surveillance (version 1.12.1)

pairedbinCUSUM: Paired binary CUSUM and its run-length computation

Description

CUSUM for paired binary data as described in Steiner et al. (1999).

Usage

pairedbinCUSUM(stsObj, control = list(range=NULL,theta0,theta1,
                                      h1,h2,h11,h22))
pairedbinCUSUM.runlength(p,w1,w2,h1,h2,h11,h22, sparse=FALSE)

Arguments

stsObj

Object of class sts containing the paired responses for each of the, say n, patients. The observed slot of stsObj is thus a $n \times 2$ matrix.

control

Control object as a list containing several parameters.

• range

{Vector of indices in the observed slot to monitor.} theta0{In-control parameters of the paired binary CUSUM.}

Value

• An sts object with observed, alarm, etc. slots trimmed to the control$range indices.

encoding

latin1

item

• p
• w1
• w2
• h1
• h2
• h11
• h22
• sparse

code

FALSE

Details

For details about the method see the Steiner et al. (1999) reference listed below. Basically, two individual CUSUMs are run based on a logistic regression model. The combined CUSUM not only signals if one of its two individual CUSUMs signals, but also if the two CUSUMs simultaneously cross the secondary limits.

References

Steiner, S. H., Cook, R. J., and Farewell, V. T. (1999), Monitoring paired binary surgical outcomes using cumulative sum charts, Statistics in Medicine, 18, pp. 69--86.

See Also

categoricalCUSUM

Examples

#Set in-control and out-of-control parameters as in paper
theta0 <- c(-2.3,-4.5,2.5)
theta1 <- c(-1.7,-2.9,2.5)

#Small helper function to compute the paired-binary likelihood
#of the length two vector yz when the true parameters are theta
dPBin <- function(yz,theta) {
    exp(dbinom(yz[1],size=1,prob=plogis(theta[1]),log=TRUE) +
    dbinom(yz[2],size=1,prob=plogis(theta[2]+theta[3]*yz[1]),log=TRUE))
}

#Likelihood ratio for all four possible configurations
p <- c(dPBin(c(0,0), theta=theta0), dPBin(c(0,1), theta=theta0),
       dPBin(c(1,0), theta=theta0), dPBin(c(1,1), theta=theta0))

#Compute ARL using non-sparse matrix operations
pairedbinCUSUM.runlength(p,w1=c(-1,37,-9,29),w2=c(-1,7),h1=70,h2=32,h11=38,h22=17)

#Sparse computations don't work on all machines (e.g. the next line
#might lead to an error. If it works this call can be considerably (!) faster
#than the non-sparse call.
pairedbinCUSUM.runlength(p,w1=c(-1,37,-9,29),w2=c(-1,7),h1=70,h2=32,
                         h11=38,h22=17,sparse=TRUE)

#Use paired binary CUSUM on the De Leval et al. (1994) arterial switch
#operation data on 104 newborn babies
data("deleval")

#Switch between death and near misses
observed(deleval) <- observed(deleval)[,c(2,1)]

#Run paired-binary CUSUM without generating alarms. 
pb.surv <- pairedbinCUSUM(deleval,control=list(theta0=theta0,
             theta1=theta1,h1=Inf,h2=Inf,h11=Inf,h22=Inf))

plot(pb.surv, xaxis.labelFormat=NULL)



######################################################################
#Scale the plots so they become comparable to the plots in Steiner et
#al. (1999). To this end a small helper function is defined.
######################################################################

######################################################################
#Log LR for conditional specification of the paired model
######################################################################
LLR.pairedbin <- function(yz,theta0, theta1) {
    #In control
    alphay0 <- theta0[1] ; alphaz0 <- theta0[2] ; beta0 <- theta0[3]
    #Out of control
    alphay1 <- theta1[1] ; alphaz1 <- theta1[2] ; beta1 <- theta1[3]
    #Likelihood ratios        
    llry <- (alphay1-alphay0)*yz[1]+log(1+exp(alphay0))-log(1+exp(alphay1))
    llrz <- (alphaz1-alphaz0)*yz[2]+log(1+exp(alphaz0+beta0*yz[1]))-
                                    log(1+exp(alphaz1+beta1*yz[1]))
    return(c(llry=llry,llrz=llrz))
}


val <- expand.grid(0:1,0:1)
table <- t(apply(val,1, LLR.pairedbin, theta0=theta0, theta1=theta1))
w1 <- min(abs(table[,1]))
w2 <- min(abs(table[,2]))
S <- upperbound(pb.surv) / cbind(rep(w1,nrow(observed(pb.surv))),w2)

#Show results
par(mfcol=c(2,1))
plot(1:nrow(deleval),S[,1],type="l",main="Near Miss",xlab="Patient No.",
     ylab="CUSUM Statistic")
lines(c(0,1e99), c(32,32),lty=2,col=2)
lines(c(0,1e99), c(17,17),lty=2,col=3)
    
plot(1:nrow(deleval),S[,2],type="l",main="Death",xlab="Patient No.",
     ylab="CUSUM Statistic")
    lines(c(0,1e99), c(70,70),lty=2,col=2)
    lines(c(0,1e99), c(38,38),lty=2,col=3)

######################################################################
# Run the CUSUM with thresholds as in Steiner et al. (1999).
# After each alarm the CUSUM statistic is set to zero and
# monitoring continues from this point. Triangles indicate alarm
# in the respective CUSUM (nearmiss or death). If in both
# simultaneously then an alarm is caued by the secondary limits.
######################################################################
pb.surv2 <- pairedbinCUSUM(deleval,control=list(theta0=theta0,
             theta1=theta1,h1=70*w1,h2=32*w2,h11=38*w1,h22=17*w2))

plot(pb.surv2, xaxis.labelFormat=NULL)