@@ -118,7 +118,7 @@ These maps illustrate the spatial heterogeneity of the cases. The incidence show
In this example, we standardized the cases distribution for population count. This simple standardization assumes that the risk of contracting the disease is similar for each person. However, assumption does not hold for all diseases and for all observed events since confounding effects can create nuisance into the interpretations (e.g., the number of childhood illness and death outcomes in a district are usually related to the age pyramid) and you should keep in mind that other standardization can be performed based on variables known to have an effect but that you don't want to analyze (e.g., sex ratio, occupations, age pyramid).
In addition, one can wonder what does an $SIR \~ 1$ means, i.e., what is the threshold to decide whether the SIR is greater, lower or equivalent to 1. The significant of the SIR can be tested globally (to determine whether or not the incidence is homogeneously distributed) and locally in each district (to determine Which district have an SIR different than 1). We won't perform these analyses in this tutorial but you can look at the function `?achisq.test()` (from `Dcluster` package [@DCluster]) and `?probmap()` (from `spdep` package [@spdep]) to compute these statistics.
In addition, one can wonder what does an SIR \~ 1 means, i.e., what is the threshold to decide whether the SIR is greater, lower or equivalent to 1. The significant of the SIR can be tested globally (to determine whether or not the incidence is homogeneously distributed) and locally in each district (to determine Which district have an SIR different than 1). We won't perform these analyses in this tutorial but you can look at the function `?achisq.test()` (from `Dcluster` package [@DCluster]) and `?probmap()` (from `spdep` package [@spdep]) to compute these statistics.
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## Cluster analysis
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@@ -138,7 +138,7 @@ Knowledge about the disease and its transmission process could orientate the cho
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### Statistic tests and distributions
In 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.
In 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 (H1), 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.
In mathematics, a probability distribution is a mathematical expression that represents what we would expect due to random chance. The choice of the probability distribution relies on the type of data you use (continuous, count, binary). In general, three distribution a used while studying disease rates, the Binomial, the Poisson and the Poisson-gamma mixture (also known as negative binomial) distributions.
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@@ -215,7 +215,7 @@ The Moran's statistics is here $I =$ `r signif(m_test$t0, 2)`. When comparing it
#### The Local Moran's I LISA test
The global Moran's test provides us a global statistical value informing whether autocorrelation occurs over the territory but does not inform on where does these correlations occurs, i.e., what is the locations of the clusters. To identify such cluster, we can decompose the Moran's I statistic to extract local information of the level of correlation of each district and its neighbors. This is called the Local Moran's I LISA statistic. Because the Local Moran's I LISA statistic test each district for autocorrelation independently, concern is raised about multiple testing limitations that increase the Type I error ($\alpha$) of the statistical tests. The use of local test should therefore be study in light of explore and describes clusters once the global test detected autocorrelation.
The global Moran's test provides us a global statistical value informing whether autocorrelation occurs over the territory but does not inform on where does these correlations occurs, i.e., what is the locations of the clusters. To identify such cluster, we can decompose the Moran's I statistic to extract local information of the level of correlation of each district and its neighbors. This is called the Local Moran's I LISA statistic. Because the Local Moran's I LISA statistic test each district for autocorrelation independently, concern is raised about multiple testing limitations that increase the Type I error ($\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 has detected autocorrelation.
For each district, we obtain a p-value based on permutations process
For each district, we obtain a p-value based on the comparison of the observed value and permutations process that draw the distribution under the null hypothesis (i.e. the distribution of cases is spatially independent).
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 `lag.listw()`):
