></span>
<span id="cb3-19"><a href="#cb3-19" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_color_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="st">"North"</span> <span class="ot">=</span> <span class="st">"#e74c3c"</span>, <span class="st">"South"</span> <span class="ot">=</span> <span class="st">"#3498db"</span>)) <span class="sc">+</span></span>
<span id="cb3-20"><a href="#cb3-20" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_fill_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="st">"North"</span> <span class="ot">=</span> <span class="st">"#e74c3c"</span>, <span class="st">"South"</span> <span class="ot">=</span> <span class="st">"#3498db"</span>)) <span class="sc">+</span></span>
<span id="cb3-21"><a href="#cb3-21" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="fu">seq</span>(<span class="dv">2000</span>, <span class="dv">2025</span>, <span class="dv">5</span>)) <span class="sc">+</span></span>
<span id="cb3-22"><a href="#cb3-22" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(</span>
<span id="cb3-23"><a href="#cb3-23" aria-hidden="true" tabindex="-1"></a>    <span class="at">title =</span> <span class="st">"North‑South divide in respiratory mortality"</span>,</span>
<span id="cb3-24"><a href="#cb3-24" aria-hidden="true" tabindex="-1"></a>    <span class="at">subtitle =</span> <span class="st">"Lines show mean rates; shaded bands show 95% confidence intervals"</span>,</span>
<span id="cb3-25"><a href="#cb3-25" aria-hidden="true" tabindex="-1"></a>    <span class="at">x =</span> <span class="st">"Year"</span>,</span>
<span id="cb3-26"><a href="#cb3-26" aria-hidden="true" tabindex="-1"></a>    <span class="at">y =</span> <span class="st">"Mean rate per 100,000"</span>,</span>
<span id="cb3-27"><a href="#cb3-27" aria-hidden="true" tabindex="-1"></a>    <span class="at">color =</span> <span class="st">""</span>,</span>
<span id="cb3-28"><a href="#cb3-28" aria-hidden="true" tabindex="-1"></a>    <span class="at">fill =</span> <span class="st">""</span></span>
<span id="cb3-29"><a href="#cb3-29" aria-hidden="true" tabindex="-1"></a>  ) <span class="sc">+</span></span>
<span id="cb3-30"><a href="#cb3-30" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme_minimal</span>() <span class="sc">+</span></span>
<span id="cb3-31"><a href="#cb3-31" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"bottom"</span>)</span>
<span id="cb3-32"><a href="#cb3-32" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb3-33"><a href="#cb3-33" aria-hidden="true" tabindex="-1"></a><span class="co"># Make interactive</span></span>
<span id="cb3-34"><a href="#cb3-34" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplotly</span>(p_ts_ci, <span class="at">tooltip =</span> <span class="fu">c</span>(<span class="st">"year"</span>, <span class="st">"region_group"</span>, <span class="st">"mean_rate"</span>, <span class="st">"ci_lower"</span>, <span class="st">"ci_upper"</span>))</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<div class="plotly html-widget html-fill-item" id="htmlwidget-37e97dfc06bd3e8f5c7b" style="width:100%;height:464px;"></div>
<script type="application/json" 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2001<br />mean_rate: 48.95298<br />region_group: NA<br />region_group: North<br />ci_lower: 45.72057<br />ci_upper: 52.18540","year: 2002<br />mean_rate: 49.00499<br />region_group: NA<br />region_group: North<br />ci_lower: 46.14605<br />ci_upper: 51.86393","year: 2003<br />mean_rate: 49.59536<br />region_group: NA<br />region_group: North<br />ci_lower: 46.53348<br />ci_upper: 52.65724","year: 2004<br />mean_rate: 45.82709<br />region_group: NA<br />region_group: North<br />ci_lower: 42.67030<br />ci_upper: 48.98387","year: 2005<br />mean_rate: 46.08503<br />region_group: NA<br />region_group: North<br />ci_lower: 43.10417<br />ci_upper: 49.06589","year: 2006<br />mean_rate: 44.40185<br />region_group: NA<br />region_group: North<br />ci_lower: 41.57633<br />ci_upper: 47.22737","year: 2007<br />mean_rate: 44.51820<br />region_group: NA<br />region_group: North<br />ci_lower: 42.13315<br />ci_upper: 46.90324","year: 2008<br />mean_rate: 45.46701<br />region_group: NA<br 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/>region_group: South<br />ci_lower: 27.20447<br />ci_upper: 31.87946","year: 2009<br />mean_rate: 28.65513<br />region_group: NA<br />region_group: South<br />ci_lower: 26.55530<br />ci_upper: 30.75496","year: 2010<br />mean_rate: 28.39157<br />region_group: NA<br />region_group: South<br />ci_lower: 26.31714<br />ci_upper: 30.46601","year: 2011<br />mean_rate: 27.80322<br />region_group: NA<br />region_group: South<br />ci_lower: 25.75722<br />ci_upper: 29.84921","year: 2012<br />mean_rate: 27.69098<br />region_group: NA<br />region_group: South<br />ci_lower: 25.56688<br />ci_upper: 29.81507","year: 2013<br />mean_rate: 27.74166<br />region_group: NA<br />region_group: South<br />ci_lower: 25.85550<br />ci_upper: 29.62782","year: 2014<br />mean_rate: 27.48367<br />region_group: NA<br />region_group: South<br />ci_lower: 25.65099<br />ci_upper: 29.31636","year: 2015<br />mean_rate: 28.87019<br />region_group: NA<br />region_group: South<br />ci_lower: 26.95212<br />ci_upper: 30.78826","year: 2016<br />mean_rate: 29.43726<br />region_group: NA<br />region_group: South<br />ci_lower: 27.63876<br />ci_upper: 31.23577","year: 2017<br />mean_rate: 28.29994<br />region_group: NA<br />region_group: South<br />ci_lower: 26.48598<br />ci_upper: 30.11390","year: 2018<br />mean_rate: 28.52179<br />region_group: NA<br />region_group: South<br />ci_lower: 26.62488<br />ci_upper: 30.41870","year: 2019<br />mean_rate: 26.26315<br />region_group: NA<br />region_group: South<br />ci_lower: 24.46019<br />ci_upper: 28.06612","year: 2020<br />mean_rate: 24.42752<br />region_group: NA<br />region_group: South<br />ci_lower: 22.52335<br />ci_upper: 26.33169","year: 2021<br />mean_rate: 23.34383<br />region_group: NA<br />region_group: South<br />ci_lower: 21.60814<br />ci_upper: 25.07952","year: 2022<br />mean_rate: 25.89396<br />region_group: NA<br />region_group: South<br />ci_lower: 24.09945<br />ci_upper: 27.68848","year: 2023<br />mean_rate: 27.85964<br />region_group: NA<br />region_group: South<br />ci_lower: 25.31203<br />ci_upper: 30.40725","year: 2024<br />mean_rate: 26.79275<br />region_group: NA<br />region_group: South<br />ci_lower: 24.28181<br />ci_upper: 29.30370","year: 2024<br />mean_rate: 26.79275<br />region_group: NA<br />region_group: South<br />ci_lower: 24.28181<br />ci_upper: 29.30370","year: 2024<br />mean_rate: 26.79275<br />region_group: NA<br />region_group: South<br />ci_lower: 24.28181<br />ci_upper: 29.30370","year: 2023<br />mean_rate: 27.85964<br />region_group: NA<br />region_group: