Don’t just count events; event at 6m worse than event at 9y

Response called failure time, survival time, event time

Ex: time until CV death, light bulb failure, pregnancy, ECG abnormality during exercise

Allow for censoring

Ex: 5y f/u study; subject still alive at 5y has failure time \(5+\)

Length of f/u can vary

Even in a well-designed randomized clinical trial, survival modeling can allow one to

Test for and describe interactions with treatment. Subgroup analyses can easily generate spurious results and they do not consider interacting factors in a dose-response manner. Once interactions are modeled, relative treatment benefits can be estimated (e.g., hazard ratios), and analyses can be done to determine if some patients are too sick or too well to have even a relative benefit.

Understand prognostic factors (strength and shape).

Model absolute clinical benefit. First, a model for the probability of surviving past time \(t\) is developed. Then differences in survival probabilities for patients on treatments A and B can be estimated. The differences will be due primarily to sickness (overall risk) of the patient and to treatment interactions.

Understand time course of treatment effect. The period of maximum effect or period of any substantial effect can be estimated from a plot of relative effects of treatment over time.

Gain power for testing treatment effects.

Adjust for imbalances in treatment allocation.

17.2 Censoring, Delayed Entry, and Truncation

Left-censoring

Interval censoring

Left-truncation (unknown subset of subjects who failed before qualifying for the study)

Delayed entry (exposure after varying periods of survival)

Choice of time zero important

Take into account waiting time bias

Usually have random type I censoring (on duration, not # events)

Must usually have non-informative censoring: censoring independent of impending failure

\(\rightarrow\) unbiased estimate of cause-specific \(\lambda(t)\) or \(S(t)\) since censoring non-informative

\(1 - S_{\text{KM}}(t)\) estimates Pr(failing from the event in absence of other events)

Method can work with dependent causes but interpretation difficult

See Larson & Dinse (1985) for joint model of time until any failure and the type of failure

See Duc & Wolbers (n.d.) for an interesting smooth semi-nonparametric cumulative incidence estimator under competing risks, incorporating a joint distribution of event time and type

where \(S(u)\) is the probability of surviving (ignoring cause of death), which equals \(\exp(-\int_{0}^{u}(\sum \lambda_{m}(x))dx)\)

Note that interpretation of cumulative incidence in the presence of competing risks can be difficult, e.g., probability of having a heart attack that precedes death

\(1-F_{m}(t) = \exp(-\int_{0}^{t}\lambda_{m}(u)du)\) only if failures due to other causes are eliminated and if the cause-specific hazard of interest remains unchanged in doing so.

where \(d_{mi}\) is the number of failures of type \(m\) at time \(t_i\) and \(n_i\) is the number of subjects at risk of failure at \(t_i\).

Margaret S. Pepe (1991), M. S. Pepe et al. (1991), Margaret S. Pepe & Mori (1993) show how to use combo of K-M estimators to estimate Pr(free of event 1 by time \(t\) given event 2 not occur by \(t\))

Suppose event 1 not a terminating event (e.g., death); even after event 1 subjects followed to find occurrences of event 2

where \(S_{12}(t)\) is the survival function for \(\min(T_{1},T_{2})\)

Estimate of Pr(subject still alive at \(t\) will be free of MI at \(t\)): \(S_{\text{KM}_{12}} / S_{\text{KM}_{2}}\)

Can also easily compute Pr(event 1 occurs by time \(t\)and that event 2 has not occurred by \(t\)) \(\rightarrow\) \(S_{2}(t) - S_{12}(t) = [1 - S_{12}(t)] - [1 - S_{2}(t)]\)

Crude survival functions come from marginal distributions, i.e. \(\Pr(T_{1} > t)\) whether or not event 2 occurs)

Estimate \(\pi_{ij}(t_{1}, t_{2})\) that subject in state \(i\) at time \(t_1\) is in state \(j\)\(t_2\) time units later

Define \(S_{KM}^{i}(t | t_{1})\) = Kaplan–Meier estimate of Pr(surviving \(t\) additional years for cohort of subjects beginning follow-up at time \(t_{1}\) in state \(i\))

where \(n_{i}(t_{1},t_{2})\) is the number of subjects in live state \(i\) at time \(t_{1}\) who are alive and uncensored \(t_{2}\) time units later, and \(n_{ij}(t_{1},t_{2})\) is the number of such subjects in state \(j\)\(t_{2}\) time units beyond \(t_{1}\).

