From e849094c8fea30bbf12fe0cfe95f2a52445473d7 Mon Sep 17 00:00:00 2001
From: "lea.douchet_ird.fr" <ldouchet@hotmail.fr>
Date: Sun, 27 Nov 2022 17:27:32 +0700
Subject: [PATCH] tes hypothesis of kulldorff
---
07-basic_statistics.qmd | 36 ++-
public/07-basic_statistics.html | 228 ++++++++++--------
.../figure-html/LocalMoransI-1.png | Bin 14830 -> 0 bytes
.../figure-html/LocalMoransI_plt-1.png | Bin 44331 -> 44382 bytes
.../figure-html/MoransI-1.png | Bin 18402 -> 17140 bytes
.../figure-html/incidence_visualization-1.png | Bin 53684 -> 0 bytes
.../figure-html/kd_test-1.png | Bin 15750 -> 15735 bytes
public/search.json | 2 +-
8 files changed, 149 insertions(+), 117 deletions(-)
delete mode 100644 public/07-basic_statistics_files/figure-html/LocalMoransI-1.png
delete mode 100644 public/07-basic_statistics_files/figure-html/incidence_visualization-1.png
diff --git a/07-basic_statistics.qmd b/07-basic_statistics.qmd
index 3e8443b..758bef2 100644
--- a/07-basic_statistics.qmd
+++ b/07-basic_statistics.qmd
@@ -43,7 +43,7 @@ mf_map(x = cases, lwd = .5, col = "#990000", pch = 20, add = TRUE)
```
-In epidemiology, the true meaning of point is very questionable. If it usually gives the location of an observation, we cannot precisely tell 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 appear 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.
+In epidemiology, the true meaning of point is very questionable. If it usually gives the location of an observation, we cannot precisely tell 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). If you can consider that the population at risk is uniformly distributed in small area (a city for example), this is likely not the case at a country scale. Considering a ratio of event compared to a population at risk is often more informative than just considering cases. Administrative divisions of countries appear 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.
```{r district_aggregate, eval = TRUE, echo = TRUE, nm = TRUE, fig.width=8, class.output="code-out", warning=FALSE, message=FALSE}
# Aggregate cases over districts
@@ -213,7 +213,7 @@ plot(m_test)
The Moran's statistics is here $I =$ `r signif(m_test$t0, 2)`. When comparing its value to the H0 distribution (built under `r m_test$R` simulations), the probability of observing such a I value under the null hypothesis, i.e. the distribution of cases is spatially independent, is $p_{value} =$ `r signif(( 1+ (sum((-abs(as.numeric(m_test$t0-mean(m_test$t))))>as.numeric(m_test$t-mean(m_test$t)))) + (sum(abs(as.numeric(m_test$t0-mean(m_test$t)))<as.numeric(m_test$t-mean(m_test$t)))) )/(m_test$R+1), 2)`. We therefore reject H0 with error risk of $\alpha = 5\%$. The distribution of cases is therefore autocorrelated across districts in Cambodia.
-#### Moran's I local test
+#### The Local Moran's I LISA test
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 correlations 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 information 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 is raised about multiple testing limitations that increase the Type I error ($\alpha$) 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.
@@ -228,7 +228,6 @@ $$I_i = \frac{(Y_i-\bar{Y})}{\sum_{i=1}^N(Y_i-\bar{Y})^2}\sum_{j=1}^Nw_{ij}(Y_j
The `localmoran()`function from the package `spdep` 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. Unfortunately, the local Moran’s test has not been implemented for Poisson distribution (population not large enough in some districts) in `spdep` package. However, Bivand **et al.** [@bivand2008applied] provided some code to manual perform the analysis using Poisson distribution and was further implemented in the course "[Spatial Epidemiology](https://mkram01.github.io/EPI563-SpatialEPI/index.html)â€.
-
```{r LocalMoransI, eval = TRUE, echo = TRUE, nm = TRUE, fig.width=8, class.output="code-out", warning=FALSE, message=FALSE}
# Step 1 - Create the standardized deviation of observed from expected
@@ -259,7 +258,6 @@ for(i in 1:nsim){
sims[, i] <- sd_lmi * wsd_lmi # this is the I(i) statistic under this iteration of null
}
-hist(sims[1,])
# Step 6 - For each county, test where the observed value ranks with respect to the null simulations
xrank <- apply(cbind(district$I_lm, sims), 1, function(x) rank(x)[1])
@@ -319,8 +317,6 @@ mf_map(x = district,
mf_layout(title = "Cluster using Local Moran's I statistic")
-
-
```
@@ -329,16 +325,35 @@ mf_layout(title = "Cluster using Local Moran's I statistic")
While Moran's indices focus on testing for autocorrelation between neighboring polygons (under the null assumption of spatial independence), 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.
-The function `kulldorff` from the package `SpatialEpi` [@SpatialEpi] is a simple tool to implement spatial-only scan statistics. Briefly, the kulldorff scan statistics scan the area for clusters using several steps:
+The function `kulldorff` from the package `SpatialEpi` [@SpatialEpi] is a simple tool to implement spatial-only scan statistics.
+
+::: callout-note
+##### Kulldorf test
+
+Under the kulldorff test, the statistics hypotheses are:
+
+- **H0**: the risk is constant over the area, i.e., there is a spatial homogeneity of the incidence.
+
+- **H1**: a particular window have higher incidence than the rest of the area , i.e., there is a spatial heterogeneity of incidence.
+
+:::
+
+
+Briefly, the kulldorff scan statistics scan the area for clusters using several steps:
1. 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 include 50% of the population).
2. It aggregates the count of events and the population at risk (or an expected count of events) inside and outside the window of observation.
-3. 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
+3. Finally, it computes the likelihood ratio and test whether the risk is equal inside versus outside the windows (H0) or greater inside the observed window (H1). The H0 distribution is estimated by simulating the distribution of counts under the null hypothesis (homogeneous risk).
4. These 3 steps are repeated for each location and each possible windows-radii.
+
+While we test the significance of a large number of observation windows, one can raise concern about multiple testing and Type I error. This approach however suggest that we are not interest in a set of signifiant cluster but only in a most-likely cluster. This **a priori** restriction eliminate concern for multpile comparison since the test is simplified to a statistically significance of one single most-likely cluster.
+
+Because we tested all-possible locations and window-radius, we can also choose to look at secondary clusters. In this case, you should keep in mind that increasing the number of secondary cluster you select, increases the risk for Type I error.
+
```{r spatialEpi, eval = TRUE, echo = TRUE, nm = TRUE, class.output="code-out", warning=FALSE, message=FALSE}
library("SpatialEpi")
@@ -370,7 +385,8 @@ kd_Wfever <- kulldorff(district_xy,
```
-All outputs are saved into an R object, here called `kd_Wfever`. Unfortunately, the package did not develop any summary and visualization of the results but we can explore the output object.
+The function plot the histogram of the distribution of log-likelihood ratio simulated under the null hypothesis that is estimated based on Monte Carlo simulations. The observed value of the most significant cluster identified from all possible scans is compared to the distribution to determine significance. All outputs are saved into an R object, here called `kd_Wfever`. Unfortunately, the package did not develop any summary and visualization of the results but we can explore the output object.
+
```{r kd_outputs, eval = TRUE, echo = TRUE, nm = TRUE, fig.width=6, class.output="code-out", warning=FALSE, message=FALSE}
names(kd_Wfever)
@@ -447,7 +463,7 @@ mf_layout(title = "Cluster using kulldorf scan statistic")
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 `strata` parameter in the `kulldorff()` function.
