maple.aft {mable}

Mable fit of AFT model with given regression coefficients

Description

Maximum approximate profile likelihood estimation of Bernstein polynomial model in accelerated failure time based on interal censored event time data with given regression coefficients which are efficient estimates provided by other semiparametric methods.

Usage

maple.aft(
  formula,
  data,
  M,
  g,
  tau = NULL,
  p = 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, 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

the given d-vector of regression coefficients.

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'.

p

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

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 support or 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. 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 selected optimal degree m

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

• coefficients the given 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, 1 indicates either the EM-like iteration for finding maximum likelihood reached the maximum iteration for at least one m or the search of an optimal degree using change-point method reached the maximum candidate degree, 2 indicates both occured, and 0 indicates a successful completion.

• 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

mable.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, 
  res1<-maple.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=res1, which="likelihood")
  plot(x=res1, y=data.frame(x=0), which="survival", model='aft', type="l", col=1, 
      add=FALSE, main="Survival Function")
  plot(x=res1, 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)