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<h1 class="title">tooth-growth-analysis</h1>
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<div class="quarto-title-meta-heading">Author</div>
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<p>Mephistopheles-0 </p>
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<section id="a-basic-statistical-analysis-of-the-toothgrowth-dataset-the-effect-of-vitamin-c-on-tooth-growth-in-guinea-pigs" class="level1">
<h1>A Basic Statistical Analysis of the ToothGrowth Dataset (The Effect of Vitamin C on Tooth Growth in Guinea Pigs)</h1>
<section id="overview" class="level2">
<h2 class="anchored" data-anchor-id="overview">Overview</h2>
<p>We’re going to analyze the ToothGrowth data in the R datasets package by:</p>
<ul>
<li><p>Load the ToothGrowth data and perform some basic exploratory data analyses</p></li>
<li><p>Provide a basic summary of the data.</p></li>
<li><p>Use confidence intervals and/or hypothesis tests to compare tooth growth by supp and dose. (Only use the techniques from class, even if there’s other approaches worth considering)</p></li>
<li><p>State your conclusions and the assumptions needed for your conclusions.</p></li>
</ul>
</section>
<section id="about-the-data" class="level2">
<h2 class="anchored" data-anchor-id="about-the-data">About the data</h2>
<p>The response is the length of odontoblasts (cells responsible for tooth growth) in 60 guinea pigs. Each animal received one of three dose levels of vitamin C (0.5, 1, and 2 mg/day) by one of two delivery methods, orange juice or ascorbic acid (a form of vitamin C and coded as <code>VC</code>).</p>
</section>
<section id="loading-the-dataset-and-the-ggplot-library" class="level2">
<h2 class="anchored" data-anchor-id="loading-the-dataset-and-the-ggplot-library">Loading the dataset and the ggplot library</h2>
<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"># Loading the ToothGrowth dataset</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(ToothGrowth)</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(ggplot2)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</section>
<section id="basic-infos" class="level2">
<h2 class="anchored" data-anchor-id="basic-infos">Basic infos</h2>
<div class="cell">
<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><span class="co"># Printing the structure of the dataset</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="fu">str</span>(ToothGrowth)</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>'data.frame': 60 obs. of 3 variables:
$ len : num 4.2 11.5 7.3 5.8 6.4 10 11.2 11.2 5.2 7 ...
$ supp: Factor w/ 2 levels "OJ","VC": 2 2 2 2 2 2 2 2 2 2 ...
$ dose: num 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 ...</code></pre>
</div>
</div>
<div class="cell">
<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"># summary for the Supplement type</span></span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(ToothGrowth<span class="sc">$</span>supp)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>OJ VC
30 30 </code></pre>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="co"># summary for the Dose</span></span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(ToothGrowth<span class="sc">$</span>dose)</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> Min. 1st Qu. Median Mean 3rd Qu. Max.
0.500 0.500 1.000 1.167 2.000 2.000 </code></pre>
</div>
</div>
<div class="cell">
<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><span class="co"># Displaying the first few rows of the dataset</span></span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span>(ToothGrowth)</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> len supp dose
1 4.2 VC 0.5
2 11.5 VC 0.5
3 7.3 VC 0.5
4 5.8 VC 0.5
5 6.4 VC 0.5
6 10.0 VC 0.5</code></pre>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Basic statistics summary</span></span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(ToothGrowth)</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> len supp dose
Min. : 4.20 OJ:30 Min. :0.500
1st Qu.:13.07 VC:30 1st Qu.:0.500
Median :19.25 Median :1.000
Mean :18.81 Mean :1.167
3rd Qu.:25.27 3rd Qu.:2.000
Max. :33.90 Max. :2.000 </code></pre>
</div>
</div>
</section>
<section id="some-dataviz" class="level2">
<h2 class="anchored" data-anchor-id="some-dataviz">Some DataViz</h2>
<div class="cell">
<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Boxplot for the variable 'len'</span></span>
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a><span class="fu">boxplot</span>(ToothGrowth<span class="sc">$</span>len, </span>
<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a> <span class="at">main=</span><span class="st">"Boxplot of Length"</span>, </span>
<span id="cb12-4"><a href="#cb12-4" aria-hidden="true" tabindex="-1"></a> <span class="at">xlab=</span><span class="st">"Supplement"</span>, </span>
<span id="cb12-5"><a href="#cb12-5" aria-hidden="true" tabindex="-1"></a> <span class="at">ylab=</span><span class="st">"Length"</span>, </span>
<span id="cb12-6"><a href="#cb12-6" aria-hidden="true" tabindex="-1"></a> <span class="at">col=</span><span class="fu">c</span>(<span class="st">"orange"</span>,<span class="st">"green"</span>,<span class="st">"blue"</span>), </span>