South<br />ci_lower: 25.31203<br />ci_upper: 30.40725","year: 2022<br />mean_rate: 25.89396<br />region_group: NA<br />region_group: South<br />ci_lower: 24.09945<br />ci_upper: 27.68848","year: 2021<br />mean_rate: 23.34383<br />region_group: NA<br />region_group: South<br />ci_lower: 21.60814<br />ci_upper: 25.07952","year: 2020<br />mean_rate: 24.42752<br />region_group: NA<br />region_group: South<br />ci_lower: 22.52335<br />ci_upper: 26.33169","year: 2019<br />mean_rate: 26.26315<br />region_group: NA<br />region_group: South<br />ci_lower: 24.46019<br />ci_upper: 28.06612","year: 2018<br />mean_rate: 28.52179<br />region_group: NA<br />region_group: South<br />ci_lower: 26.62488<br />ci_upper: 30.41870","year: 2017<br />mean_rate: 28.29994<br />region_group: NA<br />region_group: South<br />ci_lower: 26.48598<br />ci_upper: 30.11390","year: 2016<br />mean_rate: 29.43726<br />region_group: NA<br />region_group: South<br />ci_lower: 27.63876<br />ci_upper: 31.23577","year: 2015<br />mean_rate: 28.87019<br />region_group: NA<br />region_group: South<br />ci_lower: 26.95212<br />ci_upper: 30.78826","year: 2014<br />mean_rate: 27.48367<br />region_group: NA<br />region_group: South<br />ci_lower: 25.65099<br />ci_upper: 29.31636","year: 2013<br />mean_rate: 27.74166<br />region_group: NA<br />region_group: South<br />ci_lower: 25.85550<br />ci_upper: 29.62782","year: 2012<br />mean_rate: 27.69098<br />region_group: NA<br />region_group: South<br />ci_lower: 25.56688<br />ci_upper: 29.81507","year: 2011<br />mean_rate: 27.80322<br />region_group: NA<br />region_group: South<br />ci_lower: 25.75722<br />ci_upper: 29.84921","year: 2010<br />mean_rate: 28.39157<br />region_group: NA<br />region_group: South<br />ci_lower: 26.31714<br />ci_upper: 30.46601","year: 2009<br />mean_rate: 28.65513<br />region_group: NA<br />region_group: South<br />ci_lower: 26.55530<br />ci_upper: 30.75496","year: 2008<br />mean_rate: 29.54197<br />region_group: NA<br />region_group: South<br />ci_lower: 27.20447<br />ci_upper: 31.87946","year: 2007<br />mean_rate: 29.91805<br />region_group: NA<br />region_group: South<br />ci_lower: 27.46943<br />ci_upper: 32.36667","year: 2006<br />mean_rate: 30.69853<br />region_group: NA<br />region_group: South<br />ci_lower: 28.09741<br />ci_upper: 33.29965","year: 2005<br />mean_rate: 31.98650<br />region_group: NA<br />region_group: South<br />ci_lower: 29.18960<br />ci_upper: 34.78341","year: 2004<br />mean_rate: 32.15447<br />region_group: NA<br />region_group: South<br />ci_lower: 29.32881<br />ci_upper: 34.98012","year: 2003<br />mean_rate: 34.37957<br />region_group: NA<br />region_group: South<br />ci_lower: 31.32133<br />ci_upper: 37.43780","year: 2002<br />mean_rate: 33.76435<br />region_group: NA<br />region_group: South<br />ci_lower: 30.86240<br />ci_upper: 36.66629","year: 2001<br />mean_rate: 34.88668<br />region_group: NA<br />region_group: South<br />ci_lower: 31.53628<br />ci_upper: 38.23709","year: 2001<br />mean_rate: 34.88668<br />region_group: NA<br />region_group: South<br />ci_lower: 31.53628<br />ci_upper: 38.23709"],"type":"scatter","mode":"lines","line":{"width":1.8897637795275593,"color":"transparent","dash":"solid"},"fill":"toself","fillcolor":"rgba(52,152,219,0.2)","hoveron":"points","name":"(South,1)","legendgroup":"(South,1)","showlegend":true,"xaxis":"x","yaxis":"y","hoverinfo":"text","frame":null},{"x":[2001,2002,2003,2004,2005,2006,2007,2008,2009,2010,2011,2012,2013,2014,2015,2016,2017,2018,2019,2020,2021,2022,2023,2024],"y":[48.952984999999998,49.004990555555558,49.595362222222221,45.827087222222225,46.085031111111114,44.401848888888885,44.518197777777779,45.467008333333332,42.59383722222222,41.556086111111114,41.92688722222222,40.407469444444445,41.258213888888889,40.795406111111113,42.272627222222219,43.869077222222224,42.503895555555559,42.732417222222224,41.057148333333338,37.304091114083796,36.790889323365548,40.422286651291707,43.289025943318954,41.589825017186769],"text":["year: 2001<br />mean_rate: 48.95298<br />region_group: North<br />region_group: North","year: 2002<br />mean_rate: 49.00499<br />region_group: North<br />region_group: North","year: 2003<br />mean_rate: 49.59536<br />region_group: North<br />region_group: North","year: 2004<br />mean_rate: 45.82709<br />region_group: North<br />region_group: North","year: 2005<br />mean_rate: 46.08503<br />region_group: North<br />region_group: North","year: 2006<br />mean_rate: 44.40185<br />region_group: North<br />region_group: North","year: 2007<br />mean_rate: 44.51820<br />region_group: North<br />region_group: North","year: 2008<br />mean_rate: 45.46701<br />region_group: North<br />region_group: North","year: 2009<br />mean_rate: 42.59384<br />region_group: North<br />region_group: North","year: 2010<br />mean_rate: 41.55609<br />region_group: North<br />region_group: North","year: 2011<br />mean_rate: 41.92689<br />region_group: North<br />region_group: North","year: 2012<br />mean_rate: 40.40747<br />region_group: North<br />region_group: North","year: 2013<br />mean_rate: 41.25821<br />region_group: North<br />region_group: North","year: 2014<br />mean_rate: 40.79541<br />region_group: North<br />region_group: North","year: 2015<br />mean_rate: 42.27263<br />region_group: North<br />region_group: North","year: 2016<br />mean_rate: 43.86908<br />region_group: North<br />region_group: North","year: 2017<br />mean_rate: 42.50390<br />region_group: North<br />region_group: North","year: 2018<br />mean_rate: 42.73242<br />region_group: North<br />region_group: North","year: 2019<br />mean_rate: 41.05715<br />region_group: North<br />region_group: North","year: 2020<br />mean_rate: 37.30409<br />region_group: North<br />region_group: North","year: 2021<br />mean_rate: 36.79089<br />region_group: North<br />region_group: North","year: 2022<br />mean_rate: 40.42229<br />region_group: North<br />region_group: North","year: 2023<br />mean_rate: 43.28903<br />region_group: North<br />region_group: North","year: 2024<br />mean_rate: 41.58983<br />region_group: North<br />region_group: North"],"type":"scatter","mode":"lines+markers","line":{"width":3.7795275590551185,"color":"rgba(231,76,60,1)","dash":"solid"},"hoveron":"points","name":"North","legendgroup":"North","showlegend":true,"xaxis":"x","yaxis":"y","hoverinfo":"text","marker":{"autocolorscale":false,"color":"rgba(231,76,60,1)","opacity":1,"size":7.559055118110237,"symbol":"circle","line":{"width":1.8897637795275593,"color":"rgba(231,76,60,1)"}},"frame":null},{"x":[2001,2002,2003,2004,2005,2006,2007,2008,2009,2010,2011,2012,2013,2014,2015,2016,2017,2018,2019,2020,2021,2022,2023,2024],"y":[34.886684583333334,33.764346666666668,34.379565416666665,32.154465000000002,31.986504583333332,30.698527083333335,29.918051666666667,29.541965000000001,28.655130833333335,28.391575,27.803217916666668,27.690977916666668,27.74166,27.483673750000001,28.870190833333332,29.437260833333333,28.299938749999999,28.521787500000002,26.263153750000001,24.427519057009803,23.343829710231088,25.893962622189484,27.859640486020641,26.792754673544202],"text":["year: 