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Introduction.Rnw
%\VignetteIndexEntry{Introduction to the rstpm2 Package}
%\VignetteDepends{rstpm2}
%\VignetteKeyword{survival, spline}
%\VignettePackage{rstpm2}
%!\SweaveUTF8

\documentclass[nojss]{jss}

\usepackage{amsmath,amsfonts,enumitem,fancyvrb,hyperref}
\usepackage[utf8]{inputenc}
\VerbatimFootnotes
\usepackage[margin=2.6cm]{geometry} % wide margins
\usepackage{wasysym}
\usepackage{tablefootnote}

\title{Introduction to the \pkg{rstpm2} package}

\author{Mark~Clements\\Karolinska Institutet}

\Plainauthor{Mark~Clements}

\Plaintitle{Introduction to the rstpm2 package}

\Abstract{
  
  This vignette outlines the methods and provides some examples for
  generalised survival models as implemented in the \proglang{R}
  \pkg{rstpm2} package.

}

\Keywords{survival, splines}

\Plainkeywords{survival, splines}

\Address{Mark~Clements\\
  Department of Medical Epidemiology and Biostatistics\\
  Karolinska Institutet\\
  Email: \email{mark.clements@ki.se}
  }

\begin{document}

\section{Background and theory}

\emph{Generalised survival models} provide a flexible and general approach to modelling survival or time-to-event data. The survival function $S(t|x)$ to time $t$ for covariates $x$ is defined in terms of a inverse link function $G$ and a linear prediction $\eta(t,x)$, such that

\begin{align*}
  S(t|x;\,\theta) &= G(\eta(t,x;\,\theta))
\end{align*}

\noindent where $\eta$ is a function of both time $t$ and covariates $x$, with regression parameters $\theta$.  We can calculate the hazard from this function, where

\begin{align*}
  h(t|x;\,\theta) &= \frac{d}{dt} \left(-\log(S(t|x;\,\theta))\right) \\
         &= \frac{-G'(\eta(t,x;\,\theta))}{G(\eta(t,x;\,\theta))}\frac{\partial \eta(t,x;\,\theta)}{\partial t}
\end{align*}

We model using a linear predictor $\eta(t,x;\,\theta)=X(t,x)\theta$ for a design matrix $X(t,x)$. The linear predictor can be constructed in a flexible
manner, with the main constraint being that the time effects be smooth and twice differentiable. We calculate the derivative for the linear predictor using finite differences, such that
\begin{align*}
 \frac{\partial \eta(t,x;\,\theta)}{\partial t} &= \frac{\partial X(t,x)\theta}{\partial t} = \frac{X(t+\epsilon,x)-X(t-\epsilon,x)}{2\epsilon}\theta = X_D(t,x)\theta
\end{align*}
for a derivative design matrix $X_D(t,x)$. This formulation allows for considerable flexibility in the construction of the linear predictor, with possible interactions between time and covariates.

The default smoother for time using natural splines for log(time), which is the flexible parametric survival model developed by Royston and Parmar (2003) and implemented by the Stata command \verb+stpm2+~\footnote{As a technical aside, the Stata implementation uses natural splines using a truncated power basis with orthogonalisation, while the \verb+ns()+ function in \verb+R+ uses a matrix projection of B-splines. Note that we have implemented an extended \verb+nsx()+ function for natural splines that includes cure splines, centering, and a compatibility argument to use Stata \verb+stpm2+'s unusual specification of quantiles.}

\begin{table}[!ht]
  \centering
  \begin{tabular}[!ht]{llll}
    Link description & Inverse link function $G(\eta(t,x;\,\theta))$ & Interpretation & \verb+link.type+ \\ \hline
    log$-$log & $\exp(-\exp(\eta(t,x;\,\theta)))$ & Proportional hazards & \verb+"PH"+\\
    logit & $\text{expit}(-\eta(t,x;\,\theta))$ & Proportional odds & \verb+"PO"+ \\
    probit & $\Phi(-\eta(t,x;\,\theta))$ & Probit & \verb+"probit"+ \\
    log & $\exp(-\eta(t,x;\,\theta))$ & Additive hazards & \verb+"AH"+ \\
    Aranda-Ordaz & $\exp(-\log(\psi*\exp(\eta(t,x;\,\theta))+1)/\psi)$ & Aranda-Ordaz & \verb+"AO"+
  \end{tabular}
  \caption{Implemented link functions}
  \label{tab:links}
\end{table}

The models are estimated using maximum likelihood estimation (MLE) for fully parametric models, penalised MLE for penalised smoothers, maximum marginal likelihood estimation (MMLE) for parametric models with clustered data, or penalised MMLE for penalised models with clustered data. The likelihoods include left truncation, right censoring and interval censoring. For clustered data, we include Gamma frailties and normal random effects. Details on these models are available from \url{https://doi.org/10.1177/0962280216664760} and \url{https://doi.org/10.1002/sim.7451}.