-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 `scan_ep_poisson()` function in the package `scanstatistic` [@scanstatistics] for this analysis.
+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 time-period. You should look at the function `scan_ep_poisson()` function in the package `scanstatistic` [@scanstatistics] for this analysis.
:::
diff --git a/public/07-basic_statistics.html b/public/07-basic_statistics.html
index 6c97dbb..872ac41 100644
--- a/public/07-basic_statistics.html
+++ b/public/07-basic_statistics.html
@@ -242,7 +242,7 @@ div.csl-indent {
<li><a href="#test-for-spatial-autocorrelation-morans-i-test" id="toc-test-for-spatial-autocorrelation-morans-i-test" class="nav-link" data-scroll-target="#test-for-spatial-autocorrelation-morans-i-test"><span class="toc-section-number">7.2.2</span> Test for spatial autocorrelation (Moran’s I test)</a>
<ul class="collapse">
<li><a href="#the-global-morans-i-test" id="toc-the-global-morans-i-test" class="nav-link" data-scroll-target="#the-global-morans-i-test"><span class="toc-section-number">7.2.2.1</span> The global Moran’s I test</a></li>
- <li><a href="#morans-i-local-test" id="toc-morans-i-local-test" class="nav-link" data-scroll-target="#morans-i-local-test"><span class="toc-section-number">7.2.2.2</span> Moran’s I local test</a></li>
+ <li><a href="#the-local-morans-i-lisa-test" id="toc-the-local-morans-i-lisa-test" class="nav-link" data-scroll-target="#the-local-morans-i-lisa-test"><span class="toc-section-number">7.2.2.2</span> The Local Moran’s I LISA test</a></li>
</ul></li>
<li><a href="#spatial-scan-statistics" id="toc-spatial-scan-statistics" class="nav-link" data-scroll-target="#spatial-scan-statistics"><span class="toc-section-number">7.2.3</span> Spatial scan statistics</a></li>
</ul></li>
@@ -490,17 +490,17 @@ Moran’s I test
Model used when sampling: Poisson
Number of simulations: 499
Statistic: 0.1566449
- p-value : 0.012 </code></pre>
+ p-value : 0.01 </code></pre>
</div>
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(m_test)</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/MoransI-1.png" class="img-fluid" width="768"></p>
</div>
</div>
-<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>
+<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> 0.01. 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>
+<section id="the-local-morans-i-lisa-test" class="level4" data-number="7.2.2.2">
+<h4 data-number="7.2.2.2" class="anchored" data-anchor-id="the-local-morans-i-lisa-test"><span class="header-section-number">7.2.2.2</span> The Local Moran’s I LISA 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 correlations 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 information 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 is 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>
<div class="callout-note callout callout-style-default callout-captioned">
<div class="callout-header d-flex align-content-center">
@@ -546,20 +546,16 @@ Statistical test
<span id="cb12-26"><a href="#cb12-26" aria-hidden="true" tabindex="-1"></a> sims[, i] <span class="ot"><-</span> sd_lmi <span class="sc">*</span> wsd_lmi <span class="co"># this is the I(i) statistic under this iteration of null</span></span>
<span id="cb12-27"><a href="#cb12-27" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb12-28"><a href="#cb12-28" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb12-29"><a href="#cb12-29" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(sims[<span class="dv">1</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-1.png" class="img-fluid" width="768"></p>
-</div>
-<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Step 6 - For each county, test where the observed value ranks with respect to the null simulations</span></span>
-<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a>xrank <span class="ot"><-</span> <span class="fu">apply</span>(<span class="fu">cbind</span>(district<span class="sc">$</span>I_lm, sims), <span class="dv">1</span>, <span class="cf">function</span>(x) <span class="fu">rank</span>(x)[<span class="dv">1</span>])</span>
-<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Step 7 - Calculate the difference between observed rank and total possible (nsim)</span></span>
-<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a>diff <span class="ot"><-</span> nsim <span class="sc">-</span> xrank</span>
-<span id="cb13-6"><a href="#cb13-6" aria-hidden="true" tabindex="-1"></a>diff <span class="ot"><-</span> <span class="fu">ifelse</span>(diff <span class="sc">></span> <span class="dv">0</span>, diff, <span class="dv">0</span>)</span>
-<span id="cb13-7"><a href="#cb13-7" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb13-8"><a href="#cb13-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Step 8 - Assuming a uniform distribution of ranks, calculate p-value for observed</span></span>
-<span id="cb13-9"><a href="#cb13-9" aria-hidden="true" tabindex="-1"></a><span class="co"># given the null distribution generate from simulations</span></span>
-<span id="cb13-10"><a href="#cb13-10" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>pval_lm <span class="ot"><-</span> <span class="fu">punif</span>((diff <span class="sc">+</span> <span class="dv">1</span>) <span class="sc">/</span> (nsim <span class="sc">+</span> <span class="dv">1</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<span id="cb12-29"><a href="#cb12-29" aria-hidden="true" tabindex="-1"></a><span class="co"># Step 6 - For each county, test where the observed value ranks with respect to the null simulations</span></span>
+<span id="cb12-30"><a href="#cb12-30" aria-hidden="true" tabindex="-1"></a>xrank <span class="ot"><-</span> <span class="fu">apply</span>(<span class="fu">cbind</span>(district<span class="sc">$</span>I_lm, sims), <span class="dv">1</span>, <span class="cf">function</span>(x) <span class="fu">rank</span>(x)[<span class="dv">1</span>])</span>
+<span id="cb12-31"><a href="#cb12-31" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb12-32"><a href="#cb12-32" aria-hidden="true" tabindex="-1"></a><span class="co"># Step 7 - Calculate the difference between observed rank and total possible (nsim)</span></span>
+<span id="cb12-33"><a href="#cb12-33" aria-hidden="true" tabindex="-1"></a>diff <span class="ot"><-</span> nsim <span class="sc">-</span> xrank</span>
+<span id="cb12-34"><a href="#cb12-34" aria-hidden="true" tabindex="-1"></a>diff <span class="ot"><-</span> <span class="fu">ifelse</span>(diff <span class="sc">></span> <span class="dv">0</span>, diff, <span class="dv">0</span>)</span>
+<span id="cb12-35"><a href="#cb12-35" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb12-36"><a href="#cb12-36" aria-hidden="true" tabindex="-1"></a><span class="co"># Step 8 - Assuming a uniform distribution of ranks, calculate p-value for observed</span></span>
+<span id="cb12-37"><a href="#cb12-37" aria-hidden="true" tabindex="-1"></a><span class="co"># given the null distribution generate from simulations</span></span>
+<span id="cb12-38"><a href="#cb12-38" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>pval_lm <span class="ot"><-</span> <span class="fu">punif</span>((diff <span class="sc">+</span> <span class="dv">1</span>) <span class="sc">/</span> (nsim <span class="sc">+</span> <span class="dv">1</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>For each district, we obtain a p-value based on permutations process</p>
<p>A conventional way of plotting these results is to classify the districts into 5 classes based on local Moran’s I output. The classification of cluster that are significantly autocorrelated to their neighbors 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>
@@ -571,35 +567,35 @@ Statistical test
<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>
<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"># extract pvalues</span></span>
-<span id="cb14-7"><a href="#cb14-7" aria-hidden="true" tabindex="-1"></a><span class="co"># district$lm_pv <- lm_test[,5]</span></span>