2001<br />mean_rate: 34.88668<br />region_group: South<br />region_group: South","year: 2002<br />mean_rate: 33.76435<br />region_group: South<br />region_group: South","year: 2003<br />mean_rate: 34.37957<br />region_group: South<br />region_group: South","year: 2004<br />mean_rate: 32.15447<br />region_group: South<br />region_group: South","year: 2005<br />mean_rate: 31.98650<br />region_group: South<br />region_group: South","year: 2006<br />mean_rate: 30.69853<br />region_group: South<br />region_group: South","year: 2007<br />mean_rate: 29.91805<br />region_group: South<br />region_group: South","year: 2008<br />mean_rate: 29.54197<br />region_group: South<br />region_group: South","year: 2009<br />mean_rate: 28.65513<br />region_group: South<br />region_group: South","year: 2010<br />mean_rate: 28.39157<br />region_group: South<br />region_group: South","year: 2011<br />mean_rate: 27.80322<br />region_group: South<br />region_group: South","year: 2012<br />mean_rate: 27.69098<br />region_group: South<br />region_group: South","year: 2013<br />mean_rate: 27.74166<br />region_group: South<br />region_group: South","year: 2014<br />mean_rate: 27.48367<br />region_group: South<br />region_group: South","year: 2015<br />mean_rate: 28.87019<br />region_group: South<br />region_group: South","year: 2016<br />mean_rate: 29.43726<br />region_group: South<br />region_group: South","year: 2017<br />mean_rate: 28.29994<br />region_group: South<br />region_group: South","year: 2018<br />mean_rate: 28.52179<br />region_group: South<br />region_group: South","year: 2019<br />mean_rate: 26.26315<br />region_group: South<br />region_group: South","year: 2020<br />mean_rate: 24.42752<br />region_group: South<br />region_group: South","year: 2021<br />mean_rate: 23.34383<br />region_group: South<br />region_group: South","year: 2022<br />mean_rate: 25.89396<br />region_group: South<br />region_group: South","year: 2023<br />mean_rate: 27.85964<br />region_group: South<br />region_group: South","year: 2024<br />mean_rate: 26.79275<br />region_group: South<br />region_group: South"],"type":"scatter","mode":"lines+markers","line":{"width":3.7795275590551185,"color":"rgba(52,152,219,1)","dash":"solid"},"hoveron":"points","name":"South","legendgroup":"South","showlegend":true,"xaxis":"x","yaxis":"y","hoverinfo":"text","marker":{"autocolorscale":false,"color":"rgba(52,152,219,1)","opacity":1,"size":7.559055118110237,"symbol":"circle","line":{"width":1.8897637795275593,"color":"rgba(52,152,219,1)"}},"frame":null}],"layout":{"margin":{"t":40.840182648401829,"r":7.3059360730593621,"b":37.260273972602747,"l":37.260273972602747},"paper_bgcolor":"rgba(255,255,255,1)","font":{"color":"rgba(0,0,0,1)","family":"","size":14.611872146118724},"title":{"text":"North‑South divide in respiratory 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<p>North‑South divergence in respiratory mortality with 95% confidence bands</p>
</div>
</div>
<hr>
</section>
</section>
<section id="statistical-testing" class="level2">
<h2 class="anchored" data-anchor-id="statistical-testing">Statistical Testing</h2>
<section id="assumption-checks" class="level3">
<h3 class="anchored" data-anchor-id="assumption-checks">Assumption Checks</h3>
<p>Before running a t‑test, the variance ratio is calculated to compare the spread of the two groups.</p>
<p><strong>Rationale</strong>: The standard t‑test assumes equal variances. The variance ratio tests this assumption. A ratio close to 1 indicates similar spread, supporting the use of standard methods. A ratio &gt;2 or &lt;0.5 would suggest unequal variances and require adjustment.</p>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Extract data for each group</span></span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>north_data <span class="ot">&lt;-</span> north_south_data <span class="sc">|&gt;</span> </span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">filter</span>(region_group <span class="sc">==</span> <span class="st">"North"</span>) <span class="sc">|&gt;</span> </span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">pull</span>(mortality_rate)</span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a>south_data <span class="ot">&lt;-</span> north_south_data <span class="sc">|&gt;</span> </span>
<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">filter</span>(region_group <span class="sc">==</span> <span class="st">"South"</span>) <span class="sc">|&gt;</span> </span>
<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">pull</span>(mortality_rate)</span>
<span id="cb4-9"><a href="#cb4-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-10"><a href="#cb4-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Variance ratio check</span></span>
<span id="cb4-11"><a href="#cb4-11" aria-hidden="true" tabindex="-1"></a>var_north <span class="ot">&lt;-</span> <span class="fu">var</span>(north_data, <span class="at">na.rm =</span> <span class="cn">TRUE</span>)</span>
<span id="cb4-12"><a href="#cb4-12" aria-hidden="true" tabindex="-1"></a>var_south <span class="ot">&lt;-</span> <span class="fu">var</span>(south_data, <span class="at">na.rm =</span> <span class="cn">TRUE</span>)</span>
<span id="cb4-13"><a href="#cb4-13" aria-hidden="true" tabindex="-1"></a>var_ratio <span class="ot">&lt;-</span> var_north <span class="sc">/</span> var_south</span>
<span id="cb4-14"><a href="#cb4-14" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-15"><a href="#cb4-15" aria-hidden="true" tabindex="-1"></a><span class="co"># Create interpretation table</span></span>
<span id="cb4-16"><a href="#cb4-16" aria-hidden="true" tabindex="-1"></a>var_table <span class="ot">&lt;-</span> <span class="fu">data.frame</span>(</span>
<span id="cb4-17"><a href="#cb4-17" aria-hidden="true" tabindex="-1"></a>  <span class="at">Group =</span> <span class="fu">c</span>(<span class="st">"North"</span>, <span class="st">"South"</span>, <span class="st">"Ratio"</span>),</span>
<span id="cb4-18"><a href="#cb4-18" aria-hidden="true" tabindex="-1"></a>  <span class="at">Variance =</span> <span class="fu">c</span>(</span>
<span id="cb4-19"><a href="#cb4-19" aria-hidden="true" tabindex="-1"></a>    <span class="fu">round</span>(var_north, <span class="dv">2</span>),</span>
<span id="cb4-20"><a href="#cb4-20" aria-hidden="true" tabindex="-1"></a>    <span class="fu">round</span>(var_south, <span class="dv">2</span>),</span>
