coxtp {surtvep}

fit a Cox non-proportional hazards model with P-spline or Smoothing-spline, with penalization tuning parameter chosen by information criteria or cross-validation

Description

Fit a Cox non-proportional hazards model via penalized maximum likelihood.

Usage

coxtp(
  event,
  z,
  time,
  strata = NULL,
  penalty = "Smooth-spline",
  nsplines = 8,
  lambda = c(0.1, 1, 10),
  degree = 3L,
  knots = NULL,
  ties = "Breslow",
  tol = 1e-06,
  iter.max = 20L,
  method = "ProxN",
  gamma = 1e+08,
  btr = "dynamic",
  tau = 0.5,
  stop = "ratch",
  parallel = FALSE,
  threads = 2L,
  fixedstep = FALSE
)

Arguments

event

failure event response variable of length nobs, where nobs denotes the number of observations. It should be a vector containing 0 or 1.

z

input covariate matrix, with nobs rows and nvars columns; each row is an observation.

time

observed event times, which should be a vector with non-negative values.

strata

a vector of indicators for stratification. Default = NULL (i.e. no stratification group in the data), an unstratified model is implemented.

penalty

a character string specifying the spline term for the penalized Newton method. This term is added to the log-partial likelihood, and the penalized log-partial likelihood serves as the new objective function to control the smoothness of the time-varying coefficients. Default is P-spline. Three options are P-spline, Smooth-spline and NULL. If NULL, the method will be the same as coxtv (unpenalized time-varying effects models) and lambda (defined below) will be set as 0.

P-spline stands for Penalized B-spline. It combines the B-spline basis with a discrete quadratic penalty on the difference of basis coefficients between adjacent knots. When lambda goes to infinity, the time-varying effects are reduced to be constant.

Smooth-spline refers to the Smoothing-spline, the derivative-based penalties combined with B-splines. See degree for different choices. When degree=3, we use the cubic B-spline penalizing the second-order derivative, which reduces the time-varying effect to a linear term when lambda goes to infinity. When degree=2, we use the quadratic B-spline penalizing first-order derivative, which reduces the time-varying effect to a constant when lambda goes to infinity. See Wood (2017) for details.

If P-spline or Smooth-spline, then lambda is initialized as a sequence (0.1, 1, 10). Users can modify lambda. See details in lambda.

nsplines

number of basis functions in the splines to span the time-varying effects. The default value is 8. We use the R function splines::bs to generate the B-splines.

lambda

a user-specified lambda sequence as the penalization coefficients in front of the spline term specified by penalty. This is the tuning parameter for penalization. The function IC can be used to select the best tuning parameter based on the information criteria. Alternatively, cross-validation can be used via the cv.coxtp function. When lambda is 0, Newton method without penalization is fitted.

degree

degree of the piecewise polynomial for generating the B-spline basis functions—default is 3 for cubic splines. degree = 2 results in the quadratic B-spline basis functions.

If the penalty is P-spline or NULL, degree's default value is 3.

If the penalty is Smooth-spline, degree's default value is 2.

knots

the internal knot locations (breakpoints) that define the B-splines. The number of the internal knots should be nsplines-degree-1. If NULL, the locations of knots are chosen as quantiles of distinct failure time points. This choice leads to more stable results in most cases. Users can specify the internal knot locations by themselves.

ties

a character string specifying the method for tie handling. If there are no tied events, the methods are equivalent. By default "Breslow" uses the Breslow approximation, which can be faster when many ties are present. If ties = "none", no approximation will be used to handle ties.

tol

tolerance used for stopping the algorithm. See details in stop below. The default value is 1e-6.

iter.max

maximum iteration number if the stopping criterion specified by stop is not satisfied. The default value is 20.

method

a character string specifying whether to use Newton method or proximal Newton method. If Newton then Hessian is used, while the default method "ProxN" implements the proximal Newton which can be faster and more stable when there exists ill-conditioned second-order information of the log-partial likelihood. See details in Wu et al. (2022).

gamma

parameter for proximal Newton method "ProxN". The default value is 1e8.

btr

a character string specifying the backtracking line-search approach. "dynamic" is a typical way to perform backtracking line-search. See details in Convex Optimization by Boyd and Vandenberghe (2004). "static" limits Newton's increment and can achieve more stable results in some extreme cases, such as ill-conditioned second-order information of the log-partial likelihood, which usually occurs when some predictors are categorical with low frequency for some categories. Users should be careful with static, as this may lead to under-fitting.