-<span id="cb14-8"><a href="#cb14-8" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb14-9"><a href="#cb14-9" aria-hidden="true" tabindex="-1"></a><span class="co"># Classify local moran's outputs</span></span>
-<span id="cb14-10"><a href="#cb14-10" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</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>pval_lm <span class="sc">>=</span> <span class="fl">0.05</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">"blue"</span>, <span class="st">"white"</span>) , <span class="co"># "#FF755F","#7FABD3" ,</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's I statistic"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="co"># create lagged local raw_rate - in other words the average of the queen neighbors value</span></span>
+<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a><span class="co"># values are scaled (centered and reduced) to be compared to average</span></span>
+<span id="cb13-3"><a href="#cb13-3" 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="cb13-4"><a href="#cb13-4" 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="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb13-6"><a href="#cb13-6" aria-hidden="true" tabindex="-1"></a><span class="co"># extract pvalues</span></span>
+<span id="cb13-7"><a href="#cb13-7" aria-hidden="true" tabindex="-1"></a><span class="co"># district$lm_pv <- lm_test[,5]</span></span>
+<span id="cb13-8"><a href="#cb13-8" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb13-9"><a href="#cb13-9" aria-hidden="true" tabindex="-1"></a><span class="co"># Classify local moran's outputs</span></span>
+<span id="cb13-10"><a href="#cb13-10" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>lm_class <span class="ot"><-</span> <span class="cn">NA</span></span>
+<span id="cb13-11"><a href="#cb13-11" 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">$</span>lag_std <span class="sc">>=</span><span class="dv">0</span>] <span class="ot"><-</span> <span class="st">'High-High'</span></span>
+<span id="cb13-12"><a href="#cb13-12" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</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="cb13-13"><a href="#cb13-13" 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">$</span>lag_std <span class="sc">>=</span><span class="dv">0</span>] <span class="ot"><-</span> <span class="st">'Low-High'</span></span>
+<span id="cb13-14"><a href="#cb13-14" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</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="cb13-15"><a href="#cb13-15" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>lm_class[district<span class="sc">$</span>pval_lm <span class="sc">>=</span> <span class="fl">0.05</span>] <span class="ot"><-</span> <span class="st">'Non-significant'</span></span>
+<span id="cb13-16"><a href="#cb13-16" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb13-17"><a href="#cb13-17" 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="cb13-18"><a href="#cb13-18" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb13-19"><a href="#cb13-19" aria-hidden="true" tabindex="-1"></a><span class="co"># create map</span></span>
+<span id="cb13-20"><a href="#cb13-20" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_map</span>(<span class="at">x =</span> district,</span>
+<span id="cb13-21"><a href="#cb13-21" aria-hidden="true" tabindex="-1"></a> <span class="at">var =</span> <span class="st">"lm_class"</span>,</span>
+<span id="cb13-22"><a href="#cb13-22" aria-hidden="true" tabindex="-1"></a> <span class="at">type =</span> <span class="st">"typo"</span>,</span>
+<span id="cb13-23"><a href="#cb13-23" aria-hidden="true" tabindex="-1"></a> <span class="at">cex =</span> <span class="dv">2</span>,</span>
+<span id="cb13-24"><a href="#cb13-24" aria-hidden="true" tabindex="-1"></a> <span class="at">col_na =</span> <span class="st">"white"</span>,</span>
+<span id="cb13-25"><a href="#cb13-25" 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="cb13-26"><a href="#cb13-26" aria-hidden="true" tabindex="-1"></a> <span class="at">pal =</span> <span class="fu">c</span>(<span class="st">"#6D0026"</span> , <span class="st">"blue"</span>, <span class="st">"white"</span>) , <span class="co"># "#FF755F","#7FABD3" ,</span></span>
+<span id="cb13-27"><a href="#cb13-27" aria-hidden="true" tabindex="-1"></a> <span class="at">leg_title =</span> <span class="st">"Clusters"</span>)</span>
+<span id="cb13-28"><a href="#cb13-28" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb13-29"><a href="#cb13-29" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_layout</span>(<span class="at">title =</span> <span class="st">"Cluster using Local Moran's 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>
</div>
@@ -609,22 +605,42 @@ Statistical test
<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 indices focus on testing for autocorrelation between neighboring polygons (under the null assumption of spatial independence), 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>
+<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.</p>
+<div class="callout-note 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">
+Kulldorf test
+</div>
+</div>
+<div class="callout-body-container callout-body">
+<p>Under the kulldorff test, the statistics hypotheses are:</p>
+<ul>
+<li><p><strong>H0</strong>: the risk is constant over the area, i.e., there is a spatial homogeneity of the incidence.</p></li>
+<li><p><strong>H1</strong>: a particular window have higher incidence than the rest of the area , i.e., there is a spatial heterogeneity of incidence.</p></li>
+</ul>
+</div>
+</div>
+<p>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 include 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>Finally, it computes the likelihood ratio and test whether the risk is equal inside versus outside the windows (H0) or greater inside the observed window (H1). The H0 distribution is estimated by simulating the distribution of counts under the null hypothesis (homogeneous risk).</p></li>
<li><p>These 3 steps are repeated for each location and each possible windows-radii.</p></li>
</ol>
+<p>While we test the significance of a large number of observation windows, one can raise concern about multiple testing and Type I error. This approach however suggest that we are not interest in a set of signifiant cluster but only in a most-likely cluster. This <strong>a priori</strong> restriction eliminate concern for multpile comparison since the test is simplified to a statistically significance of one single most-likely cluster.</p>
+<p>Because we tested all-possible locations and window-radius, we can also choose to look at secondary clusters. In this case, you should keep in mind that increasing the number of secondary cluster you select, increases the risk for Type I error.</p>
<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>
+<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">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>
</div>
<p>The use of R spatial object is not implements 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 fall into the circle.</p>
<div class="cell" data-nm="true">
-<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="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>district_xy <span class="ot"><-</span> <span class="fu">st_centroid</span>(district) <span class="sc">%>%</span> </span>
+<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">st_coordinates</span>()</span>
+<span id="cb15-3"><a href="#cb15-3" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb15-4"><a href="#cb15-4" 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
@@ -637,20 +653,20 @@ Statistical test
</div>
<p>We can then call kulldorff function (you are strongly encouraged 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="cell" data-nm="true">
-<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">$</span>expected,</span>
