cfc.tbasis {CFC}

Cause-specific competing-risk survival analysis in time denomination

Description

Constructing cumulative incidence and event-free probability functions from cause-specific survival probabilities evaluated at fixed time points.

Usage

cfc.tbasis(p1, p2, unity.tol = 1e-06, diff.tol = 0.01,
  diff.tol.policy = c("mean", "all"), check = TRUE)

Arguments

p1

Multi-dimensional array containing survival probabilities for cause 1 (i.e. exponential of the negative integral of hazard function). First dimension must correspond to time points at which probabilities are calculated. Elements with same time, but distributed in the space of remaining dimensions, are treated independently. These diemensions can correspond, e.g., to observations or samples (in Bayesian frameworks).

p2

Multi-dimensional array containing survival probabilities for cause 2. See note for p1.

unity.tol

Tolerance for difference of survival probabilities from 1.0 at time=0.0, which is the first 'row' of arrays p1 and p2. For example, for two-dimensional arrays, we need all(abs(p1[1,]-1.0) < unity.tol), and a similar condition for p2.

diff.tol

Tolerance for change in survival probabilities from one time point to the next. Large changes lead to higher errors during numerical integration.

diff.tol.policy

If "mean", then average change in survival probabilities are compared to diff.tol. If "all", each values is compared. The latter is more strict.

check

Boolean flag indicating whether or not to check probability arrays for validity. Current validity checks are: 1) ensuring all probabilities are between 0.0 and 1.0, 2) all probabilities at time=0.0 are equal to 1.0 (see unity.tol), 3) No changes in probabilities from one time point to next are too large (see diff.tol). Dimensional consistency between p1 and p2 is always checked regardless of the value of this flag.

Details

Assuming one-dimensional p1 and p2 for clarity, the algorithm calculates cumulative incidence function for cuase 1 using a recursive formula: ci1[n+1] = ci1[n] + dci1[n], where dci1[n] = 0.5*(p2[n] + p2[n+1])*(p1[n] - p1[n+1]). The increment in cumulative incidence function for cause 2 is similarly calculated, dci2[n] = 0.5*(p1[n] + p1[n+1])*(p2[n] - p2[n+1]). These equations guarantee that dci1[n] + dci2[n] = p1[n]*p2[n] - p1[n+1]*p2[n+1]. Event-free probability is simply calculated as codeefp[n] = p1[n]*p2[n]. Taken together, this numerical integration ensures that efp[n+1] - efp[n] + dci1[n] + dci2[n] = 0.

Value

If p1 and p2 are one-dimensional arrays (i.e. vectors), a matrix with columns named "ci1", "ci2" and "efp" is returned, representing the cummulative incidence functions for cause 1 and cause 2 and the event-free probability, evaluated at same time points as p1 and p2 are provided. If p1 and p2 are multi-dimensional arrays, a list is returned with elements "ci1", "ci2" and "efp", each one with the same interpretation, and all of the same dimensions as p1 and p2.

Note

The integration algorithm described above does not require knowledge of time step. (Alternatively, using hazard functions for integration would have required specification of time step.) Since p1 and p2 are integrals (followed by exponentiation) of cause-specific hazard functions, using them directly adds to robustness of numerical integration and avoids error accumulation. The returned cumulative incidence and event-free probabilities correspond to the same time points assumed for input cause-specific probabilities.

Author(s)

Mansour T.A. Sharabiani, Alireza S. Mahani

References

Mahani A.S. and Sharabiani M.T.A. (2019). Bayesian, and Non-Bayesian, Cause-Specific Competing-Risk Analysis for Parametric and Nonparametric Survival Functions: The R Package CFC. Journal of Statistical Software, 89(9), 1-29. doi:10.18637/jss.v089.i09

Prentice et al (1978). The analysis of failure times in the presence of competing risks. Biometrics, 541-554.

See Also

cfc.pbasis

Examples

## Not run: 

# prepare data for cause-specific competing-risk analysis
data(bmt)
bmt$status1 <- 1*(bmt$cause==1)
bmt$status2 <- 1*(bmt$cause==2)
f1 <- Surv(time, status1) ~ platelet + age + tcell
f2 <- Surv(time, status2) ~ platelet + age + tcell

# sample-based bayesian weibull regression
library(BSGW)
reg1 <- bsgw(f1, bmt, ordweib = TRUE, control = bsgw.control(iter = 500, burnin = 100, nskip = 50))
reg2 <- bsgw(f2, bmt, ordweib = TRUE, control = bsgw.control(iter = 500, burnin = 100, nskip = 50))

# prediction on a uniform grid of 100 time points
# (use first 50 observations for speed)
pred1 <- predict(reg1, newdata = bmt[1:50,], tvec = 100)
pred2 <- predict(reg2, newdata = bmt[1:50,], tvec = 100)

# permuting dimensions of survival objects to conform with cfc
S1 <- aperm(pred1$smp$S, c(2,1,3))
S2 <- aperm(pred2$smp$S, c(2,1,3))

# cause-specific competing risk analysis - time mode
my.cfc <- cfc.tbasis(S1, S2)

# calculating averages across observations (e.g. patients in the study)
my.summ <- summary(my.cfc, MARGIN = c(1,2))

# plotting mean CI and event-free functions
# as well as their sampled-based confidence intervals
plot(my.summ, t = pred1$tvec)



## End(Not run)