mcf: Mean Cumulative Function for Repairable Systems.

In ReliaGrowR: Reliability Growth Analysis and Repairable Systems Modeling

Mean Cumulative Function for Repairable Systems.

Description

Computes the non-parametric Mean Cumulative Function (MCF) for recurrent event data from one or more repairable systems, using the Nelson-Aalen estimator. The MCF estimates the expected cumulative number of events per system as a function of time, properly accounting for system exposure (observation windows).

Usage

mcf(
  id = NULL,
  time = NULL,
  event = NULL,
  end_time = NULL,
  data = NULL,
  conf_level = 0.95
)

Arguments

id	A vector of system/unit identifiers. Each unique value represents a distinct system. Ignored if data is provided.
time	A numeric vector of event or censoring times. Must be positive and finite. Ignored if data is provided.
event	An optional numeric vector of event indicators: 1 for an event, 0 for censoring (end of observation). If NULL (default), all observations are treated as events.
end_time	An optional named numeric vector of end-of-observation times per system, where names correspond to system identifiers. This defines the actual exposure window for each system. When provided, a system remains in the risk set until its end_time, even if its last event occurred earlier. If NULL (default), the end of observation is inferred as the maximum time recorded for each system (from both events and censoring records).
data	An optional data frame containing columns named id, time, and optionally event and end_time.
conf_level	Confidence level for bounds (default 0.95).

Details

The MCF at time t is estimated as:

\hat{M}(t) = \sum_{t_j \le t} \frac{d_j}{n_j}

where d_j is the number of events at time t_j and n_j is the number of systems still under observation at t_j. Variance is estimated as \hat{V}(t) = \sum_{t_j \le t} d_j / n_j^2.

The risk set n_j is determined by each system's exposure window. A system is considered at risk at time t_j if its end-of-observation time (from end_time, censoring records, or last event) is \ge t_j. Specifying end_time is important when systems were observed beyond their last event – without it, the MCF may overestimate the true recurrence rate because systems with no late events are assumed to have left observation at their last event time.

Value

An object of class mcf containing:

time	Unique event times.
mcf	MCF values at each event time.
variance	Variance estimates at each event time.
lower_bounds	Lower confidence bounds.
upper_bounds	Upper confidence bounds.
conf_level	Confidence level used.
n_systems	Number of distinct systems.
n_events	Total number of events.
end_times	Named vector of end-of-observation times per system.

See Also

Other Repairable Systems Analysis: exposure(), nhpp(), overlay_nhpp(), plot.exposure(), plot.mcf(), plot.nhpp(), plot.nhpp_predict(), predict_nhpp(), print.exposure(), print.mcf(), print.nhpp(), print.nhpp_predict()

Examples

# Basic usage (end of observation inferred from last event)
id <- c(1, 1, 1, 2, 2, 3, 3, 3, 3)
time <- c(100, 300, 500, 150, 400, 50, 200, 350, 600)
result <- mcf(id, time)
print(result)
plot(result, main = "Mean Cumulative Function")

# With explicit end-of-observation times (exposure-adjusted)
end_time <- c("1" = 800, "2" = 800, "3" = 800)
result2 <- mcf(id, time, end_time = end_time)
print(result2)

df <- data.frame(id = id, time = time)
result3 <- mcf(data = df)
print(result3)

ReliaGrowR documentation built on May 22, 2026, 5:07 p.m.