-<span id="cb18-5"><a href="#cb18-5" aria-hidden="true" tabindex="-1"></a> <span class="at">pop.upper.bound =</span> <span class="fl">0.5</span>, <span class="co"># include maximum 50% of the population in a windows</span></span>
-<span id="cb18-6"><a href="#cb18-6" aria-hidden="true" tabindex="-1"></a> <span class="at">n.simulations =</span> <span class="dv">499</span>,</span>
-<span id="cb18-7"><a href="#cb18-7" aria-hidden="true" tabindex="-1"></a> <span class="at">alpha.level =</span> <span class="fl">0.2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a>kd_Wfever <span class="ot"><-</span> <span class="fu">kulldorff</span>(district_xy, </span>
+<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a> <span class="at">cases =</span> district<span class="sc">$</span>cases,</span>
+<span id="cb17-3"><a href="#cb17-3" aria-hidden="true" tabindex="-1"></a> <span class="at">population =</span> district<span class="sc">$</span>T_POP,</span>
+<span id="cb17-4"><a href="#cb17-4" aria-hidden="true" tabindex="-1"></a> <span class="at">expected.cases =</span> district<span class="sc">$</span>expected,</span>
+<span id="cb17-5"><a href="#cb17-5" 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="cb17-6"><a href="#cb17-6" aria-hidden="true" tabindex="-1"></a> <span class="at">n.simulations =</span> <span class="dv">499</span>,</span>
+<span id="cb17-7"><a href="#cb17-7" 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 develop any summary and visualization of the results but we can explore the output object.</p>
+<p>The function plot the histogram of the distribution of log-likelihood ratio simulated under the null hypothesis that is estimated based on Monte Carlo simulations. The observed value of the most significant cluster identified from all possible scans is compared to the distribution to determine significance. All outputs are saved into an R object, here called <code>kd_Wfever</code>. Unfortunately, the package did not develop any summary and visualization of the results but we can explore the output object.</p>
<div class="cell" data-nm="true">
-<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="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><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>
@@ -658,22 +674,22 @@ Statistical test
</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="sourceCode cell-code" id="cb20"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a><span class="co"># We can see which districts (r number) belong to this cluster</span></span>
+<span id="cb20-2"><a href="#cb20-2" 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="sourceCode cell-code" id="cb22"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a><span class="co"># standardized incidence ratio</span></span>
+<span id="cb22-2"><a href="#cb22-2" 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="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="co"># number of observed and expected cases in this cluster</span></span>
+<span id="cb24-2"><a href="#cb24-2" 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="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>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>
@@ -681,49 +697,49 @@ Statistical test
<p>17 districts belong to the cluster and its number of cases is 2.3 times higher than the expected number of cases.</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">$</span>secondary.clusters)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode cell-code" id="cb28"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a><span class="co"># We can see which districts (r number) belong to this cluster</span></span>
+<span id="cb28-2"><a href="#cb28-2" aria-hidden="true" tabindex="-1"></a><span class="fu">length</span>(kd_Wfever<span class="sc">$</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="sourceCode cell-code" id="cb30"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a><span class="co"># retrieve data for all secondary clusters into a table</span></span>
+<span id="cb30-2"><a href="#cb30-2" 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">$</span>secondary.clusters, <span class="st">'[['</span>, <span class="dv">5</span>), </span>
+<span id="cb30-3"><a href="#cb30-3" aria-hidden="true" tabindex="-1"></a> <span class="at">number.of.cases =</span> <span class="fu">sapply</span>(kd_Wfever<span class="sc">$</span>secondary.clusters, <span class="st">'[['</span>, <span class="dv">3</span>),</span>
+<span id="cb30-4"><a href="#cb30-4" aria-hidden="true" tabindex="-1"></a> <span class="at">expected.cases =</span> <span class="fu">sapply</span>(kd_Wfever<span class="sc">$</span>secondary.clusters, <span class="st">'[['</span>, <span class="dv">4</span>),</span>
+<span id="cb30-5"><a href="#cb30-5" aria-hidden="true" tabindex="-1"></a> <span class="at">p.value =</span> <span class="fu">sapply</span>(kd_Wfever<span class="sc">$</span>secondary.clusters, <span class="st">'[['</span>, <span class="dv">8</span>))</span>
+<span id="cb30-6"><a href="#cb30-6" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb30-7"><a href="#cb30-7" 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
-1 3.767698 16 4.246625 0.008</code></pre>
+1 3.767698 16 4.246625 0.016</code></pre>
</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">$</span>k_cluster <span class="ot"><-</span> <span class="cn">NA</span></span>
-<span id="cb33-3"><a href="#cb33-3" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb33-4"><a href="#cb33-4" aria-hidden="true" tabindex="-1"></a><span class="co"># save cluster information from kulldorff outputs</span></span>
-<span id="cb33-5"><a href="#cb33-5" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</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="sourceCode cell-code" id="cb32"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a><span class="co"># create empty column to store cluster informations</span></span>
+<span id="cb32-2"><a href="#cb32-2" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</span>k_cluster <span class="ot"><-</span> <span class="cn">NA</span></span>
+<span id="cb32-3"><a href="#cb32-3" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb32-4"><a href="#cb32-4" aria-hidden="true" tabindex="-1"></a><span class="co"># save cluster information from kulldorff outputs</span></span>
+<span id="cb32-5"><a href="#cb32-5" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</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="cb32-6"><a href="#cb32-6" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb32-7"><a href="#cb32-7" 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">$</span>secondary.clusters)){</span>
+<span id="cb32-8"><a href="#cb32-8" aria-hidden="true" tabindex="-1"></a>district<span class="sc">$</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="cb32-9"><a href="#cb32-9" 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="cb32-10"><a href="#cb32-10" aria-hidden="true" tabindex="-1"></a>}</span>
+<span id="cb32-11"><a href="#cb32-11" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb32-12"><a href="#cb32-12" aria-hidden="true" tabindex="-1"></a><span class="co">#district$k_cluster[is.na(district$k_cluster)] <- "No cluster"</span></span>
+<span id="cb32-13"><a href="#cb32-13" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb32-14"><a href="#cb32-14" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb32-15"><a href="#cb32-15" aria-hidden="true" tabindex="-1"></a><span class="co"># create map</span></span>
+<span id="cb32-16"><a href="#cb32-16" aria-hidden="true" tabindex="-1"></a><span class="fu">mf_map</span>(<span class="at">x =</span> district,</span>
+<span id="cb32-17"><a href="#cb32-17" aria-hidden="true" tabindex="-1"></a> <span class="at">var =</span> <span class="st">"k_cluster"</span>,</span>
+<span id="cb32-18"><a href="#cb32-18" aria-hidden="true" tabindex="-1"></a> <span class="at">type =</span> <span class="st">"typo"</span>,</span>
