# Non-parametric back-projection of incidence cases to exposure cases using a known incubation time as in Becker et al (1991)

`backprojNP.Rd`

The function is an implementation of the non-parametric back-projection of incidence cases to exposure cases described in Becker et al. (1991). The method back-projects exposure times from a univariate time series containing the number of symptom onsets per time unit. Here, the delay between exposure and symptom onset for an individual is seen as a realization of a random variable governed by a known probability mass function. The back-projection function calculates the expected number of exposures \(\lambda_t\) for each time unit under the assumption of a Poisson distribution, but without any parametric assumption on how the \(\lambda_t\) evolve in time.

Furthermore, the function contains a bootstrap based procedure, as
given in Yip et al (2011), which allows an indication of uncertainty
in the estimated \(\lambda_t\). The procedure is
equivalent to the suggestion in Becker and Marschner (1993). However,
the present implementation in `backprojNP`

allows only a
univariate time series, i.e. simultaneous age groups as in Becker and
Marschner (1993) are not possible.

The method in Becker et al. (1991) was originally developed for the back-projection of AIDS incidence, but it is equally useful for analysing the epidemic curve in outbreak situations of a disease with long incubation time, e.g. in order to qualitatively investigate the effect of intervention measures.

## Arguments

- sts
an object of class

`"sts"`

(or one that can be coerced to that class): contains the observed number of symptom onsets as a time series.- incu.pmf
Probability mass function (PMF) of the incubation time. The PMF is specified as a vector or matrix with the value of the PMF evaluated at \(0,...,d_max\), i.e. note that the support includes zero. The value of \(d_max\) is automatically calculated as

`length(incu.pmf)-1`

or`nrow(incu.pmf)-1`

. Note that if the sts object has more than one column, then for the backprojection the incubation time is either recycled for all components or, if it is a matrix with the same number of columns as the sts object, the \(k\)'th column of`incu.pmf`

is used for the backprojection of the \(k\)'th series.- control
A list with named arguments controlling the functionality of the non-parametric back-projection.

`k`

An integer representing the smoothing parameter to use in the smoothing step of the EMS algorithm. Needs to be an even number.

`eps`

A vector of length two representing the convergence threshold \(\epsilon\) of the EMS algorithm, see Details for further information. The first value is the threshold to use in the \(k=0\) loop, which forms the values for the parametric bootstrap. The second value is the threshold to use in the actual fit and bootstrap fitting using the specified

`k`

. If`k`

is only of length one, then this number is replicated twice.`Tmark`

Numeric with \(T'\leq T\). Upper time limit on which to base convergence, i.e. only the values \(\lambda_1,\ldots,\lambda_{T'}\) are monitored for convergence. See details.

`iter.max`

The maximum number of EM iterations to do before stopping.

`B`

Number of parametric bootstrap samples to perform from an initial k=0 fit. For each sample a back projection is performed. See Becker and Marschner (1993) for details.

`alpha`

(1-\(\alpha\))*100% confidence intervals are computed based on the percentile method.

`verbose`

(boolean). If true show extra progress and debug information.

`lambda0`

Start values for lambda. Vector needs to be of the length

`nrow(sts)`

.`eq3a.method`

A single character being either

`"R"`

or`"C"`

depending on whether the three nested loops of equation 3a in Becker et al. (1991) are to be executed as safe R code (can be extremely slow, however the implementation is not optimized for speed) or a C code (can be more than 200 times faster!). However, the C implementation is experimental and can hang R if, e.g., the time series does not go far enough back.`hookFun`

Hook function called for each iteration of the EM algorithm. The function should take a single argument

`stsbp`

of class`"stsBP"`

class. It will be have the lambda set to the current value of lambda. If no action desired just leave the function body empty (default). Additional arguments are possible.

- ...
Additional arguments are sent to the hook function.

## Details

Becker et al. (1991) specify a non-parametric back-projection algorithm based on the Expectation-Maximization-Smoothing (EMS) algorithm.

In the present implementation the algorithm iterates until $$\frac{||\lambda^{(k+1)} - \lambda^{(k)}||}{||\lambda^{(k)}||} < \epsilon$$ This is a slight adaptation of the proposals in Becker et al. (1991). If \(T\) is the length of \(\lambda\) then one can avoid instability of the algorithm near the end by considering only the \(\lambda\)'s with index \(1,\ldots,T'\).

See the references for further information.

## Value

`backprojNP`

returns an object of `"stsBP"`

.

## References

Becker NG, Watson LF and Carlin JB (1991), A method for non-parametric back-projection and its application to AIDS data, Statistics in Medicine, 10:1527-1542.

Becker NG and Marschner IC (1993), A method for estimating the age-specific relative risk of HIV infection from AIDS incidence data, Biometrika, 80(1):165-178.

Yip PSF, Lam KF, Xu Y, Chau PH, Xu J, Chang W, Peng Y, Liu Z, Xie X and Lau HY (2011), Reconstruction of the Infection Curve for SARS Epidemic in Beijing, China Using a Back-Projection Method, Communications in Statistics - Simulation and Computation, 37(2):425-433.

