Maximum approximate Bernstein/Beta likelihood estimation for accelerated failure time model based on interval censored data.
mable.aft(
formula,
data,
M,
g = NULL,
p = NULL,
tau = NULL,
x0 = NULL,
controls = mable.ctrl(),
progress = TRUE
)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
regression formula. Response must be cbind. See 'Details'.
a data frame containing variables in formula.
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.
a \(d\)-vector of regression coefficients, default is the zero vector.
an initial coefficients of Bernstein polynomial of degree m0,
default is the uniform initial.
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'.
a data frame specifying working baseline covariates on the right-hand-side of formula. See 'Details'.
Object of class mable.ctrl() specifying iteration limit
and other control options. Default is mable.ctrl.
if TRUE a text progressbar is displayed
Zhong Guan <zguan@iu.edu>
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 \(x\) 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 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.
Guan, Z. (2019) Maximum Approximate Likelihood Estimation in Accelerated Failure Time Model for Interval-Censored Data, arXiv:1911.07087.
maple.aft
# \donttest{
## Breast Cosmesis Data
g <- 0.41 #Hanson and Johnson 2004, JCGS
aft.res<-mable.aft(cbind(left, right)~treat, data=cosmesis, M=c(1, 30),
g=g, tau=100, x0=data.frame(treat="RCT"))
op<-par(mfrow=c(1,2), lwd=1.5)
plot(x=aft.res, which="likelihood")
plot(x=aft.res, y=data.frame(treat="RT"), which="survival", model='aft', type="l", col=1,
add=FALSE, main="Survival Function")
plot(x=aft.res, y=data.frame(treat="RCT"), 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)
# }
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