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<li><a href="#import-and-visualize-epidemiological-data" id="toc-import-and-visualize-epidemiological-data" class="nav-link active" data-scroll-target="#import-and-visualize-epidemiological-data"><span class="toc-section-number">7.1</span> Import and visualize epidemiological data</a></li>
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<p>This section aims at providing some basic statistical tools to study the spatial distribution of epidemiological data.</p>
<section id="import-and-visualize-epidemiological-data" class="level2" data-number="7.1">
<h2 data-number="7.1" class="anchored" data-anchor-id="import-and-visualize-epidemiological-data"><span class="header-section-number">7.1</span> Import and visualize epidemiological data</h2>
<p>In this section, we load data that reference the cases of an imaginary disease throughout Cambodia. Each point correspond to the geolocalisation of a case.</p>
<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="fu">library</span>(dplyr)</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(sf)</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="co">#Import Cambodia country border</span></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>country <span class="ot"><-</span> <span class="fu">st_read</span>(<span class="st">"data_cambodia/cambodia.gpkg"</span>, <span class="at">layer =</span> <span class="st">"country"</span>, <span class="at">quiet =</span> <span class="cn">TRUE</span>)</span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a><span class="co">#Import provincial administrative border of Cambodia</span></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a>education <span class="ot"><-</span> <span class="fu">st_read</span>(<span class="st">"data_cambodia/cambodia.gpkg"</span>, <span class="at">layer =</span> <span class="st">"education"</span>, <span class="at">quiet =</span> <span class="cn">TRUE</span>)</span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a><span class="co">#Import district administrative border of Cambodia</span></span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a>district <span class="ot"><-</span> <span class="fu">st_read</span>(<span class="st">"data_cambodia/cambodia.gpkg"</span>, <span class="at">layer =</span> <span class="st">"district"</span>, <span class="at">quiet =</span> <span class="cn">TRUE</span>)</span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a><span class="co"># Import locations of cases from an imaginary disease</span></span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a>cases <span class="ot"><-</span> <span class="fu">st_read</span>(<span class="st">"data_cambodia/cambodia.gpkg"</span>, <span class="at">layer =</span> <span class="st">"cases"</span>, <span class="at">quiet =</span> <span class="cn">TRUE</span>)</span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a>cases <span class="ot"><-</span> <span class="fu">subset</span>(cases, Disease <span class="sc">==</span> <span class="st">"W fever"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p>The first step of any statistical analysis always consists on visualizing the data to check they were correctly loaded and to observe general pattern of the cases.</p>
<div class="cell" data-nm="true">
<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"># View the cases object</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span>(cases)</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 class="code-out"><code>Simple feature collection with 6 features and 2 fields
Geometry type: MULTIPOINT
Dimension: XY
Bounding box: xmin: 255891 ymin: 1179092 xmax: 506647.4 ymax: 1467441
Projected CRS: WGS 84 / UTM zone 48N
id Disease geom
1 0 W fever MULTIPOINT ((280036.2 12841...
2 1 W fever MULTIPOINT ((451859.5 11790...
3 2 W fever MULTIPOINT ((255891 1467441))
4 5 W fever MULTIPOINT ((506647.4 12322...
5 6 W fever MULTIPOINT ((440668 1197958))
6 7 W fever MULTIPOINT ((481594.5 12714...</code></pre>
</div>
<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"># Map the cases</span></span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(mapsf)</span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_map</span>(<span class="at">x =</span> district, <span class="at">border =</span> <span class="st">"white"</span>)</span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_map</span>(<span class="at">x =</span> country,<span class="at">lwd =</span> <span class="dv">2</span>, <span class="at">col =</span> <span class="cn">NA</span>, <span class="at">add =</span> <span class="cn">TRUE</span>)</span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_map</span>(<span class="at">x =</span> cases, <span class="at">lwd =</span> .<span class="dv">5</span>, <span class="at">col =</span> <span class="st">"#990000"</span>, <span class="at">pch =</span> <span class="dv">20</span>, <span class="at">add =</span> <span class="cn">TRUE</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="07-basic_statistics_files/figure-html/cases_visualization-1.png" class="img-fluid" width="768"></p>
</div>
</div>
<p>In epidemiology, the true meaning of point is very questionable. If it usually gives the location of an observation, its not clear if this observation represents an event of interest (e.g. illness, death, …) or a person at risk (e.g. a participant that may or may not experience the disease). Considering a ratio of event compared to a population at risk is often more informative than just considering cases. Administrative divisions of countries appears as great areal units for cases aggregation since they make available data on population count and structures. In this study, we will use the district as the areal unit of the study.</p>
<div class="cell" data-nm="true">
<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"># Aggregate cases over districts</span></span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>cases <span class="ot"><-</span> <span class="fu">lengths</span>(<span class="fu">st_intersects</span>(district, cases))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p>The incidence (<span class="math inline">\(\frac{cases}{population}\)</span>) is commonly use to represent cases distribution related to population density but other indicators exists. As example, the standardized incidence ratios (SIRs) represents the deviation of observed and expected number of cases and is expressed as <span class="math inline">\(SIR = \frac{Y_i}{E_i}\)</span> with <span class="math inline">\(Y_i\)</span>, the observed number of cases and <span class="math inline">\(E_i\)</span>, the expected number of cases. In this study, we computed the expected number of cases in each district by assuming infections are homogeneously distributed across Cambodia, i.e. the incidence is the same in each district. The SIR therefore represents the deviation of incidence compared to the averaged average incidence across Cambodia.</p>