<span id="cb4-21"><a href="#cb4-21" aria-hidden="true" tabindex="-1"></a>    <span class="fu">round</span>(var_ratio, <span class="dv">2</span>)</span>
<span id="cb4-22"><a href="#cb4-22" aria-hidden="true" tabindex="-1"></a>  ),</span>
<span id="cb4-23"><a href="#cb4-23" aria-hidden="true" tabindex="-1"></a>  <span class="at">Interpretation =</span> <span class="fu">c</span>(</span>
<span id="cb4-24"><a href="#cb4-24" aria-hidden="true" tabindex="-1"></a>    <span class="st">"Spread of North mortality rates"</span>,</span>
<span id="cb4-25"><a href="#cb4-25" aria-hidden="true" tabindex="-1"></a>    <span class="st">"Spread of South mortality rates"</span>,</span>
<span id="cb4-26"><a href="#cb4-26" aria-hidden="true" tabindex="-1"></a>    <span class="fu">case_when</span>(</span>
<span id="cb4-27"><a href="#cb4-27" aria-hidden="true" tabindex="-1"></a>      var_ratio <span class="sc">&gt;</span> <span class="dv">2</span> <span class="sc">~</span> <span class="st">"&gt; 2: Variances very different (violates t-test assumption)"</span>,</span>
<span id="cb4-28"><a href="#cb4-28" aria-hidden="true" tabindex="-1"></a>      var_ratio <span class="sc">&lt;</span> <span class="fl">0.5</span> <span class="sc">~</span> <span class="st">"&lt; 0.5: Variances very different (violates t-test assumption)"</span>,</span>
<span id="cb4-29"><a href="#cb4-29" aria-hidden="true" tabindex="-1"></a>      <span class="cn">TRUE</span> <span class="sc">~</span> <span class="st">"~1: Variances similar (t-test assumption holds)"</span></span>
<span id="cb4-30"><a href="#cb4-30" aria-hidden="true" tabindex="-1"></a>    )</span>
<span id="cb4-31"><a href="#cb4-31" aria-hidden="true" tabindex="-1"></a>  )</span>
<span id="cb4-32"><a href="#cb4-32" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb4-33"><a href="#cb4-33" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-34"><a href="#cb4-34" aria-hidden="true" tabindex="-1"></a><span class="fu">kable</span>(var_table, <span class="at">caption =</span> <span class="st">"Variance ratio test: comparing spread between groups"</span>)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<table class="caption-top table table-sm table-striped small">
<caption>Variance ratio test: comparing spread between groups</caption>
<thead>
<tr class="header">
<th style="text-align: left;">Group</th>
<th style="text-align: right;">Variance</th>
<th style="text-align: left;">Interpretation</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">North</td>
<td style="text-align: right;">35.95</td>
<td style="text-align: left;">Spread of North mortality rates</td>
</tr>
<tr class="even">
<td style="text-align: left;">South</td>
<td style="text-align: right;">38.91</td>
<td style="text-align: left;">Spread of South mortality rates</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Ratio</td>
<td style="text-align: right;">0.92</td>
<td style="text-align: left;">~1: Variances similar (t-test assumption holds)</td>
</tr>
</tbody>
</table>
</div>
</div>
</section>
<section id="understanding-the-variance-ratio" class="level3">
<h3 class="anchored" data-anchor-id="understanding-the-variance-ratio">Understanding the Variance Ratio</h3>
<p>The <strong>variance ratio</strong> compares how spread out the data is in two groups:</p>
<ul>
<li>A ratio of <strong>1.0</strong> means equal variance</li>
<li>A ratio <strong>&gt;2 or &lt;0.5</strong> suggests variances are very different (violates the equal‑variance assumption of the standard t‑test)</li>
<li>A ratio <strong>close to 1</strong> (like ours at 0.92) means the groups have comparable variability</li>
</ul>
<p>Our ratio of <strong>0.92</strong> is well within the acceptable range. This means both groups have similar underlying spread, so any difference we detect is likely a real difference in central tendency, not an artefact of different variability.</p>
</section>
<section id="why-use-welchs-ttest" class="level3">
<h3 class="anchored" data-anchor-id="why-use-welchs-ttest">Why Use Welch’s t‑test?</h3>
<p>The standard t‑test assumes equal variances between groups. While our variance ratio suggests this assumption holds, I used <strong>Welch’s t‑test</strong>, which does <strong>not</strong> assume equal variances. This is now considered best practice because:</p>
<ol type="1">
<li>It gives correct results whether variances are equal or not</li>
<li>It is robust to minor violations of assumptions</li>
<li>It is recommended by statisticians as the default choice</li>
</ol>
<p>Welch’s test adjusts the degrees of freedom to account for any differences in variance, making it more reliable with real‑world data.</p>
</section>
<section id="twosample-ttest-welch" class="level3">
<h3 class="anchored" data-anchor-id="twosample-ttest-welch">Two‑Sample t‑test (Welch)</h3>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>t_test_results <span class="ot">&lt;-</span> <span class="fu">t.test</span>(mortality_rate <span class="sc">~</span> region_group, <span class="at">data =</span> north_south_data)</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Extract exact p-value</span></span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a>exact_p <span class="ot">&lt;-</span> t_test_results<span class="sc">$</span>p.value</span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a><span class="co"># Tidy the output for display</span></span>
<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a>t_test_tidy <span class="ot">&lt;-</span> <span class="fu">tidy</span>(t_test_results) <span class="sc">|&gt;</span> </span>
<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="fu">across</span>(<span class="fu">where</span>(is.numeric), <span class="sc">~</span><span class="fu">round</span>(., <span class="dv">3</span>)))</span>
<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a><span class="fu">kable</span>(t_test_tidy, <span class="at">caption =</span> <span class="st">"Welch two-sample t-test results"</span>)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<table class="caption-top table table-sm table-striped small">
<caption>Welch two-sample t-test results</caption>
<colgroup>
<col style="width: 8%">
<col style="width: 8%">
<col style="width: 8%">
<col style="width: 8%">
<col style="width: 7%">
<col style="width: 8%">
<col style="width: 8%">
<col style="width: 8%">
<col style="width: 21%">
<col style="width: 10%">
</colgroup>
<thead>
<tr class="header">
<th style="text-align: right;">estimate</th>
<th style="text-align: right;">estimate1</th>
<th style="text-align: right;">estimate2</th>
<th style="text-align: right;">statistic</th>
<th style="text-align: right;">p.value</th>
<th style="text-align: right;">parameter</th>
<th style="text-align: right;">conf.low</th>
<th style="text-align: right;">conf.high</th>
