# Difference between revisions of "Survival data"

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Comparison: | Comparison: | ||

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+ | ! Definition | ||

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+ | | Two class | ||

+ | | <math> P(Z_{case} > Z_{control}) </math> | ||

+ | | the probability that a randomly selected '''case''' will have a higher test result (marker value) than a randomly selected '''control'''. It represents a measure of concordance between the marker and the disease status. ROC curves are particularly useful for comparing the discriminatory capacity of different potential biomarkers. (Heagerty & Zheng 2005) | ||

+ | |- | ||

+ | | Survival data | ||

+ | | <math> P(\beta' Z_1 > \beta' Z_2|T_1 < T_2) </math> | ||

+ | | the probability of concordance between predicted and observed responses. The probability that the predictions for a random pair of subjects are concordant with their outcomes. (Heagerty & Zheng 2005) | ||

+ | |} | ||

p95 of Heagerty and Zheng (2005) gave a relationship of C-statistic: | p95 of Heagerty and Zheng (2005) gave a relationship of C-statistic: |

## Revision as of 18:36, 30 November 2019

## Contents

- 1 Survival data
- 1.1 Censoring
- 1.2 How to explore survival data
- 1.3 Kaplan & Meier and Nelson-Aalen: survfit.formula(), Surv()
- 1.4 Alternatives to survival function plot
- 1.5 Breslow estimate
- 1.6 Logrank test
- 1.7 Survival curve with confidence interval
- 1.8 Parametric models and survival function for censored data
- 1.9 Parametric models and likelihood function for uncensored data
- 1.10 Survival modeling
- 1.11 Weibull and Exponential model to Cox model
- 1.12 CDF follows Unif(0,1)
- 1.13 Simulate survival data
- 1.14 Standardize covariates
- 1.15 Predefined censoring rates
- 1.16 Cross validation
- 1.17 Competing risk
- 1.18 Survival rate terminology
- 1.19 Books
- 1.20 HER2-positive breast cancer

- 2 Cox proportional hazards model and the partial log-likelihood function
- 2.1 Compare the partial likelihood to the full likelihood
- 2.2 z-column (Wald statistic) from R's coxph()
- 2.3 How exactly can the Cox-model ignore exact times?
- 2.4 Partial likelihood when there are ties; hypothesis testing: Likelihood Ratio Test, Wald Test & Score Test
- 2.5 Hazard (function) and survival function
- 2.6 Check the proportional hazard (constant HR over time) assumption by cox.zph()
- 2.7 Sample size calculators
- 2.8 Extract p-values
- 2.9 Expectation of life & expected future lifetime
- 2.10 Hazard Ratio (exp^beta) vs Relative Risk
- 2.11 Piece-wise constant baseline hazard model, Poisson model and Breslow estimate
- 2.12 Estimate baseline hazard , Breslow cumulative baseline hazard , baseline survival function and the survival function
- 2.13 Predicted survival probability in Cox model: survfit.coxph(), plot.survfit() & summary.survfit( , times)
- 2.14 Loglikelihood
- 2.15 High dimensional data
- 2.16 glmnet + Cox models
- 2.17 Prognostic index/risk scores
- 2.18 Survival risk prediction
- 2.19 Assessing the performance of prediction models
- 2.19.1 Hazard ratio
- 2.19.2 D index
- 2.19.3 AUC
- 2.19.4 Concordance index/C-index/C-statistic interpretation and R packages
- 2.19.5 Integrated brier score (≈ "mean squared error" of prediction for survival data)
- 2.19.6 Kendall's tau, Goodman-Kruskal's gamma, Somers' d
- 2.19.7 C-statistics
- 2.19.8 C-statistic limitations
- 2.19.9 C-statistic applications
- 2.19.10 C-statistic vs LRT comparing nested models
- 2.19.11 Time dependent ROC curves

- 2.20 Prognostic markers vs predictive markers (and other biomarkers)
- 2.21 Computation for gene expression (microarray) data
- 2.22 Single gene vs mult-gene survival models
- 2.23 Random papers using C-index, AUC or Brier scores
- 2.24 More, Web tools

# Survival data

- A Package for Survival Analysis in S by Terry M. Therneau, 1999
- https://web.stanford.edu/~lutian/coursepdf/stat331.HTML and https://web.stanford.edu/~lutian/coursepdf/ (3 types of tests).
- http://www.stat.columbia.edu/~madigan/W2025/notes/survival.pdf.
- How to manually compute the KM curve and by R
- Estimation of parametric survival function from joint likelihood in theory and R.