<div class="cell" data-nm="true">
<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"># Compute incidence in each district (per 100 000 population)</span></span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>incidence <span class="ot"><-</span> district<span class="sc">$</span>cases<span class="sc">/</span>district<span class="sc">$</span>T_POP <span class="sc">*</span> <span class="dv">100000</span></span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Compute the global risk</span></span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a>rate <span class="ot"><-</span> <span class="fu">sum</span>(district<span class="sc">$</span>cases)<span class="sc">/</span><span class="fu">sum</span>(district<span class="sc">$</span>T_POP)</span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Compute expected number of cases </span></span>
<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>expected <span class="ot"><-</span> district<span class="sc">$</span>T_POP <span class="sc">*</span> rate</span>
<span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Compute SIR</span></span>
<span id="cb6-11"><a href="#cb6-11" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>SIR <span class="ot"><-</span> district<span class="sc">$</span>cases <span class="sc">/</span> district<span class="sc">$</span>expected</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<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">par</span>(<span class="at">mfrow =</span> <span class="fu">c</span>(<span class="dv">1</span>, <span class="dv">3</span>))</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot number of cases using proportional symbol </span></span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_map</span>(<span class="at">x =</span> district) </span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_map</span>(</span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a> <span class="at">x =</span> district, </span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a> <span class="at">var =</span> <span class="st">"cases"</span>,</span>
<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a> <span class="at">val_max =</span> <span class="dv">50</span>,</span>
<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a> <span class="at">type =</span> <span class="st">"prop"</span>,</span>
<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a> <span class="at">col =</span> <span class="st">"#990000"</span>, </span>
<span id="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></a> <span class="at">leg_title =</span> <span class="st">"Cases"</span>)</span>
<span id="cb7-11"><a href="#cb7-11" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_layout</span>(<span class="at">title =</span> <span class="st">"Number of cases of W Fever"</span>)</span>
<span id="cb7-12"><a href="#cb7-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-13"><a href="#cb7-13" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot incidence </span></span>
<span id="cb7-14"><a href="#cb7-14" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_map</span>(<span class="at">x =</span> district,</span>
<span id="cb7-15"><a href="#cb7-15" aria-hidden="true" tabindex="-1"></a> <span class="at">var =</span> <span class="st">"incidence"</span>,</span>
<span id="cb7-16"><a href="#cb7-16" aria-hidden="true" tabindex="-1"></a> <span class="at">type =</span> <span class="st">"choro"</span>,</span>
<span id="cb7-17"><a href="#cb7-17" aria-hidden="true" tabindex="-1"></a> <span class="at">pal =</span> <span class="st">"Reds 3"</span>,</span>
<span id="cb7-18"><a href="#cb7-18" aria-hidden="true" tabindex="-1"></a> <span class="at">leg_title =</span> <span class="st">"Incidence </span><span class="sc">\n</span><span class="st">(per 100 000)"</span>)</span>
<span id="cb7-19"><a href="#cb7-19" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_layout</span>(<span class="at">title =</span> <span class="st">"Incidence of W Fever"</span>)</span>
<span id="cb7-20"><a href="#cb7-20" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-21"><a href="#cb7-21" aria-hidden="true" tabindex="-1"></a><span class="co"># Plot SIRs</span></span>
<span id="cb7-22"><a href="#cb7-22" aria-hidden="true" tabindex="-1"></a><span class="co"># create breaks and associated color palette</span></span>
<span id="cb7-23"><a href="#cb7-23" aria-hidden="true" tabindex="-1"></a>break_SIR <span class="ot"><-</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="fu">exp</span>(<span class="fu">mf_get_breaks</span>(<span class="fu">log</span>(district<span class="sc">$</span>SIR), <span class="at">nbreaks =</span> <span class="dv">8</span>, <span class="at">breaks =</span> <span class="st">"pretty"</span>)))</span>
<span id="cb7-24"><a href="#cb7-24" aria-hidden="true" tabindex="-1"></a>col_pal <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">"#273871"</span>, <span class="st">"#3267AD"</span>, <span class="st">"#6496C8"</span>, <span class="st">"#9BBFDD"</span>, <span class="st">"#CDE3F0"</span>, <span class="st">"#FFCEBC"</span>, <span class="st">"#FF967E"</span>, <span class="st">"#F64D41"</span>, <span class="st">"#B90E36"</span>)</span>
<span id="cb7-25"><a href="#cb7-25" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-26"><a href="#cb7-26" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_map</span>(<span class="at">x =</span> district,</span>
<span id="cb7-27"><a href="#cb7-27" aria-hidden="true" tabindex="-1"></a> <span class="at">var =</span> <span class="st">"SIR"</span>,</span>
<span id="cb7-28"><a href="#cb7-28" aria-hidden="true" tabindex="-1"></a> <span class="at">type =</span> <span class="st">"choro"</span>,</span>
<span id="cb7-29"><a href="#cb7-29" aria-hidden="true" tabindex="-1"></a> <span class="at">breaks =</span> break_SIR, </span>
<span id="cb7-30"><a href="#cb7-30" aria-hidden="true" tabindex="-1"></a> <span class="at">pal =</span> col_pal, </span>
<span id="cb7-31"><a href="#cb7-31" aria-hidden="true" tabindex="-1"></a> <span class="at">cex =</span> <span class="dv">2</span>,</span>
<span id="cb7-32"><a href="#cb7-32" aria-hidden="true" tabindex="-1"></a> <span class="at">leg_title =</span> <span class="st">"SIR"</span>)</span>
<span id="cb7-33"><a href="#cb7-33" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_layout</span>(<span class="at">title =</span> <span class="st">"Standardized Incidence Ratio of W Fever"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p><img src="07-basic_statistics_files/figure-html/inc_visualization-1.png" class="img-fluid" width="768"></p>
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<p>These maps illustrates the spatial heterogenity of the cases. The incidence shows how the disease vary from one district to another while the SIR highlight districts that have :</p>
<ul>
<li><p>higher risk than average (SIR > 1) when standardized for population</p></li>
<li><p>lower risk than average (SIR < 1) when standardized for population</p></li>
