maple.po: Mable fit of the PO model with given regression coefficients

In mable: Maximum Approximate Bernstein/Beta Likelihood Estimation

Mable fit of the PO model with given regression coefficients

Description

Maximum approximate profile likelihood estimation of Bernstein polynomial model in proportional odds rate regression based on interal censored event time data with given regression coefficients and select an optimal degree m if coefficients are efficient estimates provided by other semiparametric methods.

Usage

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

Arguments

formula	regression formula. Response must be cbind. See 'Details'.
data	a data frame containing variables in formula.
M	a positive integer or a vector (m0, m1). If M = m or m0 = m1 = m, then m 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	right endpoint of support [0, \tau] must be greater than or equal to the maximum observed time
x0	a data frame specifying working baseline covariates on the right-hand-side of formula. 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 Generalized PO model with covariate for interval-censored failure time data: S(t|x) = S(t|x_0)^{\exp(\gamma'(x-x_0))}, where x_0 satisfies \gamma'(x-x_0)\ge 0, where x and x_0 may contain dummy variables and interaction terms. The working baseline x0 in arguments contains only the values of terms excluding dummy variables and interaction terms in the right-hand-side of formula. Thus g is the initial guess of the coefficients \gamma of x-x_0 and could be longer than x0. 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^{-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 the right endpoint of support interval. 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), i = 0, \ldots, m, 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 are 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'(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. et al. (???) Maximum Approximate Bernstein Likelihood Estimation in Generalized Proportional Odds Regression Model for Interval-Censored Data

See Also

mable.po

Examples

## Simulated Weibull data
require(icenReg)
set.seed(111)
simdata<-simIC_weib(100, model = "po", inspections = 2, 
   inspectLength = 2.5, prob_cen=1)
sp<-ic_sp(cbind(l, u) ~ x1 + x2, data = simdata, model="po") 
gt<--sp$coefficients
res0<-maple.po(cbind(l, u) ~ x1 + x2, data = simdata, M=c(1,20), g=gt, tau=6)
op<-par(mfrow=c(1,2))
plot(res0,  which=c("likelihood","change-point"))
par(op)
res1<-mable.po(cbind(l, u) ~ x1 + x2, data = simdata, M=c(1,20),
   g=gt, tau=6, x0=data.frame(x1=max(simdata$x1),x2=-1))
op<-par(mfrow=c(2,2))    
plot(res1,  which=c("likelihood","change-point")) 
plot(res0, y=data.frame(x1=0,x2=0), which="density", add=FALSE, type="l",
    xlab="Time", main="Desnity Function")
plot(res1, y=data.frame(x1=0,x2=0), which="density", add=TRUE, lty=2, col=4)
lines(xx<-seq(0, 7, len=512), dweibull(xx, 2,2), lty=3, col=2, lwd=1.5)
legend("topright", bty="n", lty=1:3, col=c(1,4,2), c(expression(hat(f)[0]),
    expression(tilde(f)[0]), expression(f[0])))
plot(res0, y=data.frame(x1=0,x2=0), which="survival", add=FALSE, type="l",
    xlab="Time", main="Survival Function")
plot(res1, y=data.frame(x1=0,x2=0), which="survival", add=TRUE, lty=2, col=4)
lines(xx, 1-pweibull(xx, 2, 2), lty=2, col=2)
legend("topright", bty="n", lty=1:3, col=c(1,4,2), c(expression(hat(S)[0]),
    expression(tilde(S)[0]), expression(S[0])))
par(op)

mable documentation built on Oct. 1, 2024, 9:06 a.m.