<<echo=FALSE,results=hide>>=

options(width=80,useFancyQuotes="UTF-8")
library(rstpm2)

@ 


\section{Syntax}

The main functions for fitting the models are \verb+stpm2+ for parametric models, possibly with clustered data, and \verb+pstpm2+ for penalised models, possibly with clustered data. A subset of the syntax for \verb+stpm2+ is:

\begin{Verbatim}
stpm2(formula, data, smooth.formula = NULL, 
      df = 3, tvc = NULL, 
      link.type=c("PH","PO","probit","AH","AO"), theta.AO=0,
      bhazard = NULL, 
      robust = FALSE, cluster = NULL, frailty = !is.null(cluster) & !robust,
      RandDist=c("Gamma","LogN"), 
      ...)
\end{Verbatim}

The \verb+formula+ has a \verb+Surv+ object on the left-hand-side and a linear predictor on the right-hand-side that does \emph{not} include time (for \verb+pstpm2+, it also does not include penalised functions). The time effects can be specified in several ways: the most general is using \verb+smooth.formula+, where the right-hand-side of the formula specifies functions for time that are smooth with respect to time. This specification can include interactions between time and covariates. As an example, \verb!smooth.formula=~nsx(log(time),df=3)+x:nsx(log(time),df=2)! specifies a baseline natural spline smoother of the log of the variable \verb+time+ used in the \verb+Surv+ object with three degrees of freedom, with an interaction between a covariate \verb+x+ and a natural spline smoother of log(\verb+time+) with two degrees of freedom. Other specifications of time effects have equivalent formulations: for example, \verb+df=3+ is equivalent to \verb!smooth.formula=~nsx(log(time),df=3)! for the variable \verb+time+. Similarly, \verb+tvc=list(x=2)+ is equivalent to \verb!smooth.formula=~x:nsx(log(time),df=2)!. Moreover, for a log-linear interaction between a covariate and time, use \verb!smooth.formula=~x:log(time)!

A current limitation of the implementation is that the dataset \verb+data+ needs to be specified.

Type of link is specified with the \verb+link.type+ argument; this defaults to a log$-$log link for proportional hazards (see Table~\ref{tab:links}). For the Aranda-Ordaz link, the fixed value of the scale term $\psi$ is specified using the \verb+theta.AO+ argument.  For relative survival, a vector for the baseline hazard can be specified using the \verb+bhazard+ argument. A vector for the clusters can be specified with the \verb+cluster+ argument. The calculation of robust standard errors can be specified with the \verb+robust=TRUE+ argument; if \verb+robust+ is false, then the model assumes a frailty or random effects model, with either a default Gamma frailty (\verb+RandDist="Gamma"+) or a normal random effect (\verb+RandDist="LogN"+, using notation from the \verb+frailtypack+ package).

The syntax for the fitting the penalised models with \verb+pstpm2+ is very similar. A subset of the arguments are:
\begin{Verbatim}
pstpm2(formula, data, smooth.formula = NULL, 
       tvc = NULL, 
       bhazard = NULL, 
       sp=NULL, 
       criterion=c("GCV","BIC"), 
       link.type=c("PH","PO","probit","AH","AO"), theta.AO=0,
       robust=FALSE,
       frailty=!is.null(cluster) & !robust, cluster = NULL, RandDist=c("Gamma","LogN"),
       ...)
\end{Verbatim}

The penalised smoothers are specified using the \verb+s()+ function from the \verb+mgcv+ package within the \verb+smooth.formula+ argument; by default, not specifying \verb+smooth.formula+ will lead to \verb!smooth.formula=~s(log(time))!. Interactions with time (both penalised and unpenalised) and penalised covariate effects should be specified using \verb+smooth.formula+. 

Note that the \verb+df+ argument is not included. By default, the smoothing parameter(s) are using the \verb+criterion+ argument; the smoothing parameters can also be fixed using the \verb+sp+ argument. 

The specifications for relative survival, link type, and clustered data follow that for the \verb+stpm2+ function.

