maple.ph {mable}

Mable fit of the PH model with given regression coefficients

Description

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

Usage

maple.ph(
  formula,
  data,
  M,
  g,
  pi0 = NULL,
  p = NULL,
  tau = Inf,
  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

pi0

Initial guess of \pi(x_0) = F(\tau_n|x_0). Without right censored data, pi0 = 1. See 'Details'.

p

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

tau

right endpoint of support [0, \tau) must be greater than or equal to the maximum observed time

x0

a working baseline covariate. 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 Cox's PH model with covariate for interval-censored failure time data: S(t|x) = S(t|x_0)^{\exp(\gamma^T(x-x_0))}, where x_0 satisfies \gamma^T(x-x_0)\ge 0. 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 [0,\tau_n] can be approximated by f_m(t|x_0; p) = \tau_n^{-1}\sum_{i=0}^m p_i\beta_{mi}(t/\tau_n), where p_i \ge 0, i = 0, \ldots, m, \sum_{i=0}^mp_i = 1-p_{m+1}, \beta_{mi}(u) is the beta denity with shapes i+1 and m-i+1, and \tau_n is 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_n] by S_m(t|x_0; p) = \sum_{i=0}^{m+1} p_i \bar B_{mi}(t/\tau_n), where \bar B_{mi}(u), i = 0, \ldots, m, is the beta survival function with shapes i+1 and m-i+1, \bar B_{m,m+1}(t) = 1, p_{m+1} = 1-\pi(x_0), and \pi(x_0) = F(\tau_n|x_0). For data without right-censored time, p_{m+1} = 1-\pi(x_0) = 0.

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 associated covariate contains d columns. The baseline x0 should chosen so that \gamma^T(x-x_0) is nonnegative 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 class 'mable_reg' object, a list with components

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

• m the selected optimal degree m

• p the estimate of p = (p_0, \dots, p_m,p_{m+1}), the coefficients of Bernstein polynomial of degree m

• coefficients the given regression coefficients of the PH model

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

• 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\}

• tau.n maximum observed time \tau_n

• tau right endpoint of support [0, \tau)

• x0 the working baseline covariates

• egx0 the value of e^{\gamma'x_0}

• convergence an integer code. 0 indicates successful completion(the iteration is convergent). 1 indicates that the maximum candidate degree had been reached in the calculation;

• delta the final convergence criterion for EM iteration;

• chpts the change-points among the candidate degrees;

• pom the p-value of the selected optimal degree m as a change-point;

Author(s)

Zhong Guan <[e-mail]>

References

Guan, Z. (2019) Maximum Approximate Bernstein Likelihood Estimation in Proportional Hazard Model for Interval-Censored Data, arXiv:1906.08882 .

See Also

mable.ph

Examples

 ## Simulated Weibull data
   require(icenReg) 
   set.seed(123)
   simdata<-simIC_weib(70, inspections = 5, inspectLength = 1)
   sp<-ic_sp(cbind(l, u) ~ x1 + x2, data = simdata)
   res0<-maple.ph(cbind(l, u) ~ x1 + x2, data = simdata, M=c(2,20), 
        g=sp$coefficients, tau=7)
   op<-par(mfrow=c(1,2))
   plot(res0,  which=c("likelihood","change-point"))
   par(op)
   res1<-mable.ph(cbind(l, u) ~ x1 + x2, data = simdata, M=res0$m, 
      g=c(.5,-.5), tau=7)
   op<-par(mfrow=c(1,2))
   plot(res1, y=data.frame(x=0, x2=0), which="density", add=FALSE, type="l", 
       xlab="Time", main="Desnity Function")
   lines(xx<-seq(0, 7, len=512), dweibull(xx, 2,2), lty=2, col=2)
   legend("topright", bty="n", lty=1:2, col=1:2, c("Estimated","True"))
   plot(res1, y=data.frame(x=0, x2=0), which="survival", add=FALSE, type="l", 
       xlab="Time", main="Survival Function")
   lines(xx, 1-pweibull(xx, 2, 2), lty=2, col=2)
   legend("topright", bty="n", lty=1:2, col=1:2, c("Estimated","True"))
   par(op)