<li><p>average risk (SIR ~ 1) when standardized for population</p></li>
</ul>
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<p>In this example, we standardized the cases distribution for population count. This simple standardization assume that the risk of contracting the disease is similar for each person. However, assumption does not hold for all diseases and for all observed events since confounding effects can create nuisance into the interpretations (e.g. the number of childhood illness and death outcomes in a district are usually related to the age pyramid) and you should keep in mind that other standardization can be performed based on variables known to have an effect but that you don’t want to analyze (e.g. sex ratio, occupations, age pyramid).</p>
<section id="cluster-analysis" class="level2" data-number="7.2">
<h2 data-number="7.2" class="anchored" data-anchor-id="cluster-analysis"><span class="header-section-number">7.2</span> Cluster analysis</h2>
<section id="general-introduction" class="level3" data-number="7.2.1">
<h3 data-number="7.2.1" class="anchored" data-anchor-id="general-introduction"><span class="header-section-number">7.2.1</span> General introduction</h3>
<p>Why studying clusters in epidemiology ? Cluster analysis help identifying unusual patterns that occurs during a given period of time. The underlying ultimate goal of such analysis is to explain the observation of such patterns. In epidemiology, we can distinguish two types of process that would explain heterogeneity in case distribution :</p>
<ul>
<li><p>The <strong>1st order effects</strong> are the spatial variations of cases distribution caused by underlying properties of environment or the population structure itself. In such process individual get infected independently from the rest of the population. Such process includes the infection through a environment at risk as, for example, air pollution, contaminated waters or soils and UV exposition. This effect assume that the observed pattern are caused by a difference in risk intensity.</p></li>
<li><p>The <strong>2nd order effects</strong> describes process of spread, contagion and diffusion of diseases caused by interactions between individuals. This includes transmission of infectious disease by proximity, but also the transmission of non-infectious disease, for example, with the diffusion of social norms within networks. This effect assume that the observed pattern are caused by correlations or co-variations.</p></li>
</ul>
<p>No statistical methods could distinguish between these competing processes since their outcome results in similar pattern of points. The cluster analysis help describing the magnitude and the location of pattern but in no way could answer the question of why such patterns occurs. It is therefore a step that help detecting cluster for description and surveillance purpose and rising hypothesis on the underlying process that will lead further investigations.</p>
<p>Knowledge about the disease and its transmission process could orientate the choice of the methods of study. We presented in this brief tutorial two methods of cluster detection, the Moran’s I test that test for spatial independence (likely related to 2nd order effects) and the scan statistics that test for homogeneous distribution (likely related 1st order effects). It relies on epidemiologist to select the tools that best serve the studied question.</p>
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<p>In statistics, problems are usually expressed by defining two hypothesis : the null hypothesis (H0), i.e. an <em>a priori</em> hypothesis of the studied phenomenon (e.g. the situation is a random) and the alternative hypothesis (HA), e.g. the situation is not random. The main principle is to measure how likely the observed situation belong to the ensemble of situation that are possible under the H0 hypothesis.</p>
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<p>In mathematics, a probability distribution is a mathematical expression that represents what we would expect due to random chance. The choice of the probability distribution relies on the type of data you use (continuous, count, binary). In general, three distribution a used while studying disease rates, the Binomial, the Poisson and the Poisson-gamma mixture (a.k.a negative binomial) distributions.</p>
<p>Many the statistical tests assume by default that data are normally distributed. It implies that your variable is continuous and that all data could easily be represented by two parameters, the mean and the variance, i.e. each value have the same level of certainty. If many measure can be assessed under the normality assumption, this is usually not the case in epidemiology with strictly positives rates and count values that 1) does not fit the normal distribution and 2) does not provide with the same degree of certainty since variances likely differ between district due to different population size, i.e. some district have very sparse data (with high variance) while other have adequate data (with lower variance).</p>
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<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"># dataset statistics</span></span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a>m_cases <span class="ot"><-</span> <span class="fu">mean</span>(district<span class="sc">$</span>incidence)</span>
<span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a>sd_cases <span class="ot"><-</span> <span class="fu">sd</span>(district<span class="sc">$</span>incidence)</span>
<span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb8-5"><a href="#cb8-5" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(district<span class="sc">$</span>incidence, <span class="at">probability =</span> <span class="cn">TRUE</span>, <span class="at">ylim =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="fl">0.4</span>), <span class="at">xlim =</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">5</span>, <span class="dv">16</span>), <span class="at">xlab =</span> <span class="st">"Number of cases"</span>, <span class="at">ylab =</span> <span class="st">"Probability"</span>, <span class="at">main =</span> <span class="st">"Histogram of observed incidence compared</span><span class="sc">\n</span><span class="st">to Normal and Poisson distributions"</span>)</span>
<span id="cb8-6"><a href="#cb8-6" aria-hidden="true" tabindex="-1"></a><span class="fu">curve</span>(<span class="fu">dnorm</span>(x, m_cases, sd_cases),<span class="at">col =</span> <span class="st">"blue"</span>, <span class="at">lwd =</span> <span class="dv">1</span>, <span class="at">add =</span> <span class="cn">TRUE</span>)</span>
