SUPPLEMENTARY MATERIALS
Proportion of symptoms in cases that are due to cancer
The CAPER study which is the source of the data used here recorded symptom presentations
in primary care in cases over the 24 months before their cancer was diagnosed. The
probability that a symptom in a patient with cancer was actually caused by cancer that we can
estimate directly from the CAPER study does not correspond with what would be observed in
a randomly chosen 2 year period. We have estimated what would be observed in practice,
based on the following assumptions:
1. The mean sojourn time of both cancers is 4 years (see discussion).
2. The rate of symptom presentation in cases during months 25-36 prior to diagnosis is
the same as during months 12-24.
3. The rate of symptom presentation in cases during months 37-48 is the same as in the
controls; in other words in cases none of the symptoms presenting in the first 12
months are caused by cancer.
Our calculation assumes that the sojourn time of lung and colorectal cancer is 4 years in
every individual. Although this is not plausible, our method will still give approximately
correct answers as long longer sojourn times are associated with proportionately longer SLTs.
However, it should be noted that the relationship between SLT and sojourn time is not
known.
WinBUGS code, data and initial values
Below we give the full WinBUGS model code, with initial values, and data for the “Any
Symptoms” investigative criteria in colorectal cancer.
Convergence occurred using the Brooks Gelman criteria (Brooks & Gelman, 1998) within
5000 iterations. Accordingly the first 10,000 samples were discarded, and we sampled 30,000
from each of two chains.
model{
for (g in 1:5) {
casy[g] <- sum(ca[g,])
casy[g] ~ dpois(thetaca[g])
thetaca[g] <- lamca[g] * nca[g]
lamca[g] ~ dunif(0,10)
cosy[g] ~ dpois(thetaco[g])
thetaco[g] <- lamco[g] * nco[g] * ntp
lamco[g] ~ dunif(0,10)
# likelihood cases
# 2-year incidence in cases
# likelihood controls
# incidence in controls, per time interval
pi[g,1:ntp] ~ ddirch(ca[g,1:ntp])
for (t in 1:ntp) { lam[g,t] <- pi[g,t] * lamca[g]
# total incidence in cases
lamc[g,t] <- max(0,lam[g,t] - lamco[g]) # cancer specific incidence
alpha[g,t] <- lamc[g,t]/sum(lamc[g,])
# SLT distribution
pc[g,t] <- lamc[g,t] / lam[g,t]
# prob symptom in case caused by cancer
}
}
for (g in 1:5) {mslt[g] <- inprod2(T[13:24],alpha[g,1:12])
mpc[g] <- sum(lamc[g,1:12]) / sum(lam[g,1:12])
anca[g] <- 100*((thetaca[g] / 2) / nca[g])
anco[g] <- 100*((thetaco[g] / 2) / nco[g])
}
mslt[6] <- inprod2(nca[1:5],mslt[1:5]) / sum(nca[1:5])
mpc[6] <- inprod2(nca[1:5],mpc[1:5]) / sum(nca[1:5])
anca[6] <- inprod2(nca[1:5],anca[1:5]) / sum(nca[1:5])
anco[6] <- inprod2(nco[1:5],anco[1:5]) / sum(nco[1:5])
# mean pr(cancer-related)
# annual incidence cases
# annual incidence controls
# average SLT all stages
# average pr(cancer-related)
# av annual incidence cases
# av annual incidence controls
for (t in 1:ntp) { alpha[6,t] <- inprod(alpha[1:5,t],nca[1:5])/sum(nca[1:5]) # av SLT distribution
pc[6,t] <- inprod(pc[1:5,t],nca[1:5])/sum(nca[1:5])}
# av probability of ca
# projection to a 4 year period, same rate in year 2 as months 1-12, nothing in year 1
for (g in 1:5) { for (t in 7:12) {xlamc[g,t] <- mean(lamc[g,1:6]) # cancer-specific incidence
xlam[g,t] <- mean(lam[g,1:6]) }
# total incidence in cases
for (t in 1:6) {xlamc[g,t] <- 0
xlam[g,t] <- lamco[g] }
for (t in 13:24) {xlamc[g,t] <- lamc[g,t-12]
xlam[g,t] <- lam[g,t-12] }
for (t in 1:24) {xalpha[g,t] <- xlamc[g,t] / sum(xlamc[g,1:24]) } # SLT distribution
mslt[g+6] <- inprod2(T[1:24],xalpha[g,1:24])
mpc[g+6] <- sum(xlamc[g,1:24]) / sum(xlam[g,1:24])
}
mslt[12] <- inprod2(nca[1:5],mslt[7:11]) / sum(nca[1:5]) # cancer related slt 4y
mpc[12] <- inprod2(nca[1:5],mpc[7:11]) / sum(nca[1:5]) # overall prob symtpms caused by cancer, 4yr
}
INITIAL VALUES
list(lamco=c(.1,.1,.1,.1,.1), lamca=c(.1,.1,.1,.1,.1))
list(lamco=c(1,1,1,1,1), lamca=c(1,1,1,1,1))
DATA
# any symptom
list(cosy=c(87,226,178,71,77), # number of symptoms in cases, Dukes A,B,C,D,Missing
ntp=12,
# number of time periods,
nco=c(230,624,500,175,215), # number of controls by stage
nca=c(46,125,100,35,43),
# number of cases by stage
T=c(47,45,43,41,39,37, 35,33,31,29,27,25, 23,21,19,17,15,13, 11,9,7,5,3,1), # average time to diagnosis
# for each time period
ca=structure(.Data=c(5,2,4,3,3,1,
6,5,7,5,16,22,
# number of symptoms in each time period, Duke’s A
6,5,6,7,4,9,
5,14,16,21,42,77,
#
“
, Duke’s B
5,6,3,6,6,8,
5,5,10,11,37,57,
#
“
, Duke’s C
2,1,1,2,4,7,
3,1,3,8,9,25,
#
“
, Duke’s D
3,3,0.1,1,2,4, 4,2,4,7,8,29),.Dim=c(5,12)))
#
“
, Missing
Reference
Brooks SP, Gelman A (1998) Alternative methods for monitoring convergence of iterative
simulations. Journal of Computational and Graphical Statistics 7: 434-455