This function calculates the the non-parametric maximum likelihood estimate for the distribution from interval
censored data using the self-consistent estimator, so the associated survival distribution generalizes
the Kaplan-Meier estimate to interval censored data. Formulas using Surv are allowed similar to survfit.
# S3 method for formula
icfit(formula, data, ...)# S3 method for default
icfit(L, R,initfit =NULL, control=icfitControl(), Lin=NULL, Rin=NULL, conf.int=FALSE, ...)
An object of class icfit (same as icsurv class, see details).
There are 4 methods for this class: plot.icfit, print.icfit, summary.icfit, and [.icfit. The last method
pulls out individual fits when the right side of the formula of the icfit call was a factor.
A list with elements:
this is the n by k matrix of indicator functions, NULL if more than one strata, not printed by default
a named numeric vector of length of pf vector (jumps in NPMLEs) for each strata, if one strata observation named NPMLE
this is max(d + u - n), see Gentleman and Geyer, 1994
number of iterations
vector of estimated probabilities of the distribution. if more than one strata, vectors are concatenated
2 by k matrix, where the ith column defines an interval corresponding to the probability, pf[i]
a logical, TRUE if normal convergence
character text message on about convergence
logical denoting whether any of the Turnbull intervals were set to zero
if conf.int=TRUE included as a list of lists for each stratum, each one having elements time, lower, upper, confMethod, conf.level
numeric vector of left endpoints of censoring interval (equivalent to first element of Surv when type='interval2', see details)
numeric vector of right endpoints of censoring interval (equivalent to second element of Surv function when type='interval2', see details)
an initial estimate as an object of class icfit or icsurv, or a character vector of the name of the function used to calculate the initial estimate (see details)
list of arguments for controling algorithm (see icfitControl)
logical vector, should L be included in the interval? (see details)
logical vector, should R be included in the interval? (see details)
a formula with response a numeric vector (which assumes no censoring) or Surv object
the right side of the formula may be 1 or a factor (which produces separate fits for each level).
an optional matrix or data frame containing the variables in the formula. By default the variables are taken from environment(formula).
logical, estimate confidence interval? For setting conf.level, etc see icfitControl. (May take very long, see Warning)
values passed to other functions
The confidence interval method can be very time consuming because it uses a modified bootstrap and the NPMLE is recalculated for each replication. That is why the default only uses 200 bootstrap replications. A message gives a crude estimate of how long the confidence interval calculation will take (it calculates a per replication value by averaging the time of the first 10 replications), but that estimate can be off by 100 percent or more because the time to calculate each bootstrap replication is quite variable.
Michael P. Fay
The icfit function fits the nonparametric maximum likelihood estimate (NPMLE) of the
distribution function for interval censored data. In the default case (when Lin=Rin=NULL)
we assume there are n (n=length(L)) failure times, and the ith one is in the interval
between L[i] and R[i]. The default is not to include L[i] in the interval unless L[i]=R[i],
and to include R[i] in the interval unless R[i]=Inf. When Lin and Rin are not NULL they describe
whether to include L and R in the associated interval. If either Lin or Rin is length 1 then it is
repeated n times, otherwise they should be logicals of length n.
The algorithm is basically an EM-algorithm applied to
interval censored data (see Turnbull, 1976); however
first we can define a set of intervals (called the Turnbull intervals)
which are the only intervals where the NPMLE may change. The Turnbull intervals are also called the
innermost intervals, and are the result of the primary reduction (see Aragon and
Eberly, 1992). The starting distribution for the E-M algorithm is given by initfit, which may be either
(1) NULL, in which case a very simple and quick starting distribution is used (see code), (2) a character vector
describing a function with inputs, L,R, Lin, Rin, and A, see for example initcomputeMLE, (3)
a list giving pf and intmap values, e.g., an icfit object. If option (2) is tried and results in an error then
the starting distribution reverts to the one used with option (1).
Convergence is defined when the maximum
reduced gradient is less than epsilon (see icfitControl), and the
Kuhn-Tucker conditions are approximately met,
otherwise a warning will result. (see Gentleman and
Geyer, 1994). There are other faster algorithms (for example see
EMICM in the package
Icens.
The output is of class icfit which is identical to the icsurv class of the
Icens package when there is only one group for which a distribution is needed.
Following that class, there is an intmap element which gives the bounds
about which each drop in the NPMLE survival function can occur.
Since the classes icfit and icsurv are so closely related, one can directly
use of initial (and faster) fits from the Icens package as input in
initfit. Note that when using a non-null initfit, the Lin and Rin values of the
initial fit are ignored. Alternatively, one may give the name of the function used to calculate the initial fit.
The function is assumed to input the transpose of the A matrix (called A in the Icens package). Options can be passed
to initfit function as a list using the initfitOpts variable in icfitControl.
The advantage of the icfit function over those in Icens package is that it allows a call similar
to that used in survfit of the survival package so that different groups may be
plotted at the same time with similar calls.
An icfit object prints as a list (see value below). A print function prints output as a list
except suppresses printing of A matrix. A summary function prints the
distribution (i.e., probabilities and the intervals where those
probability masses are known to reside) for each group in the icfit object. There is also
a plot method, see plot.icfit.
For additional references and background see Fay and Shaw (2010).
The confidence interval method is a modified bootstrap. This can be very time consuming, see warning. The method uses a percentile bootstrap confidence interval with default B=200
replicates (see icfitControl), with modifications that prevent lower intervals of 1 and upper intervals of 0. Specifically, if there are
n observations total, then at any time the largest value of the lower interval for survival is binom.test(n,n,conf.level=control()$conf.level)$conf.int[1] and analogously
the upper interval bounds using binom.test(0,n). The output (CI element of returned list) gives confidence intervals just before and just after each
assessment time (as defined by icfitControl$timeEpsilon).
Aragon, J and Eberly, D (1992). On convergence of convex minorant algorithms for distribution estimation with interval-censored data. J. of Computational and Graphical Statistics. 1: 129-140.
Fay, MP and Shaw, PA (2010). Exact and Asymptotic Weighted Logrank Tests for Interval Censored Data: The interval R package. Journal of Statistical Software. tools:::Rd_expr_doi("10.18637/jss.v036.i02"). 36 (2):1-34.
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. J. R. Statist. Soc. B 38, 290-295.
ictest, EMICM
data(bcos)
icout<-icfit(Surv(left,right,type="interval2")~treatment, data=bcos)
plot(icout)
## can pick out just one group
plot(icout[1])
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