- http://data.princeton.edu/wws509/notes/c7s1.html
- http://data.princeton.edu/pop509/ParametricSurvival.pdf Parametric survival models with covariates (logT = alpha + sigma W) p8
- Weibull p2 where T ~ Weibull and W ~ Extreme value.
- Gamma p3 where T ~ Gamma and W ~ Generalized extreme value
- Generalized gamma p4,
- log normal p4 where T ~ lognormal and W ~ N(0,1)
- log logistic p4 where T ~ log logistic and W ~ standard logistic distribution.

- http://www.math.ucsd.edu/~rxu/math284/ (good cover) Wald test
- http://www.stats.ox.ac.uk/~mlunn/
- https://www.openintro.org/download.php?file=survival_analysis_in_R&referrer=/stat/surv.php
- https://cran.r-project.org/web/packages/survival/vignettes/timedep.pdf
- https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1065034/
- Survival Analysis with R from rviews.rstudio.com
- Survival Analysis with R from bioconnector.og.

## Censoring

Sample schemes of incomplete data

- Type I censoring: the censoring time is fixed
- Type II censoring
- Random censoring
- Right censoring
- Left censoring

- Interval censoring
- Truncation

The most common is called **right censoring** and occurs when a participant does not have the event of interest during the study and thus their last observed follow-up time is less than their time to event. This can occur when a participant drops out before the study ends or when a participant is **event free** at the end of the observation period.

Definitions of common terms in survival analysis

**Event**: Death, disease occurrence, disease recurrence, recovery, or other experience of interest**Time**: The time from the beginning of an observation period (such as surgery or beginning treatment) to (i) an event, or (ii) end of the study, or (iii) loss of contact or withdrawal from the study.**Censoring / Censored observation**: If a subject does not have an event during the observation time, they are described as censored. The subject is censored in the sense that nothing is observed or known about that subject after the time of censoring. A censored subject may or may not have an event after the end of observation time.

In R, "status" should be called **event status**. status = 1 means event occurred. status = 0 means no event (censored). Sometimes the status variable has more than 2 states. We can uses "status != 0" to replace "status" in Surv() function.

- status=0/1/2 for censored, transplant and dead in survival::pbc data.
- status=0/1/2 for censored, relapse and dead in randomForestSRC::follic data.

## How to explore survival data

https://en.wikipedia.org/wiki/Survival_analysis#Survival_analysis_in_R

- Create graph of length of time that each subject was in the study

library(survival) # sort the aml data by time aml <- aml[order(aml$time),] with(aml, plot(time, type="h"))

- Create the life table survival object

summary(aml.survfit) Call: survfit(formula = Surv(time, status == 1) ~ 1, data = aml) time n.risk n.event survival std.err lower 95% CI upper 95% CI 5 23 2 0.9130 0.0588 0.8049 1.000 8 21 2 0.8261 0.0790 0.6848 0.996 9 19 1 0.7826 0.0860 0.6310 0.971 12 18 1 0.7391 0.0916 0.5798 0.942 13 17 1 0.6957 0.0959 0.5309 0.912 18 14 1 0.6460 0.1011 0.4753 0.878 23 13 2 0.5466 0.1073 0.3721 0.803 27 11 1 0.4969 0.1084 0.3240 0.762 30 9 1 0.4417 0.1095 0.2717 0.718 31 8 1 0.3865 0.1089 0.2225 0.671 33 7 1 0.3313 0.1064 0.1765 0.622 34 6 1 0.2761 0.1020 0.1338 0.569 43 5 1 0.2208 0.0954 0.0947 0.515 45 4 1 0.1656 0.0860 0.0598 0.458 48 2 1 0.0828 0.0727 0.0148 0.462

- Kaplan-Meier curve for aml with the confidence bounds.

plot(aml.survfit, xlab = "Time", ylab="Proportion surviving")

