**
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**Cervix: case - control study

with errors in covariates

Carroll, Gail and Lubin (1993) consider the problem of estimating the odds ratio of a disease d in a case-control study where the binary exposure variable is measured with error. Their example concerns exposure to herpes simplex virus (HSV) in women with invasive cervical cancer (d=1) and in controls (d=0). Exposure to HSV is measured by a relatively inaccurate western blot procedure w for 1929 of the 2044 women, whilst for 115 women, it is also measured by a refined or "gold standard'' method x. The data are given in the table below. They show a substantial amount of misclassification, as indicated by low sensitivity and specificity of w in the "complete'' data, and Carroll, Gail and Lubin also found that the degree of misclassification was significantly higher for the controls than for the cases (p=0.049 by Fisher's exact test).

They fitted a prospective logistic model to the case-control data as follows

d

logit(p

where b is the log odds ratio of disease. Since the relationship between d and x is only directly observable in the 115 women with "complete'' data, and because there is evidence of differential measurement error, the following parameters are required in order to estimate the logistic model

f

f

f

f

q = P(x=1)

The differential probability of being exposed to HSV (x=1) for cases and controls is calculated as follows

model

{

for (i in 1 : N) {

x[i] ~ dbern(q) # incidence of HSV

logit(p[i]) <- beta0C + beta * x[i] # logistic model

d[i] ~ dbern(p[i]) # incidence of cancer

x1[i] <- x[i] + 1

d1[i] <- d[i] + 1

w[i] ~ dbern(phi[x1[i], d1[i]]) # incidence of w

}

q ~ dunif(0.0, 1.0) # prior distributions

beta0C ~ dnorm(0.0, 0.00001);

beta ~ dnorm(0.0, 0.00001);

for(j in 1 : 2) {

for(k in 1 : 2){

phi[j, k] ~ dunif(0.0, 1.0)

}

}

# calculate gamma1 = P(x=1|d=0) and gamma2 = P(x=1|d=1)

gamma1 <- 1 / (1 + (1 + exp(beta0C + beta)) / (1 + exp(beta0C)) * (1 - q) / q)

gamma2 <- 1 / (1 + (1 + exp(-beta0C - beta)) / (1 + exp(-beta0C)) * (1 - q) / q)

}

Results

A 1000 update burn in followed by a further 10000 updates gave the parameter estimates