+<span id="cb32-19"><a href="#cb32-19" aria-hidden="true" tabindex="-1"></a> <span class="at">cex =</span> <span class="dv">2</span>,</span>
+<span id="cb32-20"><a href="#cb32-20" aria-hidden="true" tabindex="-1"></a> <span class="at">col_na =</span> <span class="st">"white"</span>,</span>
+<span id="cb32-21"><a href="#cb32-21" 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="cb32-22"><a href="#cb32-22" aria-hidden="true" tabindex="-1"></a> <span class="at">leg_title =</span> <span class="st">"Clusters"</span>)</span>
+<span id="cb32-23"><a href="#cb32-23" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb32-24"><a href="#cb32-24" 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>
@@ -739,7 +755,7 @@ To go further …
</div>
<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>
+<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 time-period. 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>
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diff --git a/public/search.json b/public/search.json
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"href": "07-basic_statistics.html#cluster-analysis",
"title": "7Â Basic statistics for spatial analysis",
"section": "7.2 Cluster analysis",
- "text": "7.2 Cluster analysis\n\n7.2.1 General introduction\nWhy 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:\n\nThe 1st order effects 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 an environment at risk as, for example, air pollution, contaminated waters or soils and UV exposition. This effect assume that the observed pattern is caused by a difference in risk intensity.\nThe 2nd order effects 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 is caused by correlations or co-variations.\n\nNo 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.\nKnowledge 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.\n\n\n\n\n\n\nStatistic tests and distributions\n\n\n\nIn statistics, problems are usually expressed by defining two hypotheses: the null hypothesis (H0), i.e., an a priori 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.\nIn 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 (also known as negative binomial) distributions.\nMany 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).\n\n# dataset statistics\nm_cases <- mean(district$incidence)\nsd_cases <- sd(district$incidence)\n\nhist(district$incidence, probability = TRUE, ylim = c(0, 0.4), xlim = c(-5, 16), xlab = \"Number of cases\", ylab = \"Probability\", main = \"Histogram of observed incidence compared\\nto Normal and Poisson distributions\")\ncurve(dnorm(x, m_cases, sd_cases),col = \"blue\", lwd = 1, add = TRUE)\npoints(0:max(district$incidence), dpois(0:max(district$incidence), m_cases),type = 'b', pch = 20, col = \"red\", ylim = c(0, 0.6), lty = 2)\n\nlegend(\"topright\", legend = c(\"Normal distribution\", \"Poisson distribution\", \"Observed distribution\"), col = c(\"blue\", \"red\", \"black\"),pch = c(NA, 20, NA), lty = c(1, 2, 1))\n\n\n\n\nIn this tutorial, we used the Poisson distribution in our statistical tests.\n\n\n\n\n7.2.2 Test for spatial autocorrelation (Moran’s I test)\n\n7.2.2.1 The global Moran’s I test\nA 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.\n\n\n\n\n\n\nMoran’s I test\n\n\n\nThe Moran’s statistics is:\n\\[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}\\] with:\n\n\\(N\\): the number of polygons,\n\\(w_{ij}\\): is a matrix of spatial weight with zeroes on the diagonal (i.e., \\(w_{ii}=0\\)). For example, if polygons are neighbors, the weight takes the value \\(1\\) otherwise it takes the value \\(0\\).\n\\(Y_i\\): the variable of interest,\n\\(\\bar{Y}\\): the mean value of \\(Y\\).\n\nUnder the Moran’s test, the statistics hypotheses are:\n\nH0: the distribution of cases is spatially independent, i.e., \\(I=0\\).\nH1: the distribution of cases is spatially autocorrelated, i.e., \\(I\\ne0\\).\n\n\n\nWe will compute the Moran’s statistics using spdep(R. Bivand et al. 2015) and Dcluster(Gómez-Rubio et al. 2015) packages. spdep package provides a collection of functions to analyze spatial correlations of polygons and works with sp objects. In this example, we use poly2nb() and nb2listw(). These functions respectively detect the neighboring polygons and assign weight corresponding to \\(1/\\#\\ of\\ neighbors\\). Dcluster package provides a set of functions for the detection of spatial clusters of disease using count data.\n\nlibrary(spdep) # Functions for creating spatial weight, spatial analysis\nlibrary(DCluster) # Package with functions for spatial cluster analysis\n\nqueen_nb <- poly2nb(district) # Neighbors according to queen case\nq_listw <- nb2listw(queen_nb, style = 'W') # row-standardized weights\n\n# Moran's I test\nm_test <- moranI.test(cases ~ offset(log(expected)), \n data = district,\n model = 'poisson',\n R = 499,\n listw = q_listw,\n n = length(district$cases), # number of regions\n S0 = Szero(q_listw)) # Global sum of weights\nprint(m_test)\n\nMoran's I test of spatial autocorrelation \n\n Type of boots.: parametric \n Model used when sampling: Poisson \n Number of simulations: 499 \n Statistic: 0.1566449 \n p-value : 0.012 \n\nplot(m_test)\n\n\n\n\nThe Moran’s statistics is here \\(I =\\) 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 \\(p_{value} =\\) 0.012. We therefore reject H0 with error risk of \\(\\alpha = 5\\%\\). The distribution of cases is therefore autocorrelated across districts in Cambodia.\n\n\n7.2.2.2 Moran’s I local test\nThe 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 correlations 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 information 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 is raised about multiple testing limitations that increase the Type I error (\\(\\alpha\\)) 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.\n\n\n\n\n\n\nStatistical test\n\n\n\nFor each district \\(i\\), the Local Moran’s I statistics is:\n\\[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\\]\n\n\nThe localmoran()function from the package spdep 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. Unfortunately, the local Moran’s test has not been implemented for Poisson distribution (population not large enough in some districts) in spdep package. However, Bivand et al. (R. S. Bivand et al. 2008) provided some code to manual perform the analysis using Poisson distribution and was further implemented in the course “Spatial Epidemiologyâ€.\n\n# Step 1 - Create the standardized deviation of observed from expected\nsd_lm <- (district$cases - district$expected) / sqrt(district$expected)\n\n# Step 2 - Create a spatially lagged version of standardized deviation of neighbors\nwsd_lm <- lag.listw(q_listw, sd_lm)\n\n# Step 3 - the local Moran's I is the product of step 1 and step 2\ndistrict$I_lm <- sd_lm * wsd_lm\n\n# Step 4 - setup parameters for simulation of the null distribution\n\n# Specify number of simulations to run\nnsim <- 499\n\n# Specify dimensions of result based on number of regions\nN <- length(district$expected)\n\n# Create a matrix of zeros to hold results, with a row for each county, and a column for each simulation\nsims <- matrix(0, ncol = nsim, nrow = N)\n\n# Step 5 - Start a for-loop to iterate over simulation columns\nfor(i in 1:nsim){\n y <- rpois(N, lambda = district$expected) # generate a random event count, given expected\n sd_lmi <- (y - district$expected) / sqrt(district$expected) # standardized local measure\n wsd_lmi <- lag.listw(q_listw, sd_lmi) # standardized spatially lagged measure\n sims[, i] <- sd_lmi * wsd_lmi # this is the I(i) statistic under this iteration of null\n}\n\nhist(sims[1,])\n\n\n\n# Step 6 - For each county, test where the observed value ranks with respect to the null simulations\nxrank <- apply(cbind(district$I_lm, sims), 1, function(x) rank(x)[1])\n\n# Step 7 - Calculate the difference between observed rank and total possible (nsim)\ndiff <- nsim - xrank\ndiff <- ifelse(diff > 0, diff, 0)\n\n# Step 8 - Assuming a uniform distribution of ranks, calculate p-value for observed\n# given the null distribution generate from simulations\ndistrict$pval_lm <- punif((diff + 1) / (nsim + 1))\n\nFor each district, we obtain a p-value based on permutations process\nA conventional way of plotting these results is to classify the districts into 5 classes based on local Moran’s I output. The classification of cluster that are significantly autocorrelated to their neighbors 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 lag.listw()):\n\nDistricts that have higher-than-average rates in both index regions and their neighbors and showing statistically significant positive values for the local \\(I_i\\) statistic are defined as High-High (hotspot of the disease)\nDistricts that have lower-than-average rates in both index regions and their neighbors and showing statistically significant positive values for the local \\(I_i\\) statistic are defined as Low-Low (cold spot of the disease).