<span id="cb8-7"><a href="#cb8-7" aria-hidden="true" tabindex="-1"></a><span class="fu">points</span>(<span class="dv">0</span><span class="sc">:</span><span class="fu">max</span>(district<span class="sc">$</span>incidence), <span class="fu">dpois</span>(<span class="dv">0</span><span class="sc">:</span><span class="fu">max</span>(district<span class="sc">$</span>incidence), m_cases),<span class="at">type =</span> <span class="st">'b'</span>, <span class="at">pch =</span> <span class="dv">20</span>, <span class="at">col =</span> <span class="st">"red"</span>, <span class="at">ylim =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="fl">0.6</span>), <span class="at">lty =</span> <span class="dv">2</span>)</span>
<span id="cb8-8"><a href="#cb8-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb8-9"><a href="#cb8-9" aria-hidden="true" tabindex="-1"></a><span class="fu">legend</span>(<span class="st">"topright"</span>, <span class="at">legend =</span> <span class="fu">c</span>(<span class="st">"Normal distribution"</span>, <span class="st">"Poisson distribution"</span>, <span class="st">"Observed distribution"</span>), <span class="at">col =</span> <span class="fu">c</span>(<span class="st">"blue"</span>, <span class="st">"red"</span>, <span class="st">"black"</span>),<span class="at">pch =</span> <span class="fu">c</span>(<span class="cn">NA</span>, <span class="dv">20</span>, <span class="cn">NA</span>), <span class="at">lty =</span> <span class="fu">c</span>(<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">1</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p><img src="07-basic_statistics_files/figure-html/distribution-1.png" class="img-fluid" width="576"></p>
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<p>In this tutorial, we used the poisson distribution in our statistical tests.</p>
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</section>
<section id="test-for-spatial-autocorrelation-morans-i-test" class="level3" data-number="7.2.2">
<h3 data-number="7.2.2" class="anchored" data-anchor-id="test-for-spatial-autocorrelation-morans-i-test"><span class="header-section-number">7.2.2</span> Test for spatial autocorrelation (Moran’s I test)</h3>
<section id="the-global-morans-i-test" class="level4" data-number="7.2.2.1">
<h4 data-number="7.2.2.1" class="anchored" data-anchor-id="the-global-morans-i-test"><span class="header-section-number">7.2.2.1</span> The global Moran’s I test</h4>
<p>A popular test for spatial autocorrelation is the Moran’s test. This test tells us whether nearby units tend to exhibit similar incidences. It ranges from -1 to +1. A value of -1 denote that units with low rates are located near other units with high rates, while a Moran’s I value of +1 indicates a concentration of spatial units exhibiting similar rates.</p>
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Moran’s I test
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<p>The Moran’s statistics is :</p>
<p><span class="math display">\[I = \frac{N}{\sum_{i=1}^N\sum_{j=1}^Nw_{ij}}\frac{\sum_{i=1}^N\sum_{j=1}^Nw_{ij}(Y_i-\bar{Y})(Y_j - \bar{Y})}{\sum_{i=1}^N(Y_i-\bar{Y})^2}\]</span> with :</p>
<li><p><span class="math inline">\(N\)</span>: the number of polygons,</p></li>
<li><p><span class="math inline">\(w_{ij}\)</span>: is a matrix of spatial weight with zeroes on the diagonal (i.e., <span class="math inline">\(w_{ii}=0\)</span>). For example, if polygons are neighbors, the weight takes the value <span class="math inline">\(1\)</span> otherwise it take the value <span class="math inline">\(0\)</span>.</p></li>
<li><p><span class="math inline">\(Y_i\)</span>: the variable of interest,</p></li>
<li><p><span class="math inline">\(\bar{Y}\)</span>: the mean value of <span class="math inline">\(Y\)</span>.</p></li>
<p>Under the Moran’s test, the statistics hypothesis are :</p>
<ul>
<li><p><strong>H0</strong> : the distribution of cases is spatially independent, i.e. <span class="math inline">\(I=0\)</span>.</p></li>
<li><p><strong>H1</strong>: the distribution of cases is spatially autocorrelated, i.e. <span class="math inline">\(I\ne0\)</span>.</p></li>
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<p>We will compute the Moran’s statistics using <code>spdep</code><span class="citation" data-cites="spdep">(<a href="references.html#ref-spdep" role="doc-biblioref">Bivand et al. 2015</a>)</span> and <code>Dcluster</code><span class="citation" data-cites="DCluster">(<a href="references.html#ref-DCluster" role="doc-biblioref">Gómez-Rubio et al. 2015</a>)</span> packages. <code>spdep</code> package provides a collection of functions to analyze spatial correlations of polygons and works with sp objects. In this example, we use <code>poly2nb()</code> and <code>nb2listw()</code>. These function respectively detect the neighboring polygons and assign weight corresponding to <span class="math inline">\(1/\#\ of\ neighbors\)</span>. <code>Dcluster</code> package provides a set of functions for the detection of spatial clusters of disease using count data.</p>
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<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="fu">library</span>(spdep) <span class="co"># Functions for creating spatial weight, spatial analysis</span></span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(DCluster) <span class="co"># Package with functions for spatial cluster analysis</span></span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a>queen_nb <span class="ot"><-</span> <span class="fu">poly2nb</span>(district) <span class="co"># Neighbors according to queen case</span></span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a>q_listw <span class="ot"><-</span> <span class="fu">nb2listw</span>(queen_nb, <span class="at">style =</span> <span class="st">'W'</span>) <span class="co"># row-standardized weights</span></span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Moran's I test</span></span>
<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a>m_test <span class="ot"><-</span> <span class="fu">moranI.test</span>(cases <span class="sc">~</span> <span class="fu">offset</span>(<span class="fu">log</span>(expected)), </span>
<span id="cb9-9"><a href="#cb9-9" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> district,</span>
<span id="cb9-10"><a href="#cb9-10" aria-hidden="true" tabindex="-1"></a> <span class="at">model =</span> <span class="st">'poisson'</span>,</span>
<span id="cb9-11"><a href="#cb9-11" aria-hidden="true" tabindex="-1"></a> <span class="at">R =</span> <span class="dv">499</span>,</span>
<span id="cb9-12"><a href="#cb9-12" aria-hidden="true" tabindex="-1"></a> <span class="at">listw =</span> q_listw,</span>
<span id="cb9-13"><a href="#cb9-13" aria-hidden="true" tabindex="-1"></a> <span class="at">n =</span> <span class="fu">length</span>(district<span class="sc">$</span>cases), <span class="co"># number of regions</span></span>
<span id="cb9-14"><a href="#cb9-14" aria-hidden="true" tabindex="-1"></a> <span class="at">S0 =</span> <span class="fu">Szero</span>(q_listw)) <span class="co"># Global sum of weights</span></span>
<span id="cb9-15"><a href="#cb9-15" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(m_test)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre class="code-out"><code>Moran's I test of spatial autocorrelation