<th style="text-align: left;">method</th>
<th style="text-align: left;">alternative</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">14.1</td>
<td style="text-align: right;">43.121</td>
<td style="text-align: right;">29.021</td>
<td style="text-align: right;">35.549</td>
<td style="text-align: right;">0</td>
<td style="text-align: right;">907.421</td>
<td style="text-align: right;">13.322</td>
<td style="text-align: right;">14.879</td>
<td style="text-align: left;">Welch Two Sample t-test</td>
<td style="text-align: left;">two.sided</td>
</tr>
</tbody>
</table>
</div>
</div>
</section>
<section id="understanding-p-value" class="level3">
<h3 class="anchored" data-anchor-id="understanding-p-value">Understanding p-value</h3>
<p>The p-value from this test is <strong>4.327337e-174</strong>. This is <strong>not zero</strong>: it is an extremely small number that R displays in scientific notation.</p>
<table class="caption-top table">
<thead>
<tr class="header">
<th>Representation</th>
<th>Meaning</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><strong>4.327337e-174</strong></td>
<td>4.33 × 10<sup>-174</sup></td>
</tr>
<tr class="even">
<td><strong>In words</strong></td>
<td>4.33 divided by a 1 with 174 zeros</td>
</tr>
<tr class="odd">
<td><strong>Statistical convention</strong></td>
<td>p &lt; 0.001</td>
</tr>
</tbody>
</table>
<p>To put this in context: there are about 10<sup>80</sup> atoms in the observable universe. This result is about 100 billion billion times more unlikely than randomly picking the correct atom. There is <strong>no realistic doubt</strong> that the North‑South difference is real.</p>
</section>
<section id="effect-size-cohens-d" class="level3">
<h3 class="anchored" data-anchor-id="effect-size-cohens-d">Effect Size (Cohen’s d)</h3>
<p>Statistical significance doesn’t always mean practical significance. Cohen’s d measures the size of the difference in standard deviation units, with confidence intervals to show precision.</p>
<div class="cell" data-tbl-cap="Cohen's d effect size">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Calculate Cohen's d with confidence intervals</span></span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>cohens_d_results <span class="ot">&lt;-</span> <span class="fu">cohens_d</span>(mortality_rate <span class="sc">~</span> region_group, </span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a>                             <span class="at">data =</span> north_south_data,</span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a>                             <span class="at">ci =</span> <span class="fl">0.95</span>)</span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a><span class="co"># cohens_d_results |&gt; </span></span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a><span class="co">#   mutate(</span></span>
<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a><span class="co">#     across(where(is.numeric), ~round(., 2))</span></span>
<span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a><span class="co">#   ) |&gt; </span></span>
<span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a><span class="co">#   kable(caption = "Cohen's d results with confidence intervals")</span></span>
<span id="cb6-11"><a href="#cb6-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-12"><a href="#cb6-12" aria-hidden="true" tabindex="-1"></a><span class="co"># Extract values for reporting</span></span>
<span id="cb6-13"><a href="#cb6-13" aria-hidden="true" tabindex="-1"></a>d_value <span class="ot">&lt;-</span> cohens_d_results<span class="sc">$</span>Cohens_d</span>
<span id="cb6-14"><a href="#cb6-14" aria-hidden="true" tabindex="-1"></a>d_ci_low <span class="ot">&lt;-</span> cohens_d_results<span class="sc">$</span>CI_low</span>
<span id="cb6-15"><a href="#cb6-15" aria-hidden="true" tabindex="-1"></a>d_ci_high <span class="ot">&lt;-</span> cohens_d_results<span class="sc">$</span>CI_high</span>
<span id="cb6-16"><a href="#cb6-16" aria-hidden="true" tabindex="-1"></a>d_interpretation <span class="ot">&lt;-</span> <span class="fu">interpret_cohens_d</span>(d_value)</span>
<span id="cb6-17"><a href="#cb6-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-18"><a href="#cb6-18" aria-hidden="true" tabindex="-1"></a><span class="co"># Create a clean table for display</span></span>
<span id="cb6-19"><a href="#cb6-19" aria-hidden="true" tabindex="-1"></a>d_table <span class="ot">&lt;-</span> <span class="fu">data.frame</span>(</span>
<span id="cb6-20"><a href="#cb6-20" aria-hidden="true" tabindex="-1"></a>  <span class="at">Measure =</span> <span class="fu">c</span>(</span>
<span id="cb6-21"><a href="#cb6-21" aria-hidden="true" tabindex="-1"></a>    <span class="st">"Cohen's d"</span>,</span>
<span id="cb6-22"><a href="#cb6-22" aria-hidden="true" tabindex="-1"></a>    <span class="st">"95% CI (lower)"</span>,</span>
<span id="cb6-23"><a href="#cb6-23" aria-hidden="true" tabindex="-1"></a>    <span class="st">"95% CI (upper)"</span>,</span>
<span id="cb6-24"><a href="#cb6-24" aria-hidden="true" tabindex="-1"></a>    <span class="st">"Effect size interpretation"</span></span>
<span id="cb6-25"><a href="#cb6-25" aria-hidden="true" tabindex="-1"></a>  ),</span>
<span id="cb6-26"><a href="#cb6-26" aria-hidden="true" tabindex="-1"></a>  <span class="at">Value =</span> <span class="fu">c</span>(</span>
<span id="cb6-27"><a href="#cb6-27" aria-hidden="true" tabindex="-1"></a>    <span class="fu">round</span>(d_value, <span class="dv">2</span>),</span>
<span id="cb6-28"><a href="#cb6-28" aria-hidden="true" tabindex="-1"></a>    <span class="fu">round</span>(d_ci_low, <span class="dv">2</span>),</span>
<span id="cb6-29"><a href="#cb6-29" aria-hidden="true" tabindex="-1"></a>    <span class="fu">round</span>(d_ci_high, <span class="dv">2</span>),</span>
<span id="cb6-30"><a href="#cb6-30" aria-hidden="true" tabindex="-1"></a>    d_interpretation</span>
<span id="cb6-31"><a href="#cb6-31" aria-hidden="true" tabindex="-1"></a>  )</span>
<span id="cb6-32"><a href="#cb6-32" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb6-33"><a href="#cb6-33" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-34"><a href="#cb6-34" aria-hidden="true" tabindex="-1"></a><span class="fu">kable</span>(d_table, <span class="at">caption =</span> <span class="st">"Cohen's effect size with confidence intervals"</span>)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<table class="caption-top table table-sm table-striped small">
<caption>Cohen’s d effect size</caption>
<thead>
<tr class="header">
<th style="text-align: left;">Measure</th>
<th style="text-align: left;">Value</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">Cohen’s d</td>
<td style="text-align: left;">2.3</td>
</tr>
<tr class="even">
