SISMIDSpatialStatisticsinEpidemiologyand PublicHealth...

SISMID Spatial Statistics in Epidemiology andPublic Health

2015 R Notes: Infectious Disease Data

Jon WakefieldDepartments of Statistics and Biostatistics, University of

Washington

2015-07-21

Lower Saxony Measles Data

As a first example of the Held et al. (2005) approach we examinedata on measles considered in this paper

These data are in the surveillance package.

The data consist of weekly measles counts over 2 years, for each of15 neighborhoods of Lower Saxony in Germany.

The figure below shows the study region.

Included in the dataset are a 15× 15 matrix of 0/1 entriesindicating which areas share a common boundary.

There is also the population fraction that is contained in each area.

Lower Saxony Measles: Reading in the Data

library(surveillance)library(RColorBrewer)data(measles.weser)# Shapefile for Germanydata <- url("http://faculty.washington.edu/jonno/SISMIDmaterial/DEU_adm3.RData")load(data)names(gadm)## [1] "PID" "ID_0" "ISO" "NAME_0" "ID_1"## [6] "NAME_1" "ID_2" "NAME_2" "ID_3" "NAME_3"## [11] "NL_NAME_3" "VARNAME_3" "TYPE_3" "ENGTYPE_3"xxx <- gadm[which(gadm$NAME_2 == "Weser-Ems" & gadm$ID_3 !=

249), ]xxx$HHH <- c(3458, 3404, 3459, 3460, 3461, 3462, 3451,

3452, 3453, 3402, 3454, 3455, 3456, 3457, 3403)dat <- data.frame(t(measles.weser$observed)[c(11, 3,

12, 13, 14, 15, 4, 5, 6, 1, 7, 8, 9, 10, 2), ])names(dat) <- paste("wk", 1:104, sep = "")attributes(xxx)$data <- cbind(attributes(xxx)$data,

dat)weser <- xxx


Plotting a map of the study region.

save(weser, file = "~Weser_Data.RData")totals <- colSums(measles.weser$observed)totals <- totals[c(11, 3, 12, 13, 14, 15, 4, 5, 6,

1, 7, 8, 9, 10, 2)]nclr <- 5brks <- unique(round(quantile(totals, probs = seq(0,

1, 1/10))), digits = 1)plotclr <- colorRampPalette(brewer.pal(nclr, "BuPu"))(52)plot(xxx)text(coordinates(xxx), labels = xxx$HHH, cex = 0.8,

col = 1, font = 2)

Visualizing Spatial Data

3458

34043459

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Lower Saxony Measles DataWe next plot the data over time# plot of no. infected in each of 15# neighborhoodsplot(measles.weser, type = observed ~ time,

legend.opts = NULL)

time

No.

of I

nfec

ted

2001

I

2001

III

2002

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2003

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Plot of number of infected by neighbourhood

plot(measles.weser, as.one = FALSE, xaxis.years = FALSE)


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Lower Saxony Measles Data# Map of cases in week 70plot(xxx)plot(xxx, col = plotclr[findInterval(dat[,

70], 0:51, all.inside = T)], add = T)

Lower Saxony Measles DataWe first fit the model

µi ,t = λAR︸︷︷︸exp(αAR

0 )

yi ,t−1 + λNE︸︷︷︸exp(αNE

0 )

∑j∈ne(i)

yj,t−1 + NitλENit ,

with endemic term:

log(λENit ) = αEN

0 + α1t + γ sin(ωt) + δ cos(ωt)

where,

I λAR is the epidemic force,I λNE is the neighborhood effect,I Nit are (possibly standardized) population counts in area i at

time t,I λEN

it is the endemic term,I α1 is a slope parameter describing the large scale endemic

temporal trend,I γ and δ are seasonal parameters and do not vary across areas,ω = (2π)/52.

Lower Saxony Measles Data# Get the data in the right format ('sts' class)measles <- disProg2sts(measles.weser)str(measles)## Formal class 'sts' [package "surveillance"] with 13 slots## ..@ epoch : int [1:104] 1 2 3 4 5 6 7 8 9 10 ...## ..@ freq : num 52## ..@ start : num [1:2] 2001 1## ..@ observed : int [1:104, 1:15] 0 0 0 0 0 0 0 0 0 0 ...## .. ..- attr(*, "dimnames")=List of 2## .. .. ..$ : NULL## .. .. ..$ : chr [1:15] "3402" "3403" "3404" "3451" ...## ..@ state : num [1:104, 1:15] 0 0 0 0 0 0 0 0 0 0 ...## .. ..- attr(*, "dimnames")=List of 2## .. .. ..$ : NULL## .. .. ..$ : chr [1:15] "3402" "3403" "3404" "3451" ...## ..@ alarm : logi [1:104, 1:15] NA NA NA NA NA NA ...## .. ..- attr(*, "dimnames")=List of 2## .. .. ..$ : NULL## .. .. ..$ : chr [1:15] "3402" "3403" "3404" "3451" ...## ..@ upperbound : logi [1:104, 1:15] NA NA NA NA NA NA ...## .. ..- attr(*, "dimnames")=List of 2## .. .. ..$ : NULL## .. .. ..$ : chr [1:15] "3402" "3403" "3404" "3451" ...## ..@ neighbourhood : int [1:15, 1:15] 0 0 0 0 1 0 0 0 0 1 ...## .. ..- attr(*, "dimnames")=List of 2## .. .. ..$ : chr [1:15] "3402" "3403" "3404" "3451" ...## .. .. ..$ : chr [1:15] "3402" "3403" "3404" "3451" ...## ..@ populationFrac: num [1:104, 1:15] 0.0224 0.0224 0.0224 0.0224 0.0224 ...## .. ..- attr(*, "dimnames")=List of 2## .. .. ..$ : NULL## .. .. ..$ : chr [1:15] "3402" "3403" "3404" "3451" ...## ..@ map :Formal class 'SpatialPolygons' [package "sp"] with 4 slots## .. .. ..@ polygons : list()## .. .. ..@ plotOrder : int(0)## .. .. ..@ bbox : num[0 , 0 ]## .. .. ..@ proj4string:Formal class 'CRS' [package "sp"] with 1 slot## .. .. .. .. ..@ projargs: chr ""## ..@ control : list()## ..@ epochAsDate : logi FALSE## ..@ multinomialTS : logi FALSE