Type of boots.: parametric
Model used when sampling: Poisson
Number of simulations: 499
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Statistic: 0.1566449
p-value : 0.012 </code></pre>
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<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">plot</span>(m_test)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p><img src="07-basic_statistics_files/figure-html/MoransI-1.png" class="img-fluid" width="768"></p>
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<p>The Moran’s statistics is here <span class="math inline">\(I =\)</span> 0.16. When comparing its value to the H0 distribution (built under 499 simulations), the probability of observing such a I value under the null hypothesis, i.e. the distribution of cases is spatially independent, is <span class="math inline">\(p_{value} =\)</span> 0.012. We therefore reject H0 with error risk of <span class="math inline">\(\alpha = 5\%\)</span>. The distribution of cases is therefore autocorrelated across districts in Cambodia.</p>
</section>
<section id="morans-i-local-test" class="level4" data-number="7.2.2.2">
<h4 data-number="7.2.2.2" class="anchored" data-anchor-id="morans-i-local-test"><span class="header-section-number">7.2.2.2</span> Moran’s I local test</h4>
<p>The global Moran’s test provides us a global statistical value informing whether autocorrelation occurs over the territory but does not inform on where does these correlation occurs, i.e. what is the locations of the clusters. To identify such cluster we can decompose the Moran’s I statistic to extract local informations of the level of correlation of each district and its neighbors. This is called the Local Moran’s I LISA statistic. Because the Local Moran’s I LISA statistic test each district for autocorrelation independently, concern are raised about multiple testing limitations that increase the Type I error (<span class="math inline">\(\alpha\)</span>) of the statistical tests. The use of local test should therefore be study in light of explore and describes clusters once the global test detected autocorrelation.</p>
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Statistical test
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<p>For each district <span class="math inline">\(i\)</span>, the Moran’s statistics is :</p>
<p><span class="math display">\[I_i = \frac{(Y_i-\bar{Y})}{\sum_{i=1}^N(Y_i-\bar{Y})^2}\sum_{j=1}^Nw_{ij}(Y_j - \bar{Y}) \text{ with } I = \sum_{i=1}^NI_i/N\]</span></p>
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<p>The <code>localmoran()</code>function from the package <code>spdep</code> treats the variable of interest as if it was normally distributed. In some cases, this assumption could be reasonable for incidence rate, especially when the areal units of analysis have sufficiently large population count suggesting that the values have similar level of variances.</p>
<div class="cell" data-nm="true">
<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>lm_test <span class="ot"><-</span> <span class="fu">localmoran</span>(<span class="at">x =</span> district<span class="sc">$</span>incidence,</span>
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a> <span class="at">listw =</span> q_listw) </span>
<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb12-4"><a href="#cb12-4" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(lm_test)</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 class="code-out"><code> Ii E.Ii Var.Ii Z.Ii
Min. :-2.23556 Min. :-8.892e-02 Min. :0.000008 Min. :-4.0207
1st Qu.:-0.04241 1st Qu.:-3.153e-03 1st Qu.:0.015478 1st Qu.:-0.3209
Median : 0.07664 Median :-2.853e-03 Median :0.085746 Median : 0.4380
Mean : 0.15664 Mean :-5.102e-03 Mean :0.256203 Mean : 0.3632
3rd Qu.: 0.28011 3rd Qu.:-4.032e-04 3rd Qu.:0.152437 3rd Qu.: 1.0690
Max. : 3.04441 Max. :-2.400e-07 Max. :5.265316 Max. : 4.7356
Pr(z != E(Ii))
Min. :0.0000022
1st Qu.:0.2244163
Median :0.4315420
Mean :0.4601943
3rd Qu.:0.6979413
Max. :0.9939022 </code></pre>
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<p><code>localmoran()</code>returns for each district fives outputs : the local moran indice <span class="math inline">\(I_i\)</span>, the is the expected value of the Moran’s I statistic under the null hypothesis <span class="math inline">\(E_i\)</span>, the variance of each local moran’s I statistics Var.Ii, the standardized deviation of the local Moran’s I statistic Z.Ii and Pr(Z>0) is the p-value.</p>
<p>A conventional way of plotting these results is to classify the district into 5 classes based on local Moran’s I outputs. Classification is performed based on a comparison of the scaled incidence in the district compared to the scaled weighted averaged incidence of it neighboring districts (computed with <code>lag.listw()</code>) :</p>
<ul>
<li><p>Districts that have higher-than-average rates in both index regions and their neighbors and showing statistically significant positive values for the local <span class="math inline">\(I_i\)</span> statistic are defined as <strong>High-High</strong> (hotspot of the disease)</p></li>
<li><p>Districts that have lower-than-average rates in both index regions and their neighbors adn showing statistically significant positive values for the local <span class="math inline">\(I_i\)</span> statistic are defined as <strong>Low-Low</strong> (coldspot of the disease).</p></li>
<li><p>Districts that have higher-than-average rates in the index regions and lower-than-average rates in their neighbors, and showing statistically significant negative values for the local <span class="math inline">\(I_i\)</span> statistic are defined as <strong>High-Low</strong>(outlier with high incidence in an area with low incidence).</p></li>
<li><p>Districts that have lower-than-average rates in the index regions and higher-than-average rates in their neighbors, and showing statistically significant negative values for the local <span class="math inline">\(I_i\)</span> statistic are defined as <strong>Low-High</strong>(outlier of low incidence in area with high incidence).</p></li>
<li><p>Districts with non-significant values for the <span class="math inline">\(I_i\)</span> statistic are defined as <strong>Non-significant</strong>.</p></li>
</ul>
<p>In this example, we adjusted for multiple testing using bonferroni correction, i.e. the p-value are multiplied by the number of tests.</p>
<div class="cell" data-nm="true">
<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="co"># create lagged local raw_rate - in other words the average of the queen neighbors value</span></span>
<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a><span class="co"># values are scaled (centered and reduced) to be compared to average</span></span>
<span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>lag_std <span class="ot"><-</span> <span class="fu">scale</span>(<span class="fu">lag.listw</span>(q_listw, <span class="at">var =</span> district<span class="sc">$</span>incidence))</span>