\subsection{Post-estimation}

One of the strengths of these models is varied post-estimation. Most of the estimators are described in Tables~\ref{tab:condpostest} and \ref{tab:margpostest}. These estimators are typically calculated from the \verb+predict+ function or from \verb+plot+ function calls. All of these calls require that the \verb+newdata+ argument is specified (in contrast to prediction in the \verb+survival+ package, which defaults to the average of each covariate). 

For contrasts (e.g. survival differences, hazard ratios), the \verb+newdata+ argument is the ``unexposed'' group, while the exposed group is defined by either: (i) a unit change in a variable in \verb+newdata+ as defined by the \verb+var+ argument (e.g. \verb+var="x"+ for variable \verb+x+); or (ii) an \verb+exposed+ function that takes a data-frame and returns the ``exposed'' group (e.g. \verb+exposed=function(data) transform(data, x=1)+). The latter mechanism is quite general and allows for standardised survival, standardised hazards, and attributable fractions under possibly counterfactual exposures.

\begin{table}[!ht]
  \centering
\begin{minipage}{\linewidth}
  \begin{tabular}[!ht]{lll}
    Description & Formulation\footnote{Notation: $x^*$ is a covariate pattern for the ``exposed'' group; $X^*$ is a set of possibly counterfactual covariates; $E_X(g(X))$ is the expectation or average of $g(X)$ across the set $X$; $X_0^*$ and $X_1^*$ are sets of possibly counterfactual covariates for the ``unexposed'' and ``exposed'' sets, respectively.} & \verb+type+ \\ \hline
    Conditional link & $\eta(t,x;\hat\theta)$ & \verb+"link"+ \\
    Conditional survival & $S(t|x;\hat\theta) = G(\eta(t,x;\hat\theta))$ & \verb+"surv"+ \\
    Conditional odds & $\text{Odds}(t|x;\hat\theta)=S(t|x;\hat\theta)/(1-S(t|x;\hat\theta))$ & \verb+"odds"+ \\
    Conditional failure & $1-S(t|x;\hat\theta)$ & \verb+"fail"+ \\
    Conditional cumulative hazard & $H(t|x;\hat\theta) = -\log G(\eta(t,x;\hat\theta))$ & \verb+"cumhaz"+ \\
    Conditional density & $f(t|x;\hat\theta) = G'(\eta(t,x;\hat\theta))\frac{\partial \eta(t,x;\hat\theta)}{\partial t}$ & \verb+"density"+ \\
    Conditional hazard & $h(t|x;\hat\theta) = \frac{G'(\eta(t,x;\hat\theta))}{G(\eta(t,x;\hat\theta))}\frac{\partial \eta(t,x;\hat\theta)}{\partial t}$ & \verb+"hazard"+ \\
    Conditional log hazard & $\log h(t|x;\hat\theta)$ & \verb+"loghazard"+ \\
    Conditional survival differences & $S(t|x^*;\hat\theta)-S(t|x;\hat\theta)$ & \verb+"survdiff"+ \\
    Conditional hazard differences & $h(t|x^*;\hat\theta)-h(t|x;\hat\theta)$ & \verb+"hazdiff"+ \\
    Conditional hazard ratios & $h(t|x^*;\hat\theta)/h(t|x;\hat\theta)$ & \verb+"hr"+ \\
    Conditional odds ratios & $\text{Odds}(t|x^*;\hat\theta)/\text{Odds}(t|x;\hat\theta)$ & \verb+"or"+ \\
    Restricted mean survival time & $\int_0^tS(u|x;\hat\theta)du$ & \verb+"rmst"+ \\
    Standardised survival & $E_{X^*}S(t|X^*;\hat\theta)$ & \verb+"meansurv"+ \\
    Standardised survival differences & $E_{X_1^*}S(t|X_1^*;\hat\theta)-E_{X_0^*}S(t|X_0^*;\hat\theta)$ & \verb+"meansurvdiff"+ \\
    Standardised hazard & $h_{X^*}(t|X^*;\hat\theta) = \frac{E_{X^*}(S(t|X^*;\hat\theta)h(t|X^*;\hat\theta))}{E_{X^*}(S(t|X^*;\hat\theta))}$ & \verb+"meanhaz"+ \\
    Standardised hazard ratio & $h_{X_1^*}(t|X_1^*;\hat\theta)/h_{X_0^*}(t|X_0^*\hat\theta)$ & \verb+"meanhazdiff"+ \\
    Attributable fraction & $\frac{E_{X^*}S(t|X^*;\hat\theta)--E_{X}S(t|X;\hat\theta)}{1-E_{X}S(t|X;\hat\theta)}$ & \verb+"af"+ \\
  \end{tabular}
\end{minipage}
\caption{Types of conditional post-estimators}
  \label{tab:condpostest}
\end{table}


