Flexible parametric survival analysis with the elastic net

Parametric survival analysis for proportional hazards regression and competing risk models.

This toolbox implements flexible parametric survival analysis for the proportional hazards regression model with optional right censoring. There are three main functions:

1. flexph(): parametric proportional hazards regression;
2. flexphreg(): parametric proportional hazards regression with elastic net regularization;
3. flexphcr(): parametric proportional hazards regression for competing risk data.

We have five kinds of parametric baseline hazard functions: exponential, Weibull, natural cubic splines, log-logistic and integrated splines (or i-splines); see below for mathematical details.

Let t = (t_1, ..., t_n) denote n time-to-event data points. We model the hazard function at time t by:

h(t) = h_0(t) exp(z'*beta), t > 0, beta \in R^p

where z is a vector of covariates and h_0(t) is the baseline hazard function to be discussed below. The cumulative hazard function is

H(t) = H_0(t) exp(z'*beta)

where H_0(t) is the cumulative baseline hazard. The survival function is then

S(t) = exp(-H(t)) = exp(-H_0(t))^exp(z'*beta) = S_0(t)^exp(z'*beta)

where S_0(t)=exp(-H_0(t)) is the baseline survival function. Following Royston and Mahesh (2002), we write the log of the cumulative hazard function H(t) as:

log H(t) = log H_0(t) + z'*beta = s(t) + z'*beta

where s(t) is a parametric log cumulative baseline hazard function which may depend on other parameters. The type of s(t) is selected with the argument modeltype. Possible options are:

(1) 'exp': exponential baseline hazard [exponential regression]

Log cumulative baseline hazard: s(t) = gamma_0 + log(t)

(2) 'natural': natural cubic splines

Baseline cumulative hazard is fitted using natural cubic splines with m internal knots (k_min,k_1,...k_m,k_max). Let x = log(t). We have:

s(t) = gam0 + gam1 x + gam2 v_1(x) + ...

where the j-th basis function v_j(x) is:

v_j(x) = max(0,x-k_j)^3 - lam_j max(0,x-k_min)^3 - (1-lam_j) max(0,x-k_max)^3

lam_j = (k_max - k_) / (k_max - k_min).

The argument m>0 determines the number of knots. For more technical details, please see Royston and Mahesh (2002).

(3) 'log-logistic: log-logistic baseline hazard

(4) 'weibull': Weibull baseline hazard [Weibull regression]

Log cumulative baseline hazard: s(t) = gamma_0 + gamma_1 x

(5) 'ispline': integrated splines.

All functions allo right censoring with the argument d = (d_1, ..., d_n) where d(i)=0 implying a censored data point.

For examples of usage, please see example[1-10].m.