- Create aml life tables broken out by treatment (x, "Maintained" vs. "Not maintained")

surv.by.aml.rx <- survfit(Surv(time, status == 1) ~ x, data = aml) summary(surv.by.aml.rx) Call: survfit(formula = Surv(time, status == 1) ~ x, data = aml) x=Maintained time n.risk n.event survival std.err lower 95% CI upper 95% CI 9 11 1 0.909 0.0867 0.7541 1.000 13 10 1 0.818 0.1163 0.6192 1.000 18 8 1 0.716 0.1397 0.4884 1.000 23 7 1 0.614 0.1526 0.3769 0.999 31 5 1 0.491 0.1642 0.2549 0.946 34 4 1 0.368 0.1627 0.1549 0.875 48 2 1 0.184 0.1535 0.0359 0.944 x=Nonmaintained time n.risk n.event survival std.err lower 95% CI upper 95% CI 5 12 2 0.8333 0.1076 0.6470 1.000 8 10 2 0.6667 0.1361 0.4468 0.995 12 8 1 0.5833 0.1423 0.3616 0.941 23 6 1 0.4861 0.1481 0.2675 0.883 27 5 1 0.3889 0.1470 0.1854 0.816 30 4 1 0.2917 0.1387 0.1148 0.741 33 3 1 0.1944 0.1219 0.0569 0.664 43 2 1 0.0972 0.0919 0.0153 0.620 45 1 1 0.0000 NaN NA NA

- Plot KM plot broken out by treatment

plot(surv.by.aml.rx, xlab = "Time", ylab="Survival", col=c("black", "red"), lty = 1:2, main="Kaplan-Meier Survival vs. Maintenance in AML") legend(100, .6, c("Maintained", "Not maintained"), lty = 1:2, col=c("black", "red"))

- Perform the log rank test using the R function survdiff().

surv.diff.aml <- survdiff(Surv(time, status == 1) ~ x, data=aml) surv.diff.aml Call: survdiff(formula = Surv(time, status == 1) ~ x, data = aml) N Observed Expected (O-E)^2/E (O-E)^2/V x=Maintained 11 7 10.69 1.27 3.4 x=Nonmaintained 12 11 7.31 1.86 3.4 Chisq= 3.4 on 1 degrees of freedom, p= 0.07

### Summary statistics

- Kaplan-Meier Method and Log-Rank Test
- Statistics
- Table of status vs treatment (with proportion)
- Table of treatment vs training/test

- Life table
- summary(survfit(Surv(time, status) ~ 1)) or summary(survfit(Surv(time, status) ~ treatment))
- KMsurv::lifetab()

- Coxph function and visualize them using the ggforest package

### Some public data

package | data (sample size) |
---|---|

survival | pbc (418), ovarian (26), aml/leukemia (23), colon (1858), lung (228), veteran (137) |

pec | GBSG2 (686), cost (518) |

randomForestSRC | follic (541) |

KMsurv | A LOT. tongue (80) |

survivalROC | mayo (312) |

survAUC | NA |

## Kaplan & Meier and Nelson-Aalen: survfit.formula(), Surv()

- Landmarks
- Kaplan-Meier: 1958
- Nelson: 1969
- Cox and Brewlow: 1972 S(t) = exp(-Lambda(t))
- Aalen: 1978 Lambda(t)

- D distinct times . At time there are events. Let be the number of individuals who are at risk at time . The quantity provides an estimate of the conditional probability that an individual who survives to just prior to time experiences the event at time . The
**KM estimator of the survival function**and the**Nelson-Aalen estimator of the cumulative hazard**(their relationship is given below) are define as follows ():

