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<h1 class="title toc-ignore">Diamonds</h1>
<h4 class="author">Stuart Leeds</h4>
<h4 class="date">19/03/2022</h4>
</div>
<div id="introduction" class="section level1">
<h1>Introduction</h1>
<p>Are diamonds really a girls best friend? Are they even forever? They
are certainly expensive; and they last a long time. Diamonds are the
hardest known natural material and are developed deep underground over
millennia of immense pressure and temperature. See the <a
href="https://en.wikipedia.org/w/index.php?title=Diamond&oldid=1073530080"><em>Wikipedia</em></a>
entry for general information about diamonds.</p>
<p>The purpose of this short article is to analyse a dataset of diamond
information using various <code>R</code> packages. We are interested in
the descriptive statistics of the dataset; the correlation between price
and the main variables; and whether the relationship between carat and
the price of diamonds is mediated by cut, color and clarity.</p>
</div>
<div id="data" class="section level1">
<h1>Data</h1>
<p>The <code>diamonds</code> data are found in <code>ggplot2</code>
<span class="citation">(<a href="#ref-R-ggplot2">Wickham et al.,
2024</a>)</span> and consist of almost 54,000 round-cut diamonds with
the following variables:</p>
<blockquote>
<ol style="list-style-type: decimal">
<li><p><strong>Carat</strong> <em>(weight of the diamond):</em> 0.2 to
5.01ct.</p></li>
<li><p><strong>Cut</strong> <em>(quality of the cut):</em> Fair, Good,
Very Good, Premium, Ideal.</p></li>
<li><p><strong>Color:</strong> [Best] D, E, F, G, H, I, J
[Worst].</p></li>
<li><p><strong>Clarity</strong> <em>(how clear the diamond is):</em>
[Worst] I1, SI2, SI1, VS2, VS1, VVS2, VVS1, IF [Best].</p></li>
<li><p><strong>Depth</strong> <em>(total depth percentage):</em> 43 to
79%, calculated with either <span
class="math display">\[\frac{z}{mean(x, y)}\times 100 \quad
\text{or}\quad \frac{2
\times z}{(x + y)}\times 100\]</span></p></li>
<li><p><strong>Table</strong> <em>(width of top of diamond relative to
widest point [percentage of diameter]):</em> 43 to 95%.</p></li>
<li><p><strong>Price</strong> <em>(US$):</em> 326 to 18823.</p>
<p>The variables below are used to calculate <em>Depth (Item 5)</em>,
and are not used in any analysis in this article:<br />
<strong>x</strong> <em>(length):</em> 0 to 10.74mm.<br />
<strong>y</strong> <em>(width):</em> 0 to 58.9mm.<br />
<strong>z</strong> <em>(depth):</em> 0 to 31.8mm.</p></li>
</ol>
<p><small><em>information from <code>?diamonds</code></em>.</small></p>
</blockquote>
<table class="kable_wrapper lightable-classic table" style="font-family: "Arial Narrow", "Source Sans Pro", sans-serif; margin-left: auto; margin-right: auto; margin-left: auto; margin-right: auto;border-bottom: 0;">
<thead>
<tr>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: left; " colspan="1">
<div style="border-bottom: 1px solid #111111; margin-bottom: -1px; ">
<span class="math inline">\(Cut\)</span>
</div>
</th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: left; " colspan="1">
<div style="border-bottom: 1px solid #111111; margin-bottom: -1px; ">
<span class="math inline">\(Color\)</span>
</div>
</th>
<th style="padding-bottom:0; padding-left:3px;padding-right:3px;text-align: left; " colspan="1">
<div style="border-bottom: 1px solid #111111; margin-bottom: -1px; ">
<span class="math inline">\(Clarity\)</span>
</div>
</th>
</tr>
</thead>
<caption>
<b>Table 1.</b><br><i>Number of Occurances and Average Price for each
Categorical Variable</i>
</caption>
<tbody>
<tr>
<td>
<table>
<thead>
<tr>
<th style="text-align:left;">
</th>
<th style="text-align:right;">
N
</th>
<th style="text-align:left;">
Percent
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
Fair
</td>
<td style="text-align:right;">
1,610
</td>
<td style="text-align:left;">