<span id="cb14-4"><a href="#cb14-4" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>incidence_std <span class="ot"><-</span> <span class="fu">scale</span>(district<span class="sc">$</span>incidence)</span>
<span id="cb14-5"><a href="#cb14-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb14-6"><a href="#cb14-6" aria-hidden="true" tabindex="-1"></a><span class="co"># adjust for pvalues</span></span>
<span id="cb14-7"><a href="#cb14-7" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>lm_pv <span class="ot"><-</span> <span class="fu">p.adjust</span>(lm_test[,<span class="dv">5</span>], <span class="st">"bonferroni"</span>)</span>
<span id="cb14-8"><a href="#cb14-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb14-9"><a href="#cb14-9" aria-hidden="true" tabindex="-1"></a><span class="co"># Classify local moran's outputs</span></span>
<span id="cb14-10"><a href="#cb14-10" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>lm_class <span class="ot"><-</span> <span class="cn">NA</span></span>
<span id="cb14-11"><a href="#cb14-11" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>lm_class[district<span class="sc">$</span>incidence_std <span class="sc">>=</span><span class="dv">0</span> <span class="sc">&</span> district<span class="sc">$</span>lag_std <span class="sc">>=</span><span class="dv">0</span>] <span class="ot"><-</span> <span class="st">'High-High'</span></span>
<span id="cb14-12"><a href="#cb14-12" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>lm_class[district<span class="sc">$</span>incidence_std <span class="sc"><=</span><span class="dv">0</span> <span class="sc">&</span> district<span class="sc">$</span>lag_std <span class="sc"><=</span><span class="dv">0</span>] <span class="ot"><-</span> <span class="st">'Low-Low'</span></span>
<span id="cb14-13"><a href="#cb14-13" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>lm_class[district<span class="sc">$</span>incidence_std <span class="sc"><=</span><span class="dv">0</span> <span class="sc">&</span> district<span class="sc">$</span>lag_std <span class="sc">>=</span><span class="dv">0</span>] <span class="ot"><-</span> <span class="st">'Low-High'</span></span>
<span id="cb14-14"><a href="#cb14-14" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>lm_class[district<span class="sc">$</span>incidence_std <span class="sc">>=</span><span class="dv">0</span> <span class="sc">&</span> district<span class="sc">$</span>lag_std <span class="sc"><=</span><span class="dv">0</span>] <span class="ot"><-</span> <span class="st">'High-Low'</span></span>
<span id="cb14-15"><a href="#cb14-15" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>lm_class[district<span class="sc">$</span>lm_pv <span class="sc">>=</span> <span class="fl">0.5</span>] <span class="ot"><-</span> <span class="st">'Non-significant'</span></span>
<span id="cb14-16"><a href="#cb14-16" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb14-17"><a href="#cb14-17" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>lm_class <span class="ot"><-</span> <span class="fu">factor</span>(district<span class="sc">$</span>lm_class, <span class="at">levels=</span><span class="fu">c</span>(<span class="st">"High-High"</span>, <span class="st">"Low-Low"</span>, <span class="st">"High-Low"</span>, <span class="st">"Low-High"</span>, <span class="st">"Non-significant"</span>) )</span>
<span id="cb14-18"><a href="#cb14-18" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb14-19"><a href="#cb14-19" aria-hidden="true" tabindex="-1"></a><span class="co"># create map</span></span>
<span id="cb14-20"><a href="#cb14-20" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_map</span>(<span class="at">x =</span> district,</span>
<span id="cb14-21"><a href="#cb14-21" aria-hidden="true" tabindex="-1"></a> <span class="at">var =</span> <span class="st">"lm_class"</span>,</span>
<span id="cb14-22"><a href="#cb14-22" aria-hidden="true" tabindex="-1"></a> <span class="at">type =</span> <span class="st">"typo"</span>,</span>
<span id="cb14-23"><a href="#cb14-23" aria-hidden="true" tabindex="-1"></a> <span class="at">cex =</span> <span class="dv">2</span>,</span>
<span id="cb14-24"><a href="#cb14-24" aria-hidden="true" tabindex="-1"></a> <span class="at">col_na =</span> <span class="st">"white"</span>,</span>
<span id="cb14-25"><a href="#cb14-25" aria-hidden="true" tabindex="-1"></a> <span class="co">#val_order = c("High-High", "Low-Low", "High-Low", "Low-High", "Non-significant") ,</span></span>
<span id="cb14-26"><a href="#cb14-26" aria-hidden="true" tabindex="-1"></a> <span class="at">pal =</span> <span class="fu">c</span>(<span class="st">"#6D0026"</span> , <span class="st">"#7FABD3"</span> , <span class="st">"white"</span>) , <span class="co"># "blue", "#FF755F",</span></span>
<span id="cb14-27"><a href="#cb14-27" aria-hidden="true" tabindex="-1"></a> <span class="at">leg_title =</span> <span class="st">"Clusters"</span>)</span>
<span id="cb14-28"><a href="#cb14-28" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb14-29"><a href="#cb14-29" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_layout</span>(<span class="at">title =</span> <span class="st">"Cluster using Local moran'I statistic"</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="07-basic_statistics_files/figure-html/LocalMoransI_plt-1.png" class="img-fluid" width="768"></p>
</section>
<section id="spatial-scan-statistics" class="level3" data-number="7.2.3">
<h3 data-number="7.2.3" class="anchored" data-anchor-id="spatial-scan-statistics"><span class="header-section-number">7.2.3</span> Spatial scan statistics</h3>
<p>While Moran’s indice focuses on testing for autocorrelation between neighboring polygons (under the null assumption of spatial independance), the spatial scan statistic aims at identifying an abnormal higher risk in a given region compared to the risk outside of this region (under the null assumption of homogeneous distribution). The conception of a cluster is therefore different between the two methods.</p>
<p>The function <code>kulldorff</code> from the package <code>SpatialEpi</code> <span class="citation" data-cites="SpatialEpi">(<a href="references.html#ref-SpatialEpi" role="doc-biblioref">Kim and Wakefield 2010</a>)</span> is a simple tool to implement spatial-only scan statistics. Briefly, the kulldorff scan statistics scan the area for clusters using several steps:</p>
<ol type="1">
<li><p>It create a circular window of observation by defining a single location and an associated radius of the windows varying from 0 to a large number that depends on population distribution (largest radius could includes 50% of the population).</p></li>
<li><p>It aggregates the count of events and the population at risk (or an expected count of events) inside and outside the window of observation.</p></li>
