Royston/Parmar spline survival distribution

Probability density, distribution, quantile, random generation, hazard, cumulative hazard, mean and restricted mean functions for the Royston/Parmar spline model. These functions have all parameters of the distribution collected together in a single argument gamma. For the equivalent functions with one argument per parameter, see Survsplinek.

Usage

dsurvspline(
  x,
  gamma,
  beta = 0,
  X = 0,
  knots = c(-10, 10),
  scale = "hazard",
  timescale = "log",
  spline = "rp",
  offset = 0,
  log = FALSE
)

psurvspline(
  q,
  gamma,
  beta = 0,
  X = 0,
  knots = c(-10, 10),
  scale = "hazard",
  timescale = "log",
  spline = "rp",
  offset = 0,
  lower.tail = TRUE,
  log.p = FALSE
)

qsurvspline(
  p,
  gamma,
  beta = 0,
  X = 0,
  knots = c(-10, 10),
  scale = "hazard",
  timescale = "log",
  spline = "rp",
  offset = 0,
  lower.tail = TRUE,
  log.p = FALSE
)

rsurvspline(
  n,
  gamma,
  beta = 0,
  X = 0,
  knots = c(-10, 10),
  scale = "hazard",
  timescale = "log",
  spline = "rp",
  offset = 0
)

Hsurvspline(
  x,
  gamma,
  beta = 0,
  X = 0,
  knots = c(-10, 10),
  scale = "hazard",
  timescale = "log",
  spline = "rp",
  offset = 0
)

hsurvspline(
  x,
  gamma,
  beta = 0,
  X = 0,
  knots = c(-10, 10),
  scale = "hazard",
  timescale = "log",
  spline = "rp",
  offset = 0
)

rmst_survspline(
  t,
  gamma,
  beta = 0,
  X = 0,
  knots = c(-10, 10),
  scale = "hazard",
  timescale = "log",
  spline = "rp",
  offset = 0,
  start = 0
)

mean_survspline(
  gamma,
  beta = 0,
  X = 0,
  knots = c(-10, 10),
  scale = "hazard",
  timescale = "log",
  spline = "rp",
  offset = 0
)

Arguments

x, q, t

Vector of times.

gamma

Parameters describing the baseline spline function, as described in flexsurvspline. This may be supplied as a vector with number of elements equal to the length of knots, in which case the parameters are common to all times. Alternatively a matrix may be supplied, with rows corresponding to different times, and columns corresponding to knots.

beta

Vector of covariate effects. Not supported and ignored since version 2.3, and this argument will be removed in 2.4.

X

Matrix of covariate values. Not supported and ignored since version 2.3, and this argument will be removed in 2.4.

knots

Locations of knots on the axis of log time, supplied in increasing order. Unlike in flexsurvspline, these include the two boundary knots. If there are no additional knots, the boundary locations are not used. If there are one or more additional knots, the boundary knots should be at or beyond the minimum and maximum values of the log times. In flexsurvspline these are exactly at the minimum and maximum values.

This may in principle be supplied as a matrix, in the same way as for gamma, but in most applications the knots will be fixed.

scale

"hazard", "odds", or "normal", as described in flexsurvspline. With the default of no knots in addition to the boundaries, this model reduces to the Weibull, log-logistic and log-normal respectively. The scale must be common to all times.

timescale

"log" or "identity" as described in flexsurvspline.

spline

"rp" to use the natural cubic spline basis described in Royston and Parmar. "splines2ns" to use the alternative natural cubic spline basis from the splines2 package (Wang and Yan 2021), which may be better behaved due to the basis being orthogonal.

offset

An extra constant to add to the linear predictor \(\eta\). Not supported and ignored since version 2.3, and this argument will be removed in 2.4.

log, log.p

Return log density or probability.

lower.tail

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

p

Vector of probabilities.

n

Number of random numbers to simulate.

start

Optional left-truncation time or times. The returned restricted mean survival will be conditioned on survival up to this time.

Value

dsurvspline gives the density, psurvspline gives the distribution function, hsurvspline gives the hazard and Hsurvspline gives the cumulative hazard, as described in flexsurvspline.

qsurvspline gives the quantile function, which is computed by crude numerical inversion (using qgeneric).

rsurvspline generates random survival times by using qsurvspline on a sample of uniform random numbers. Due to the numerical root-finding involved in qsurvspline, it is slow compared to typical random number generation functions.

References

Royston, P. and Parmar, M. (2002). Flexible parametric proportional-hazards and proportional-odds models for censored survival data, with application to prognostic modelling and estimation of treatment effects. Statistics in Medicine 21(1):2175-2197.

Wang W, Yan J (2021). Shape-Restricted Regression Splines with R Package splines2. Journal of Data Science, 19(3), 498-517.

See also

flexsurvspline.

Author

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

Examples

## reduces to the weibull
regscale <- 0.786; cf <- 1.82
a <- 1/regscale; b <- exp(cf)
dweibull(1, shape=a, scale=b)
#> [1] 0.1137858
dsurvspline(1, gamma=c(log(1 / b^a), a)) # should be the same
#> [1] 0.1137858

## reduces to the log-normal
meanlog <- 1.52; sdlog <- 1.11
dlnorm(1, meanlog, sdlog) 
#> [1] 0.1407338
dsurvspline(1, gamma = c(-meanlog/sdlog, 1/sdlog), scale="normal")
#> [1] 0.1407338
# should be the same