tau

a positive scalar used to control the step size inside the backtracking line-search. The default value is 0.5.

stop

a character string specifying the stopping rule to determine convergence. "incre" means we stop the algorithm when Newton's increment is less than the tol. See details in Convex Optimization (Chapter 10) by Boyd and Vandenberghe (2004). "relch" means we stop the algorithm when the (loglik(m)-loglik(m-1))/(loglik(m)) is less than the tol, where loglik(m) denotes the log-partial likelihood at iteration step m. "ratch" means we stop the algorithm when (loglik(m)-loglik(m-1))/(loglik(m)-loglik(0)) is less than the tol. "all" means we stop the algorithm when all the stopping rules ("incre", "relch", "ratch") are met. The default value is ratch. If iter.max is achieved, it overrides any stop rule for algorithm termination.

parallel

if TRUE, then the parallel computation is enabled. The number of threads in use is determined by threads.

threads

an integer indicating the number of threads to be used for parallel computation. The default value is 2. If parallel is false, then the value of threads has no effect.

fixedstep

if TRUE, the algorithm will be forced to run iter.max steps regardless of the stopping criterion specified.

Details

The sequence of models implied by lambda.spline is fit by the (proximal) Newton method. The objective function is

loglik - P_{\lambda},

where P_{\lambda} is a penalty matrix for P-spline or Smooth-spline. The \lambda is the tuning parameter (See details in lambda). Users can define the initial sequence. The function IC below provides different information criteria to choose the tuning parameter \lambda. Another function cv.coxtp uses the cross-validation to choose the tuning parameter.

Value

A list of objects with S3 class "coxtp". The length is the same as that of lambda; each represents the model output with each value of the tuning parameter lambda.

call

the call that produced this object.

beta

the estimated time-varying coefficient for each predictor at each unique time. It is a matrix of dimension len_unique_t by nvars, where len_unique_t is the length of unique observed event times.

bases

the basis matrix used in model fitting. If ties="none", the dimension of the basis matrix is nvars by nsplines; if ties="Breslow", the dimension is len_unique_t by nsplines. The matrix is constructed using the bs::splines function.

ctrl.pts

estimated coefficient of the basis matrix of dimension nvars by nsplines. Each row represents a covariate's coefficient on the nsplines-dimensional basis functions.

Hessian

the Hessian matrix of the log-partial likelihood, of which the dimension is nsplines * nvars by nsplines * nvars.

internal.knots

the internal knot locations (breakpoints) that define the B-splines.

nobs

number of observations.

penalty

the spline term penalty specified by user.

theta.list

the history of ctrl.pts of length m (the length of algorithm iterations), including ctrl.pts for each algorithm iteration.

VarianceMatrix

the variance matrix of the estimated coefficients of the basis matrix, which is the inverse of the negative Hessian matrix.

References

Boyd, S., and Vandenberghe, L. (2004) Convex optimization. Cambridge University Press.

Gray, R. J. (1992) Flexible methods for analyzing survival data using splines, with applications to breast cancer prognosis. Journal of the American Statistical Association, 87(420): 942-951.

Gray, R. J. (1994) Spline-based tests in survival analysis. Biometrics, 50(3): 640-652.

Luo, L., He, K., Wu, W., and Taylor, J. M. (2023) Using information criteria to select smoothing parameters when analyzing survival data with time-varying coefficient hazard models. Statistical Methods in Medical Research, in press.

Perperoglou, A., le Cessie, S., and van Houwelingen, H. C. (2006) A fast routine for fitting Cox models with time varying effects of the covariates. Computer Methods and Programs in Biomedicine, 81(2): 154-161.

Wu, W., Taylor, J. M., Brouwer, A. F., Luo, L., Kang, J., Jiang, H., and He, K. (2022) Scalable proximal methods for cause-specific hazard modeling with time-varying coefficients. Lifetime Data Analysis, 28(2): 194-218.

Wood, S. N. (2017) P-splines with derivative based penalties and tensor product smoothing of unevenly distributed data. Statistics and Computing, 27(4): 985-989.

See Also

IC, cv.coxtp plot, get.tvcoef and baseline.

Examples

data(ExampleData)
z <- ExampleData$z
time  <- ExampleData$time
event <- ExampleData$event

lambda  = c(0,1)
fit   <- coxtp(event = event, z = z, time = time, lambda=lambda)