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<section id="задание-общих-параметров" class="level3">
<h3 class="anchored" data-anchor-id="задание-общих-параметров">Задание общих параметров</h3>
<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><span class="co"># Общие параметры</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>significance_level <span class="ot"><-</span> <span class="fl">0.05</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>power <span class="ot"><-</span> <span class="fl">0.2</span></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a>dropout_rate <span class="ot"><-</span> <span class="fl">0.1</span></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a><span class="co"># Квантили нормального распределения</span></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a>z_critical <span class="ot"><-</span> <span class="fu">abs</span>(<span class="fu">qnorm</span>(<span class="dv">1</span> <span class="sc">-</span> significance_level <span class="sc">/</span> <span class="dv">2</span>))</span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a>z_power <span class="ot"><-</span> <span class="fu">abs</span>(<span class="fu">qnorm</span>(power))</span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Выводим квантили</span></span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a>z_critical</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 1.959964</code></pre>
</div>
<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>z_power</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0.8416212</code></pre>
</div>
</div>
</section>
<section id="задание-1" class="level2">
<h2 class="anchored" data-anchor-id="задание-1">Задание 1</h2>
<p>Рассчитайте выборку для исследования терапевтической эквивалентности для двухпериодного cross-over дизайна. Из предыдущих исследований известно, что дисперсия составляет 20% ( <em>m = 0.20 ), а разница средних составляет −10% ( = </em>{} - _{} ). Клиницисты сообщают нам, что клинически значимая разница составляет 25% ( = 0.25 ).</p>
<p>Используйте следующую формулу: <span class="math display">\[
n_1 = n_2 = \frac{(Z_{\alpha / 2} + Z_\beta)^2 \sigma_m^2}{2 * (\delta - |\varepsilon|)^2}, \quad n = n_1 + n_2
\]</span></p>
<div class="cell">
<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><span class="co"># Параметры для расчета</span></span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>variance <span class="ot"><-</span> <span class="fl">0.20</span> <span class="co"># Дисперсия</span></span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>mean_difference <span class="ot"><-</span> <span class="fl">0.1</span> <span class="co"># Разница средних</span></span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a>clinically_significant_diff <span class="ot"><-</span> <span class="fl">0.25</span> <span class="co"># Клинически значимая разница</span></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"># Расчет размера выборки</span></span>
<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a>numerator <span class="ot"><-</span> (z_critical <span class="sc">+</span> z_power)<span class="sc">^</span><span class="dv">2</span> <span class="sc">*</span> variance</span>
<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a>denominator <span class="ot"><-</span> <span class="dv">2</span> <span class="sc">*</span> (clinically_significant_diff <span class="sc">-</span> <span class="fu">abs</span>(mean_difference))<span class="sc">^</span><span class="dv">2</span></span>
<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a>sample_size_base <span class="ot"><-</span> numerator <span class="sc">/</span> denominator</span>
<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-11"><a href="#cb5-11" aria-hidden="true" tabindex="-1"></a><span class="co"># Корректировка на drop-out rate</span></span>
<span id="cb5-12"><a href="#cb5-12" aria-hidden="true" tabindex="-1"></a>sample_size_adjusted <span class="ot"><-</span> sample_size_base <span class="sc">/</span> (<span class="dv">1</span> <span class="sc">-</span> dropout_rate)</span>
<span id="cb5-13"><a href="#cb5-13" aria-hidden="true" tabindex="-1"></a>sample_size_per_group <span class="ot"><-</span> <span class="fu">ceiling</span>(sample_size_adjusted)</span>
<span id="cb5-14"><a href="#cb5-14" aria-hidden="true" tabindex="-1"></a>total_sample_size <span class="ot"><-</span> sample_size_per_group <span class="sc">*</span> <span class="dv">2</span></span>