\begin{table}[!ht]
  \centering
\begin{minipage}{\linewidth}
  \begin{tabular}[!ht]{lll}
    Description & Formulation\footnote{Notation: $Z$ is a random effect or frailty; $x^*$ is a covariate pattern for the ``exposed'' group; $X^*$ is a set of possibly counterfactual covariates; $E_X(g(X))$ is the expectation or average of $g(X)$ across the set $X$; $X_0^*$ and $X_1^*$ are sets of possibly counterfactual covariates for the ``unexposed'' and ``exposed'' sets, respectively.} & \verb+type+ \\ \hline
    Marginal survival & $S_M(t|x;\hat\theta) = E_ZG(\eta(t,x,Z;\hat\theta))$ & \verb+"margsurv"+ \\
    Marginal hazard & $h_M(t|x;\hat\theta) = E_Z(h(t,x,Z;\hat\theta))$ & \verb+"marghaz"+ \\
    Marginal survival differences & $S_M(t|x^*;\hat\theta)-S_M(t|x;\hat\theta)$ & \verb+"margsurvdiff"+ \\
    Marginal hazard ratios & $h_M(t|x^*;\hat\theta)/h_M(t|x;\hat\theta)$ & \verb+"marghr"+ \\
    Standardised marginal survival & $E_ZE_{X^*}S(t|X^*,Z;\hat\theta)$ & \verb+"meanmargsurv"+ \\
    Standardised marginal survival differences & $E_ZE_{X^*}S(t|X^*,Z;\hat\theta)-E_ZE_{X}S(t|X,Z;\hat\theta)$ & \verb+"meansurvdiff"+ \\
    Attributable fraction & $\frac{E_ZE_{X^*}S(t|X^*,Z;\hat\theta)-E_ZE_{X}S(t|X,Z;\hat\theta)}{1-E_ZE_{X}S(t|X,Z;\hat\theta)}$ & \verb+"af"+ \\
  \end{tabular}
\end{minipage}
\caption{Types of post-estimators for clustered data}
  \label{tab:margpostest}
\end{table}

Standard errors for the post-estimators are calculated on a possibly transformed scale using the delta method. For the delta method, the partial derivatives of the post-estimators are calculated either directly or using finite differences.

\begin{table}
  \centering
  \begin{minipage}{\textwidth}
    \begin{tabular}{lp{1.5cm}p{1.5cm}p{1.5cm}p{1.5cm}p{1.5cm}p{1.5cm}}
  Functionality & Uncorre-lated param. & Uncorre-lated penal. & Param. gamma frailty & Penal. gamma frailty & Param. normal random effects & Penal. normal random effects  \\ \hline
  Multiple links & \XBox& \XBox& \XBox& \XBox& \XBox& \XBox \\
  Right censoring & \XBox& \XBox& \XBox& \XBox& \XBox& \XBox \\
  Left truncation & \XBox& \XBox& \XBox\footnote{Gradients not currently implemented.}& \XBox$^a$& \XBox$^a$& \XBox$^a$ \\
  Interval censoring & \XBox& \XBox& \Square& \Square& \XBox& \XBox \\
  Time-varying effects & \XBox& \XBox& \XBox& \XBox& \XBox& \XBox \\
  Excess hazards & \XBox& \XBox& \XBox& \XBox& \XBox& \XBox \\
  Conditional estimators\footnote{Estimators including survival, survival differences, hazards, hazard ratios, hazard differences, density, odds and odds ratios.} &\XBox &\XBox & \XBox& \XBox& \XBox& \XBox \\
  Conditional standardisation\footnote{Standardised estimators include mean survival, mean survival differences, mean hazards and attributable fractions.} &\XBox &\XBox & \XBox& \XBox& \XBox& \XBox \\
  Frailty/random effects variance & & & \XBox& \XBox& \XBox& \XBox \\
  Marginal estimators\footnote{Marginal estimators include survival, survival differences, hazards, hazard ratios and hazard differences.} & & & \XBox& \XBox& \XBox& \XBox \\
  Marginal standardisation\footnote{Marginal standardised estimators include mean survival, mean survival differences and attributable fractions.} & & & \XBox& \XBox& \Square& \Square \\
  Random intercept & & &\XBox &\XBox & \XBox& \XBox \\
  Random slope & & & & & \XBox& \XBox \\
  Multiple random effects & & & & & \Square& \Square \\
    \end{tabular}
  \end{minipage}
  \caption{Functionality for the different generalised survival models}
\end{table}


