We start with the setup for simulation.
We simulate (log) survival times and marker
values from a bivariate normal distribution with m = (0, 10)^{T}, r = -0.70, and variance of 1 (please
note that time is in log scale if not mentioned
otherwise). As per the convention, r is
taken to be negative so that higher marker value indicates
smaller event time. Censoring
time is generated independently as normal, so
that, about 40% of the observations are censored. Using this data we
estimate ROC curves, AUCs at each of TargetTimes ranging from 7.5
to 12.5. Let *n* denote the sample size and *Nsim*
denote the number of repetitions. For each of *Nsim* number of
repetitions, we generate data for *n* individuals and repeat this
procedure *Nsim* times, to get *Nsim* set of AUCs
values at each of TargetTimes. Finally, we estimate the
integrated AUC (iAUC) from the estimated AUC and the
Kaplan-Meier estimate of the survival time.

As discussed in the paper by Heagerty and Zheng (2005),
*Biometrics*, **61**, 92 -- 105, the true positive (TP)
and false positive based on a bivariate normal distribution of
Marker(M) and Time(T) can be expressed as follows:

TP_{t}^{I}(c) = P(M > c | T =
t) = F(r (log(t) -
c)/
((1-r^{2})^{1/2}))

FP_{t}^{D}(c) = P(M > c | T >
t) =
S_{2}^{N} (c, log(t); r)/F(-log(t))

The integrated AUC (iAUC or concordance index)
is given by: C = sin ^{-1}(-r)/p+0.5

The "true" AUC trace and concordance index is obtained as
follows. At each of
the TargetTimes, we evaluated the TP and FP using the known
parametric values and obtained AUC by numerically integrating
the area under ROC curve. True concordance index is evaluated at
the known r.

The AUC curves are estimates using five
methods. We first use MLEs of the parameter along with the known
parametric form of TP and FP to estimate the bivariate normal
AUC. The iAUC is estimated using the MLE of the correlation
coefficient of marker and event time. Estimation for MLEs is
discussed below while the details of the rest of
the methods (Cox, LocalCox, Schoenfeld with order = 0 and order
= 1) are discussed in the aforementioned paper and the
risksetROC package is used to estimate AUC traces at TargetTimes
as well as iAUC.

The MLEs are estimated as follows. First note
that we have observed marker values and (censored) event
times. Observed marker mean and variance is used as MLEs for
marker distribution. To estimate the expectation and variance
from the censored event time we fit a parametric survival model
for event time with normal error and marker as the only covariate:

T|M = b_{0}
+ b_{1} M + Error

so, E(T) = b_{0}
+ b_{1} E(M)

and V(T) = b_{1}^{2}
V(M) + Error Variance

Also, b_{1} = r (V(T)/V(M))^{1/2}

so, E(T) = b

and V(T) = b

Also, b

Once the estimates of b_{0}, b_{1} and error variance are
obtained, we estimated the mean and variance of event time and
correlation using the above relations. R package *
survival* and function *survreg* was used to estimate
the regression coefficients.

Two output files are out1.txt
and out2.txt, the first
contains the average AUC values at each TargetTimes (averaged over
simulations) while the second contains the iAUC values over
simulations. The plot associated with out1.txt is figure 1
while figure 2 plots the iAUC traces over simulations. Please see the documentation file for further details of
the functions and more background.

mail to: Paramita Saha Last modified: Mon Aug 27 20:31:21 PDT 2007