M-spline survival distribution

Probability density, distribution, quantile, random generation, hazard, cumulative hazard, mean and restricted mean functions for the M-spline time-to-event model. This can optionally be combined with a cure probability, and / or with a known background hazard trajectory that is a piecewise-constant function of time.

Usage

psurvmspline(
  q,
  alpha,
  coefs,
  knots,
  degree = 3,
  lower.tail = TRUE,
  log.p = FALSE,
  pcure = 0,
  offsetH = 0,
  backhaz = NULL,
  bsmooth = TRUE
)

Hsurvmspline(
  x,
  alpha,
  coefs,
  knots,
  degree = 3,
  log = FALSE,
  pcure = 0,
  offsetH = 0,
  backhaz = NULL,
  bsmooth = TRUE
)

hsurvmspline(
  x,
  alpha,
  coefs,
  knots,
  degree = 3,
  log = FALSE,
  pcure = 0,
  offseth = 0,
  backhaz = NULL,
  bsmooth = TRUE
)

dsurvmspline(
  x,
  alpha,
  coefs,
  knots,
  degree = 3,
  log = FALSE,
  pcure = 0,
  offseth = 0,
  offsetH = 0,
  backhaz = NULL,
  bsmooth = TRUE
)

qsurvmspline(
  p,
  alpha,
  coefs,
  knots,
  degree = 3,
  lower.tail = TRUE,
  log.p = FALSE,
  pcure = 0,
  offsetH = 0,
  backhaz = NULL,
  bsmooth = TRUE
)

rsurvmspline(
  n,
  alpha,
  coefs,
  knots,
  degree = 3,
  pcure = 0,
  offsetH = 0,
  backhaz = NULL,
  bsmooth = TRUE
)

rmst_survmspline(
  t,
  alpha,
  coefs,
  knots,
  degree = 3,
  pcure = 0,
  offsetH = 0,
  backhaz = NULL,
  bsmooth = TRUE
)

mean_survmspline(
  alpha,
  coefs,
  knots,
  degree = 3,
  pcure = 0,
  offsetH = 0,
  backhaz = NULL,
  bsmooth = TRUE
)

Arguments

alpha

Log hazard scale parameter.

coefs

Spline basis coefficients. These should sum to 1, otherwise they are normalised internally to sum to 1. Supplied either as a vector with one element per basis term, or a matrix with one column per basis term, and rows for alternative values of the coefficients (in vectorised usage of this function). If an array is supplied, it is collapsed into a matrix with number of columns equal to the final dimension of the array.

knots

Locations of knots on the axis of time, supplied in increasing order. These include the two boundary knots.

In vectorised usage of these functions, the knots and degree must be the same for all alternative times and parameter values.

degree

Spline polynomial degree. Can only be changed from the default of 3 if bsmooth is FALSE.

lower.tail

logical; if TRUE (default), probabilities are \(P(X \le x)\), otherwise, \(P(X > x)\).

pcure

Probability of "cure", which defaults to zero. If this is non-zero, this defines a "mixture cure" version of the M-spline model.

offsetH

Constant to be added to the cumulative hazard.

backhaz

A data frame that defines the background hazard as a piecewise-constant function of time. This should have two columns, named "time" and "hazard". Each row gives the "background" hazard between the specified time and the next time. The first element of "time" should be 0, and the final row specifies the hazard at all times greater than the last element of "time".

bsmooth

If TRUE then the function is constrained to also have zero derivative and second derivative at the boundary.

x, q, t

Vector of times.

log, log.p

Return log density or probability.

offseth

Constant to be added to the hazard, e.g. representing a "background" risk of death from causes other than the cause of interest.

p

Vector of probabilities.

n

Number of random numbers to simulate.

Value

dsurvmspline gives the density, psurvmspline gives the distribution function, hsurvmspline gives the hazard and Hsurvmspline gives the cumulative hazard.

qsurvmspline gives the quantile function, which is computed by numerical inversion.

rsurvmspline generates random survival times by using qsurvmspline on a sample of uniform random numbers.

Details

These are the same as the M-splines used to model survival data in rstanarm, except that an additional assumption is made that the hazard is constant beyond the boundary knots at its value at the boundary. This gives a continuous but non-smooth function.

The "cure" model can be interpreted in two different ways. These result in identical probability distribution functions for the event time, hence they are indistinguishable from data:

(a) a model where everyone follows the same hazard trajectory that is decreasing through time, with a higher rate of decrease for higher pcure.

(b) a model where a person either has a constant zero hazard at all times, or a hazard that follows a parametric model (M-spline in this case). The probability that a person has a zero hazard is pcure. This is the "mixture model" interpretation.

In the "background hazard" model, the overall hazard is defined as a sum of the background hazard and a cause-specific hazard. The cause-specific hazard is modelled with the M-spline model, and the background hazard is assumed to be a known piecewise-constant function defined by backhaz.

If both backhaz and pcure are supplied, then the cure probablity applies only to the cause-specific hazard. That is, the cause-specific hazard decreases to zero, and the overall hazard converges towards the background hazard, as time increases.

References

Ramsay, J. O. (1988). Monotone regression splines in action. Statistical Science, 3(4): 425-441.

Brilleman, S. L., Elci, E. M., Novik, J. B., & Wolfe, R. (2020). Bayesian survival analysis using the rstanarm R package. arXiv preprint arXiv:2002.09633.

Wang, W., Yan, J. (2021). Shape-restricted regression splines with R package splines2. Journal of Data Science_, 19(3), 498-517.

Author

Christopher Jackson chris.jackson@mrc-bsu.cam.ac.uk