\section{Examples: Independent survival analysis}

We begin with some simple proportional hazard models using the
brcancer dataset. We can fit the models using very similar syntax to
coxph, except that we need to specify the degrees of freedom for the
baseline smoother. Typical values for df are 3-6. For this model the
model parameters include an intercept term, time-invariant log-hazard
ratios, and parameters for the baseline smoother. The default for the
baseline smoother is to use the nsx function, which is a limited
extension to the splines::ns function, with log of the time effect.

<<>>=

fit <- stpm2(Surv(rectime,censrec==1)~hormon,
             data=brcancer, df=4)
summary(fit)

## utility to exponentiate the ith components
expi <- function(x,i=1:length(x)) { x[i] <- exp(x[i]); x }
fit.cox <- coxph(Surv(rectime,censrec==1)~hormon, data=brcancer)
rbind(coxph=coef(summary(fit.cox)),
      stpm2=expi(coef(summary(fit))["hormon",c(1,1,2:4)],2))

@ 

We see that the hazard ratios are very similar to the coxph model. The
model fit can also be used to estimate a variety of parameters. For
example, we can easily estimate survival for a given parameter.

<<fig=TRUE>>=

# tms=seq(0,10,length=301)[-1]
plot(fit,newdata=data.frame(hormon=1), 
     xlab="Time since diagnosis (days)")
lines(fit, newdata=data.frame(hormon=0), col=2, lty=2)
legend("topright", c("hormon=1","hormon=0"),lty=1:2,col=1:2,bty="n")

@ 

We can also calculate the hazards.

<<fig=TRUE>>=

# tms=seq(0,10,length=301)[-1]
plot(fit,newdata=data.frame(hormon=1), type="hazard",
     xlab="Time since diagnosis (days)", ylim=c(0,8e-4))
lines(fit, newdata=data.frame(hormon=0), col=2, lty=2, type="hazard")
legend("topright", c("hormon=1","hormon=0"),lty=1:2,col=1:2,bty="n")

@ 

Usefully, we can also estimate survival differences and hazard
differences. We define the survival differences using a reference
covariate pattern using the newdata argument, and then define an
exposed function which takes the newdata and transforms for the
'exposed' covariate pattern. As an example

<<fig=TRUE>>=

plot(fit,newdata=data.frame(hormon=0), type="hdiff",
     exposed=function(data) transform(data, hormon=1),
     main="hormon=1 compared with hormon=0",
     xlab="Time since diagnosis (days)")

@ 

<<fig=TRUE>>=

plot(fit,newdata=data.frame(hormon=0), type="sdiff",
     exposed=function(data) transform(data, hormon=1),
     main="hormon=1 compared with hormon=0",
     xlab="Time since diagnosis (days)")

@ 


\section{Mean survival}

This has a useful interpretation for causal inference.

$E_Z(S(t|Z,X=1))-E_Z(S(t|Z,X=0))$

\begin{verbatim}
fit <- stpm2(...)
predict(fit,type="meansurv",newdata=data)
\end{verbatim}

\section{Cure models}

For cure, we use the melanoma dataset used by Andersson and colleagues
for cure models with Stata's stpm2 (see
\url{http://www.pauldickman.com/survival/}).

Initially, we merge the patient data with the all cause mortality rates.

<<echo=FALSE,results=hide>>=

options(width=80,useFancyQuotes="UTF-8")
library(rstpm2)

@ 
<<>>=

popmort2 <- transform(rstpm2::popmort,exitage=age,exityear=year,age=NULL,year=NULL)
colon2 <- within(rstpm2::colon, {
  status <- ifelse(surv_mm>120.5,1,status)
  tm <- pmin(surv_mm,120.5)/12
  exit <- dx+tm*365.25
  sex <- as.numeric(sex)
  exitage <- pmin(floor(age+tm),99)
  exityear <- floor(yydx+tm)
  ##year8594 <- (year8594=="Diagnosed 85-94")
})
colon2 <- merge(colon2,popmort2)