str(kidney) 'data.frame': 76 obs. of 7 variables: $ id : num 1 1 2 2 3 3 4 4 5 5 ... $ time : num 8 16 23 13 22 28 447 318 30 12 ... $ status : num 1 1 1 0 1 1 1 1 1 1 ... $ age : num 28 28 48 48 32 32 31 32 10 10 ... $ sex : num 1 1 2 2 1 1 2 2 1 1 ... $ disease: Factor w/ 4 levels "Other","GN","AN",..: 1 1 2 2 1 1 1 1 1 1 ... $ frail : num 2.3 2.3 1.9 1.9 1.2 1.2 0.5 0.5 1.5 1.5 ... kidney[order(kidney$time), c("time", "status")] kidney[kidney$time == 13, ] # one is dead and the other is alive length(unique(kidney$time)) # 60 sfit <- survfit(Surv(time, status) ~ 1, data = kidney) sfit Call: survfit(formula = Surv(time, status) ~ 1, data = kidney) n events median 0.95LCL 0.95UCL 76 58 78 39 152 str(sfit) List of 13 $ n : int 76 $ time : num [1:60] 2 4 5 6 7 8 9 12 13 15 ... $ n.risk : num [1:60] 76 75 74 72 71 69 65 64 62 60 ... $ n.event : num [1:60] 1 0 0 0 2 2 1 2 1 2 ... $ n.censor : num [1:60] 0 1 2 1 0 2 0 0 1 0 ... $ surv : num [1:60] 0.987 0.987 0.987 0.987 0.959 ... $ type : chr "right" length(unique(kidney$time)) # [1] 60 all(sapply(sfit$time, function(tt) sum(kidney$time >= tt)) == sfit$n.risk) # TRUE all(sapply(sfit$time, function(tt) sum(kidney$status[kidney$time == tt])) == sfit$n.event) # TRUE all(sapply(sfit$time, function(tt) sum(1-kidney$status[kidney$time == tt])) == sfit$n.censor) # TRUE all(cumprod(1 - sfit$n.event/sfit$n.risk) == sfit$surv) # FALSE range(abs(cumprod(1 - sfit$n.event/sfit$n.risk) - sfit$surv)) # [1] 0.000000e+00 1.387779e-17 summary(sfit) time n.risk n.event survival std.err lower 95% CI upper 95% CI 2 76 1 0.987 0.0131 0.96155 1.000 7 71 2 0.959 0.0232 0.91469 1.000 8 69 2 0.931 0.0297 0.87484 0.991 ... 511 3 1 0.042 0.0288 0.01095 0.161 536 2 1 0.021 0.0207 0.00305 0.145 562 1 1 0.000 NaN NA NA

- Note that the KM estimate is
**left-continuous**step function with the intervals closed at left and open at right. For for a certain*j*, we have where is the number people who have an event during the interval and is the number of people at risk just before the beginning of the interval . - The product-limit estimator can be constructed by using a
*reduced-sample*approach. We can estimate the for . because S(0)=1 and, for a discrete distribution, . **Self consistency**. If we had no censored observations, the estimator of the survival function at a time*t*is the proportion of observations which are larger than*t*, that is, .- Curves are plotted in the same order as they are listed by print (which gives a 1 line summary of each). For example, -1 < 1 and 'Maintenance' < 'Nonmaintained'. That means, the labels list in the legend() command should have the same order as the curves.
- Kaplan and Meier is used to give an estimator of the survival function S(t)
- Nelson-Aalen estimator is for the cumulative hazard H(t). Note that and as t goes to infinity. So there is a constraint on the hazard function, see Wikipedia.

Note that S(t) is related to H(t) by or . The two estimators are similar (see example 4.1A and 4.1B from Klein and Moeschberge).

The Nelson-Aalen estimator has two primary uses in analyzing data

- Selecting between parametric models for the time to event
- Crude estimates of the hazard rate h(t). This is related to the estimation of the survival function in Cox model. See 8.6 of Klein and Moeschberge.

The Kaplan–Meier estimator (the product limit estimator) is an estimator for estimating the survival function from lifetime data. In medical research, it is often used to measure the fraction of patients living for a certain amount of time after treatment.

Note that

**The "+" sign in the KM curves means censored observations (this convention matches with the output of Surv() function) and a long vertical line (not '+') means there is a dead observation at that time.**

> aml[1:5,] time status x 1 9 1 Maintained 2 13 1 Maintained 3 13 0 Maintained 4 18 1 Maintained 5 23 1 Maintained > Surv(aml$time, aml$status)[1:5,] [1] 9 13 13+ 18 23

**If the last observation (longest survival time) is dead, the survival curve will goes down to zero. Otherwise, the survival curve will remain flat from the last event time.**

Usually the KM curve of treatment group is higher than that of the control group.

