STAND (version 2.0)

icfit: Calculates the Self-Consistent Estimate of Survival from Interval Censored Data

Description

This function calculates the self-consistent estimate of survival for interval censored data. (i.e., the nonparametric maximum likelihood estimate that generalizes the Kaplan-Meier estimate to interval censored data). The censoring is such that if the ith observation fails at $x$, we only observe that $L[i] < x \le R[i]$. Data may be entered with "exact" values, i.e., $L[i] = x = R[i]$. In that case the $L[i]$ is changed internally to $L[i]*$ which is the next lower of any of the observed endpoints (unless $R[i] = 0$ then an error results).

Usage

icfit(L, R, initp = NA, minerror = 1e-06, maxcount = 1000)

Arguments

L

a vector of the left endpoints of the interval

R

a vector of the right endpoints of the interval

initp

a vector with an initial estimate of the density function. This vector should sum to 1 and have a length equal to the number of unique values of L and R combined. If initp = NA (default) then an initial value is estimated from the data.

minerror

The minimum error for convergence purposes. The EM algorithm stops when error < minerror, where error is the maximum of the reduced gradients (see Gentleman and Geyer, 1994). Default = 1e-06.

maxcount

the maximum number of iterations. Default is 10000.

Value

Returns a list with the following elements:

u

a vector of Lagrange multipliers. If there are any negative values of u the Kuhn-Tucker conditions for convergence are not met. If this happens a warning will result.

error

this is the maximum of the reduced gradients. If convergence is correct then error < minerror and all values of u are nonnegative, otherwise a warning results.

count

number of iterations of the self-consistent algorithm (i.e., EM-algorithm)

time

a vector of times (see details)

p

a vector of probabilities, all except the last values are associated with the time vector above, i.e., $p[i] = Prob (X = time[i])$. The last value is associated with time==Inf. (see details).

surv

a vector of survival values associated with the time vector above, i.e., $surv[i] = Prob (X > time[i] )$

Details

The algorithm is basically an EM-algorithm applied to interval censored data (see Turnbull, 1976); however, first there is a primary reduction (See Aragon and Eberly, 1992). Convergence is defined when the maximum reduced gradient is less than minerror, and the Kuhn-Tucker conditions are approximately met, otherwise a warning will result. (see Gentleman and Geyer, 1994). There may be other faster algorithms, but they are more complicated (see Aragon and Eberly, 1992). The output for time is sort(unique(c(0,L,R,Inf))) without the Inf. The output for p keeps the value related to Inf so that p may be inserted into initp for another run. The outputs for p and surv act as if the jumps in the survival curve happen at the largest of the possible times (see Gentleman and Geyer, 1994, Table 2, for a more accurate way to present p).

References

Aragon, J. and Eberly, D. (1992), "On Convergence of Convex Minorant Algorithms for Distribution Estimation with Interval-Censored Data," Journal of Computational and Graphical Statistics. 1: 129-140.

Gentleman, R. and Geyer, C. J. (1994), "Maximum Likelihood for Interval Censored Data: Consistency and Computation," Biometrika, 81, 618-623.

Turnbull, B. W. (1976), "The Empirical Distribution Function with Arbitrarily Grouped, Censored and Truncated Data," Journal of the Royal Statistical Society, Series B,(Methodological), 38(3), 290-295.

See Also

plekm , qq.lnorm

Examples

# Calculate and plot a Kaplan-Meier type curve for interval censored data.
# This is S(x) = 1 - F(x) and is the sample estimate of the probability
# of exceeding x.  The filmbadge data is used as an example.
data(filmbadge)
out <- icfit(filmbadge$dlow,filmbadge$dhigh)
icplot(out$surv, out$time,XLAB="Dose",YLAB="Exceedance Probability")