<span id="cb5-15"><a href="#cb5-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-16"><a href="#cb5-16" aria-hidden="true" tabindex="-1"></a><span class="co"># Вывод результатов</span></span>
<span id="cb5-17"><a href="#cb5-17" aria-hidden="true" tabindex="-1"></a><span class="fu">cat</span>(<span class="st">'Размер выборки для каждой группы:'</span>, sample_size_per_group, <span class="st">'</span><span class="sc">\n</span><span class="st">'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Размер выборки для каждой группы: 39 </code></pre>
</div>
<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><span class="fu">cat</span>(<span class="st">'Общий размер выборки:'</span>, total_sample_size, <span class="st">'</span><span class="sc">\n</span><span class="st">'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Общий размер выборки: 78 </code></pre>
</div>
</div>
</section>
<section id="задание-2" class="level2">
<h2 class="anchored" data-anchor-id="задание-2">Задание 2</h2>
<p>Рассчитайте выборку для гипотезы non-inferiority для двухвыборочного параллельного дизайна. Пусть клинически значимая разница ( = -0.1 ), то есть мы ожидаем разницу не меньше 10%, а долю ответов для тестового препарата ( p_2 = 0.65 ), в то время как нам известно из предыдущих исследований, что доля ответов у препарата сравнения составляет ( p_1 = 0.85 ). Соотношение групп равно ( k = 1 ).</p>
<p>Используйте следующую формулу: <span class="math display">\[
n_1 = n_2 = \frac{(Z_{\alpha / 2} + Z_\beta)^2 (p_1^*(1 - p_1) + p_2^*(1 - p_2))}{(p_1 - p_2 - \delta)^2}, \quad n = n_1 + n_2
\]</span></p>
<div class="cell">
<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"># Клинически значимая разница и доля ответов</span></span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a>non_inferiority_margin <span class="ot"><-</span> <span class="sc">-</span><span class="fl">0.1</span></span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a>response_rate_test <span class="ot"><-</span> <span class="fl">0.65</span></span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a>response_rate_reference <span class="ot"><-</span> <span class="fl">0.85</span></span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a><span class="co"># Расчет размера выборки</span></span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a>numerator <span class="ot"><-</span> (z_critical <span class="sc">+</span> z_power)<span class="sc">^</span><span class="dv">2</span> <span class="sc">*</span> (response_rate_reference <span class="sc">*</span> (<span class="dv">1</span> <span class="sc">-</span> response_rate_reference) <span class="sc">+</span> response_rate_test <span class="sc">*</span> (<span class="dv">1</span> <span class="sc">-</span> response_rate_test))</span>
<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a>denominator <span class="ot"><-</span> (response_rate_reference <span class="sc">-</span> response_rate_test <span class="sc">-</span> non_inferiority_margin)<span class="sc">^</span><span class="dv">2</span></span>
<span id="cb9-9"><a href="#cb9-9" aria-hidden="true" tabindex="-1"></a>sample_size_base <span class="ot"><-</span> numerator <span class="sc">/</span> denominator</span>
<span id="cb9-10"><a href="#cb9-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-11"><a href="#cb9-11" aria-hidden="true" tabindex="-1"></a><span class="co"># Корректировка на drop-out rate</span></span>
<span id="cb9-12"><a href="#cb9-12" aria-hidden="true" tabindex="-1"></a>sample_size_adjusted <span class="ot"><-</span> sample_size_base <span class="sc">/</span> (<span class="dv">1</span> <span class="sc">-</span> dropout_rate)</span>
<span id="cb9-13"><a href="#cb9-13" aria-hidden="true" tabindex="-1"></a>sample_size_per_group <span class="ot"><-</span> <span class="fu">ceiling</span>(sample_size_adjusted)</span>
<span id="cb9-14"><a href="#cb9-14" aria-hidden="true" tabindex="-1"></a>total_sample_size <span class="ot"><-</span> sample_size_per_group <span class="sc">*</span> <span class="dv">2</span></span>
<span id="cb9-15"><a href="#cb9-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-16"><a href="#cb9-16" aria-hidden="true" tabindex="-1"></a><span class="co"># Вывод результатов</span></span>