@ 
For comparisons, we fit the relative survival model without and with cure. 
%% <<results=hide>>=
<<>>=

fit0 <- stpm2(Surv(tm,status %in% 2:3)~I(year8594=="Diagnosed 85-94"),
              data=colon2,
              bhazard=colon2$rate, df=5)

@ 
<<>>=

summary(fit <- stpm2(Surv(tm,status %in% 2:3)~I(year8594=="Diagnosed 85-94"),
                     data=colon2,
                     bhazard=colon2$rate,
                     df=5,cure=TRUE))
predict(fit,head(colon2),se.fit=TRUE)

@ 
The estimate for the year parameter from the model without cure is within three significant
figures with that in Stata. For the predictions, the Stata model gives:
\begin{verbatim}
     +---------------------------------+
     |      surv   surv_lci   surv_uci |
     |---------------------------------|
  1. | .86108264   .8542898   .8675839 |
  2. | .79346526   .7850106   .8016309 |
  3. | .69674037   .6863196   .7068927 |
  4. | .86108264   .8542898   .8675839 |
  5. | .82212425   .8143227   .8296332 |
     |---------------------------------|
  6. | .86108264   .8542898   .8675839 |
     +---------------------------------+
\end{verbatim}
We can estimate the proportion of failures prior to the last event time:
<<>>=

newdata.eof <- data.frame(year8594 = unique(colon2$year8594),
                          tm=10)
1-predict(fit0, newdata.eof, type="surv", se.fit=TRUE)
1-predict(fit, newdata.eof, type="surv", se.fit=TRUE)
predict(fit, newdata.eof, type="haz", se.fit=TRUE)

@ 
We can plot the predicted survival estimates:
\begin{center}
<<fig=TRUE>>=

tms=seq(0,10,length=301)[-1]
plot(fit0,newdata=data.frame(year8594 = "Diagnosed 85-94", tm=tms), ylim=0:1,
     xlab="Time since diagnosis (years)", ylab="Relative survival")
plot(fit0,newdata=data.frame(year8594 = "Diagnosed 75-84",tm=tms),
     add=TRUE,line.col="red",rug=FALSE)
## warnings: Predicted hazards less than zero for cure
plot(fit,newdata=data.frame(year8594 = "Diagnosed 85-94",tm=tms),
     add=TRUE,ci=FALSE,lty=2,rug=FALSE)
plot(fit,newdata=data.frame(year8594="Diagnosed 75-84",tm=tms),
     add=TRUE,rug=FALSE,line.col="red",ci=FALSE,lty=2)
legend("topright",c("85-94 without cure","75-84 without cure",
                    "85-94 with cure","75-84 with cure"),
       col=c(1,2,1,2), lty=c(1,1,2,2), bty="n")

@
\end{center}

And the hazard curves:

\begin{center}
<<fig=TRUE>>=

plot(fit0,newdata=data.frame(year8594 = "Diagnosed 85-94", tm=tms), 
     ylim=c(0,0.5), type="hazard",
     xlab="Time since diagnosis (years)",ylab="Excess hazard")
plot(fit0,newdata=data.frame(year8594 = "Diagnosed 75-84", tm=tms),
     type="hazard",
     add=TRUE,line.col="red",rug=FALSE)
plot(fit,newdata=data.frame(year8594 = "Diagnosed 85-94", tm=tms),
     type="hazard",
     add=TRUE,ci=FALSE,lty=2,rug=FALSE)
plot(fit,newdata=data.frame(year8594="Diagnosed 75-84", tm=tms),
     type="hazard",
     add=TRUE,rug=FALSE,line.col="red",ci=FALSE,lty=2)
legend("topright",c("85-94 without cure","75-84 without cure",
                    "85-94 with cure","75-84 with cure"),
       col=c(1,2,1,2), lty=c(1,1,2,2), bty="n")

@
\end{center}


\section{Potential limitations and next steps}

\begin{itemize}
\item The calculation of the matrix $X_D(t,x)$ using finite differences can be inaccurate for small event times. The functions \verb+stpm2+ and \verb+pstpm2+ raise errors when the event times are less than 1e-4. One working solution is to scale the event times (e.g. multiply by 1000) for analysis. TODO: investigate whether we can calculate $X_D(t,x)$ more accurately using the \verb+numDeriv+ package.
  \item TODO: Extend the generalised survival models to use multiple random effects.
  \item TODO: Extend the generalised survival models to use automatic differentiation.
\end{itemize}


\end{document}

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