17.6.4 Joint Analysis of Time and Severity of an Event

Can give more weight to an event that occurs earlier or is more severe

Ex: time until first headache / severity of headache

“joint hazard function”, ordinal category \(j\)

\[
\lambda_{j}(t) = \lambda(t) \pi_{j}(t),
\]

Allows shift in dist. of response severity as \(t \uparrow\)

17.6.5 Analysis of Multiple Events

Ex: MI, ulcer, pregnancy, infection

Analysis of time to first event may lose info

Specialized multivariate failure time methods exist

Simpler to model marginal distributions

One record per event per subject

Can have # previous events as covariable

Correct variances for intra-subject correlation using clustered sandwich estimator

Method can handle multiple events of differing types

17.6.6 Generalization

Longitudinal ordinal models based on a discrete time state transition Markov process provide a unified framework for most of the contexts discussed in this chapter. See the last chapter of these RMS course notes and also this.

17.7 R Functions

event.chart in Hmisc draws a variety of charts for displaying raw survival time data, for both single and multiple events per subject (see also event.history, event.chart2)

Analyses in this chapter can be done as special cases of the Cox model

Particular functions for this chapter (no covariables) from Therneau:

Surv function: Combines time to event variable and event/censoring indicator into a single survival time matrix object

Right censoring: Surv(y, event); event is event/censoring indicator, usually coded TRUE/FALSE, 0=censored 1=event or 1=censored 2=event. If the event status variable has other coding, e.g., 3 means death, use Surv(y, s==3).

survfit: Kaplan–Meier and other nonparametric survival curves using the survival package

npsurv: rms package wrapper for survfit

Code

units(y) <-"Month"# Default is "Day" - used for axis labels, etc.survfit(Surv(y, event) ~ svar1 + svar2 + ... , data, subset,na.action=na.delete,type=c("kaplan-meier", "fleming-harrington", "fh2"),error=c("greenwood", "tsiatis"), se.fit=TRUE,conf.int=.95,conf.type=c("log", "plain", "log-log"))

If there are no stratification variables (svar1, …), omit them. To print a table of estimates, use

Code

f <-survfit(...)print(f) # print brief summary of fsummary(f, times, censored=FALSE)

For failure times stored in days, use

Code

f <-survfit(Surv(futime, event) ~ sex)summary(f, seq(30,180,by=30))

This invokes plot.survfit. More options: use npsurv and survplot in rms for other options that include automatic curve labeling and showing the number of subjects at risk at selected times. See code for Figure 17.6 above.

Stratified estimates, with four treatments distinguished by line type and curve labels, could be drawn by

bootkm function in Hmisc bootstraps Kaplan–Meier survival estimates or Kaplan–Meier estimates of quantiles of the survival time distribution. It is easy to use bootkm to compute for example a nonparametric confidence interval for the ratio of median survival times for two groups.

Berridge, D. M., & Whitehead, J. (1991). Analysis of failure time data with ordinal categories of response. Stat Med, 10, 1703–1710. https://doi.org/10.1002/sim.4780101108

Duc, A. N., & Wolbers, M. (n.d.). Smooth semi-nonparametric (SNP) estimation of the cumulative incidence function. Stat Med, n/a. https://doi.org/10.1002/sim.7331

Larson, M. G., & Dinse, G. E. (1985). A mixture model for the regression analysis of competing risks data. Appl Stat, 34, 201–211.

Pepe, Margaret S. (1991). Inference for events with dependent risks in multiple endpoint studies. J Am Stat Assoc, 86, 770–778.

Pepe, M. S., Longton, G., & Thornquist, M. (1991). A qualifier Q for the survival function to describe the prevalence of a transient condition. Stat Med, 10, 413–421.