The Y-axis (the probability that a member from a given population will have a lifetime exceeding time) is often called

- Cumulative probability
- Cumulative survival
- Percent survival
- Probability without event
- Proportion alive/surviving
- Survival
- Survival probability

File:KMcurve.png File:KMcurve cumhaz.png

> library(survival) > str(aml$x) Factor w/ 2 levels "Maintained","Nonmaintained": 1 1 1 1 1 1 1 1 1 1 ... > plot(leukemia.surv <- survfit(Surv(time, status) ~ x, data = aml[7:17,] ) , lty=2:3, mark.time = TRUE) # a (small) subset, mark.time is used to show censored obs > aml[7:17,] time status x 7 31 1 Maintained 8 34 1 Maintained 9 45 0 Maintained 10 48 1 Maintained 11 161 0 Maintained 12 5 1 Nonmaintained 13 5 1 Nonmaintained 14 8 1 Nonmaintained 15 8 1 Nonmaintained 16 12 1 Nonmaintained 17 16 0 Nonmaintained > legend(100, .9, c("Maintenance", "No Maintenance"), lty = 2:3) # lty: 2=dashed, 3=dotted > title("Kaplan-Meier Curves\nfor AML Maintenance Study") # Cumulative hazard plot # Lambda(t) = -log(S(t)); # see https://en.wikipedia.org/wiki/Survival_analysis # http://statweb.stanford.edu/~olshen/hrp262spring01/spring01Handouts/Phil_doc.pdf plot(leukemia.surv <- survfit(Surv(time, status) ~ x, data = aml[7:17,] ) , lty=2:3, mark.time = T, fun="cumhaz", ylab="Cumulative Hazard")

- Kaplan-Meier estimator from the wikipedia.
- Two papers this and this to describe steps to calculate the KM estimate.
- Estimating a survival probability in R

# https://www.lexjansen.com/pharmasug/2011/CC/PharmaSUG-2011-CC16.pdf mydata <- data.frame(time=c(3,6,8,12,12,21),status=c(1,1,0,1,1,1)) km <- survfit(Surv(time, status)~1, data=mydata) plot(km, mark.time = T) survest <- stepfun(km$time, c(1, km$surv)) plot(survest) > str(km) List of 13 $ n : int 6 $ time : num [1:5] 3 6 8 12 21 $ n.risk : num [1:5] 6 5 4 3 1 $ n.event : num [1:5] 1 1 0 2 1 $ n.censor : num [1:5] 0 0 1 0 0 $ surv : num [1:5] 0.833 0.667 0.667 0.222 0 $ type : chr "right" $ std.err : num [1:5] 0.183 0.289 0.289 0.866 Inf $ upper : num [1:5] 1 1 1 1 NA $ lower : num [1:5] 0.5827 0.3786 0.3786 0.0407 NA $ conf.type: chr "log" $ conf.int : num 0.95 > class(survest) [1] "stepfun" "function" > survest Step function Call: stepfun(km$time, c(1, km$surv)) x[1:5] = 3, 6, 8, 12, 21 6 plateau levels = 1, 0.83333, 0.66667, ..., 0.22222, 0 > str(survest) function (v) - attr(*, "class")= chr [1:2] "stepfun" "function" - attr(*, "call")= language stepfun(km$time, c(1, km$surv))

### Multiple curves

Curves/groups are ordered. The first color in the palette is used to color the first level of the factor variable. This is same idea as ggsurvplot in the survminer package. This affects parameters like **col** and **lty** in plot() function. For example,

- 1<2
- 'c' < 't'
- 'control' < 'treatment'
- 'Control' < 'Treatment'
- 'female' < 'male'.

For **legend()**, the first category in legend argument will appear at the top of the legend box.

