*! version 1.6.6 22Jul2016
help stpm2
also see: stpm, stpm2 postestimation

Title

stpm2 -- Flexible parametric survival models

Syntax

stpm2 [varlist] [if] [in] [, options]

Model
bhazard(varname) invokes relative survival models where varname
holds the expected mortality rate (hazard) at the time of death

bknots(numlist) boundary knots for baseline

bknotstvc(knots list) boundary knots for time-dependent effects

cure fit a cure model

df(#) degrees of freedom for baseline hazard function

dftvc(df_list) degrees of freedom for each time-dependent effect

failconvlininit automatically try lininit option if convergence fails

knots(numlist) knot locations for baseline hazard

knotstvc(numlist) knot locations for time-dependent effects

knscale(scale) scale for user-defined knots (default scale is time)

noconstant suppress constant term

rcsbaseoff do not include baseline spline variables

noorthog do not use orthogonal transformation of splines variables

scale(scalename) specifies the scale on which the survival model is
to be fitted

stratify(varlist) for backward comapatibility with stpm

theta(est|#) for backward comapatibility with stpm

tvc(varlist) varlist of time varying effects

Reporting
alleq report all equations

eform exponentiate coefficients

keepcons do not drop constraints used in ml routine

level(#) set confidence level; default is level(95)

showcons list constraints in output

Max options
constheta(#) constrain value of theta when using Aranda-Ordaz family of link functions

initstrata(varlist) stratification variables for initial values

inittheta(#) initial value of theta (default 1: log cumulative odds scale)

lininit obtain initial values by first fitting a linear function of ln(time)

maximize_options control the maximization process; seldom used

You must stset your data before using stpm2; see [ST] stset.

fweights, iweights, and pweights may be specified using stset; [ST] stset.

Description

stpm2 fits flexible parametric survival models (Royston-Parmar models). stpm2
can be used with single- or multiple-record or single- or multiple-failure st data.
Survival models can be fitted on the log cumulative hazard scale, the log cumulative
odds scale, the standard normal deviate (probit) scale, or on a scale defined by the
value of theta using the Aranda-Ordaz family of link functions.

stpm2 can fit the same models as stpm, but is more flexible in that it does
not force the knots for time-dependent effects to be the same as those used
for the baseline distribution function. In addition, stpm2 can fit relative survival
models by use of the bhazard() option. Post-estimation commands have been extended
over what is available in stpm. stpm2 is noticeably faster than stpm.

See [ST] streg for other (standard) parametric survival models.

Options

Model

bhazard(varname) is used when fitting relative survival models.
varname gives the expected mortality rate at the time of death/censoring.
stpm2 gives an error message when there are missing values of varname,
since this usually indicates that an error has occurred when merging the
expected mortality rates.

bknots(knotslist) knotslist is a two-element numlist giving
the boundary knots. By default these are located at the minimum and maximum
of the uncensored survival times. They are specified on the scale defined
by knscale().

bknotstvc(knotslist) knotslist gives the boundary knots for
any time-dependent effects. By default these are the same as for the bknots
option. They are specified on the scale defined by knscale().

For example,

bknotstvc(x1 0.01 10 x2 0.01 8)

cure is used when fitting cure models. It forces the cumulative hazard to
be constant after the last knot. When the df() option is used together with
the cure option the internal knots are placed evenly according to centiles of the
distribution of the uncensored log survival times except one that is placed at the
95th centile. Cure models can only be used when modelling on the log cumulative
hazard scale (scale(hazard)

df(#) specifies the degrees of freedom for the restricted
cubic spline function used for the baseline function. # must be between
1 and 10, but usually a value between 1 and 4 is sufficient, with 3 being the
default. The knots() option is not applicable if the df() option
is specified. The knots are placed at the following centiles of the
distribution of the uncensored log survival times:

df knots Centile positions

1 0 (no knots)
2 1 50
3 2 33 67
4 3 25 50 75
5 4 20 40 60 80
6 5 17 33 50 67 83
7 6 14 29 43 57 71 86
8 7 12.5 25 37.5 50 62.5 75 87.5
9 8 11.1 22.2 33.3 44.4 55.6 66.7 77.8 88.9
10 9 10 20 30 40 50 60 70 80 90

Note that these are interior knots and there are also boundary knots
placed at the minimum and maximum of the distribution of uncensored survival
times.