Pepe, Margaret S., & Mori, M. (1993). Kaplan–Meier, marginal or conditional probability curves in summarizing competing risks failure time data? Stat Med, 12, 737–751.

Strauss, D., & Shavelle, R. (1998). An extended Kaplan–Meier estimator and its applications. Stat Med, 17, 971–982.

estimation of transition probabilities of an individual in state i at time x being in state j at a subsequent time t;dead state and multiple live states;prognostic chart;generalized uninformative censoring;multistate Kaplan-Meier estimator

```{r include=FALSE}require(Hmisc)options(qproject='rms', prType='html')require(qreport)getRs('qbookfun.r')hookaddcap()knitr::set_alias(w = 'fig.width', h = 'fig.height', cap = 'fig.cap', scap ='fig.scap')```<!-- NEW: base graphics -> ggplot2 --># Introduction to Survival Analysis## Background* Use when time to occurrence of event is important* Don't just count events; event at 6m worse than event at 9y* Response called _failure time, survival time, event time_* Ex: time until CV death, light bulb failure, pregnancy, ECG abnormality during exercise* Allow for censoring* Ex: 5y f/u study; subject still alive at 5y has failure time $5+$* Length of f/u can vary* Even in a well-designed randomized clinical trial, survival modeling can allow one to1. Test for and describe interactions with treatment. Subgroup analysescan easily generate spurious results and they do not consider interactingfactors in a dose-response manner. Once interactions are modeled,relative treatment benefits can be estimated (e.g., hazard ratios), andanalyses can be done to determine if some patients are too sick or too well to have even a relative benefit.1. Understand prognostic factors (strength and shape).1. Model absolute clinical benefit. First, a model for the probability of surviving past time $t$ is developed. Then differences in survival probabilities for patients on treatments A and B can be estimated. The differences will be due primarily to sickness (overall risk) of the patient and to treatment interactions.1. Understand time course of treatment effect. The period of maximum effect or period of any substantial effect can be estimated from a plot of relative effects of treatment over time.1. Gain power for testing treatment effects.1. Adjust for imbalances in treatment allocation.## Censoring, Delayed Entry, and Truncation* Left-censoring* Interval censoring* Left-truncation (unknown subset of subjects who failed before qualifying for the study)* Delayed entry (exposure after varying periods of survival)* Choice of time zero important* Take into account _waiting time bias_* Usually have random type I censoring (on duration, not # events)* Must usually have _non-informative censoring_: <br> censoring independent of impending failure* Intention-to-treat is a preventative measure## Notation, Survival, and Hazard Functions* $T$: response variable$$S(t) = \Pr(T > t) = 1 - F(t)$$```{r fun-example,h=2.5,w=3.5,cap='Survival function'}#| label: fig-surv-fun-examplett <- c(seq(.0001,.002,by=.001),seq(.002,.02,by=.001), seq(.02,1,by=.01))# Note: bb,dd not stated in usual Weibull formaa <- .5bb <- -.5cc <- 10dd <- 4cumhaz <- (aa/(bb+1))*tt^(bb + 1) + (cc/(dd+1))*tt^(dd + 1)survival <- exp(-cumhaz)hazard <- ifelse(tt>.001, aa*tt^bb + cc*tt^dd, NA)ggplot(mapping=aes(x=tt, y=survival)) + geom_line() + xlab(expression(t)) + ylab('Survival Function')``````{r cumhaz-example,h=2.5,w=3.5,cap='Cumulative hazard function'}#| label: fig-surv-cumhaz-exampleggplot(mapping=aes(x=tt, y=cumhaz)) + geom_line() + xlab(expression(t)) + ylab('Cumulative Hazard Function')```* Hazard function (force of mortality; instantaneous event rate)* $\approx \Pr($event will occur in small interval around $t$ given has not occurred before $t$)* Very useful for learning about mechanisms and forces of risk over time```{r hazard-example,h=2.5,w=3.5,cap='Hazard function'}#| label: fig-surv-hazard-exampleggplot(mapping=aes(x=tt, y=hazard)) + geom_line() + xlab(expression(t)) + ylab('Hazard Function')```$$\lambda(t) = \lim_{u \rightarrow 0} \frac{\Pr(t < T \leq t+u | T>t)}{u},$$which using the law of conditional probability becomes\begin{array}{ccc}\lambda (t) &=& \lim_{u \rightarrow 0} \frac{\Pr(t < T \leq t+u) /\Pr(T>t)}{u} \nonumber \\&=& \lim_{u \rightarrow 0} \frac{[F(t+u)-F(t)]/u}{S(t)} \nonumber \\&=& \frac{\partial F(t)/\partial t}{S(t)} \\&=& \frac{f(t)}{S(t)} , \nonumber\end{array}$$\frac{\partial \log S(t)}{\partial t} = \frac{\partial S(t)/\partial t}{S(t)} = -\frac{f(t)}{S(t)},$$$$\lambda (t) = -\frac{\partial \log S(t)}{\partial t},$$$$\int_{0}^{t}\lambda(v)dv = -\log S(t) .$$$$\Lambda(t) = -\log S(t) ,$$$$S(t) = \exp[-\Lambda(t)] .$$* Expected value of $\Lambda(T) = 1$$$T_{q} = S^{-1}(1-q) .$$$$T_{0.5} = S^{-1}(0.5) .$$\begin{array}{ccc}T_{q} &=& \Lambda^{-1}[-\log(1-q)] ~\text{and as a special case,} \nonumber \\T_{.5} &=& \Lambda^{-1}(\log 2) .\end{array}$$\mu = \int_{0}^{\infty}S(v)dv$$* Event time for subject $i$: $T_{i}$* Censoring time: $C_{i}$* Event indicator:\begin{array}{ccc}e_{i} &=& 1~~\text{if the event was observed}~~(T_{i} \leq C_{i}) , \nonumber \\ &=& 0~~\text{if the response was censored}~~(T_{i} > C_{i}) \end{array}* The observed response is$$Y_{i} = \min(T_{i}, C_{i}) ,$$```{r out.width='85%',echo=FALSE}#| label: fig-surv-censored-data#| fig-cap: "Some censored data. Circles denote events."knitr::include_graphics('surv-censored-data.png')```## Homogeneous Failure Time Distributions* Exponential distribution: constant hazard\begin{array}{ccc}\Lambda(t) &=& \lambda t {\rm \ \ and} \nonumber \\S(t) &=& \exp(-\Lambda(t)) = \exp(-\lambda t)\end{array}$$T_{0.5} = \log(2)/\lambda$$* Weibull distribution\begin{array}{ccc}\lambda(t) &=& \alpha \gamma t^{\gamma -1} \nonumber \\\Lambda(t) &=& \alpha t^{\gamma} \\S(t) &=& \exp(-\alpha t^{\gamma}) \nonumber\end{array}```{r weibull-shapes,h=2.75,w=3.5,cap='Some Weibull hazard functions with $\\alpha=1$ and various values of $\\gamma$'}#| label: fig-surv-weibull-shapestt <- seq(1e-3, 1.2, length=100)w <- NULLa <- 1for(b in c(.5, 1, 2, 4)) w <- rbind(w, data.frame(gamma=b, tt=tt, y= a * b * tt ^ (b-1)))ggplot(w, aes(x=tt, y=y, color=factor(gamma))) + geom_line() + xlab(expression(t)) + ylab('Hazard Function') + guides(color=guide_legend(title=expression(gamma))) + theme(legend.position='bottom')```$$T_{0.5} =[(\log 2)/\alpha]^{1/\gamma}$$* The restricted cubic spline hazard model with $k$ knots is$$\lambda_{k}(t) = a + bt + \sum_{j=1}^{k-2} \gamma_{j} w_{j}(t)$$## Nonparametric Estimation of $S$ and $\Lambda$### Kaplan--Meier Estimator* No censoring $\rightarrow$$$S_{n}(t) =[{\rm number\ of\ } T_{i} > t]/n$$* Kaplan--Meier (product-limit) estimator| Day | No. Subjects at Risk| Deaths | Censored | Cumulative Survival ||-----|-----|-----|-----|-----|| 12 | 100 | 1 | 0 | $99/100=.99$ || 30 | 99 | 2 | 1 | $97/99 \times 99/100 = .97$ || 60 | 96 | 0 | 3 | $96/96 \times .97 = .97$ || 72 | 93 | 3 | 0 | $90/93 \times .97 = .94$ || . | . | . | . | . || . | . | . | . | . |$$S_{{\rm KM}}(t) = \prod_{i:t_{i}<t}(1-d_{i}/n_{i})$$* The Kaplan--Meier estimator of $\Lambda(t)$ is $\Lambda_\text{KM}(t) = -\log S_{{\rm KM}}(t)$.* Simple example\begin{array}{c}1 \ \ 3\ \ 3\ \ 6^{+}\ \ 8^{+}\ \ 9\ \ 10^{+} . \nonumber\end{array}| $i$ | $t_{i}$ | $n_{i}$ | $d_{i}$ | $(n_{i}-d_{i})/n_{i}$ ||-----|-----|-----|-----|-----|| 1 | 1 | 7 | 1 | 6/7 || 2 | 3 | 6 | 2 | 4/6 || 3 | 9 | 2 | 1 | 1/2 |\begin{array}{ccc}S_{\text{KM}}(t) &=& 1, ~~~~0 \le t<1 \nonumber \\ &=& 6/7=.85,~~1 \le t<3 \nonumber \\ &=& (6/7)(4/6)=.57,~~ 3 \le t<9 \\ &=& (6/7)(4/6)(1/2)=.29,~~9 \le t<10 . \nonumber\end{array}```{r km-example,scap='Kaplan-Meier and Nelson--Aalen estimates',cap='Kaplan--Meier product-limit estimator with 0.95 confidence bands. The Altschuler--Nelson--Aalen--Fleming--Harrington estimator is depicted with the dashed lines.'}#| label: fig-surv-km-examplerequire(rms)spar(bty='l')tt <- c(1,3,3,6,8,9,10)stat <- c(1,1,1,0,0,1,0)S <- Surv(tt, stat)survplot(npsurv(S ~ 1), conf="bands", n.risk=TRUE, xlab="t")survplot(npsurv(S ~ 1, type="fleming-harrington", conf.int=FALSE), add=TRUE, lty=3)```$$\text{Var}(\log \Lambda_{{\rm KM}}(t)) = \frac{\sum_{i:t_{i} < t} d_{i} / [n_{i}(n_{i}-d_{i})]} {\{\sum_{i:t_{i}<t} \log[(n_{i}-d_{i})/n_{i}]\}^{2}}$$$$S_{\text{KM}}(t)^{\exp(\pm zs)}$$### Altschuler--Nelson Estimator\begin{array}{ccc}\hat{\Lambda}(t) &=& \sum_{i:t_{i}<t}\frac{d_{i}}{n_{i}} \nonumber \\S_{\Lambda}(t) &=& \exp(-\hat{\Lambda}(t))\end{array}## Analysis of Multiple Endpoints* Cancer trial: recurrence of disease or death* CV trial: nonfatal MI or death* Analyze usual way but watch out for differing risk factors* Analyzing multiple causes of terminating event $\rightarrow$ + Cause-specific hazards, censor on cause not currently analyzed + Not assume mechanism for cause removal or correlations of causes + Problem if apply to a setting where causes are removed differently* More complex if explicitly handle mixture of nonfatal outcomes with fatal outcome### Competing Risks* Events independent $\rightarrow$ analyze separately, censoring others* $\rightarrow$ unbiased estimate of cause-specific $\lambda(t)$ or $S(t)$ since censoring non-informative* $1 - S_{\text{KM}}(t)$ estimates Pr(failing from the event in absence of other events)* Method can work with dependent causes but interpretation difficult* See @lar85mix for joint model of time until any failure and the type of failure* See @duc17smo for an interesting smooth semi-nonparametric cumulative incidence estimator under competing risks, incorporating a joint distribution of event time and type### Competing Dependent Risks* Ordinary K-M estimator biased* Suppose cause $m$ is of interest* Cause-specific hazard function:$$\lambda_{m}(t) = \lim_{u \rightarrow 0} \frac{\Pr(\text{fail from cause}~m~ \text{in}~ [t, t+u) | \text{alive at} ~t)}{u}$$* The _cumulative incidence function_ or probability of failure from cause $m$ by time $t$ is given by$$F_{m}(t) = \int_{0}^{t} \lambda_{m}(u)S(u)du$$where $S(u)$ is the probability of surviving (ignoring cause of death),which equals $\exp(-\int_{0}^{u}(\sum \lambda_{m}(x))dx)$* Note that interpretation of cumulative incidence in the presence of competing risks can be difficult, e.g., probability of having a heart attack _that precedes death_* $1-F_{m}(t) = \exp(-\int_{0}^{t}\lambda_{m}(u)du)$ only if failures due to other causes are eliminated and if the cause-specific hazard of interest remains unchanged in doing so.* Nonparametric estimator of $F_{m}(t)$:$$\hat{F}_{m}(t) = \sum_{i:t_{i}\leq t}\frac{d_{mi}}{n_{i}} S_{\text{KM}}(t_{i-1})$$where $d_{mi}$ is the number of failures of type $m$ at time $t_i$ and$n_i$ is the number of subjects at risk of failure at $t_i$.* @pep91inf, @pep91qua, @pep93kap show how to use combo of K-M estimators to estimate Pr(free of event 1 by time $t$ given event 2 not occur by $t$)* Suppose event 1 not a terminating event (e.g., death); even after event 1 subjects followed to find occurrences of event 2* $\Pr(T_{1} > t | T_{2} > t)$:\begin{array}{ccc}\Pr(T_{1} > t | T_{2} > t) &=& \frac{\Pr(T_{1} > t ~\text{and} T_{2} > t)}{\Pr(T_{2} > t)} \nonumber \\&=& \frac{S_{12}(t)}{S_{2}(t)},\end{array}where $S_{12}(t)$ is the survival function for $\min(T_{1},T_{2})$* Estimate of Pr(subject still alive at $t$ will be free of MI at $t$): <br>$S_{\text{KM}_{12}} / S_{\text{KM}_{2}}$* Can also easily compute Pr(event 1 occurs by time $t$ _and_ that event 2 has not occurred by $t$) $\rightarrow$ <br> $S_{2}(t) - S_{12}(t) = [1 - S_{12}(t)] - [1 - S_{2}(t)]$* _Crude survival functions_ come from _marginal distributions_, i.e. $\Pr(T_{1} > t)$ whether or not event 2 occurs)\begin{array}{ccc}S_{c}(t) &=& 1 - F_{1}(t) \nonumber \\F_{1}(t) &=& \Pr(T_{1} \leq t)\end{array}* $F_{1}(t)$ is _crude incidence function_* $T_{1} < t \rightarrow$ occurrence of event 1 is part of the prob. being computed* Event 2 terminating $\rightarrow$ some subjects can never suffer event 1 $\rightarrow$ <br> crude survival fctn. for $T_{1}$ never drops to zero* Crude survival fctn interpreted as surv. dist. of $W$ where $W=T_{1}$ if $T_{1} < T_{2}$ and $W=\infty$ otherwise### State Transitions and Multiple Types of Nonfatal EventsMultiple live states, one absorbing state (all-cause mortality)* @str98ext extended Kaplan--Meier estimator* Estimate $\pi_{ij}(t_{1}, t_{2})$ that subject in state $i$ at time $t_1$ is in state $j$ $t_2$ time units later* Define $S_{KM}^{i}(t | t_{1})$ = Kaplan--Meier estimate of Pr(surviving $t$ additional years for cohort of subjects beginning follow-up at time $t_{1}$ in state $i$)* Then$$\pi_{ij}(t_{1},t_{2}) = \frac{n_{ij}(t_{1},t_{2})}{n_{i}(t_{1},t_{2})}S_{KM}^{i}(t_{2} | t_{1})$$where $n_{i}(t_{1},t_{2})$ is the number of subjects in live state $i$at time $t_{1}$ who are alive and uncensored $t_{2}$ time units later, and$n_{ij}(t_{1},t_{2})$ is the number of such subjects in state $j$$t_{2}$ time units beyond $t_{1}$.