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    "title": "7  Basic statistics for spatial analysis",
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    "text": "This section aims at providing some basic statistical tools to study the spatial distribution of epidemiological data."
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    "title": "7  Basic statistics for spatial analysis",
    "section": "7.1 Import and visualize epidemiological data",
    "text": "7.1 Import and visualize epidemiological data\nIn this section, we load data that reference the cases of an imaginary disease throughout Cambodia. Each point correspond to the geolocalisation of a case.\n\nlibrary(dplyr)\nlibrary(sf)\n\n#Import Cambodia country border\ncountry <- st_read(\"data_cambodia/cambodia.gpkg\", layer = \"country\", quiet = TRUE)\n#Import provincial administrative border of Cambodia\neducation <- st_read(\"data_cambodia/cambodia.gpkg\", layer = \"education\", quiet = TRUE)\n#Import district administrative border of Cambodia\ndistrict <- st_read(\"data_cambodia/cambodia.gpkg\", layer = \"district\", quiet = TRUE)\n\n# Import locations of cases from an imaginary disease\ncases <- st_read(\"data_cambodia/cambodia.gpkg\", layer = \"cases\", quiet = TRUE)\ncases <- subset(cases, Disease == \"W fever\")\n\nThe first step of any statistical analysis always consists on visualizing the data to check they were correctly loaded and to observe general pattern of the cases.\n\n# View the cases object\nhead(cases)\n\nSimple feature collection with 6 features and 2 fields\nGeometry type: MULTIPOINT\nDimension:     XY\nBounding box:  xmin: 255891 ymin: 1179092 xmax: 506647.4 ymax: 1467441\nProjected CRS: WGS 84 / UTM zone 48N\n  id Disease                           geom\n1  0 W fever MULTIPOINT ((280036.2 12841...\n2  1 W fever MULTIPOINT ((451859.5 11790...\n3  2 W fever  MULTIPOINT ((255891 1467441))\n4  5 W fever MULTIPOINT ((506647.4 12322...\n5  6 W fever  MULTIPOINT ((440668 1197958))\n6  7 W fever MULTIPOINT ((481594.5 12714...\n\n# Map the cases\nlibrary(mapsf)\n\nmf_map(x = district, border = \"white\")\nmf_map(x = country,lwd = 2, col = NA, add = TRUE)\nmf_map(x = cases, lwd = .5, col = \"#990000\", pch = 20, add = TRUE)\n\n\n\n\nIn epidemiology, the true meaning of point is very questionable. If it usually gives the location of an observation, its not clear if this observation represents an event of interest (e.g. illness, death, …) or a person at risk (e.g. a participant that may or may not experience the disease). Considering a ratio of event compared to a population at risk is often more informative than just considering cases. Administrative divisions of countries appears as great areal units for cases aggregation since they make available data on population count and structures. In this study, we will use the district as the areal unit of the study.\n\n# Aggregate cases over districts\ndistrict$cases <- lengths(st_intersects(district, cases))\n\nThe incidence (\\(\\frac{cases}{population}\\)) is commonly use to represent cases distribution related to population density but other indicators exists. As example, the standardized incidence ratios (SIRs) represents the deviation of observed and expected number of cases and is expressed as \\(SIR = \\frac{Y_i}{E_i}\\) with \\(Y_i\\), the observed number of cases and \\(E_i\\), the expected number of cases. In this study, we computed the expected number of cases in each district by assuming infections are homogeneously distributed across Cambodia, i.e. the incidence is the same in each district. The SIR therefore represents the deviation of incidence compared to the averaged average incidence across Cambodia.\n\n# Compute incidence in each district (per 100 000 population)\ndistrict$incidence <- district$cases/district$T_POP * 100000\n\n# Compute the global risk\nrate <- sum(district$cases)/sum(district$T_POP)\n\n# Compute expected number of cases \ndistrict$expected <- district$T_POP * rate\n\n# Compute SIR\ndistrict$SIR <- district$cases / district$expected\n\n\npar(mfrow = c(1, 3))\n# Plot number of cases using proportional symbol \nmf_map(x = district) \nmf_map(\n  x = district, \n  var = \"cases\",\n  val_max = 50,\n  type = \"prop\",\n  col = \"#990000\", \n  leg_title = \"Cases\")\nmf_layout(title = \"Number of cases of W Fever\")\n\n# Plot incidence \nmf_map(x = district,\n       var = \"incidence\",\n       type = \"choro\",\n       pal = \"Reds 3\",\n       leg_title = \"Incidence \\n(per 100 000)\")\nmf_layout(title = \"Incidence of W Fever\")\n\n# Plot SIRs\n# create breaks and associated color palette\nbreak_SIR <- c(0, exp(mf_get_breaks(log(district$SIR), nbreaks = 8, breaks = \"pretty\")))\ncol_pal <- c(\"#273871\", \"#3267AD\", \"#6496C8\", \"#9BBFDD\", \"#CDE3F0\", \"#FFCEBC\", \"#FF967E\", \"#F64D41\", \"#B90E36\")\n\nmf_map(x = district,\n       var = \"SIR\",\n       type = \"choro\",\n       breaks = break_SIR, \n       pal = col_pal, \n       cex = 2,\n       leg_title = \"SIR\")\nmf_layout(title = \"Standardized Incidence Ratio of W Fever\")\n\n\n\n\nThese maps illustrates the spatial heterogenity of the cases. The incidence shows how the disease vary from one district to another while the SIR highlight districts that have :\n\nhigher risk than average (SIR > 1) when standardized for population\nlower risk than average (SIR < 1) when standardized for population\naverage risk (SIR ~ 1) when standardized for population\n\nIn this example, we standardized the cases distribution for population count. This simple standardization assume that the risk of contracting the disease is similar for each person. However, assumption does not hold for all diseases and for all observed events since confounding effects can create nuisance into the interpretations (e.g. the number of childhood illness and death outcomes in a district are usually related to the age pyramid) and you should keep in mind that other standardization can be performed based on variables known to have an effect but that you don’t want to analyze (e.g. sex ratio, occupations, age pyramid)."
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    "section": "7.2 Cluster analysis",
    "text": "7.2 Cluster analysis\nSince this W fever seems to have a heterogeneous distribution across Cambodia, it would be interesting to study where excess of cases appears, i.e. to identify clusters of the disease. The definition of clusters emcompass many XXXXXXX\nThe first question is to wonder if data are auto correlated or spatially independent, i.e. study if neighboring districts are likely to have similar incidence.\n\n7.2.1 Test for spatial autocorrelation (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\nStatistical test\n\n\n\nIn statistics, problems are usually expressed by defining two hypothesis : 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.\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 take 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 hypothesis 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 and Dcluster 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 function 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 :  0.014 \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} =\\) 0.014. We therefore reject H0 with error risk of \\(\\alpha = 5\\%\\). The distribution of cases is therefore autocorrelated across districts in Cambodia.\n\n\n\n\n\n\nStatistic distributions\n\n\n\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 (a.k.a negative binomial) distributions.\nThe default Global Moran’s I test assume data are normally distributed. It implies that the mean However, in epidemiology, rates and count values are usually not normally distributed and their variance is not homogeneous across the districts since the size of population at risk differs. to be the same since more variability occurs when we study smaller populations.\nWhile many measures may be appropriately assessed under the normality assumptions of the previous Global Moran’s I, in general disease rates are not best assessed this way. This is because the rates themselves may not be normally distributed, but also because the variance of each rate likely differs because of different size population at risk. For example the previous test assumed that we had the same level of certainty about the rate in each county, when in fact some counties have very sparse data (with high variance) and others have adequate data (with relatively lower variance).\n\n# dataset statistics\nm_cases <- mean(district$cases)\nsd_cases <- sd(district$cases)\n\ncurve(dnorm(x, m_cases, sd_cases), from = -5, to = 16, ylim = c(0, 0.4), col = \"blue\", lwd = 1, \n      xlab = \"Number of cases\", ylab = \"Probability\", main = \"Histogram of observed data compared\\nto Normal and Poisson distributions\")\npoints(0:max(district$cases), dpois(0:max(district$cases), m_cases),type = 'b ', pch = 20, col = \"red\", ylim = c(0, 0.6), lty = 2)\nhist(district$cases,  add = TRUE, probability = TRUE)\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\n\n\n\n\n7.2.2 Spatial scan statistics\nWhile Moran’s indice focuses on testing for autocorrelation between neighboring polygons (under the null assumption of spatial independance), the spatial scan statistic aims at identifying an abnormal higher risk in a given region compared to the risk outside of this region (under the null assumption of homogeneous distribution). The conception of a cluster is therefore different between the two methods.\nThe function kulldorf from the package SpatialEpiis a simple tool to implement spatial-only scan statistics. Briefly, the kulldorf 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 includes 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 repeted for each location and each possible windows-radii.