sft (version 0.0-4)

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

Description

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

Usage

estimateNAK(RT, CR)

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 respsone time data, this corresponds to 1=correct, 0=incorrect.

Value

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

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 occured up to and including the event. If G(t) is the number events that have occured up to and including t, then the N-A esitmator of the cumulative reverse hazard function is given by: $$K(t) = -\sum_{s \in {{\rm Event Times} > t}} \frac{1}{G(s)}$$

References

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

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

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
 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')