<td style="text-align: left;">95% CI (lower)</td>
<td style="text-align: left;">2.13</td>
</tr>
<tr class="odd">
<td style="text-align: left;">95% CI (upper)</td>
<td style="text-align: left;">2.46</td>
</tr>
<tr class="even">
<td style="text-align: left;">Effect size interpretation</td>
<td style="text-align: left;">large</td>
</tr>
</tbody>
</table>
</div>
</div>
</section>
<section id="what-the-effect-size-tells-us" class="level3">
<h3 class="anchored" data-anchor-id="what-the-effect-size-tells-us">What the Effect Size Tells Us</h3>
<p>Cohen’s d of <strong>2.3</strong> (95% CI: 2.13 to 2.46) is a <strong>large</strong> effect size. This means:</p>
<ul>
<li>The North is <strong>2.4 standard deviations higher</strong> than the South</li>
<li>There is minimal overlap between the two distributions</li>
<li>This is not just statistically significant: it’s a <strong>substantial public health gap</strong></li>
</ul>
<p>The confidence interval tells us we can be 95% confident the true effect size lies between 2.13 and 2.46. Even the lower bound represents a large effect, confirming the robustness of our finding.</p>
<hr>
</section>
</section>
<section id="confidence-intervals" class="level2">
<h2 class="anchored" data-anchor-id="confidence-intervals">Confidence Intervals</h2>
<p>The confidence interval around Cohen’s d shows the range of plausible values for the true effect size.</p>
<p><strong>Rationale</strong>: A confidence interval provides more information than a point estimate alone. It indicates the precision of the estimate and whether the effect could be small, medium, or large.</p>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a>group_ci <span class="ot">&lt;-</span> north_south_data <span class="sc">|&gt;</span> </span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">group_by</span>(region_group) <span class="sc">|&gt;</span> </span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">summarise</span>(</span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a>    <span class="at">mean_rate =</span> <span class="fu">mean</span>(mortality_rate, <span class="at">na.rm =</span> <span class="cn">TRUE</span>),</span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a>    <span class="at">se =</span> <span class="fu">sd</span>(mortality_rate, <span class="at">na.rm =</span> <span class="cn">TRUE</span>) <span class="sc">/</span> <span class="fu">sqrt</span>(<span class="fu">n</span>()),</span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a>    <span class="at">ci_lower =</span> mean_rate <span class="sc">-</span> <span class="fl">1.96</span> <span class="sc">*</span> se,</span>
<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a>    <span class="at">ci_upper =</span> mean_rate <span class="sc">+</span> <span class="fl">1.96</span> <span class="sc">*</span> se,</span>
<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a>    <span class="at">.groups =</span> <span class="st">"drop"</span></span>
<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a>  )</span>
<span id="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-11"><a href="#cb7-11" aria-hidden="true" tabindex="-1"></a>p_ci <span class="ot">&lt;-</span></span>
<span id="cb7-12"><a href="#cb7-12" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ggplot</span>(</span>
<span id="cb7-13"><a href="#cb7-13" aria-hidden="true" tabindex="-1"></a>    group_ci,</span>
<span id="cb7-14"><a href="#cb7-14" aria-hidden="true" tabindex="-1"></a>    <span class="fu">aes</span>(<span class="at">x =</span> region_group, <span class="at">y =</span> mean_rate, <span class="at">color =</span> region_group)</span>
<span id="cb7-15"><a href="#cb7-15" aria-hidden="true" tabindex="-1"></a>  ) <span class="sc">+</span></span>
<span id="cb7-16"><a href="#cb7-16" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">size =</span> <span class="dv">3</span>) <span class="sc">+</span></span>
<span id="cb7-17"><a href="#cb7-17" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_errorbar</span>(<span class="fu">aes</span>(<span class="at">ymin =</span> ci_lower, <span class="at">ymax =</span> ci_upper), <span class="at">width =</span> <span class="fl">0.2</span>, <span class="at">linewidth =</span> <span class="dv">1</span>) <span class="sc">+</span></span>
<span id="cb7-18"><a href="#cb7-18" aria-hidden="true" tabindex="-1"></a>  <span class="fu">scale_color_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="st">"North"</span> <span class="ot">=</span> <span class="st">"#e74c3c"</span>, <span class="st">"South"</span> <span class="ot">=</span> <span class="st">"#3498db"</span>)) <span class="sc">+</span></span>
<span id="cb7-19"><a href="#cb7-19" aria-hidden="true" tabindex="-1"></a>  <span class="fu">labs</span>(</span>
<span id="cb7-20"><a href="#cb7-20" aria-hidden="true" tabindex="-1"></a>    <span class="at">title =</span> <span class="st">"Group means with 95% confidence intervals"</span>,</span>
<span id="cb7-21"><a href="#cb7-21" aria-hidden="true" tabindex="-1"></a>    <span class="at">subtitle =</span> <span class="st">"Non-overlapping intervals suggest a genuine difference"</span>,</span>
<span id="cb7-22"><a href="#cb7-22" aria-hidden="true" tabindex="-1"></a>    <span class="at">x =</span> <span class="st">""</span>,</span>
<span id="cb7-23"><a href="#cb7-23" aria-hidden="true" tabindex="-1"></a>    <span class="at">y =</span> <span class="st">"Mean rate per 100,000"</span>,</span>
<span id="cb7-24"><a href="#cb7-24" aria-hidden="true" tabindex="-1"></a>    <span class="at">color =</span> <span class="st">""</span></span>
<span id="cb7-25"><a href="#cb7-25" aria-hidden="true" tabindex="-1"></a>  ) <span class="sc">+</span></span>
<span id="cb7-26"><a href="#cb7-26" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme_minimal</span>() <span class="sc">+</span></span>
<span id="cb7-27"><a href="#cb7-27" aria-hidden="true" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">"none"</span>)</span>
<span id="cb7-28"><a href="#cb7-28" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-29"><a href="#cb7-29" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplotly</span>(p_ci, <span class="at">tooltip =</span> <span class="fu">c</span>(<span class="st">"region_group"</span>, <span class="st">"mean_rate"</span>, <span class="st">"ci_lower"</span>, <span class="st">"ci_upper"</span>))</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<div class="plotly html-widget html-fill-item" id="htmlwidget-2db337b2062eb5db2711" style="width:100%;height:464px;"></div>
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<p>Group means with 95% confidence intervals</p>
</div>
</div>
<hr>
</section>
<section id="summary-of-findings" class="level2">
<h2 class="anchored" data-anchor-id="summary-of-findings">Summary of Findings</h2>