\n\n\nlibrary(\"SpatialEpi\")\n\nThe use of R spatial object is not implementes in kulldorf() function. It uses instead matrix of xy coordinates that represents the centroids of the districts. A given district is included into the observed circular window if its centroids falls into the circle.\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 encourage to call ?kulldorf 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$expected,\n                pop.upper.bound = 0.5, # include maximum 50% of the population in a windows\n                n.simulations = 499,\n                alpha.level = 0.2)\n\n\n\n\nAll outputs are saved into the R object kd_Wfever. Unfortunately the package did not developed any summary and visualization of the results but we can explore the output object.\n\nnames(kd_Wfever)\n\n[1] \"most.likely.cluster\" \"secondary.clusters\"  \"type\"               \n[4] \"log.lkhd\"            \"simulated.log.lkhd\" \n\n\nFirst, we can focus on the most likely cluster and explore its characteristics.\n\n# We can see which districts (r number) belong to this cluster\nkd_Wfever$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$expected.cases\n\n[1] 52.97195\n\n\n17 districts belong to the cluster and its number of cases is 2.3 times higher than the expected number of case.\nSimilarly, we could study the secondary clusters. Results are saved in a list.\n\n# We can see which districts (r number) belong to this cluster\nlength(kd_Wfever$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   0.012\n\n\nWe only have one secondary cluster composed of one district.\n\n# create empty column to store cluster informations\ndistrict$k_cluster <- NA\n\n# save cluster informations from kulldorff outputs\ndistrict$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# create map\nmf_map(x = district,\n       var = \"k_cluster\",\n       type = \"typo\",\n       cex = 2,\n       leg_title = \"Clusters\")\nmf_layout(title = \"Cluster using kulldorf scan statistic\")"
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    "text": "Historically, 4 packages make it possible to import, manipulate and transform spatial data:\n\nThe package rgdal (Bivand, Keitt, and Rowlingson 2022) which is an interface between R and the GDAL (GDAL/OGR contributors, n.d.) and PROJ (PROJ contributors 2021) libraries allow you to import and export spatial data (shapefiles for example) and also to manage cartographic projections\n\nThe package sp (E. J. Pebesma and Bivand 2005) provides class and methods for vector spatial data in R. It allows displaying background maps, inspectiong an attribute table etc.\n\nThe package rgeos (Bivand and Rundel 2021) gives access to the GEOS spatial operations library and therefore makes classic GIS operations available: calculation of surfaces or perimeters, calculation of distances, spatial aggregations, buffer zones, intersections, etc.\n\nThe package raster (Hijmans 2022a) is dedicated to the import, manipulation and modeling of raster data.\n\nToday, the main developments concerning vector data have moved away from the old 3 (sp, rgdal, rgeos) to rely mainly on the package sf ((E. Pebesma 2018a), (E. Pebesma 2018b)). In this manual we will rely exclusively on this package to manipulate vector data.\nThe packages stars (E. Pebesma 2021) and terra (Hijmans 2022b) come to replace the package raster for processing raster data. We have chosen to use the package here terra for its proximity to the raster."
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    "section": "1.2 The package sf",
    "text": "1.2 The package sf\n The package sf was released in late 2016 by Edzer Pebesma (also author of sp). Its goal is to combine the feature of sp, rgeos and rgdal in a single, more ergonomic package. This package offers simple objects (following the simple feature standard) which are easier to manipulate. Particular attention has been paid to the compatibility of the package with the pipe syntax and the operators of the tidyverse.\nsf directly uses the GDAL, GEOS and PROJ libraries.\n\n\n\n\n\nFrom r-spatial.org\n\n\n\n\n\n\nWebsite of package sf : Simple Features for R\n\n\n\n\n1.2.1 Format of spatial objects sf\n\n\n\n\n\nObjectssf are objects in data.frame which one of the columns contains geometries. This column is the class of sfc (simple feature column) and each individual of the column is a sfg (simple feature geometry). This format is very practical insofa as the data and the geometries are intrinsically linked in the same object.\n\n\n\n\n\n\nThumbnail describing the simple feature format: Simple Features for R\n\n\n\n\n\n\n\n\n\nTip\n\n\n\nA benchmark of vector processing libraries is available here."
