## Sunday, June 1, 2014

### Autocorrelation in project Tycho's measles data

Project Tycho includes data from all weekly notifiable disease reports for the United States dating back to 1888. These data are freely available to anybody interested.I have looked at Ptoject Tycho's measles data before, general look, incidence, some high incidence data and correlation between states. After a detour, it is now time to look at the autocorrelations in these data. These show positive correlation at three years.

### Data

first steps, see correlation between states

#### Preparation for autocorrelation

As detailed before, the data contain weekly counts. Summer has less incidence, winter more. For this reason calender year is abolished and a shifted year used (named cycle), which runs from summer to summer. The data, real world as they are, contain plenty of missings. Arbitrarily chosen, if a cycle has data from at least 40 weeks, then I will us this particular year.
r7 <- aggregate(
r6\$incidence,
list(cycle=r6\$cycle,
State=r6\$State),
function(x)
if(length(x)>40)
sum(x) else
NA)

To calculate an autocorrelation, a set of consecutive years are needed. Again arbitrarily chosen, 15 years is the minimum. As a first attempt I just kicked out all missing years. Since that resulted in sufficient states with data, no attempts to refine were made. As additional item the number of data points is stored. All in a nice list.
la <- lapply(levels(r7\$State),function(x) {
datain <- r7[r7\$State==x,]
datain <- datain[complete.cases(datain),]
if (nrow(datain)==(1+max(datain\$cycle)-min(datain\$cycle)) &
nrow(datain)>15)
aa <- acf(datain\$x,plot=FALSE,lag.max=6) else aa <- TRUE
list(aa=aa,nr=nrow(datain))
})

To make a plot, the autocorrelations are pulled out and it is all stuck in a dataframe.
la2 <- la[which(sapply(la,function(x) class(x\$aa))=='acf')]
scfs <- as.data.frame(t(sapply(la2,function(x) as.numeric(x\$aa\$acf))))
scfs\$state <- levels(r7\$State)[sapply(la,function(x) class(x\$aa)=='acf')]
scfs\$n <- sapply(la2,function(x) x\$nr)

And a reshape prior to plotting.
tc <- reshape(scfs,
idvar=c('state','n'),
varying=list(names(scfs[1:7])),
timevar='lag',
times=0:6,
direction='long',
v.names='acf')

### Plot

The plot shows a somewhat negative correlation after one year and a positive correlation after 3 years. The only state not to show the positive correlation after 3 years had much less data, hence for conclusion I ignore that result.
library(ggplot2)
ggplot(tc,
aes(y=acf, x=lag,group=state,col=state) )+
geom_line(aes(size = n)) +
scale_size(range = c(0.5, 3))+
theme(text=element_text(family='Arial'))

### References

Willem G. van Panhuis, John Grefenstette, Su Yon Jung, Nian Shong Chok, Anne Cross, Heather Eng, Bruce Y Lee, Vladimir Zadorozhny, Shawn Brown, Derek Cummings, Donald S. Burke. Contagious Diseases in the United States from 1888 to the present. NEJM 2013; 369(22): 2152-2158.