<p>The summary table consolidates all key statistics: group means, difference, confidence intervals, p‑values, effect size, and variance ratio.</p>
<p><strong>Rationale</strong>: A single table allows readers to see all results at once, facilitating comparison and interpretation.</p>
<div class="cell" data-tbl-cap="Statistical summary">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb8"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a>mean_north <span class="ot">&lt;-</span> <span class="fu">mean</span>(north_data, <span class="at">na.rm =</span> <span class="cn">TRUE</span>)</span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a>mean_south <span class="ot">&lt;-</span> <span class="fu">mean</span>(south_data, <span class="at">na.rm =</span> <span class="cn">TRUE</span>)</span>
<span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a>mean_diff <span class="ot">&lt;-</span> mean_north <span class="sc">-</span> mean_south</span>
<span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb8-5"><a href="#cb8-5" aria-hidden="true" tabindex="-1"></a><span class="fu">data.frame</span>(</span>
<span id="cb8-6"><a href="#cb8-6" aria-hidden="true" tabindex="-1"></a>  <span class="at">Measure =</span> <span class="fu">c</span>(</span>
<span id="cb8-7"><a href="#cb8-7" aria-hidden="true" tabindex="-1"></a>    <span class="st">"North mean rate"</span>,</span>
<span id="cb8-8"><a href="#cb8-8" aria-hidden="true" tabindex="-1"></a>    <span class="st">"South mean rate"</span>,</span>
<span id="cb8-9"><a href="#cb8-9" aria-hidden="true" tabindex="-1"></a>    <span class="st">"Difference"</span>,</span>
<span id="cb8-10"><a href="#cb8-10" aria-hidden="true" tabindex="-1"></a>    <span class="st">"95% CI (lower)"</span>,</span>
<span id="cb8-11"><a href="#cb8-11" aria-hidden="true" tabindex="-1"></a>    <span class="st">"95% CI (upper)"</span>,</span>
<span id="cb8-12"><a href="#cb8-12" aria-hidden="true" tabindex="-1"></a>    <span class="st">"p-value (exact)"</span>,</span>
<span id="cb8-13"><a href="#cb8-13" aria-hidden="true" tabindex="-1"></a>    <span class="st">"p-value (reported)"</span>,</span>
<span id="cb8-14"><a href="#cb8-14" aria-hidden="true" tabindex="-1"></a>    <span class="st">"Cohen's d"</span>,</span>
<span id="cb8-15"><a href="#cb8-15" aria-hidden="true" tabindex="-1"></a>    <span class="st">"d 95% CI"</span>,</span>
<span id="cb8-16"><a href="#cb8-16" aria-hidden="true" tabindex="-1"></a>    <span class="st">"Effect size"</span>,</span>
<span id="cb8-17"><a href="#cb8-17" aria-hidden="true" tabindex="-1"></a>    <span class="st">"Variance ratio"</span></span>
<span id="cb8-18"><a href="#cb8-18" aria-hidden="true" tabindex="-1"></a>  ),</span>
<span id="cb8-19"><a href="#cb8-19" aria-hidden="true" tabindex="-1"></a>  <span class="at">Value =</span> <span class="fu">c</span>(</span>
<span id="cb8-20"><a href="#cb8-20" aria-hidden="true" tabindex="-1"></a>    <span class="fu">paste0</span>(<span class="fu">round</span>(<span class="fu">mean</span>(north_data, <span class="at">na.rm =</span> <span class="cn">TRUE</span>), <span class="dv">1</span>), <span class="st">" per 100k"</span>),</span>
<span id="cb8-21"><a href="#cb8-21" aria-hidden="true" tabindex="-1"></a>    <span class="fu">paste0</span>(<span class="fu">round</span>(<span class="fu">mean</span>(south_data, <span class="at">na.rm =</span> <span class="cn">TRUE</span>), <span class="dv">1</span>), <span class="st">" per 100k"</span>),</span>
<span id="cb8-22"><a href="#cb8-22" aria-hidden="true" tabindex="-1"></a>    <span class="fu">paste0</span>(<span class="fu">round</span>(mean_diff, <span class="dv">1</span>), <span class="st">" per 100k"</span>),</span>
<span id="cb8-23"><a href="#cb8-23" aria-hidden="true" tabindex="-1"></a>    <span class="fu">paste0</span>(<span class="fu">round</span>(t_test_tidy<span class="sc">$</span>conf.low, <span class="dv">1</span>), <span class="st">" per 100k"</span>),</span>
<span id="cb8-24"><a href="#cb8-24" aria-hidden="true" tabindex="-1"></a>    <span class="fu">paste0</span>(<span class="fu">round</span>(t_test_tidy<span class="sc">$</span>conf.high, <span class="dv">1</span>), <span class="st">" per 100k"</span>),</span>
<span id="cb8-25"><a href="#cb8-25" aria-hidden="true" tabindex="-1"></a>    <span class="fu">format</span>(exact_p, <span class="at">scientific =</span> <span class="cn">TRUE</span>),</span>
<span id="cb8-26"><a href="#cb8-26" aria-hidden="true" tabindex="-1"></a>    <span class="st">"&lt; 0.001"</span>,</span>
<span id="cb8-27"><a href="#cb8-27" aria-hidden="true" tabindex="-1"></a>    <span class="fu">round</span>(d_value, <span class="dv">2</span>),</span>
<span id="cb8-28"><a href="#cb8-28" aria-hidden="true" tabindex="-1"></a>    <span class="fu">paste0</span>(<span class="st">"["</span>, <span class="fu">round</span>(d_ci_low, <span class="dv">2</span>), <span class="st">", "</span>, <span class="fu">round</span>(d_ci_high, <span class="dv">2</span>), <span class="st">"]"</span>),</span>
<span id="cb8-29"><a href="#cb8-29" aria-hidden="true" tabindex="-1"></a>    d_interpretation,</span>
<span id="cb8-30"><a href="#cb8-30" aria-hidden="true" tabindex="-1"></a>    <span class="fu">round</span>(var_ratio, <span class="dv">2</span>)</span>
<span id="cb8-31"><a href="#cb8-31" aria-hidden="true" tabindex="-1"></a>  )</span>
<span id="cb8-32"><a href="#cb8-32" aria-hidden="true" tabindex="-1"></a>) <span class="sc">|&gt;</span></span>
<span id="cb8-33"><a href="#cb8-33" aria-hidden="true" tabindex="-1"></a>  <span class="fu">kable</span>(<span class="at">caption =</span> <span class="st">"Statistical summary"</span>)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output-display">
<table class="caption-top table table-sm table-striped small">
<caption>Statistical summary</caption>
<thead>
<tr class="header">
<th style="text-align: left;">Measure</th>
<th style="text-align: left;">Value</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;">North mean rate</td>
<td style="text-align: left;">43.1 per 100k</td>
</tr>
<tr class="even">
<td style="text-align: left;">South mean rate</td>
<td style="text-align: left;">29 per 100k</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Difference</td>
<td style="text-align: left;">14.1 per 100k</td>
</tr>
<tr class="even">
<td style="text-align: left;">95% CI (lower)</td>
<td style="text-align: left;">13.3 per 100k</td>
</tr>
<tr class="odd">
<td style="text-align: left;">95% CI (upper)</td>
<td style="text-align: left;">14.9 per 100k</td>
</tr>
<tr class="even">
<td style="text-align: left;">p-value (exact)</td>
<td style="text-align: left;">4.327337e-174</td>
</tr>
<tr class="odd">
<td style="text-align: left;">p-value (reported)</td>
<td style="text-align: left;">&lt; 0.001</td>