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    "section": "1.3 Package mapsf",
    "text": "1.3 Package mapsf\nThe free R software spatial ecosystem is rich, dynamic and mature and several packages allow to import, process and represent spatial data. The package mapsf (Giraud 2022) relies on this ecosystem to integrate the creation of quality thematic maps into processing chains with R.\nOther packages can be used to make thematic maps. The package ggplot2 (Wickham 2016), in association with the package ggspatial (Dunnington 2021), allows for example to display spatial objects and to make simple thematic maps. The package tmap (Tennekes 2018) is dedicated to the creation of thematic maps, it uses a syntax close to that of ggplot2 (sequence of instructions combined with the ‘+’ sign). Documentation and tutorials for using these two packages are readily available on the web.\nHere, we will mainly use the package mapsf whose functionalities are quite complete and the handling rather simple. In addition, the package is relatively light.\n\nmapsf allows you to create most of the types of map usually used in statistical cartography (choropleth maps, typologies, proportional or graduated symbols, etc.). For each type of map, several parameters are used to customize the cartographic representation. These parameters are the same as those found in the usual GIS or cartography software (for example, the choice of discretizations and color palettes, the modification of the size of the symbols or the customization of the legends). Associated with the data representation functions, other functions are dedicated to cartographic dressing (themes or graphic charters, legends, scales, orientation arrows, title, credits, annotations, etc.), the creation of boxes or the exporting maps.\nmapsf is the successor of cartography (Giraud and Lambert 2016), it offers the same main functionalities while being lighter and more ergonomic.\nTo use this package several sources can be consulted:\n\nThe package documentation accessible on the internet or directly in R (?mapsf),\nA cheat sheet,\n\n\n\n\n\n\n\nThe vignettes associated with the package show sample scripts,\nThe R Geomatics blog which provides resources and examples related to the package and more generally to the R spatial ecosystem."
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    "section": "1.4 The package terra",
    "text": "1.4 The package terra\n The package terra was release in early 2020 by Robert J. Hijmans (also author of raster). Its objective is to propose methods of treatment and analysis of raster data. This package is very similar to the package raster; but it has more features, it’s easier to use, and it’s faster.\n\n\n\n\n\n\nWebsite of package terra : Spatial Data Science with R and “terra”\n\n\n\n\n\n\n\n\n\nTip\n\n\n\nA benchmark of raster processing libraries is available here.\n\n\n\n\n\n\nBivand, Roger, Tim Keitt, and Barry Rowlingson. 2022. “Rgdal: Bindings for the ’Geospatial’ Data Abstraction Library.” https://CRAN.R-project.org/package=rgdal.\n\n\nBivand, Roger, and Colin Rundel. 2021. “Rgeos: Interface to Geometry Engine - Open Source (’GEOS’).” https://CRAN.R-project.org/package=rgeos.\n\n\nDunnington, Dewey. 2021. “Ggspatial: Spatial Data Framework for Ggplot2.” https://CRAN.R-project.org/package=ggspatial.\n\n\nGDAL/OGR contributors. n.d. GDAL/OGR Geospatial Data Abstraction Software Library. Open Source Geospatial Foundation. https://gdal.org.\n\n\nGiraud, Timothée. 2022. “Mapsf: Thematic Cartography.” https://CRAN.R-project.org/package=mapsf.\n\n\nGiraud, Timothée, and Nicolas Lambert. 2016. “Cartography: Create and Integrate Maps in Your r Workflow” 1. https://doi.org/10.21105/joss.00054.\n\n\nHijmans, Robert J. 2022a. “Raster: Geographic Data Analysis and Modeling.” https://CRAN.R-project.org/package=raster.\n\n\n———. 2022b. “Terra: Spatial Data Analysis.” https://CRAN.R-project.org/package=terra.\n\n\nPebesma, Edzer. 2018a. “Simple Features for r: Standardized Support for Spatial Vector Data” 10. https://doi.org/10.32614/RJ-2018-009.\n\n\n———. 2018b. “Simple Features for R: Standardized Support for Spatial Vector Data.” The R Journal 10 (1): 439. https://doi.org/10.32614/rj-2018-009.\n\n\n———. 2021. “Stars: Spatiotemporal Arrays, Raster and Vector Data Cubes.” https://CRAN.R-project.org/package=stars.\n\n\nPebesma, Edzer J., and Roger S. Bivand. 2005. “Classes and Methods for Spatial Data in r” 5. https://CRAN.R-project.org/doc/Rnews/.\n\n\nPROJ contributors. 2021. PROJ Coordinate Transformation Software Library. Open Source Geospatial Foundation. https://proj.org/.\n\n\nTennekes, Martijn. 2018. “Tmap: Thematic Maps in r” 84. https://doi.org/10.18637/jss.v084.i06.\n\n\nWickham, Hadley. 2016. “Ggplot2: Elegant Graphics for Data Analysis.” https://ggplot2.tidyverse.org."
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