<li><p>Finally, it computes the likelihood ratio to test whether the risk is equal inside versus outside the windows (H0) or greater inside the observed window</p></li>
<li><p>These 3 steps are repeted for each location and each possible windows-radii.</p></li>
</ol>
<div class="cell" data-nm="true">
<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">library</span>(<span class="st">"SpatialEpi"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<p>The use of R spatial object is not implementes in <code>kulldorff()</code> function. It uses instead matrix of xy coordinates that represents the centroids of the districts. A given district is included into the observed circular window if its centroids falls into the circle.</p>
<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>district_xy <span class="ot"><-</span> <span class="fu">st_centroid</span>(district) <span class="sc">%>%</span> </span>
<span id="cb16-2"><a href="#cb16-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">st_coordinates</span>()</span>
<span id="cb16-3"><a href="#cb16-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb16-4"><a href="#cb16-4" aria-hidden="true" tabindex="-1"></a><span class="fu">head</span>(district_xy)</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 class="code-out"><code> X Y
1 330823.3 1464560
2 749758.3 1541787
3 468384.0 1277007
4 494548.2 1215261
5 459644.2 1194615
6 360528.3 1516339</code></pre>
</div>
</div>
<p>We can then call kulldorff function (you are strongly encourage to call <code>?kulldorff</code> to properly call the function). The <code>alpha.level</code> threshold filter for the secondary clusters that will be retained. The most-likely cluster will be saved whatever its significance.</p>
<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>kd_Wfever <span class="ot"><-</span> <span class="fu">kulldorff</span>(district_xy, </span>
<span id="cb18-2"><a href="#cb18-2" aria-hidden="true" tabindex="-1"></a> <span class="at">cases =</span> district<span class="sc">$</span>cases,</span>
<span id="cb18-3"><a href="#cb18-3" aria-hidden="true" tabindex="-1"></a> <span class="at">population =</span> district<span class="sc">$</span>T_POP,</span>
<span id="cb18-4"><a href="#cb18-4" aria-hidden="true" tabindex="-1"></a> <span class="at">expected.cases =</span> district<span class="sc">$</span>expected,</span>
<span id="cb18-5"><a href="#cb18-5" aria-hidden="true" tabindex="-1"></a> <span class="at">pop.upper.bound =</span> <span class="fl">0.5</span>, <span class="co"># include maximum 50% of the population in a windows</span></span>
<span id="cb18-6"><a href="#cb18-6" aria-hidden="true" tabindex="-1"></a> <span class="at">n.simulations =</span> <span class="dv">499</span>,</span>
<span id="cb18-7"><a href="#cb18-7" aria-hidden="true" tabindex="-1"></a> <span class="at">alpha.level =</span> <span class="fl">0.2</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="07-basic_statistics_files/figure-html/kd_test-1.png" class="img-fluid" width="576"></p>
</div>
</div>
<p>All outputs are saved into an R object, here called <code>kd_Wfever</code>. Unfortunately the package did not developed any summary and visualization of the results but we can explore the output object.</p>
<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="fu">names</span>(kd_Wfever)</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 class="code-out"><code>[1] "most.likely.cluster" "secondary.clusters" "type"
[4] "log.lkhd" "simulated.log.lkhd" </code></pre>
</div>
</div>
<p>First, we can focus on the most likely cluster and explore its characteristics.</p>
<div class="cell" data-nm="true">
<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="co"># We can see which districts (r number) belong to this cluster</span></span>
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a>kd_Wfever<span class="sc">$</span>most.likely.cluster<span class="sc">$</span>location.IDs.included</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre class="code-out"><code> [1] 48 93 66 180 133 29 194 118 50 144 31 141 3 117 22 43 142</code></pre>
</div>
<div class="sourceCode cell-code" id="cb23"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a><span class="co"># standardized incidence ratio</span></span>
<span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a>kd_Wfever<span class="sc">$</span>most.likely.cluster<span class="sc">$</span>SMR</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre class="code-out"><code>[1] 2.303106</code></pre>
</div>
<div class="sourceCode cell-code" id="cb25"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a><span class="co"># number of observed and expected cases in this cluster</span></span>
<span id="cb25-2"><a href="#cb25-2" aria-hidden="true" tabindex="-1"></a>kd_Wfever<span class="sc">$</span>most.likely.cluster<span class="sc">$</span>number.of.cases</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre class="code-out"><code>[1] 122</code></pre>
</div>
<div class="sourceCode cell-code" id="cb27"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a>kd_Wfever<span class="sc">$</span>most.likely.cluster<span class="sc">$</span>expected.cases</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre class="code-out"><code>[1] 52.97195</code></pre>
</div>
</div>
<p>17 districts belong to the cluster and its number of cases is 2.3 times higher than the expected number of case.</p>
<p>Similarly, we could study the secondary clusters. Results are saved in a list.</p>
<div class="cell" data-nm="true">
<div class="sourceCode cell-code" id="cb29"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="co"># We can see which districts (r number) belong to this cluster</span></span>
<span id="cb29-2"><a href="#cb29-2" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span>(kd_Wfever<span class="sc">$</span>secondary.clusters)</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 class="code-out"><code>[1] 1</code></pre>
</div>
<div class="sourceCode cell-code" id="cb31"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a><span class="co"># retrieve data for all secondary clusters into a table</span></span>
<span id="cb31-2"><a href="#cb31-2" aria-hidden="true" tabindex="-1"></a>df_secondary_clusters <span class="ot"><-</span> <span class="fu">data.frame</span>(<span class="at">SMR =</span> <span class="fu">sapply</span>(kd_Wfever<span class="sc">$</span>secondary.clusters, <span class="st">'[['</span>, <span class="dv">5</span>), </span>
<span id="cb31-3"><a href="#cb31-3" aria-hidden="true" tabindex="-1"></a> <span class="at">number.of.cases =</span> <span class="fu">sapply</span>(kd_Wfever<span class="sc">$</span>secondary.clusters, <span class="st">'[['</span>, <span class="dv">3</span>),</span>
<span id="cb31-4"><a href="#cb31-4" aria-hidden="true" tabindex="-1"></a> <span class="at">expected.cases =</span> <span class="fu">sapply</span>(kd_Wfever<span class="sc">$</span>secondary.clusters, <span class="st">'[['</span>, <span class="dv">4</span>),</span>