Associations of Age and Sex on Clinical Outcome and Incubation Period of Shiga toxin-producing Escherichia coli O104:H4 Infections, 2011 (2013), Werber D, King LA, Müller L, Follin P, Buchholz U, Bernard H, Rosner BM, Ethelberg S, de Valk H, Höhle M, American Journal of Epidemiology, 178(6):984-992.

## Note

The method is still experimental. A proper plot routine for
`stsBP`

objects is currently missing.

## Examples

```
#Generate an artificial outbreak of size n starting at time t0 and being of length
n <- 1e3 ; t0 <- 23 ; l <- 10
#PMF of the incubation time is an interval censored gamma distribution
#with mean 15 truncated at 25.
dmax <- 25
inc.pmf <- c(0,(pgamma(1:dmax,15,1.4) - pgamma(0:(dmax-1),15,1.4))/pgamma(dmax,15,1.4))
#Function to sample from the incubation time
rincu <- function(n) {
sample(0:dmax, size=n, replace=TRUE, prob=inc.pmf)
}
#Sample time of exposure and length of incubation time
set.seed(123)
exposureTimes <- t0 + sample(x=0:(l-1),size=n,replace=TRUE)
symptomTimes <- exposureTimes + rincu(n)
#Time series of exposure (truth) and symptom onset (observed)
X <- table( factor(exposureTimes,levels=1:(max(symptomTimes)+dmax)))
Y <- table( factor(symptomTimes,levels=1:(max(symptomTimes)+dmax)))
#Convert Y to an sts object
Ysts <- sts(Y)
#Plot the outbreak
plot(Ysts, xaxis.labelFormat=NULL, legend=NULL)
#Add true number of exposures to the plot
lines(1:length(Y)+0.2,X,col="red",type="h",lty=2)
#Helper function to show the EM step
plotIt <- function(cur.sts) {
plot(cur.sts,xaxis.labelFormat=NULL, legend.opts=NULL,ylim=c(0,140))
}
#Call non-parametric back-projection function with hook function but
#without bootstrapped confidence intervals
bpnp.control <- list(k=0,eps=rep(0.005,2),iter.max=rep(250,2),B=-1,hookFun=plotIt,verbose=TRUE)
#Fast C version (use argument: eq3a.method="C")!
sts.bp <- backprojNP(Ysts, incu.pmf=inc.pmf,
control=modifyList(bpnp.control,list(eq3a.method="C")), ylim=c(0,max(X,Y)))
#Show result
plot(sts.bp,xaxis.labelFormat=NULL,legend=NULL,lwd=c(1,1,2),lty=c(1,1,1),main="")
lines(1:length(Y)+0.2,X,col="red",type="h",lty=2)
#Do the convolution for the expectation
mu <- matrix(0,ncol=ncol(sts.bp),nrow=nrow(sts.bp))
#Loop over all series
for (j in 1:ncol(sts.bp)) {
#Loop over all time points
for (t in 1:nrow(sts.bp)) {
#Convolution, note support of inc.pmf starts at zero (move idx by 1)
i <- seq_len(t)
mu[t,j] <- sum(inc.pmf[t-i+1] * upperbound(sts.bp)[i,j],na.rm=TRUE)
}
}
#Show the fit
lines(1:nrow(sts.bp)-0.5,mu[,1],col="green",type="s",lwd=3)
#Non-parametric back-projection including boostrap CIs
bpnp.control2 <- modifyList(bpnp.control, list(hookFun=NULL, k=2,
B=10, # in practice, use B >= 1000 !
eq3a.method="C"))
sts.bp2 <- backprojNP(Ysts, incu.pmf=inc.pmf, control=bpnp.control2)
######################################################################
# Plot the result. This is currently a manual routine.
# ToDo: Need to specify a plot method for stsBP objects which also
# shows the CI.
#
# Parameters:
# stsBP - object of class stsBP which is to be plotted.
######################################################################
plot.stsBP <- function(stsBP) {
maxy <- max(observed(stsBP),upperbound(stsBP),stsBP@ci,na.rm=TRUE)
plot(upperbound(stsBP),type="n",ylim=c(0,maxy), ylab="Cases",xlab="time")
if (!all(is.na(stsBP@ci))) {
polygon( c(1:nrow(stsBP),rev(1:nrow(stsBP))),
c(stsBP@ci[2,,1],rev(stsBP@ci[1,,1])),col="lightgray")
}
lines(upperbound(stsBP),type="l",lwd=2)
legend(x="topright",c(expression(lambda[t])),lty=c(1),col=c(1),fill=c(NA),border=c(NA),lwd=c(2))
invisible()
}
#Plot the result of k=0 and add truth for comparison. No CIs available
plot.stsBP(sts.bp)
lines(1:length(Y),X,col=2,type="h")
#Same for k=2
plot.stsBP(sts.bp2)
lines(1:length(Y),X,col=2,type="h")
```