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.

## 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
```