\nDistricts 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 \\(I_i\\) statistic are defined as High-Low(outlier with high incidence in an area with low incidence).\nDistricts 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 \\(I_i\\) statistic are defined as Low-High (outlier of low incidence in area with high incidence).\nDistricts with non-significant values for the \\(I_i\\) statistic are defined as Non-significant.\n\n\n# create lagged local raw_rate - in other words the average of the queen neighbors value\n# values are scaled (centered and reduced) to be compared to average\ndistrict$lag_std <- scale(lag.listw(q_listw, var = district$incidence))\ndistrict$incidence_std <- scale(district$incidence)\n\n# extract pvalues\n# district$lm_pv <- lm_test[,5]\n\n# Classify local moran's outputs\ndistrict$lm_class <- NA\ndistrict$lm_class[district$incidence_std >=0 & district$lag_std >=0] <- 'High-High'\ndistrict$lm_class[district$incidence_std <=0 & district$lag_std <=0] <- 'Low-Low'\ndistrict$lm_class[district$incidence_std <=0 & district$lag_std >=0] <- 'Low-High'\ndistrict$lm_class[district$incidence_std >=0 & district$lag_std <=0] <- 'High-Low'\ndistrict$lm_class[district$pval_lm >= 0.05] <- 'Non-significant'\n\ndistrict$lm_class <- factor(district$lm_class, levels=c(\"High-High\", \"Low-Low\", \"High-Low\", \"Low-High\", \"Non-significant\") )\n\n# create map\nmf_map(x = district,\n var = \"lm_class\",\n type = \"typo\",\n cex = 2,\n col_na = \"white\",\n #val_order = c(\"High-High\", \"Low-Low\", \"High-Low\", \"Low-High\", \"Non-significant\") ,\n pal = c(\"#6D0026\" , \"blue\", \"white\") , # \"#FF755F\",\"#7FABD3\" ,\n leg_title = \"Clusters\")\n\nmf_layout(title = \"Cluster using Local Moran's I statistic\")\n\n\n\n\n\n\n\n7.2.3 Spatial scan statistics\nWhile Moran’s indices focus on testing for autocorrelation between neighboring polygons (under the null assumption of spatial independence), 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.\nThe function kulldorff from the package SpatialEpi (Kim and Wakefield 2010) is a simple tool to implement spatial-only scan statistics. Briefly, the kulldorff scan statistics scan the area for clusters using several steps:\n\nIt 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 include 50% of the population).\nIt aggregates the count of events and the population at risk (or an expected count of events) inside and outside the window of observation.\nFinally, it computes the likelihood ratio to test whether the risk is equal inside versus outside the windows (H0) or greater inside the observed window\nThese 3 steps are repeated for each location and each possible windows-radii.\n\n\nlibrary(\"SpatialEpi\")\n\nThe use of R spatial object is not implements in kulldorff() 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 fall into the circle.\n\ndistrict_xy <- st_centroid(district) %>% \n st_coordinates()\n\nhead(district_xy)\n\n X Y\n1 330823.3 1464560\n2 749758.3 1541787\n3 468384.0 1277007\n4 494548.2 1215261\n5 459644.2 1194615\n6 360528.3 1516339\n\n\nWe can then call kulldorff function (you are strongly encouraged to call ?kulldorff to properly call the function). The alpha.level threshold filter for the secondary clusters that will be retained. The most-likely cluster will be saved whatever its significance.\n\nkd_Wfever <- kulldorff(district_xy, \n cases = district$cases,\n population = district$T_POP,\n expected.cases = district$expected,\n pop.upper.bound = 0.5, # include maximum 50% of the population in a windows\n n.simulations = 499,\n alpha.level = 0.2)\n\n\n\n\nAll outputs are saved into an R object, here called kd_Wfever. Unfortunately, the package did not develop any summary and visualization of the results but we can explore the output object.\n\nnames(kd_Wfever)\n\n[1] \"most.likely.cluster\" \"secondary.clusters\" \"type\" \n[4] \"log.lkhd\" \"simulated.log.lkhd\" \n\n\nFirst, we can focus on the most likely cluster and explore its characteristics.\n\n# We can see which districts (r number) belong to this cluster\nkd_Wfever$most.likely.cluster$location.IDs.included\n\n [1] 48 93 66 180 133 29 194 118 50 144 31 141 3 117 22 43 142\n\n# standardized incidence ratio\nkd_Wfever$most.likely.cluster$SMR\n\n[1] 2.303106\n\n# number of observed and expected cases in this cluster\nkd_Wfever$most.likely.cluster$number.of.cases\n\n[1] 122\n\nkd_Wfever$most.likely.cluster$expected.cases\n\n[1] 52.97195\n\n\n17 districts belong to the cluster and its number of cases is 2.3 times higher than the expected number of cases.\nSimilarly, we could study the secondary clusters. Results are saved in a list.\n\n# We can see which districts (r number) belong to this cluster\nlength(kd_Wfever$secondary.clusters)\n\n[1] 1\n\n# retrieve data for all secondary clusters into a table\ndf_secondary_clusters <- data.frame(SMR = sapply(kd_Wfever$secondary.clusters, '[[', 5), \n number.of.cases = sapply(kd_Wfever$secondary.clusters, '[[', 3),\n expected.cases = sapply(kd_Wfever$secondary.clusters, '[[', 4),\n p.value = sapply(kd_Wfever$secondary.clusters, '[[', 8))\n\nprint(df_secondary_clusters)\n\n SMR number.of.cases expected.cases p.value\n1 3.767698 16 4.246625 0.008\n\n\nWe only have one secondary cluster composed of one district.\n\n# create empty column to store cluster informations\ndistrict$k_cluster <- NA\n\n# save cluster information from kulldorff outputs\ndistrict$k_cluster[kd_Wfever$most.likely.cluster$location.IDs.included] <- 'Most likely cluster'\n\nfor(i in 1:length(kd_Wfever$secondary.clusters)){\ndistrict$k_cluster[kd_Wfever$secondary.clusters[[i]]$location.IDs.included] <- paste(\n 'Secondary cluster', i, sep = '')\n}\n\n#district$k_cluster[is.na(district$k_cluster)] <- \"No cluster\"\n\n\n# create map\nmf_map(x = district,\n var = \"k_cluster\",\n type = \"typo\",\n cex = 2,\n col_na = \"white\",\n pal = mf_get_pal(palette = \"Reds\", n = 3)[1:2],\n leg_title = \"Clusters\")\n\nmf_layout(title = \"Cluster using kulldorf scan statistic\")\n\n\n\n\n\n\n\n\n\n\nTo go further …\n\n\n\nIn 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 strata parameter in the kulldorff() function.\nIn 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 scan_ep_poisson() function in the package scanstatistic (Allévius 2018) for this analysis.\n\n\n\n\n\n\nAllévius, Benjamin. 2018. “Scanstatistics: Space-Time Anomaly Detection Using Scan Statistics.†Journal of Open Source Software 3 (25): 515.\n\n\nBivand, Roger S, Edzer J Pebesma, Virgilio Gómez-Rubio, and Edzer Jan Pebesma. 2008. Applied Spatial Data Analysis with r. Vol. 747248717. Springer.\n\n\nBivand, Roger, Micah Altman, Luc Anselin, Renato Assunção, Olaf Berke, Andrew Bernat, and Guillaume Blanchet. 2015. “Package ‘Spdep’.†The Comprehensive R Archive Network.\n\n\nGómez-Rubio, Virgilio, Juan Ferrándiz-Ferragud, Antonio López-QuıÌlez, et al. 2015. “Package ‘DCluster’.â€\n\n\nKim, Albert Y, and Jon Wakefield. 2010. “R Data and Methods for Spatial Epidemiology: The SpatialEpi Package.†Dept of Statistics, University of Washington."