<span id="cb31-5"><a href="#cb31-5" aria-hidden="true" tabindex="-1"></a> <span class="at">p.value =</span> <span class="fu">sapply</span>(kd_Wfever<span class="sc">$</span>secondary.clusters, <span class="st">'[['</span>, <span class="dv">8</span>))</span>
<span id="cb31-6"><a href="#cb31-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb31-7"><a href="#cb31-7" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(df_secondary_clusters)</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 class="code-out"><code> SMR number.of.cases expected.cases p.value
</div>
</div>
<p>We only have one secondary cluster composed of one district.</p>
<div class="cell" data-nm="true">
<div class="sourceCode cell-code" id="cb33"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb33-1"><a href="#cb33-1" aria-hidden="true" tabindex="-1"></a><span class="co"># create empty column to store cluster informations</span></span>
<span id="cb33-2"><a href="#cb33-2" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>k_cluster <span class="ot"><-</span> <span class="cn">NA</span></span>
<span id="cb33-3"><a href="#cb33-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb33-4"><a href="#cb33-4" aria-hidden="true" tabindex="-1"></a><span class="co"># save cluster informations from kulldorff outputs</span></span>
<span id="cb33-5"><a href="#cb33-5" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>k_cluster[kd_Wfever<span class="sc">$</span>most.likely.cluster<span class="sc">$</span>location.IDs.included] <span class="ot"><-</span> <span class="st">'Most likely cluster'</span></span>
<span id="cb33-6"><a href="#cb33-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb33-7"><a href="#cb33-7" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span>(i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(kd_Wfever<span class="sc">$</span>secondary.clusters)){</span>
<span id="cb33-8"><a href="#cb33-8" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>k_cluster[kd_Wfever<span class="sc">$</span>secondary.clusters[[i]]<span class="sc">$</span>location.IDs.included] <span class="ot"><-</span> <span class="fu">paste</span>(</span>
<span id="cb33-9"><a href="#cb33-9" aria-hidden="true" tabindex="-1"></a> <span class="st">'Secondary cluster'</span>, i, <span class="at">sep =</span> <span class="st">''</span>)</span>
<span id="cb33-10"><a href="#cb33-10" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb33-11"><a href="#cb33-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb33-12"><a href="#cb33-12" aria-hidden="true" tabindex="-1"></a><span class="co">#district$k_cluster[is.na(district$k_cluster)] <- "No cluster"</span></span>
<span id="cb33-13"><a href="#cb33-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb33-14"><a href="#cb33-14" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb33-15"><a href="#cb33-15" aria-hidden="true" tabindex="-1"></a><span class="co"># create map</span></span>
<span id="cb33-16"><a href="#cb33-16" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_map</span>(<span class="at">x =</span> district,</span>
<span id="cb33-17"><a href="#cb33-17" aria-hidden="true" tabindex="-1"></a> <span class="at">var =</span> <span class="st">"k_cluster"</span>,</span>
<span id="cb33-18"><a href="#cb33-18" aria-hidden="true" tabindex="-1"></a> <span class="at">type =</span> <span class="st">"typo"</span>,</span>
<span id="cb33-19"><a href="#cb33-19" aria-hidden="true" tabindex="-1"></a> <span class="at">cex =</span> <span class="dv">2</span>,</span>
<span id="cb33-20"><a href="#cb33-20" aria-hidden="true" tabindex="-1"></a> <span class="at">col_na =</span> <span class="st">"white"</span>,</span>
<span id="cb33-21"><a href="#cb33-21" aria-hidden="true" tabindex="-1"></a> <span class="at">pal =</span> <span class="fu">mf_get_pal</span>(<span class="at">palette =</span> <span class="st">"Reds"</span>, <span class="at">n =</span> <span class="dv">3</span>)[<span class="dv">1</span><span class="sc">:</span><span class="dv">2</span>],</span>
<span id="cb33-22"><a href="#cb33-22" aria-hidden="true" tabindex="-1"></a> <span class="at">leg_title =</span> <span class="st">"Clusters"</span>)</span>
<span id="cb33-23"><a href="#cb33-23" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb33-24"><a href="#cb33-24" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_layout</span>(<span class="at">title =</span> <span class="st">"Cluster using kulldorf scan statistic"</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="07-basic_statistics_files/figure-html/plt_clusters-1.png" class="img-fluid" width="768"></p>
</div>
</div>
<div class="callout-tip callout callout-style-default callout-captioned">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon"></i>
</div>
<div class="callout-caption-container flex-fill">
To go futher …
</div>
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<div class="callout-body-container callout-body">
<p>In this example, the expected number of cases was defined using the population count but note that standardization over other variables as age could also be implemented with the <code>strata</code> parameter in the <code>kulldorff()</code> function.</p>
<p>In addition, this cluster analysis was performed solely using the spatial scan but you should keep in mind that this method of cluster detection can be implemented for spatio-temporal data as well where the cluster definition is an abnormal number of cases in a delimited spatial area and during a given period of time. The windows of observation are therefore defined for a different center, radius and period of time. You should look at the function <code>scan_ep_poisson()</code> function in the package <code>scanstatistic</code> <span class="citation" data-cites="scanstatistics">(<a href="references.html#ref-scanstatistics" role="doc-biblioref">Allévius 2018</a>)</span> for this analysis.</p>
</div>
</div>
<div id="refs" class="references csl-bib-body hanging-indent" role="doc-bibliography" style="display: none">
<div id="ref-scanstatistics" class="csl-entry" role="doc-biblioentry">
Allévius, Benjamin. 2018. <span>“Scanstatistics: Space-Time Anomaly Detection Using Scan Statistics.”</span> <em>Journal of Open Source Software</em> 3 (25): 515.
</div>
<div id="ref-spdep" class="csl-entry" role="doc-biblioentry">
Bivand, Roger, Micah Altman, Luc Anselin, Renato Assunção, Olaf Berke, Andrew Bernat, and Guillaume Blanchet. 2015. <span>“Package <span>‘Spdep’</span>.”</span> <em>The Comprehensive R Archive Network</em>.
</div>
<div id="ref-DCluster" class="csl-entry" role="doc-biblioentry">
Gómez-Rubio, Virgilio, Juan Ferrándiz-Ferragud, Antonio López-Quı́lez, et al. 2015. <span>“Package <span>‘DCluster’</span>.”</span>
</div>
<div id="ref-SpatialEpi" class="csl-entry" role="doc-biblioentry">
Kim, Albert Y, and Jon Wakefield. 2010. <span>“R Data and Methods for Spatial Epidemiology: The SpatialEpi Package.”</span> <em>Dept of Statistics, University of Washington</em>.
</div>
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