IC2MM {MMAD}

MM algorithm based on the AD method for case II interval-censored data

Description

The IC2MM function is used to calculate the case II interval-censored data model. A failure time study that consists of n independent subjects from a homogeneous population with survival function S_{(t)}. Let T_{i} denote the survival time, and i=1, \ldots, n. Suppose that interval-censored data on the T_i are observed and given by

Y_{obs} = \{ (L_{i}, R_{i}]; i=1, \ldots, n \}

where T_i \in (L_{i}, R_{i}] . Let \{s_i \}_{j=0}^{m} denote the unique ordered elements of {0, L_{i}, R_{i}, i=1, \ldots, n } . Take \alpha_{ij} = I(s_{j} \in (L_{i}, R_{i}] ) and p_{j}= S(s_{j-1}) - S(s_{j}), j= 1, \ldots, m . The log-likelihood function is

\ell( {p} | Y_{obs}) = \sum_{i=1}^{n} \log (S(L_{i}) - S(R_{i}) ) = \sum_{i=1}^{n} \log \left( \sum_{j=1}^{m} \alpha_{ij} p_{j} \right)

where {p} = (p_1, \ldots, p_m)^{T} and \sum_{j=1}^{m} p_{j} = 1 , p_{j} \geqslant 0.

Usage

IC2MM(formula, data, ...)

Arguments

formula

A formula object, which contains on the left hand side an object of type = 'interval2' of the type Surv e.g. formula=Surv(L,R, type = 'interval2') ~ 1

data

A data.frame in which to interpret the variables named in the formula.

...

Additional arguments, e.g. control=IC2Control()

Details

The IC2MM function allows the distributions for multiple strata of dataset to be stored as one IC2 object, e.g. data=bcos.

Value

An object of class IC2MM that contains the following fields: error: convergence result; strata: dimensions of each df_tab; s: unique ordered elements of {0, L_{i}, R_{i}, Inf} , if more than one strata, elements are concatenated; S: the survival function, if more than one strata, values are concatenated; df_tab: the dataframe of survival intervals and survival probabilities for each interval, if more than one strata, dataframes are concatenated.

References

Tian G.L., Huang X.F. and Xu, J.(2019). 'An assembly and decomposition approach for constructing separable minorizing functions in a class of MM algorithms.' Statistica Sinica 29(2), 961-982.

See Also

IC2Pro

Examples

library(survival)
L <- c(1.5, 0.1, 1.5, 0.5, 0.4, 0.2, 0.9, 0.2, 0.08, 1.9)
R <- c(2.1, 2.9, 2.7, 1.9, 1.3, 1.4, 2.3, 0.5, 1.5, 4.6 )
data <- data.frame(L, R)
IC2MM(Surv(L,R, type = 'interval2') ~ 1, data )

IC2MM(Surv(L,R, type = 'interval2') ~ 1, data, control=IC2Control(Pdigits=2) )