flexsurvspline(formula, data, k=0, knots=NULL, bknots=NULL,
scale="hazard", ...)Surv function, and any covariates are given on the
right-hand side. Forformula.
If not given, the variables should be in the working environment.k=0 gives
a Weibull, log-logistic or lognormal model, if "scale" is
"hazard", "odds" or "normal" respectively. k is equivalk above.
Either k or knots must be
specified. If both are specified, knots override"hazard", the log cumulative hazard is modelled as a spline
function of log time.
If "odds", the log cumulative odds is modelled as a spline
function of log time.
If "normal", $-\Phi^{-1}(Sflexsurvreg, for example, anc, inits,
fixedpars, weights, subset, na.actio"flexsurvreg" with the same elements as
described in flexsurvreg, and including extra components
describing the spline model. See in particular:scale of the model, hazard, odds or normal."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 dllogis from the 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.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 $(x - a)_+ = max(0, x - a)$.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.## 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
splw2 <- flexsurvreg(Surv(recyrs, censrec) ~ group, data=bc, dist="weibull")Run the code above in your browser using DataLab