When the cure option is used df must be between 3 and 11 and the default
location of the knots are as follows.

df knots Centile positions

3 2 50 95
4 3 33 67 95
5 4 25 50 75 95
6 5 20 40 60 80 95
7 6 17 33 50 67 83 95
8 7 14 29 43 57 71 86 95
9 8 12.5 25 37.5 50 62.5 75 87.5 95
10 9 11.1 22.2 33.3 44.4 55.6 66.7 77.8 88.9 95
11 10 10 20 30 40 50 60 70 80 90 95

dftvc(df_list) gives the degrees of freedom for time-dependent effects
in df_list. The potential degrees of freedom are listed under the
df() option. With 1 degree of freedom a linear effect of log time is fitted.
If there is more than one time-dependent effect and different degress of freedom
are requested for each time-dependent effect then the following syntax applies:

dftvc(x1:3 x2:2 1)

This will use 3 degrees of freedom for x1, 2 degrees of freedom for
x2 and 1 degree of freedom for all remaining time-dependent effects.

failconvlininit automatically tries the lininit option of the
model fails to converge.

initstrata(varlist) By default stpm2 fits a Cox model to obtain initial values.
The initstrata() option fits a stratified model for the initial values and may useful when fitting competing risks models or to allow the estimated baseline hazard to vary between the specified strata.

knots(# [# ...]) specifies knot locations for the baseline distribution
function, as opposed to the default locations set by df(). Note that
the locations of the knots are placed on the scale defined by knscale.
However, the scale used by the restricted cubic spline function is always
log time. Default knot positions are determined by the df() option.

knotstvc(knotslist) defines numlist knotslist as the location
of the interior knots for time-dependent effects. If different knots
are required for different time-dependent effects the option is
specified, for example, as follows:

knotstvc(x1 1 2 3 x2 1.5 3.5)

knscale(scale) sets the scale on which user-defined knots are specified.
knscale(time) denotes the original time scale, knscale(log) the
log time scale and knscale(centile) specifies that the knots
are taken to be centile positions in the distribution of the uncensored log
survival times. The default is knscale(time).

noconstant;
see [ST] estimation options.

noorthog suppresses orthogonal transformation of spline variables.

rcsbaseoff drops baseline spline variables from the model. With this option
you will generally want to specify your baseline separatly in two or more strata. For example,
the following code will fit a separate baseline hazard for males and females.

stpm2 males females, scale(hazard) tvc(males females) dftvc(3) nocons rcsbaseoff

Note that identical fitted values would be obtained if using the following.

stpm2 females, df(3) scale(hazard) tvc(females) dftvc(3)

scale(scalename) specifies on which scale the survival model is to be
fitted.

scale(hazard) fits a model on the log cumulative hazard scale,
i.e. the scale of ln(-ln S(t)). If no time-dependent effects are specified,
the resulting model has proportional hazards.

scale(odds) fits a model on the log cumulative odds scale,
i.e. ln((1 - S(t))/S(t)). If no time-dependent effects
are specified then this is a gives a proportional odds model.

scale(normal) fits a model on the normal equivalent deviate
scale (i.e. a probit link for the survival function, invnorm(1 - S(t))).

scale(theta) fits a model on a scale defined by the value of theta
for the Aranda-Ordaz family of link functions, i.e.
ln((S(t)^(-theta) - 1)/theta). Note that theta = 1 corresponds to a
proportional odds model and theta = 0 to a proportional
cumulative hazards model.

stratify(varlist) is provided for compatibility with stpm.
Members of varlist are modelled with time-dependent effects. See
the tvc() and dftvc() options for stpm2's way of
specifying time-dependent effects.

theta(est|#) is provided for compatibility with
stpm. est requests that theta be estimated, whereas #
fixes theta to #. See constheta() and inittheta() for
stpm2's way of specifying theta.

tvc(varlist) gives the name of the variables that are time-dependent.
Time-dependent effects are fitted using restricted cubic splines.
The degrees of freedom are specified using the dftvc() option.