<span id="cb9-17"><a href="#cb9-17" aria-hidden="true" tabindex="-1"></a><span class="fu">cat</span>(<span class="st">'Размер выборки для каждой группы:'</span>, sample_size_per_group, <span class="st">'</span><span class="sc">\n</span><span class="st">'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Размер выборки для каждой группы: 35 </code></pre>
</div>
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="fu">cat</span>(<span class="st">'Общий размер выборки:'</span>, total_sample_size, <span class="st">'</span><span class="sc">\n</span><span class="st">'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Общий размер выборки: 70 </code></pre>
</div>
</div>
</section>
<section id="задание-3" class="level2">
<h2 class="anchored" data-anchor-id="задание-3">Задание 3</h2>
<p>Рассчитайте выборку для гипотезы equality для следующего исследования. Мы хотим сравнить новую терапию инфекции, присоединяющейся в больничных условиях у пациентов с ожогами, с золотым стандартом, основываясь на данных, анализируемых с помощью регрессии Кокса. Пусть отношение рисков «золотой стандарт / новая терапия», hazard ratio, HR = 2. Мы предполагаем, что 80% пациентов ( d = 0.8 ) могут столкнуться с этим заболеванием. Соотношения групп терапии равны ( p_1 = p_2 = 0.5 ).</p>
<p>Используйте следующую формулу: $$ n_1 = n_2 = , n = n_1 + n_2 $$</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Параметры для расчета</span></span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a>hazard_ratio <span class="ot"><-</span> <span class="dv">2</span></span>
<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a>disease_incidence <span class="ot"><-</span> <span class="fl">0.8</span> <span class="co"># Процент пациентов, сталкивающихся с заболеванием</span></span>
<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a>group_ratio <span class="ot"><-</span> <span class="fl">0.5</span> <span class="co"># Соотношение групп</span></span>
<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-6"><a href="#cb13-6" aria-hidden="true" tabindex="-1"></a><span class="co"># Расчет размера выборки</span></span>
<span id="cb13-7"><a href="#cb13-7" aria-hidden="true" tabindex="-1"></a>numerator <span class="ot"><-</span> (z_critical <span class="sc">+</span> z_power)<span class="sc">^</span><span class="dv">2</span></span>
<span id="cb13-8"><a href="#cb13-8" aria-hidden="true" tabindex="-1"></a>denominator <span class="ot"><-</span> (<span class="fu">log</span>(hazard_ratio))<span class="sc">^</span><span class="dv">2</span> <span class="sc">*</span> group_ratio <span class="sc">*</span> group_ratio <span class="sc">*</span> disease_incidence</span>
<span id="cb13-9"><a href="#cb13-9" aria-hidden="true" tabindex="-1"></a>sample_size_base <span class="ot"><-</span> numerator <span class="sc">/</span> denominator</span>
<span id="cb13-10"><a href="#cb13-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-11"><a href="#cb13-11" aria-hidden="true" tabindex="-1"></a><span class="co"># Корректировка на drop-out rate</span></span>
<span id="cb13-12"><a href="#cb13-12" aria-hidden="true" tabindex="-1"></a>sample_size_adjusted <span class="ot"><-</span> sample_size_base <span class="sc">/</span> (<span class="dv">1</span> <span class="sc">-</span> dropout_rate)</span>
<span id="cb13-13"><a href="#cb13-13" aria-hidden="true" tabindex="-1"></a>sample_size_per_group <span class="ot"><-</span> <span class="fu">ceiling</span>(sample_size_adjusted)</span>
<span id="cb13-14"><a href="#cb13-14" aria-hidden="true" tabindex="-1"></a>total_sample_size <span class="ot"><-</span> sample_size_per_group <span class="sc">*</span> <span class="dv">2</span></span>
<span id="cb13-15"><a href="#cb13-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-16"><a href="#cb13-16" aria-hidden="true" tabindex="-1"></a><span class="co"># Вывод результатов</span></span>
<span id="cb13-17"><a href="#cb13-17" aria-hidden="true" tabindex="-1"></a><span class="fu">cat</span>(<span class="st">'Размер выборки для каждой группы:'</span>, sample_size_per_group, <span class="st">'</span><span class="sc">\n</span><span class="st">'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Размер выборки для каждой группы: 91 </code></pre>
</div>
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="fu">cat</span>(<span class="st">'Общий размер выборки:'</span>, total_sample_size, <span class="st">'</span><span class="sc">\n</span><span class="st">'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>Общий размер выборки: 182 </code></pre>
</div>
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