<span id="cb12-7"><a href="#cb12-7" aria-hidden="true" tabindex="-1"></a> <span class="at">border=</span><span class="st">"brown"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="tooth-growth-analysis_files/figure-html/unnamed-chunk-7-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(ToothGrowth<span class="sc">$</span>len, </span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a> <span class="at">main=</span><span class="st">"Histogram of Length"</span>, </span>
<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a> <span class="at">xlab=</span><span class="st">"Length"</span>, </span>
<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a> <span class="at">ylab=</span><span class="st">"Frequency"</span>, </span>
<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a> <span class="at">col=</span><span class="st">"skyblue"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="tooth-growth-analysis_files/figure-html/unnamed-chunk-8-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">density</span>(ToothGrowth<span class="sc">$</span>len), </span>
<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a> <span class="at">main=</span><span class="st">"Density Plot of Length"</span>, </span>
<span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a> <span class="at">xlab=</span><span class="st">"Length"</span>, </span>
<span id="cb14-4"><a href="#cb14-4" aria-hidden="true" tabindex="-1"></a> <span class="at">ylab=</span><span class="st">"Density"</span>, </span>
<span id="cb14-5"><a href="#cb14-5" aria-hidden="true" tabindex="-1"></a> <span class="at">col=</span><span class="st">"red"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="tooth-growth-analysis_files/figure-html/unnamed-chunk-9-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Q-Q plot for checking normality of the variable 'len'</span></span>
<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a><span class="fu">qqnorm</span>(ToothGrowth<span class="sc">$</span>len, </span>
<span id="cb15-3"><a href="#cb15-3" aria-hidden="true" tabindex="-1"></a> <span class="at">main=</span><span class="st">"Q-Q Plot of Length"</span>)</span>
<span id="cb15-4"><a href="#cb15-4" aria-hidden="true" tabindex="-1"></a><span class="fu">qqline</span>(ToothGrowth<span class="sc">$</span>len)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="tooth-growth-analysis_files/figure-html/unnamed-chunk-10-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></a><span class="fu">cor</span>(ToothGrowth[<span class="fu">sapply</span>(ToothGrowth, is.numeric)])</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> len dose
len 1.0000000 0.8026913
dose 0.8026913 1.0000000</code></pre>
</div>
</div>
<p>We can see that the length of the tooth and the dose of vitamin C have a high correlation.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb18"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a>plot1 <span class="ot"><-</span> <span class="fu">ggplot</span>(ToothGrowth, <span class="fu">aes</span>(<span class="at">x =</span> <span class="fu">factor</span>(dose), <span class="at">y =</span> len, <span class="at">fill =</span> <span class="fu">factor</span>(dose)))</span>
<span id="cb18-2"><a href="#cb18-2" aria-hidden="true" tabindex="-1"></a>plot1 <span class="sc">+</span> <span class="fu">geom_boxplot</span>() <span class="sc">+</span> <span class="fu">xlab</span>(<span class="st">"dosage"</span>) <span class="sc">+</span> <span class="fu">ylab</span>(<span class="st">"length"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="tooth-growth-analysis_files/figure-html/unnamed-chunk-12-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p>again, the boxplot clearly shows that as the dosage of vitamin C increases the length increases also.</p>
</section>
<section id="compare-tooth-growth-by-dose-and-supplement-type-using-confidence-intervals-and-testing." class="level2">
<h2 class="anchored" data-anchor-id="compare-tooth-growth-by-dose-and-supplement-type-using-confidence-intervals-and-testing.">Compare Tooth Growth by Dose and Supplement Type using Confidence Intervals and Testing.</h2>
<section id="the-95-confidence-interval-of-tooth-growth" class="level3">
<h3 class="anchored" data-anchor-id="the-95-confidence-interval-of-tooth-growth">The 95% Confidence Interval of Tooth Growth</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The mean of the tooth length</span></span>
<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a>mu <span class="ot"><-</span> <span class="fu">mean</span>(ToothGrowth<span class="sc">$</span>len)</span>
<span id="cb19-3"><a href="#cb19-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb19-4"><a href="#cb19-4" aria-hidden="true" tabindex="-1"></a><span class="co"># The standard deviation of the tooth length</span></span>
<span id="cb19-5"><a href="#cb19-5" aria-hidden="true" tabindex="-1"></a>sigma <span class="ot"><-</span> <span class="fu">sd</span>(ToothGrowth<span class="sc">$</span>len)</span>
<span id="cb19-6"><a href="#cb19-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb19-7"><a href="#cb19-7" aria-hidden="true" tabindex="-1"></a><span class="co"># the 95% confidence interval</span></span>