Reporting

alleq reports all equations used by ml. The models are fitted by using
various constraints for parameters associated with the derivatives of the
spline functions. These parameters are generally not of interest and thus
are not shown by default. In addition, an extra equation is used when fitting
delayed entry models, and again this is not shown by default.

eform reports the exponentiated coefficents. For models on the log
cumulative hazard scale scale(hazard) this gives hazard ratios if
the covariate is not-time dependent. Similarly, for models on the log
cumulative odds scale scale(odds) this option will give odds ratios
for non time-dependent effects.

keepcons prevents the constraints imposed by stpm2 on the
derivatives of the spline function when fitting delayed entry models
being dropped. By default, the constraints are dropped.

level(#) specifies the confidence level, as a percentage, for confidence
intervals. The default is level(95) or as set by set level.

showcons The constraints used by stpm2 for the derivatives of
the spline function and when fitting delayed entry models are not listed by
default. Use of this option lists them in the output.

Max options

constheta(#) constrains the value of theta, i.e. it is treated as a known
constant.

inittheta(#) gives an initial value for theta in the Aranda-Ordaz
family of link functions.

lininit This obtains initial values by fitting only the first spline
basis function (i.e. a linear function of log survival time).
This option is seldom needed.

maximize_options; difficult, technique(algorithm_spec),
iterate(#), [no]log, trace, gradient,
showstep, hessian, shownrtolerance, tolerance(#),
ltolerance(#) gtolerance(#), nrtolerance(#),
nonrtolerance, from(init_specs); see [R] maximize. These
options are seldom used, but the difficult option may be useful if there
are convergence problems when fitting models that use Aranda-Ordaz family of link
functions.

Remarks

Let t denote time. stpm2 works by first calculating the survival function
after fitting a Cox proportional hazards model. The procedure is
illustrated for proportional hazards models, specified by option
scale(hazard). S(t) is converted to an estimate of the log cumulative hazard
function Z(t) by the formula

Z(t) = ln(-ln S(t))

This estimate of Z(t) is then smoothed on ln(t) using regression splines with
knots placed at certain quantiles of the distribution of t. The knot positions
are chosen automatically if the spline complexity is specified by the df()
option, or manually by way of the knots() option. (Note that the knots
are placed on values of ln(t), not t.) Denote the predicted values of the log cumulative
hazard function by Z_hat(t). The density function f(t) is

f(t) = -dS(t)/dt = dS/dZ_hat dZ_hat/dt = S(t) exp(Z_hat) dZ_hat(t)/dt

dZ_hat(t)/dt is computed from the regression coefficients of the fitted spline
function. The estimated survival function is calculated as

S_hat(t) = exp(-exp Z_hat(t)).

The hazard function is calculated as f(t)/S_hat(t).

If varlist is specified, the baseline survival function (i.e. at zero values
of the covariates) is used instead of the survival function of the raw
observations. With df(1) a Weibull model is fitted.

With scale(normal), smoothing is of the Normal quantile function,
invnorm(1 - S(t)), instead of the log cumulative hazard function. With
df(1) a lognormal model is fitted.

With scale(odds), smoothing is of the log odds of failure function,
ln((1 - S(t))/S(t)), instead of the log cumulative hazard function. With
df(1) a log-logistic model is fitted.

Estimation is performed by maximum likelihood. Optimisation uses the
default technique (nr, meaning Stata's version of Newton-Raphson
iteration.

Examples

Setup

webuse brcancer

stset rectime, failure(censrec = 1)

Proportional hazards model

stpm2 hormon, scale(hazard) df(4) eform

Proportional odds model

stpm2 hormon, scale(odds) df(4) eform

Time-dependent effects on cumulative hazard scale

stpm2 hormon, scale(hazard) df(4) tvc(hormon) dftvc(3)

User defined knots at centiles of uncensored event times

stpm2 hormon, scale(hazard) knots(20 50 80) knscale(centile)

Author

Paul Lambert, University of Leicester, UK.
(paul.lambert@leicester.ac.uk)

The option to fit cure models was implemented by Therese Andersson, Karolinska Institutet, Stockholm, Sweden
(therese.m-l.andersson@ki.se)

Various other additions and suggestions by Patrick Royston, MRC Clinical Trials Unit, London, UK.
(pr@ctu.mrc.ac.uk)

References

P. C. Lambert and P. Royston. Further development of flexible parametric
models for survival analysis. Stata Journal 2009;9:265-290

C. P. Nelson, P. C. Lambert, I. B. Squire and D. R. Jones.
Flexible parametric models for relative survival, with application
in coronary heart disease. Statistics in Medicine 2007;26:5486-5498

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

P. Royston, P.C. Lambert. Flexible parametric survival analysis in Stata:
Beyond the Cox model StataPress, 2011

Also see

Online: [ST] stpm2 postestimation;
[ST] stset,
stpm