# Paired binary CUSUM and its run-length computation

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### Description

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

### Usage

 1 2 3 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. rangeVector of indices in the observed slot to monitor. theta0In-control parameters of the paired binary CUSUM. theta1Out-of-control parameters of the paired binary CUSUM. h1Primary control limit (=threshold) of 1st CUSUM. h2Primary control limit (=threshold) of 2nd CUSUM. h11Secondary limit for 1st CUSUM. h22Secondary limit for 2nd CUSUM. p Vector giving the probability of the four different possible states, i.e. c((death=0,near-miss=0),(death=1,near-miss=0), (death=0,near-miss=1),(death=1,near-miss=1)). w1 The parameters w1 and w2 are the sample weights vectors for the two CUSUMs, see eqn. (2) in the paper. We have that w1 is equal to deaths w2 As for w1 h1 decision barrier for 1st individual cusums h2 decision barrier for 2nd cusums h11 together with h22 this makes up the joing decision barriers h22 together with h11 this makes up the joing decision barriers sparse Boolean indicating whether to use sparse matrix computations from the Matrix library (usually much faster!). Default: 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.

### Value

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

### Author(s)

S. Steiner and M. Höhle

### 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.

categoricalCUSUM
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 #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 ## Not run: pairedbinCUSUM.runlength(p,w1=c(-1,37,-9,29),w2=c(-1,7),h1=70,h2=32,h11=38,h22=17) ## End(Not run) #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. ## Not run: pairedbinCUSUM.runlength(p,w1=c(-1,37,-9,29),w2=c(-1,7),h1=70,h2=32, h11=38,h22=17,sparse=TRUE) ## End(Not run) #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)