<span id="cb19-8"><a href="#cb19-8" aria-hidden="true" tabindex="-1"></a>interval <span class="ot"><-</span> mu <span class="sc">+</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">1</span>,<span class="dv">1</span>) <span class="sc">*</span> <span class="fu">qnorm</span>(<span class="fl">0.975</span>) <span class="sc">*</span> sigma <span class="sc">/</span> <span class="fu">sqrt</span>(<span class="fu">length</span>(ToothGrowth))</span>
<span id="cb19-9"><a href="#cb19-9" aria-hidden="true" tabindex="-1"></a>interval</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] 10.15748 27.46919</code></pre>
</div>
</div>
</section>
<section id="the-t-test-by-dose-and-supp" class="level3">
<h3 class="anchored" data-anchor-id="the-t-test-by-dose-and-supp">The T-Test by dose and supp</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="fu">t.test</span>(len <span class="sc">~</span> supp, <span class="at">data =</span> ToothGrowth)</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>
Welch Two Sample t-test
data: len by supp
t = 1.9153, df = 55.309, p-value = 0.06063
alternative hypothesis: true difference in means between group OJ and group VC is not equal to 0
95 percent confidence interval:
-0.1710156 7.5710156
sample estimates:
mean in group OJ mean in group VC
20.66333 16.96333 </code></pre>
</div>
</div>
<p>thus, we can’t reject the null hypothesis .</p>
</section>
<section id="t-test-of-the-dose-level." class="level3">
<h3 class="anchored" data-anchor-id="t-test-of-the-dose-level.">T-Test of the Dose level.</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb23"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a><span class="co"># create subsets of small, medium, and large dosage</span></span>
<span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a>dose1 <span class="ot"><-</span> <span class="fu">subset</span>(ToothGrowth, dose <span class="sc">==</span> <span class="fl">0.5</span>)</span>
<span id="cb23-3"><a href="#cb23-3" aria-hidden="true" tabindex="-1"></a>dose2 <span class="ot"><-</span> <span class="fu">subset</span>(ToothGrowth, dose <span class="sc">==</span> <span class="dv">1</span>)</span>
<span id="cb23-4"><a href="#cb23-4" aria-hidden="true" tabindex="-1"></a>dose3 <span class="ot"><-</span> <span class="fu">subset</span>(ToothGrowth, dose <span class="sc">==</span> <span class="dv">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<section id="t-test-for-small-dasage" class="level4">
<h4 class="anchored" data-anchor-id="t-test-for-small-dasage">T-Test for small dasage</h4>
<div class="cell">
<div class="sourceCode cell-code" id="cb24"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="fu">t.test</span>(len<span class="sc">~</span>supp, <span class="at">data =</span> dose1)</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>
Welch Two Sample t-test
data: len by supp
t = 3.1697, df = 14.969, p-value = 0.006359
alternative hypothesis: true difference in means between group OJ and group VC is not equal to 0
95 percent confidence interval:
1.719057 8.780943
sample estimates:
mean in group OJ mean in group VC
13.23 7.98 </code></pre>
</div>
</div>
</section>
<section id="t-test-for-medium-dosage" class="level4">
<h4 class="anchored" data-anchor-id="t-test-for-medium-dosage">T-Test for medium dosage</h4>
<div class="cell">
<div class="sourceCode cell-code" id="cb26"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a><span class="fu">t.test</span>(len<span class="sc">~</span>supp, <span class="at">data =</span> dose2)</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>
Welch Two Sample t-test
data: len by supp
t = 4.0328, df = 15.358, p-value = 0.001038
alternative hypothesis: true difference in means between group OJ and group VC is not equal to 0
95 percent confidence interval:
2.802148 9.057852
sample estimates:
mean in group OJ mean in group VC
22.70 16.77 </code></pre>
</div>
</div>
</section>
<section id="t-test-for-large-dosage" class="level4">
<h4 class="anchored" data-anchor-id="t-test-for-large-dosage">T-Test for large dosage</h4>
<div class="cell">
<div class="sourceCode cell-code" id="cb28"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a><span class="fu">t.test</span>(len<span class="sc">~</span>supp, <span class="at">data =</span> dose3)</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>
Welch Two Sample t-test
data: len by supp
t = -0.046136, df = 14.04, p-value = 0.9639
alternative hypothesis: true difference in means between group OJ and group VC is not equal to 0
95 percent confidence interval:
-3.79807 3.63807
sample estimates:
mean in group OJ mean in group VC
26.06 26.14 </code></pre>
</div>
</div>
<ul>
<li><p>for the 1st and the 2nd tests we have quietly small p-value, so we can reject the null hypothesis.</p></li>
<li><p>for the 3rd test, we have a high p-value of 0. 9639, thus we cannot reject the null hypothesis.</p></li>
</ul>
</section>
</section>
</section>
<section id="conclusion" class="level2">
<h2 class="anchored" data-anchor-id="conclusion">Conclusion</h2>
<ul>
<li><p>The tooth length growth has nothing to do with the supplements.</p></li>
<li><p>for the 0.5 and 1 dosages, Orange Juice (OJ) has a higher effect on the tooth length comparing the Vitamin C (VC).</p></li>
<li><p>for the dosage 2, there is no such difference between the OJ and VC supplements effect.</p></li>
</ul>
</section>
</section>
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