3.0%
</td>
</tr>
<tr>
<td style="text-align:left;">
Good
</td>
<td style="text-align:right;">
4,906
</td>
<td style="text-align:left;">
9.1%
</td>
</tr>
<tr>
<td style="text-align:left;">
Very Good
</td>
<td style="text-align:right;">
12,082
</td>
<td style="text-align:left;">
22.4%
</td>
</tr>
<tr>
<td style="text-align:left;">
Premium
</td>
<td style="text-align:right;">
13,791
</td>
<td style="text-align:left;">
25.6%
</td>
</tr>
<tr>
<td style="text-align:left;">
Ideal
</td>
<td style="text-align:right;">
21,551
</td>
<td style="text-align:left;">
40.0%
</td>
</tr>
</tbody>
</table>
</td>
<td>
<table>
<thead>
<tr>
<th style="text-align:left;">
</th>
<th style="text-align:right;">
N
</th>
<th style="text-align:left;">
Percent
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
D
</td>
<td style="text-align:right;">
6,775
</td>
<td style="text-align:left;">
12.6%
</td>
</tr>
<tr>
<td style="text-align:left;">
E
</td>
<td style="text-align:right;">
9,797
</td>
<td style="text-align:left;">
18.2%
</td>
</tr>
<tr>
<td style="text-align:left;">
F
</td>
<td style="text-align:right;">
9,542
</td>
<td style="text-align:left;">
17.7%
</td>
</tr>
<tr>
<td style="text-align:left;">
G
</td>
<td style="text-align:right;">
11,292
</td>
<td style="text-align:left;">
20.9%
</td>
</tr>
<tr>
<td style="text-align:left;">
H
</td>
<td style="text-align:right;">
8,304
</td>
<td style="text-align:left;">
15.4%
</td>
</tr>
<tr>
<td style="text-align:left;">
I
</td>
<td style="text-align:right;">
5,422
</td>
<td style="text-align:left;">
10.1%
</td>
</tr>
<tr>
<td style="text-align:left;">
J
</td>
<td style="text-align:right;">
2,808
</td>
<td style="text-align:left;">
5.2%
</td>
</tr>
</tbody>
</table>
</td>
<td>
<table>
<thead>
<tr>
<th style="text-align:left;">
</th>
<th style="text-align:right;">
N
</th>
<th style="text-align:left;">
Percent
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
I1
</td>
<td style="text-align:right;">
741
</td>
<td style="text-align:left;">
1.4%
</td>
</tr>
<tr>
<td style="text-align:left;">
SI2
</td>
<td style="text-align:right;">
9,194
</td>
<td style="text-align:left;">
17.0%
</td>
</tr>
<tr>
<td style="text-align:left;">
SI1
</td>
<td style="text-align:right;">
13,065
</td>
<td style="text-align:left;">
24.2%
</td>
</tr>
<tr>
<td style="text-align:left;">
VS2
</td>
<td style="text-align:right;">
12,258
</td>
<td style="text-align:left;">
22.7%
</td>
</tr>
<tr>
<td style="text-align:left;">
VS1
</td>
<td style="text-align:right;">
8,171
</td>
<td style="text-align:left;">
15.1%
</td>
</tr>
<tr>
<td style="text-align:left;">
VVS2
</td>
<td style="text-align:right;">
5,066
</td>
<td style="text-align:left;">
9.4%
</td>
</tr>
<tr>
<td style="text-align:left;">
VVS1
</td>
<td style="text-align:right;">
3,655
</td>
<td style="text-align:left;">
6.8%
</td>
</tr>
<tr>
<td style="text-align:left;">
IF
</td>
<td style="text-align:right;">
1,790
</td>
<td style="text-align:left;">
3.3%
</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
<tfoot>
<tr>
<td style="padding: 0; " colspan="100%">
<span style="font-style: italic;"><strong>Note. </strong> </span>
<sup></sup> <em>Color</em>- D (Absolutely colorless or icy-white); E
& F (Colorless); G - J (Near colorless). <sup></sup>
<em>Clarity</em>- I1 - I3 (Included); SI2 & SI1 (Slightly included);
VS2 & VS1 (Very slightly included); VVS2 & VVS1 (Very very
slightly included); IF (Internally flawless); [FL (Flawless), not listed
here]. <sup></sup> See <em>Brilliant Earth</em> for further details of
variable identification (ID) categories: <a
href="https://www.brilliantearth.com/diamond-cuts/"><em>Cut</em></a>; <a
href="https://www.brilliantearth.com/diamond-color/"><em>Color</em></a>;
<a
href="https://www.brilliantearth.com/diamond-clarity/"><em>Clarity</em></a>
</td>
</tr>
</tfoot>
</table>
<p><br />
</p>