### Joint Analysis of Time and Severity of an Event* Can give more weight to an event that occurs earlier or is more severe* @ber91ana* Ordinal scale for severity* Severity measured at time of (single) event* Ex: time until first headache / severity of headache* "joint hazard function", ordinal category $j$$$\lambda_{j}(t) = \lambda(t) \pi_{j}(t),$$* Allows shift in dist. of response severity as $t \uparrow$### Analysis of Multiple Events* Ex: MI, ulcer, pregnancy, infection* Analysis of time to first event may lose info* Specialized multivariate failure time methods exist* Simpler to model _marginal distributions_* One record per event per subject* Can have # previous events as covariable* Correct variances for intra-subject correlation using clustered sandwich estimator* Method can handle multiple events of differing types### GeneralizationLongitudinal ordinal models based on a discrete time state transition Markov process provide a unified framework for most of the contexts discussed in this chapter. See the last chapter of these RMS course notes and also [this](https://www.fharrell.com/talk/cmstat).## R Functions* `event.chart` in `Hmisc` draws a variety of charts for displaying raw survival time data, for both single and multiple events per subject (see also `event.history, event.chart2`)* Analyses in this chapter can be done as special cases of the Cox model* Particular functions for this chapter (no covariables) from Therneau:* `Surv` function: Combines time to event variable and event/censoring indicator into a single survival time matrix object* Right censoring: `Surv(y, event)`; `event` is event/censoring indicator, usually coded `TRUE/FALSE`, 0=censored 1=event or 1=censored 2=event. If the event status variable has other coding, e.g., 3 means death, use `Surv(y, s==3)`.* `survfit`: Kaplan--Meier and other nonparametric survival curves using the `survival` package* `npsurv`: `rms` package wrapper for `survfit````{r eval=FALSE}units(y) <- "Month" # Default is "Day" - used for axis labels, etc.survfit(Surv(y, event) ~ svar1 + svar2 + ... , data, subset, na.action=na.delete, type=c("kaplan-meier", "fleming-harrington", "fh2"), error=c("greenwood", "tsiatis"), se.fit=TRUE, conf.int=.95, conf.type=c("log", "plain", "log-log"))```If there are no stratification variables (`svar1`, ...), omit them.To print a table of estimates, use```{r eval=FALSE}f <- survfit(...)print(f) # print brief summary of fsummary(f, times, censored=FALSE)```For failure times stored in days, use```{r eval=FALSE}f <- survfit(Surv(futime, event) ~ sex)summary(f, seq(30,180,by=30))```to print monthly estimates.To plot the object returned by `survfit`, use```{r eval=FALSE}plot(f, conf.int=TRUE, mark.time=TRUE, mark=3, col=1, lty=1, lwd=1, cex=1, log=FALSE, xscale=1, yscale=1, xlab="", ylab="", xaxs="S", ...)```This invokes `plot.survfit`. More options: use `npsurv` and`survplot` in `rms`for other options that include automatic curve labeling and showingthe number of subjects at risk at selected times.See code for @fig-surv-km-example above.Stratified estimates, with four treatmentsdistinguished by line type and curve labels, could be drawn by```{r eval=FALSE}require(rms)units(y) <- "Year"f <- npsurv(Surv(y, stat) ~ treatment)survplot(f, ylab="Fraction Pain-Free")```* `bootkm` function in `Hmisc` bootstrapsKaplan--Meier survival estimates or Kaplan--Meier estimates ofquantiles of the survival time distribution. It is easy to use`bootkm` to compute for example a nonparametric confidence intervalfor the ratio of median survival times for two groups.```{r echo=FALSE}saveCap('17')```