CoxMM {MMAD}

MM algorithm based on AD technology for Cox model

Description

Let T_i, C_i and X_i = (x_{i1},\cdots, x_{iq})^T denote the, survival time, the censoring time and a q dimension vector of coefficients for the i-th individual, respectively. And assume the censoring time C_i is independent of the survival time T_i are mutually independent, and I_i = I(T_{i} \leqslant C_{i}) is the censoring indicator. Then the instantaneous hazard rate function of T_i is

\lambda(t|X_i)=\lambda_{0}(t) \exp(X_{i}^{T} \beta)

where \lambda_{0}(.) is a baseline hazard rate and \beta = (\beta_1, \cdots, \beta_q)^{T} is a vector of regression parameters. We denote \Lambda as the accumulative hazard rate. Then the observed data likelihood function is

L(\alpha | Y_{obs}) = \prod_{i=1}^n (\lambda_{0}(t_i) \exp(X_{i}^{T} \beta))^{I_i} \exp(-\Lambda(t_i) \exp(X_{i}^{T} \beta))

where \alpha = (\beta, \Lambda). The CoxMM function is used to calculate the Cox model.

Usage

CoxMM(formula, data, beta = NULL, Maxiter = 2000, convergence = 1e-06, ...)

Arguments

formula

A formula object, which contains on the left hand side an object of the type Surv and on the right hand side is the terms, e.g. formula=Surv(time, status) ~ x.

data

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

beta

A vector of unknown regression parameters, default is NULL. If is NULL, then make all beta=0 during calculation.

Maxiter

The maximum number of iterations is specified by default as 2000.

convergence

Specify the convergence criterion, the default is 1e-6.

...

Additional arguments

Details

The CoxMM function is used to calculate the Cox model using MM algorithms based on AD technology. EM algorithms rely on the fact that, after profiling out the nonparametric component \Lambda, the resulting function is concave. However, when this assumption does not hold, maximizing the resulting function using Newton’s method becomes difficult, especially when there are a large number of covariates. MM algorithms can avoid the concavity requirement and bypass the need for Newton method and matrix inversion.

Value

An object of class CoxMM that contains the following fields: the Time, total amount of observations, total number of failure events, the variable name, the \beta, the \lambda, the \Lambda, convergence result, the log likelihood value, the standard deviation of the estimated \beta, the likelihood-based 95% confidence interval for the \beta.

References

D.R. Cox.(1972). 'Regression models and life tables.' Journal of the Royal Statistical Society(Series B) 34(2), 187-220.

Zhang L.L. and Huang X.F.(2022). 'On MM algorithms for Cox model with right-censored data.' In International Conference on Cloud Computing, Internet of Things, and Computer Applications (CICA 2022) 12303, 29-38.

Examples

library(survival)
CoxMM(Surv(time, status) ~ age + sex, lung)