<div class="figure" style="text-align: center">
<img src="diamonds_files/figure-html/show_cat_figs-1.png" alt="__Figure 1.__ _Number of Occurances for each Categorical Variable_" width="33.3%" /><img src="diamonds_files/figure-html/show_cat_figs-2.png" alt="__Figure 1.__ _Number of Occurances for each Categorical Variable_" width="33.3%" /><img src="diamonds_files/figure-html/show_cat_figs-3.png" alt="__Figure 1.__ _Number of Occurances for each Categorical Variable_" width="33.3%" />
<p class="caption">
<strong>Figure 1.</strong> <em>Number of Occurances for each Categorical
Variable</em>
</p>
</div>
</div>
<div id="method" class="section level1">
<h1>Method</h1>
</div>
<div id="results" class="section level1">
<h1>Results</h1>
<div id="correlation" class="section level2">
<h2>Correlation</h2>
<p>Correlational analysis shows the relationships between variables
whether significant, or otherwise (see Figure 2 below). As expected all
the main variables in the diamond data set are significantly correlated
with price. Of the significant positive relationships, <em>carat</em> is
the strongest (<span class="math inline">\(r=\)</span> <span
class="math inline">\(0.92\)</span>, <span class="math inline">\(p<
.001\)</span>), with weak correlations with price found for
<em>color</em> (<span class="math inline">\(r=\)</span> <span
class="math inline">\(0.17\)</span>, <span class="math inline">\(p<
.001\)</span>). The remaining variables have negative, weak, yet
significant correlations with price: <em>clarity</em> (<span
class="math inline">\(r=\)</span> <span
class="math inline">\(-0.15\)</span>, <span class="math inline">\(p<
.001\)</span>) and <em>cut</em> (<span class="math inline">\(r=\)</span>
<span class="math inline">\(-0.05\)</span>, <span
class="math inline">\(p< .001\)</span>).</p>
<div class="figure" style="text-align: center">
<img src="diamonds_files/figure-html/corr%20plot-1.png" alt="__Figure 2.__ _Correlation Plot of Diamonds Data with the Main Variables_" width="672" />
<p class="caption">
<strong>Figure 2.</strong> <em>Correlation Plot of Diamonds Data with
the Main Variables</em>
</p>
</div>
<p><br />
</p>
<p>The variance of each correlation with price can be explained with
<span class="math inline">\(r^2\)</span> (see Figure 3 below). For
example, <span class="math inline">\(85\%\)</span> of the variance in
the price of diamonds can be explained by <em>carat</em> (<span
class="math inline">\(r^2=\)</span> <span
class="math inline">\(0.85\)</span>). The same principle applies to the
other correlations: <em>color</em> (<span
class="math inline">\(r^2=\)</span> <span
class="math inline">\(0.03\)</span> explains <span
class="math inline">\(3\%\)</span> of the variance in price);
<em>clarity</em> (<span class="math inline">\(r^2=\)</span> <span
class="math inline">\(0.02\)</span>; <span
class="math inline">\(2\%\)</span>); and <em>cut</em> (<span
class="math inline">\(r^2=\)</span> <span
class="math inline">\(0.003\)</span>); <span
class="math inline">\(0.3\%\)</span>).</p>
<p><br />
</p>
<div class="figure" style="text-align: center">
<img src="diamonds_files/figure-html/correlation%20venns-1.png" alt="__Figure 3.__ _Percentage of Variance in Price (shown in green) and each Variable_" width="25%" /><img src="diamonds_files/figure-html/correlation%20venns-2.png" alt="__Figure 3.__ _Percentage of Variance in Price (shown in green) and each Variable_" width="25%" /><img src="diamonds_files/figure-html/correlation%20venns-3.png" alt="__Figure 3.__ _Percentage of Variance in Price (shown in green) and each Variable_" width="25%" /><img src="diamonds_files/figure-html/correlation%20venns-4.png" alt="__Figure 3.__ _Percentage of Variance in Price (shown in green) and each Variable_" width="25%" />
<p class="caption">
<strong>Figure 3.</strong> <em>Percentage of Variance in Price (shown in
green) and each Variable</em>