Lower Saxony Measles Data# Endemic component: inear trend and 1 pair of# seasonal termsf.end1 <- addSeason2formula(f = ~1 + t, S = 1, period = 52)model1 <- list(ar = list(f = ~1), ne = list(f = ~1,

weights = neighbourhood(measles)), end = list(f = f.end1,offset = population(measles)), family = "NegBin1") # NegBin1 has a single

# overdisperion parameterres.hhh1 <- hhh4(measles, control = model1)summary(res.hhh1)#### Call:## hhh4(stsObj = measles, control = model1)#### Coefficients:## Estimate Std. Error## ar.1 -0.472565 0.143066## ne.1 -4.474443 0.420570## end.1 0.343990 0.244113## end.t 0.004128 0.003956## end.sin(2 * pi * t/52) 0.931667 0.175207## end.cos(2 * pi * t/52) -0.241281 0.156280## overdisp 2.823871 0.369716#### Log-likelihood: -1002.05## AIC: 2018.1## BIC: 2055.5#### Number of units: 15## Number of time points: 103


Hence, the mean model is

E [Yi,t+1|yit , yjt ] = exp(−0.47)yit + exp(−4.47)∑

j∈ne(i)yjt

+ exp [0.34+ 0.0041× t + 0.93 sin(ωt)− 0.24 cos(ωt)]

The variance is

var(Yi,t+1|yit) = E [Yit |yit ](1+ 2.28× E [Yit |yit ]).

The fits are different in each of the areas due to the two epidemiccomponents.

Lower Saxony Measles Data# Plot for the fifth area:plot(res.hhh1, unit = 5, col = c("#FFA50095",

"#0000FF75", "#D9D9D9"), legend = TRUE)

2001.0 2001.5 2002.0 2002.5 2003.0

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spatiotemporalautoregressiveendemic


y <- measles@observed[2:104, ] # Notice no firstmu1 <- fitted(res.hhh1, "pearson")time <- matrix(rep(measles@epoch, 15), nrow = 104,

ncol = 15)time <- time[-1, ]res1 <- (y - mu1)/sqrt(mu1 * (1 + 2.28 * mu1))par(mfrow = c(1, 3))plot(mu1 ~ y, xlab = "Observed", ylab = "Fitted", type = "n")for (i in 1:15) {

points(mu1[, i] ~ y[, i], col = i)}abline(0, 1)plot(res1 ~ mu1, xlab = "Fitted", ylab = "Residuals",

type = "n")for (i in 1:15) {

points(res1[, i] ~ mu1[, i], col = i)}abline(0, 0)plot(res1 ~ time, xlab = "Time (week)", ylab = "Residuals",

type = "n")for (i in 1:15) {

points(res1[, i] ~ time[, i], col = i)}abline(0, 0)


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Residuals do not look good!

But residuals are very difficult to examine when there are lots ofobserved counts that are zero


We now extend the model to add area-specific intercepts to theendemic component.

# Endemic Component: region-specific intercepts,# linear trend and 1 pair of seasonal termsf.end2 <- addSeason2formula(f = ~-1 + fe(1, which = rep(TRUE,

ncol(measles))) + t, S = 1, period = 52)# Autoregressive Comp, Neighbourhood Effect,# Endemic compmodel2 <- list(ar = list(f = ~1), ne = list(f = ~1,

weights = neighbourhood(measles)), end = list(f = f.end2,offset = population(measles)), family = "NegBin1")