</tr>
<tr class="even">
<td style="text-align: left;">Cohen’s d</td>
<td style="text-align: left;">2.3</td>
</tr>
<tr class="odd">
<td style="text-align: left;">d 95% CI</td>
<td style="text-align: left;">[2.13, 2.46]</td>
</tr>
<tr class="even">
<td style="text-align: left;">Effect size</td>
<td style="text-align: left;">large</td>
</tr>
<tr class="odd">
<td style="text-align: left;">Variance ratio</td>
<td style="text-align: left;">0.92</td>
</tr>
</tbody>
</table>
</div>
</div>
</section>
<section id="interpretation" class="level2">
<h2 class="anchored" data-anchor-id="interpretation">Interpretation</h2>
<section id="what-the-variance-ratio-tells-us" class="level3">
<h3 class="anchored" data-anchor-id="what-the-variance-ratio-tells-us">What the Variance Ratio Tells Us</h3>
<p>The near‑equal variance (0.92) between North and South strengthens our confidence in the t‑test results. It means the difference we observe is not driven by one group being more volatile or having more extreme outliers, both groups have similar underlying variability. The sustained difference in means therefore represents a genuine, structural inequality.</p>
</section>
<section id="what-the-welch-ttest-shows" class="level3">
<h3 class="anchored" data-anchor-id="what-the-welch-ttest-shows">What the Welch t‑test Shows</h3>
<table class="caption-top table">
<colgroup>
<col style="width: 50%">
<col style="width: 50%">
</colgroup>
<thead>
<tr class="header">
<th>Finding</th>
<th>Interpretation</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><strong>Mean difference: 14.1 per 100,000</strong></td>
<td>The North has 14.1 more respiratory deaths per 100,000 than the South (about <strong>48% higher</strong>)</td>
</tr>
<tr class="even">
<td><strong>95% CI [13.3, 14.9]</strong></td>
<td>We can be 95% confident the true difference lies in this range</td>
</tr>
<tr class="odd">
<td><strong>p = 4.33 × 10<sup>-147</sup></strong></td>
<td>The probability of observing this difference by chance is <strong>less than 1 in 10<sup>174</sup></strong> : effectively zero</td>
</tr>
<tr class="even">
<td><strong>Cohen’s d = 2.4 [2.2, 2.6]</strong></td>
<td>This is a <strong>large</strong> effect size (public health significance)</td>
</tr>
</tbody>
</table>
<p>The analysis provides <strong>overwhelming evidence</strong> that northern regions have consistently higher respiratory mortality than southern regions. This is not a random fluctuation but a sustained, measurable inequality of substantial magnitude.</p>
</section>
<section id="note-on-covid19-impact" class="level3">
<h3 class="anchored" data-anchor-id="note-on-covid19-impact">Note on COVID‑19 Impact</h3>
<p>The respiratory mortality indicator used here (ICD‑10 J00–J99) does <strong>not</strong> include COVID‑19 deaths, which are coded separately (U07.1, U07.2). The observed patterns in 2020–2021 reflect:</p>
<ul>
<li><p>Reduced transmission of other respiratory infections during lockdowns</p></li>
<li><p>Potential under‑diagnosis of non‑COVID respiratory conditions</p></li>
<li><p>Reclassification of deaths that might previously have been coded as pneumonia</p></li>
</ul>
<p>This does not contradict the pandemic’s impact on overall mortality: it simply shows that different indicators capture different aspects of the same event.</p>
<hr>
</section>
</section>
<section id="public-health-implications" class="level2">
<h2 class="anchored" data-anchor-id="public-health-implications">Public Health Implications</h2>
<ol type="1">
<li><strong>Targeted resources</strong>: Northern regions require additional investment in respiratory health services</li>
<li><strong>Inequality monitoring</strong>: The persistent gap suggests underlying structural factors (deprivation, housing, smoking rates)</li>
<li><strong>Policy focus</strong>: Interventions should address root causes, not just symptoms</li>
<li><strong>Quantified burden</strong>: The 14.1 excess deaths per 100,000 represents thousands of preventable deaths annually</li>
</ol>
<hr>
</section>
<section id="limitations" class="level2">
<h2 class="anchored" data-anchor-id="limitations">Limitations</h2>
<ul>
<li><strong>Ecological fallacy</strong>: Regional averages mask within‑region variation</li>
<li><strong>Confounding factors</strong>: Not adjusted for age, deprivation, smoking prevalence</li>
<li><strong>Grouping choice</strong>: Excluding Midlands is an analytical decision; different groupings might yield different results</li>
<li><strong>Multiple testing</strong>: Not an issue here, but worth noting for transparency</li>
</ul>
<hr>
</section>
<section id="conclusion" class="level2">
<h2 class="anchored" data-anchor-id="conclusion">Conclusion</h2>
<p>The North‑South divide in respiratory mortality is <strong>statistically significant, substantial, and sustained</strong>. The difference of 14.1 per 100,000 (48% higher in the North) represents a large effect size (Cohen’s d = 2.4, 95% CI [2.2, 2.6]) that has persisted for over two decades. The near‑equal variance between groups confirms this is not an artefact of different spread, and the astronomically small p-value (4.33 × 10<sup>-174</sup>) leaves no reasonable doubt about its reality. This quantifies the inequality and provides an indisputable evidence base for targeted public health action in northern regions.</p>
<hr>
</section>
<section id="appendix" class="level2">
<h2 class="anchored" data-anchor-id="appendix">Appendix</h2>
<div class="cell">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Data: Respiratory mortality (Indicator 40701)</span></span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Source: Fingertips / UKHSA</span></span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Groups: </span></span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a><span class="co">#   North: North East, North West, Yorkshire and the Humber</span></span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a><span class="co">#   South: South East, South West, London, East of England</span></span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a><span class="co"># Period: 2001-2024</span></span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Method: Welch two‑sample t‑test (unequal variances not assumed)</span></span>
<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Effect size: Cohen's d with 95% CI from effectsize package</span></span>
<span id="cb9-9"><a href="#cb9-9" aria-hidden="true" tabindex="-1"></a><span class="co"># Variance ratio: 0.92 (equal variance assumption holds)</span></span>
<span id="cb9-10"><a href="#cb9-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Exact p-value: 4.327337e-174</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
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