+ "text": "7.2 Cluster analysis\n\n7.2.1 General introduction\nWhy 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:\n\nThe 1st order effects 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 an environment at risk as, for example, air pollution, contaminated waters or soils and UV exposition. This effect assume that the observed pattern is caused by a difference in risk intensity.\nThe 2nd order effects 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 is caused by correlations or co-variations.\n\nNo 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.\nKnowledge 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.\n\n\n\n\n\n\nStatistic tests and distributions\n\n\n\nIn statistics, problems are usually expressed by defining two hypotheses: the null hypothesis (H0), i.e., an a priori 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.\nIn 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 (also known as negative binomial) distributions.\nMany 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).\n\n# dataset statistics\nm_cases <- mean(district$incidence)\nsd_cases <- sd(district$incidence)\n\nhist(district$incidence, probability = TRUE, ylim = c(0, 0.4), xlim = c(-5, 16), xlab = \"Number of cases\", ylab = \"Probability\", main = \"Histogram of observed incidence compared\\nto Normal and Poisson distributions\")\ncurve(dnorm(x, m_cases, sd_cases),col = \"blue\", lwd = 1, add = TRUE)\npoints(0:max(district$incidence), dpois(0:max(district$incidence), m_cases),type = 'b', pch = 20, col = \"red\", ylim = c(0, 0.6), lty = 2)\n\nlegend(\"topright\", legend = c(\"Normal distribution\", \"Poisson distribution\", \"Observed distribution\"), col = c(\"blue\", \"red\", \"black\"),pch = c(NA, 20, NA), lty = c(1, 2, 1))\n\n\n\n\nIn this tutorial, we used the Poisson distribution in our statistical tests.\n\n\n\n\n7.2.2 Test for spatial autocorrelation (Moran’s I test)\n\n7.2.2.1 The global Moran’s I test\nA 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.\n\n\n\n\n\n\nMoran’s I test\n\n\n\nThe Moran’s statistics is:\n\\[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}\\] with:\n\n\\(N\\): the number of polygons,\n\\(w_{ij}\\): is a matrix of spatial weight with zeroes on the diagonal (i.e., \\(w_{ii}=0\\)). For example, if polygons are neighbors, the weight takes the value \\(1\\) otherwise it takes the value \\(0\\).\n\\(Y_i\\): the variable of interest,\n\\(\\bar{Y}\\): the mean value of \\(Y\\).\n\nUnder the Moran’s test, the statistics hypotheses are:\n\nH0: the distribution of cases is spatially independent, i.e., \\(I=0\\).\nH1: the distribution of cases is spatially autocorrelated, i.e., \\(I\\ne0\\).\n\n\n\nWe will compute the Moran’s statistics using spdep(R. Bivand et al. 2015) and Dcluster(Gómez-Rubio et al. 2015) packages. spdep package provides a collection of functions to analyze spatial correlations of polygons and works with sp objects. In this example, we use poly2nb() and nb2listw(). These functions respectively detect the neighboring polygons and assign weight corresponding to \\(1/\\#\\ of\\ neighbors\\). Dcluster package provides a set of functions for the detection of spatial clusters of disease using count data.\n\nlibrary(spdep) # Functions for creating spatial weight, spatial analysis\nlibrary(DCluster) # Package with functions for spatial cluster analysis\n\nqueen_nb <- poly2nb(district) # Neighbors according to queen case\nq_listw <- nb2listw(queen_nb, style = 'W') # row-standardized weights\n\n# Moran's I test\nm_test <- moranI.test(cases ~ offset(log(expected)), \n data = district,\n model = 'poisson',\n R = 499,\n listw = q_listw,\n n = length(district$cases), # number of regions\n S0 = Szero(q_listw)) # Global sum of weights\nprint(m_test)\n\nMoran's I test of spatial autocorrelation \n\n Type of boots.: parametric \n Model used when sampling: Poisson \n Number of simulations: 499 \n Statistic: 0.1566449 \n p-value : 0.01 \n\nplot(m_test)\n\n\n\n\nThe Moran’s statistics is here \\(I =\\) 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 \\(p_{value} =\\) 0.01. We therefore reject H0 with error risk of \\(\\alpha = 5\\%\\). The distribution of cases is therefore autocorrelated across districts in Cambodia.\n\n\n7.2.2.2 The Local Moran’s I LISA test\nThe 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 correlations 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 information 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 is raised about multiple testing limitations that increase the Type I error (\\(\\alpha\\)) 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.\n\n\n\n\n\n\nStatistical test\n\n\n\nFor each district \\(i\\), the Local Moran’s I statistics is:\n\\[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\\]\n\n\nThe localmoran()function from the package spdep 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. Unfortunately, the local Moran’s test has not been implemented for Poisson distribution (population not large enough in some districts) in spdep package. However, Bivand et al. (R. S. Bivand et al. 2008) provided some code to manual perform the analysis using Poisson distribution and was further implemented in the course “Spatial Epidemiologyâ€.\n\n# Step 1 - Create the standardized deviation of observed from expected\nsd_lm <- (district$cases - district$expected) / sqrt(district$expected)\n\n# Step 2 - Create a spatially lagged version of standardized deviation of neighbors\nwsd_lm <- lag.listw(q_listw, sd_lm)\n\n# Step 3 - the local Moran's I is the product of step 1 and step 2\ndistrict$I_lm <- sd_lm * wsd_lm\n\n# Step 4 - setup parameters for simulation of the null distribution\n\n# Specify number of simulations to run\nnsim <- 499\n\n# Specify dimensions of result based on number of regions\nN <- length(district$expected)\n\n# Create a matrix of zeros to hold results, with a row for each county, and a column for each simulation\nsims <- matrix(0, ncol = nsim, nrow = N)\n\n# Step 5 - Start a for-loop to iterate over simulation columns\nfor(i in 1:nsim){\n y <- rpois(N, lambda = district$expected) # generate a random event count, given expected\n sd_lmi <- (y - district$expected) / sqrt(district$expected) # standardized local measure\n wsd_lmi <- lag.listw(q_listw, sd_lmi) # standardized spatially lagged measure\n sims[, i] <- sd_lmi * wsd_lmi # this is the I(i) statistic under this iteration of null\n}\n\n# Step 6 - For each county, test where the observed value ranks with respect to the null simulations\nxrank <- apply(cbind(district$I_lm, sims), 1, function(x) rank(x)[1])\n\n# Step 7 - Calculate the difference between observed rank and total possible (nsim)\ndiff <- nsim - xrank\ndiff <- ifelse(diff > 0, diff, 0)\n\n# Step 8 - Assuming a uniform distribution of ranks, calculate p-value for observed\n# given the null distribution generate from simulations\ndistrict$pval_lm <- punif((diff + 1) / (nsim + 1))\n\nFor each district, we obtain a p-value based on permutations process\nA conventional way of plotting these results is to classify the districts into 5 classes based on local Moran’s I output. The classification of cluster that are significantly autocorrelated to their neighbors 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 lag.listw()):\n\nDistricts that have higher-than-average rates in both index regions and their neighbors and showing statistically significant positive values for the local \\(I_i\\) statistic are defined as High-High (hotspot of the disease)\nDistricts that have lower-than-average rates in both index regions and their neighbors and showing statistically significant positive values for the local \\(I_i\\) statistic are defined as Low-Low (cold spot of the disease).\nDistricts 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 \\(I_i\\) statistic are defined as High-Low(outlier with high incidence in an area with low incidence).\nDistricts 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 \\(I_i\\) statistic are defined as Low-High (outlier of low incidence in area with high incidence).\nDistricts with non-significant values for the \\(I_i\\) statistic are defined as Non-significant.