# Fit the modelres.hhh2 <- hhh4(measles, control = model2)

Lower Saxony Measles Datasummary(res.hhh2)#### Call:## hhh4(stsObj = measles, control = model2)#### Coefficients:## Estimate Std. Error## ar.1 -0.69791 0.14835## ne.1 -6.68692 2.13455## end.t 0.01204 0.00360## end.sin(2 * pi * t/52) 1.32238 0.16243## end.cos(2 * pi * t/52) -0.58866 0.13454## end.1.3402 2.04845 0.33483## end.1.3403 -0.48083 0.41065## end.1.3404 -1.91449 0.65148## end.1.3451 -1.21596 0.70807## end.1.3452 0.74223 0.33867## end.1.3453 -0.19656 0.44766## end.1.3454 0.07133 0.33151## end.1.3455 -2.16494 1.27427## end.1.3456 -2.10119 0.76601## end.1.3457 2.25367 0.27666## end.1.3458 -0.81222 0.48152## end.1.3459 -0.59600 0.37366## end.1.3460 -1.26344 0.54192## end.1.3461 0.31475 0.39120## end.1.3462 -0.66236 0.61236## overdisp 1.73842 0.22807#### Log-likelihood: -910.27## AIC: 1862.53## BIC: 1974.73#### Number of units: 15## Number of time points: 103

Lower Saxony Measles Data# Plot for the fifth area:plot(res.hhh2, unit = 5, col = c("#FFA50095",


2001.0 2001.5 2002.0 2002.5 2003.0

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Lower Saxony Measles Data#mu2 <- fitted(res.hhh2)res2 <- (y - mu2)/sqrt(mu2 * (1 + 1.74 * mu2))par(mfrow = c(1, 3))plot(mu2 ~ y, xlab = "Observed", ylab = "Fitted", type = "n")for (i in 1:15) {


type = "n")for (i in 1:15) {


type = "n")for (i in 1:15) {



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Lower Saxony Measles Data# Plot for the 5th area:plot(res.hhh2, unit = 5, col = c("#FFA50095",


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#mu2 <- fitted(res.hhh2)res2 <- (y - mu2)/sqrt(mu2 * (1 + 1.74 * mu2))par(mfrow = c(1, 3))plot(mu2 ~ y, xlab = "Observed", ylab = "Fitted", type = "n")for (i in 1:15) {


type = "n")for (i in 1:15) {


type = "n")for (i in 1:15) {



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Southern Germany Influenza Data

These data were analyzed by Paul and Held (2011) and consist ofweekly counts of influenza cases in 140 areas of Southern Germanyover 2001–2008.

Southern Germany Influenza Datadata(fluBYBW)# Time series of countsplot(fluBYBW, type = observed ~ time, legend.opts = NULL)

time

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# Map of Total number of Cases in 2001plot(fluBYBW[year(fluBYBW) == 2001, ], type = observed ~

1 | unit, labels = FALSE)

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Southern Germany Influenza DataThe code below gives a number of increasingly complex models,including spatial conditional auroregressive (CAR) random effects.

Model is

E [Yi ,t+1|yit , yjt ] = µi ,t+1

varv(Yi ,t+1|yit , yjt) = µi ,t+1(1+ µi ,t+1/ψ)

with

µi ,t+1 = λARi yit + λNE

it∑

j∈ne(i)yjt + Nitλ

ENit

log(λENit ) = αEN

0i + α1t + γ sin(ωt) + δ cos(ωt)

with the possibility of allowing log λARi , log λNE

i and log λENit to

contain independent or spatial random effects.

Southern Germany Influenza Data

First, model is basically the measles model, except that area effectsare taken as random effects.

# Endemic Component: linear trend, seasonal terms,# iid random effects Center time variable and put# on year scale: I((t-208)/100)flu.end <- addSeason2formula(~-1 + ri(type = "iid",

corr = "all") + I((t - 208)/100), S = 3, period = 52)# Make Weight matrix for neighbourhood effectswji <- neighbourhood(fluBYBW)/rowSums(neighbourhood(fluBYBW))# Model specificationmod.flu <- list(end = list(f = flu.end, offset = population(fluBYBW)),

ar = list(f = ~1), ne = list(f = ~-1 + ri(type = "iid",corr = "all"), weights = wji), family = "NegBin1")

#res.flu1 <- hhh4(fluBYBW, mod.flu)

Southern Germany Influenza Datasummary(res.flu1)#### Call:## hhh4(stsObj = fluBYBW, control = mod.flu)#### Random effects:## Var Corr## ne.ri(iid) 0.9645## end.ri(iid) 0.5066 0.5652#### Fixed effects:## Estimate Std. Error## ar.1 -0.89193 0.03676## ne.ri(iid) -1.51746 0.10350## end.I((t - 208)/100) 0.57368 0.02366## end.sin(2 * pi * t/52) 2.17870 0.09808## end.cos(2 * pi * t/52) 2.33738 0.12134## end.sin(4 * pi * t/52) 0.45161 0.10427## end.cos(4 * pi * t/52) -0.37668 0.09404## end.sin(6 * pi * t/52) 0.30145 0.06452## end.cos(6 * pi * t/52) -0.24798 0.06294## end.ri(iid) 0.22345 0.10222## overdisp 1.08439 0.03392#### Penalized log-likelihood: -18696.6## Marginal log-likelihood: -343.59#### Number of units: 140## Number of time points: 415

Southern Germany Influenza Data# A plot with the fit for area 8111:plot(res.flu1, unit = 77, col = c("#FFA50095",


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