</p>
</div>
<p><br />
</p>
</div>
<div id="linear-regression" class="section level2">
<h2>Linear Regression</h2>
<table class=" lightable-classic table" style="font-family: "Arial Narrow", "Source Sans Pro", sans-serif; margin-left: auto; margin-right: auto; width: auto !important; margin-left: auto; margin-right: auto;">
<caption>
<b>Table 2.</b><br><i>Linear Regression Showing the Effect Size Between
Variables and Price.</i>
</caption>
<thead>
<tr>
<th style="text-align:left;">
<span class="math inline">\(Predictor\)</span>
</th>
<th style="text-align:left;">
<span class="math inline">\(b\)</span>
</th>
<th style="text-align:left;">
<span class="math inline">\(95\% \space CI\)</span>
</th>
<th style="text-align:left;">
<span class="math inline">\(t\)</span>
</th>
<th style="text-align:left;">
<span class="math inline">\(\mathit{df}\)</span>
</th>
<th style="text-align:left;">
<span class="math inline">\(p\)</span>
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
Intercept
</td>
<td style="text-align:left;">
-4,661.17
</td>
<td style="text-align:left;">
[-4,715.25, -4,607.10]
</td>
<td style="text-align:left;">
-168.94
</td>
<td style="text-align:left;">
53935
</td>
<td style="text-align:left;">
< .001
</td>
</tr>
<tr>
<td style="text-align:left;">
Carat
</td>
<td style="text-align:left;">
8,783.77
</td>
<td style="text-align:left;">
[8,758.90, 8,808.65]
</td>
<td style="text-align:left;">
692.09
</td>
<td style="text-align:left;">
53935
</td>
<td style="text-align:left;">
< .001
</td>
</tr>
<tr>
<td style="text-align:left;">
Cut
</td>
<td style="text-align:left;">
155.70
</td>
<td style="text-align:left;">
[146.17, 165.23]
</td>
<td style="text-align:left;">
32.01
</td>
<td style="text-align:left;">
53935
</td>
<td style="text-align:left;">
< .001
</td>
</tr>
<tr>
<td style="text-align:left;">
Color
</td>
<td style="text-align:left;">
-319.67
</td>
<td style="text-align:left;">
[-326.14, -313.20]
</td>
<td style="text-align:left;">
-96.81
</td>
<td style="text-align:left;">
53935
</td>
<td style="text-align:left;">
< .001
</td>
</tr>
<tr>
<td style="text-align:left;">
Clarity
</td>
<td style="text-align:left;">
524.84
</td>
<td style="text-align:left;">
[517.93, 531.76]
</td>
<td style="text-align:left;">
148.80
</td>
<td style="text-align:left;">
53935
</td>
<td style="text-align:left;">
< .001
</td>
</tr>
</tbody>
</table>
<p><br />
</p>
<p>A significant regression model was found (<span
class="math inline">\(F(4, 53935) = 127,219.09\)</span>, <span
class="math inline">\(p < .001\)</span>) with <span
class="math inline">\(R^2_{adj} = .90\)</span></p>
<p><br />
</p>
</div>
<div id="mediation" class="section level2">
<h2>Mediation</h2>
<div class="figure" style="text-align: center">
<img src="diamonds_files/figure-html/med%20calc-1.png" alt="__Figure 4.__ _Mediation Plot Between Carat and Price Mediated by Cut, Color and Clarity_" width="672" />
<p class="caption">
<strong>Figure 4.</strong> <em>Mediation Plot Between Carat and Price
Mediated by Cut, Color and Clarity</em>
</p>
</div>
<p><br />
</p>
</div>
<div id="discussion" class="section level2">
<h2>Discussion</h2>
<p><br />
</p>
</div>
<div id="conclusion" class="section level2">
<h2>Conclusion</h2>
<p><br />
</p>
</div>
<div id="references" class="section level2">
<h2>References</h2>
<div id="refs" class="references csl-bib-body hanging-indent"
entry-spacing="0" line-spacing="2">
<div id="ref-R-ggplot2" class="csl-entry">
Wickham, H., Chang, W., Henry, L., Pedersen, T. L., Takahashi, K.,
Wilke, C., Woo, K., Yutani, H., Dunnington, D., & van den Brand, T.
(2024). <em>ggplot2: Create elegant data visualisations using the
grammar of graphics</em>. <a
href="https://ggplot2.tidyverse.org">https://ggplot2.tidyverse.org</a>
</div>
</div>
<p><br />
</p>
</div>
</div>
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