sft (version 2.4)

estimateNAK: Neslon-Aalen Estimator of the Reverse Cumulative Hazard Function

Description

Computes the Nelson-Aalen estimator of a reverse cumulative hazard function.

Usage

estimateNAK(RT, CR)

Value

K

A function of class "stepfun" that returns the Nelson-Aalen estimator of the cumulative reverse hazard function.

Var

A function of class "stepfun" that returns estimated variance of the Nelson-Aalen estimator of the cumulative reverse hazard function.

Arguments

RT

A vector of times at which an event occurs (e.g., a vector of response times).

CR

A vector of status indicators, 1=normal, 0=censored. For response time data, this corresponds to 1=correct, 0=incorrect.

Author

Joe Houpt <[e-mail]>

Details

The Nelson-Aalen estimator of the cumulative reverse hazard function is a step function with jumps at each event time. The jump size is given by the number of events that have occurred up to and including the event. If G(t) is the number events that have occurred up to and including t, then the N-A estimator of the cumulative reverse hazard function is given by: $$K(t) = -\sum_{s \in \{{\rm Event Times} > t\}} \frac{1}{G(s)}$$

References

Aalen, O. O., Borgan, O., & Gjessing, H. K. (2008). Survival and event history analysis: A process point of view. New York: Springer.

Houpt, J.W. & Townsend, J.T. (2012). Statistical Measures for Workload Capacity Analysis. Journal of Mathematical Psychology, 56, 341-355.

See Also

estimateNAH stepfun

Examples

x <- rexp(50, rate=.5)
censoring <- runif(50) < .90
K.NA <- estimateNAK(x, censoring)

# Plot the estimated cumulative reverse hazard function
plot(K.NA$K, 
  main="Cumulative Reverse Hazard Function\n X ~ Exp(.5)    n=50", 
  xlab="X", ylab="K(x)")

# Plot 95% Confidence intervals
times <- seq(0,10, length.out=100)
lines(times, K.NA$K(times) + sqrt(K.NA$Var(times))*qnorm(1-.05/2), lty=2)
lines(times, K.NA$K(times) - sqrt(K.NA$Var(times))*qnorm(1-.05/2), lty=2)

# Plot the true cumulative reverse hazard function
lines(times, log(pexp(times, .5)), col='red')