mable.aft {mable}

Mable fit of Accelerated Failure Time Model

Description

Maximum approximate Bernstein/Beta likelihood estimation for accelerated failure time model based on interval censored data.

Usage

mable.aft(
  formula,
  data,
  M,
  g = NULL,
  p = NULL,
  tau = NULL,
  x0 = NULL,
  controls = mable.ctrl(),
  progress = TRUE
)

Arguments

formula

regression formula. Response must be cbind. See 'Details'.

data

a dataset

M

a positive integer or a vector (m0, m1). If M = m0 or m0 = m1 = m, then m0 is a preselected degree. If m0 < m1 it specifies the set of consective candidate model degrees m0:m1 for searching an optimal degree, where m1-m0>3.

g

a d-vector of regression coefficients, default is the zero vector.

p

an initial coefficients of Bernstein polynomial of degree m0, default is the uniform initial.

tau

the right endpoint of the support or truncation interval [0,\tau) of the baseline density. Default is NULL (unknown), otherwise if tau is given then it is taken as a known value of \tau. See 'Details'.

x0

a working baseline covariate x_0, default is zero vector. See 'Details'.

controls

Object of class mable.ctrl() specifying iteration limit and other control options. Default is mable.ctrl.

progress

if TRUE a text progressbar is displayed

Details

Consider the accelerated failure time model with covariate for interval-censored failure time data: S(t|x) = S(t \exp(\gamma^T(x-x_0))|x_0), where x_0 is a baseline covariate. Let f(t|x) and F(t|x) = 1-S(t|x) be the density and cumulative distribution functions of the event time given X = x, respectively. Then f(t|x_0) on a truncation interval [0, \tau] can be approximated by f_m(t|x_0; p) = \tau^{-1}\sum_{i=0}^m p_i\beta_{mi}(t/\tau), where p_i\ge 0, i = 0, \ldots, m, \sum_{i=0}^mp_i=1, \beta_{mi}(u) is the beta denity with shapes i+1 and m-i+1, and \tau is larger than the largest observed time, either uncensored time, or right endpoint of interval/left censored, or left endpoint of right censored time. So we can approximate S(t|x_0) on [0, \tau] by S_m(t|x_0; p) = \sum_{i=0}^{m} p_i \bar B_{mi}(t/\tau), where \bar B_{mi}(u) is the beta survival function with shapes i+1 and m-i+1.

Response variable should be of the form cbind(l, u), where (l,u) is the interval containing the event time. Data is uncensored if l = u, right censored if u = Inf or u = NA, and left censored data if l = 0. The truncation time tau and the baseline x0 should be chosen so that S(t|x)=S(t \exp(\gamma^T(x-x_0))|x_0) on [\tau, \infty) is negligible for all the observed x.

The search for optimal degree m stops if either m1 is reached or the test for change-point results in a p-value pval smaller than sig.level.

Value

A list with components

• m the given or selected optimal degree m

• p the estimate of p = (p_0, ..., p_m), the coefficients of Bernstein polynomial of degree m

• coefficients the estimated regression coefficients of the AFT model

• SE the standard errors of the estimated regression coefficients

• z the z-scores of the estimated regression coefficients

• mloglik the maximum log-likelihood at an optimal degree m

• tau.n maximum observed time \tau_n

• tau right endpoint of trucation interval [0, \tau)

• x0 the working baseline covariates

• egx0 the value of e^{\gamma^T x_0}

• convergence an integer code: 0 indicates a successful completion; 1 indicates that the search of an optimal degree using change-point method reached the maximum candidate degree; 2 indicates that the matimum iterations was reached for calculating \hat p and \hat\gamma with the selected degree m, or the divergence of the last EM-like iteration for p or the divergence of the last (quasi) Newton iteration for \gamma; 3 indicates 1 and 2.

• delta the final delta if m0 = m1 or the final pval of the change-point for searching the optimal degree m;

and, if m0<m1,

• M the vector (m0, m1), where m1 is the last candidate when the search stoped

• lk log-likelihoods evaluated at m \in \{m_0, \ldots, m_1\}

• lr likelihood ratios for change-points evaluated at m \in \{m_0+1, \ldots, m_1\}

• pval the p-values of the change-point tests for choosing optimal model degree

• chpts the change-points chosen with the given candidate model degrees

Author(s)

Zhong Guan <[e-mail]>

References

Guan, Z. (2019) Maximum Approximate Likelihood Estimation in Accelerated Failure Time Model for Interval-Censored Data, arXiv:1911.07087.

See Also

maple.aft

Examples

## Breast Cosmesis Data
  bcos=cosmesis
  bcos2<-data.frame(bcos[,1:2], x=1*(bcos$treat=="RCT"))
  g <- 0.41 #Hanson and  Johnson 2004, JCGS
  aft.res<-mable.aft(cbind(left, right)~x, data=bcos2, M=c(1, 30), g=g, tau=100, x0=1)
  op<-par(mfrow=c(1,2), lwd=1.5)
  plot(x=aft.res, which="likelihood")
  plot(x=aft.res, y=data.frame(x=0), which="survival", model='aft', type="l", col=1, 
      add=FALSE, main="Survival Function")
  plot(x=aft.res, y=data.frame(x=1), which="survival", model='aft', lty=2, col=1)
  legend("bottomleft", bty="n", lty=1:2, col=1, c("Radiation Only", "Radiation and Chemotherapy"))
  par(op)