R/spline.R
flexsurvspline.Rd
Flexible parametric modelling of time-to-event data using the spline model of Royston and Parmar (2002).
flexsurvspline( formula, data, weights, bhazard, rtrunc, subset, k = 0, knots = NULL, bknots = NULL, scale = "hazard", timescale = "log", ... )
formula | A formula expression in conventional R linear modelling
syntax. The response must be a survival object as returned by the
specifies a model where the log cumulative hazard (by default, see
If there are no covariates, specify Time-varying covariate effects can be specified using the method described
in Therefore a model with one internal spline knot, where the equivalents of
the Weibull shape and scale parameters, but not the higher-order term
or alternatively (and more safely, see
|
---|---|
data | A data frame in which to find variables supplied in
|
weights | Optional variable giving case weights. |
bhazard | Optional variable giving expected hazards for relative survival models. |
rtrunc | Optional variable giving individual right-truncation times (see |
subset | Vector of integers or logicals specifying the subset of the observations to be used in the fit. |
k | Number of knots in the spline. The default |
knots | Locations of knots on the axis of log time (or time, see
|
bknots | Locations of boundary knots, on the axis of log time (or
time, see |
scale | If If If |
timescale | If |
... | Any other arguments to be passed to or through
|
A list of class "flexsurvreg"
with the same elements as
described in flexsurvreg
, and including extra components
describing the spline model. See in particular:
Number of knots.
Location of knots on the log time axis.
The scale
of the model, hazard, odds or normal.
Matrix of maximum likelihood estimates and confidence limits.
Spline coefficients are labelled "gamma..."
, and covariate effects
are labelled with the names of the covariates.Coefficients gamma1,gamma2,...
here are the equivalent of
s0,s1,...
in Stata streg
, and gamma0
is the equivalent
of the xb
constant term. To reproduce results, use the
noorthog
option in Stata, since no orthogonalisation is performed on
the spline basis here.In the Weibull model, for example, gamma0,gamma1
are
-shape*log(scale), shape
respectively in dweibull
or
flexsurvreg
notation, or (-Intercept/scale
,
1/scale
) in survreg
notation.In the log-logistic model with shape a
and scale b
(as in
eha::dllogis
from the eha package), 1/b^a
is
equivalent to exp(gamma0)
, and a
is equivalent to
gamma1
.In the log-normal model with log-scale mean mu
and standard
deviation sigma
, -mu/sigma
is equivalent to gamma0
and
1/sigma
is equivalent to gamma1
.
The maximised log-likelihood. This will differ from Stata, where the sum of the log uncensored survival times is added to the log-likelihood in survival models, to remove dependency on the time scale.
This function works as a wrapper around flexsurvreg
by
dynamically constructing a custom distribution using
dsurvspline
, psurvspline
and
unroll.function
.
In the spline-based survival model of Royston and Parmar (2002), a transformation \(g(S(t,z))\) of the survival function is modelled as a natural cubic spline function of log time \(x = \log(t)\) plus linear effects of covariates \(z\).
$$g(S(t,z)) = s(x, \bm{\gamma}) + \bm{\beta}^T \mathbf{z}$$
The proportional hazards model (scale="hazard"
) defines
\(g(S(t,\mathbf{z})) = \log(-\log(S(t,\mathbf{z}))) =
\log(H(t,\mathbf{z}))\), the
log cumulative hazard.
The proportional odds model (scale="odds"
) defines
\(g(S(t,\mathbf{z})) \)\( =
\log(S(t,\mathbf{z})^{-1} - 1)\), the log
cumulative odds.
The probit model (scale="normal"
) defines \(g(S(t,\mathbf{z})) =
\)\( -\Phi^{-1}(S(t,\mathbf{z}))\), where \(\Phi^{-1}()\) is the inverse normal
distribution function qnorm
.
With no knots, the spline reduces to a linear function, and these models are equivalent to Weibull, log-logistic and lognormal models respectively.
The spline coefficients \(\gamma_j: j=1, 2 \ldots \), which are called the "ancillary parameters" above, may also be modelled as linear functions of covariates \(\mathbf{z}\), as
$$\gamma_j(\mathbf{z}) = \gamma_{j0} + \gamma_{j1}z_1 + \gamma_{j2}z_2 + ... $$
giving a model where the effects of covariates are arbitrarily flexible functions of time: a non-proportional hazards or odds model.
Natural cubic splines are cubic splines constrained to be linear beyond boundary knots \(k_{min},k_{max}\). The spline function is defined as
$$s(x,\bm{\gamma}) = \gamma_0 + \gamma_1 x + \gamma_2 v_1(x) + \ldots + $$$$ \gamma_{m+1} v_m(x)$$
where \(v_j(x)\) is the \(j\)th basis function
$$v_j(x) = (x - k_j)^3_+ - \lambda_j(x - k_{min})^3_+ - (1 - $$$$ \lambda_j) (x - k_{max})^3_+$$
$$\lambda_j = \frac{k_{max} - k_j}{k_{max} - k_{min}}$$
and \((x - a)_+ = max(0, x - a)\).
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.
Jackson, C. (2016). flexsurv: A Platform for Parametric Survival Modeling in R. Journal of Statistical Software, 70(8), 1-33. doi:10.18637/jss.v070.i08
flexsurvreg
for flexible survival modelling using
general parametric distributions.
plot.flexsurvreg
and lines.flexsurvreg
to plot
fitted survival, hazards and cumulative hazards from models fitted by
flexsurvspline
and flexsurvreg
.
Christopher Jackson <chris.jackson@mrc-bsu.cam.ac.uk>
## Best-fitting model to breast cancer data from Royston and Parmar (2002) ## One internal knot (2 df) and cumulative odds scale spl <- flexsurvspline(Surv(recyrs, censrec) ~ group, data=bc, k=1, scale="odds") ## Fitted survival plot(spl, lwd=3, ci=FALSE)## Simple Weibull model fits much less well splw <- flexsurvspline(Surv(recyrs, censrec) ~ group, data=bc, k=0, scale="hazard") lines(splw, col="blue", ci=FALSE)## Alternative way of fitting the Weibull if (FALSE) { splw2 <- flexsurvreg(Surv(recyrs, censrec) ~ group, data=bc, dist="weibull") }