\n\n\n# create lagged local raw_rate - in other words the average of the queen neighbors value\n# values are scaled (centered and reduced) to be compared to average\ndistrict$lag_std <- scale(lag.listw(q_listw, var = district$incidence))\ndistrict$incidence_std <- scale(district$incidence)\n\n# extract pvalues\n# district$lm_pv <- lm_test[,5]\n\n# Classify local moran's outputs\ndistrict$lm_class <- NA\ndistrict$lm_class[district$incidence_std >=0 & district$lag_std >=0] <- 'High-High'\ndistrict$lm_class[district$incidence_std <=0 & district$lag_std <=0] <- 'Low-Low'\ndistrict$lm_class[district$incidence_std <=0 & district$lag_std >=0] <- 'Low-High'\ndistrict$lm_class[district$incidence_std >=0 & district$lag_std <=0] <- 'High-Low'\ndistrict$lm_class[district$pval_lm >= 0.05] <- 'Non-significant'\n\ndistrict$lm_class <- factor(district$lm_class, levels=c(\"High-High\", \"Low-Low\", \"High-Low\", \"Low-High\", \"Non-significant\") )\n\n# create map\nmf_map(x = district,\n var = \"lm_class\",\n type = \"typo\",\n cex = 2,\n col_na = \"white\",\n #val_order = c(\"High-High\", \"Low-Low\", \"High-Low\", \"Low-High\", \"Non-significant\") ,\n pal = c(\"#6D0026\" , \"blue\", \"white\") , # \"#FF755F\",\"#7FABD3\" ,\n leg_title = \"Clusters\")\n\nmf_layout(title = \"Cluster using Local Moran's I statistic\")\n\n\n\n\n\n\n\n7.2.3 Spatial scan statistics\nWhile Moran’s indices focus on testing for autocorrelation between neighboring polygons (under the null assumption of spatial independence), 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.\nThe function kulldorff from the package SpatialEpi (Kim and Wakefield 2010) is a simple tool to implement spatial-only scan statistics.\n\n\n\n\n\n\nKulldorf test\n\n\n\nUnder the kulldorff test, the statistics hypotheses are:\n\nH0: the risk is constant over the area, i.e., there is a spatial homogeneity of the incidence.\nH1: a particular window have higher incidence than the rest of the area , i.e., there is a spatial heterogeneity of incidence.\n\n\n\nBriefly, the kulldorff scan statistics scan the area for clusters using several steps:\n\nIt 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 include 50% of the population).\nIt aggregates the count of events and the population at risk (or an expected count of events) inside and outside the window of observation.\nFinally, it computes the likelihood ratio and test whether the risk is equal inside versus outside the windows (H0) or greater inside the observed window (H1). The H0 distribution is estimated by simulating the distribution of counts under the null hypothesis (homogeneous risk).\nThese 3 steps are repeated for each location and each possible windows-radii.\n\nWhile we test the significance of a large number of observation windows, one can raise concern about multiple testing and Type I error. This approach however suggest that we are not interest in a set of signifiant cluster but only in a most-likely cluster. This a priori restriction eliminate concern for multpile comparison since the test is simplified to a statistically significance of one single most-likely cluster.\nBecause we tested all-possible locations and window-radius, we can also choose to look at secondary clusters. In this case, you should keep in mind that increasing the number of secondary cluster you select, increases the risk for Type I error.\n\nlibrary(\"SpatialEpi\")\n\nThe use of R spatial object is not implements in kulldorff() 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 fall into the circle.\n\ndistrict_xy <- st_centroid(district) %>% \n st_coordinates()\n\nhead(district_xy)\n\n X Y\n1 330823.3 1464560\n2 749758.3 1541787\n3 468384.0 1277007\n4 494548.2 1215261\n5 459644.2 1194615\n6 360528.3 1516339\n\n\nWe can then call kulldorff function (you are strongly encouraged to call ?kulldorff to properly call the function). The alpha.level threshold filter for the secondary clusters that will be retained. The most-likely cluster will be saved whatever its significance.\n\nkd_Wfever <- kulldorff(district_xy, \n cases = district$cases,\n population = district$T_POP,\n expected.cases = district$expected,\n pop.upper.bound = 0.5, # include maximum 50% of the population in a windows\n n.simulations = 499,\n alpha.level = 0.2)\n\n\n\n\nThe function plot the histogram of the distribution of log-likelihood ratio simulated under the null hypothesis that is estimated based on Monte Carlo simulations. The observed value of the most significant cluster identified from all possible scans is compared to the distribution to determine significance. All outputs are saved into an R object, here called kd_Wfever. Unfortunately, the package did not develop any summary and visualization of the results but we can explore the output object.\n\nnames(kd_Wfever)\n\n[1] \"most.likely.cluster\" \"secondary.clusters\" \"type\" \n[4] \"log.lkhd\" \"simulated.log.lkhd\" \n\n\nFirst, we can focus on the most likely cluster and explore its characteristics.\n\n# We can see which districts (r number) belong to this cluster\nkd_Wfever$most.likely.cluster$location.IDs.included\n\n [1] 48 93 66 180 133 29 194 118 50 144 31 141 3 117 22 43 142\n\n# standardized incidence ratio\nkd_Wfever$most.likely.cluster$SMR\n\n[1] 2.303106\n\n# number of observed and expected cases in this cluster\nkd_Wfever$most.likely.cluster$number.of.cases\n\n[1] 122\n\nkd_Wfever$most.likely.cluster$expected.cases\n\n[1] 52.97195\n\n\n17 districts belong to the cluster and its number of cases is 2.3 times higher than the expected number of cases.\nSimilarly, we could study the secondary clusters. Results are saved in a list.\n\n# We can see which districts (r number) belong to this cluster\nlength(kd_Wfever$secondary.clusters)\n\n[1] 1\n\n# retrieve data for all secondary clusters into a table\ndf_secondary_clusters <- data.frame(SMR = sapply(kd_Wfever$secondary.clusters, '[[', 5), \n number.of.cases = sapply(kd_Wfever$secondary.clusters, '[[', 3),\n expected.cases = sapply(kd_Wfever$secondary.clusters, '[[', 4),\n p.value = sapply(kd_Wfever$secondary.clusters, '[[', 8))\n\nprint(df_secondary_clusters)\n\n SMR number.of.cases expected.cases p.value\n1 3.767698 16 4.246625 0.016\n\n\nWe only have one secondary cluster composed of one district.\n\n# create empty column to store cluster informations\ndistrict$k_cluster <- NA\n\n# save cluster information from kulldorff outputs\ndistrict$k_cluster[kd_Wfever$most.likely.cluster$location.IDs.included] <- 'Most likely cluster'\n\nfor(i in 1:length(kd_Wfever$secondary.clusters)){\ndistrict$k_cluster[kd_Wfever$secondary.clusters[[i]]$location.IDs.included] <- paste(\n 'Secondary cluster', i, sep = '')\n}\n\n#district$k_cluster[is.na(district$k_cluster)] <- \"No cluster\"\n\n\n# create map\nmf_map(x = district,\n var = \"k_cluster\",\n type = \"typo\",\n cex = 2,\n col_na = \"white\",\n pal = mf_get_pal(palette = \"Reds\", n = 3)[1:2],\n leg_title = \"Clusters\")\n\nmf_layout(title = \"Cluster using kulldorf scan statistic\")\n\n\n\n\n\n\n\n\n\n\nTo go further …\n\n\n\nIn 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 strata parameter in the kulldorff() function.\nIn 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 time-period. You should look at the function scan_ep_poisson() function in the package scanstatistic (Allévius 2018) for this analysis.\n\n\n\n\n\n\nAllévius, Benjamin. 2018. “Scanstatistics: Space-Time Anomaly Detection Using Scan Statistics.†Journal of Open Source Software 3 (25): 515.\n\n\nBivand, Roger S, Edzer J Pebesma, Virgilio Gómez-Rubio, and Edzer Jan Pebesma. 2008. Applied Spatial Data Analysis with r. Vol. 747248717. Springer.\n\n\nBivand, Roger, Micah Altman, Luc Anselin, Renato Assunção, Olaf Berke, Andrew Bernat, and Guillaume Blanchet. 2015. “Package ‘Spdep’.†The Comprehensive R Archive Network.\n\n\nGómez-Rubio, Virgilio, Juan Ferrándiz-Ferragud, Antonio López-QuıÌlez, et al. 2015. “Package ‘DCluster’.â€\n\n\nKim, Albert Y, and Jon Wakefield. 2010. “R Data and Methods for Spatial Epidemiology: The SpatialEpi Package.†Dept of Statistics, University of Washington."
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