### Inverse Probability of Censoring Weighted (IPCW)

- Inverse probability weighting from Wikipedia
- R packages
- kmcens() from survC1 package
- get.censoring.weights() from TreatmentSelection package (not sure)

- Consistent Estimation of the Expected Brier Score in General Survival Models with Right‐Censored Event Times Gerds et al 2006.
- https://www.bmj.com/content/352/bmj.i189 Examples are considered.
- Correcting for Noncompliance and Dependent Censoring in an AIDS Clinical Trial with Inverse Probability of Censoring Weighted (IPCW) Log‐Rank Tests by James M. Robins, Biometrics 2000.
- The Kaplan–Meier Estimator as an Inverse-Probability-of-Censoring Weighted Average by Satten 2001. IPCW.

The plots below show by flipping the status variable, we can accurately *recover* the survival function of the censoring variable. See the R code here for superimposing the true exponential distribution on the KM plot of the censoring variable.

require(survival) n = 10000 beta1 = 2; beta2 = -1 lambdaT = 1 # baseline hazard lambdaC = 2 # hazard of censoring set.seed(1234) x1 = rnorm(n,0) x2 = rnorm(n,0) # true event time # T = rweibull(n, shape=1, scale=lambdaT*exp(-beta1*x1-beta2*x2)) # Wrong T = Vectorize(rweibull)(n=1, shape=1, scale=lambdaT*exp(-beta1*x1-beta2*x2)) # method 1: exponential censoring variable C <- rweibull(n, shape=1, scale=lambdaC) time = pmin(T,C) status <- 1*(T <= C) mean(status) summary(T) summary(C) par(mfrow=c(2,1), mar = c(3,4,2,2)+.1) status2 <- 1-status plot(survfit(Surv(time, status2) ~ 1), ylab="Survival probability", main = 'Exponential censoring time') # method 2: uniform censoring variable C <- runif(n, 0, 21) time = pmin(T,C) status <- 1*(T <= C) status2 <- 1-status plot(survfit(Surv(time, status2) ~ 1), ylab="Survival probability", main = "Uniform censoring time")

### stepfun() and plot.stepfun()

- Draw cumulative hazards using stepfun()
- For KM curve case, see an example above.

### Survival curves with number at risk at bottom: survminer package

R function survminer::ggsurvplot()

- http://www.sthda.com/english/articles/24-ggpubr-publication-ready-plots/81-ggplot2-easy-way-to-mix-multiple-graphs-on-the-same-page/#mix-table-text-and-ggplot
- http://r-addict.com/2016/05/23/Informative-Survival-Plots.html

Paper examples

### Life table

- https://www.r-bloggers.com/veterinary-epidemiologic-research-modelling-survival-data-non-parametric-analyses/
- lifetab()

## Alternatives to survival function plot

https://www.rdocumentation.org/packages/survival/versions/2.43-1/topics/plot.survfit
The **fun** argument, a transformation of the survival curve

- fun = "event" or "F": f(y) = 1-y; it calculates P(T < t). This is like a t-year risk (Blanche 2018).
- fun = "cumhaz": cumulative hazard function (f(y) = -log(y)); it calculates H(t). See Intuition for cumulative hazard function.

## Breslow estimate

- http://support.sas.com/documentation/cdl/en/statug/68162/HTML/default/viewer.htm#statug_lifetest_details03.htm
- Breslow estimate is the exponentiation of the negative Nelson-Aalen estimate of the cumulative hazard function

## Logrank test

- Logrank test is a hypothesis test to compare the survival distributions of two samples. The logrank test statistic compares estimates of the hazard functions of the two groups at each observed event time.
- On null hypotheses in survival analysis Stensrud 2019
- Efficiency of two sample tests via the restricted mean survival time for analyzing event time observations Tian 2017

## Survival curve with confidence interval

http://www.sthda.com/english/wiki/survminer-r-package-survival-data-analysis-and-visualization

## Parametric models and survival function for censored data

Assume the CDF of survival time *T* is and the CDF of the censoring time *C* is ,

- http://www.stat.columbia.edu/~madigan/W2025/notes/survival.pdf#page=23
- http://www.ms.uky.edu/~mai/sta635/LikelihoodCensor635.pdf#page=2 survival function of
- https://web.stanford.edu/~lutian/coursepdf/unit2.pdf#page=3 joint density of
- http://data.princeton.edu/wws509/notes/c7.pdf#page=6
- Special case:
*T*follows Log normal distribution and*C*follows .

### R

- flexsurv package.
- Parametric survival modeling which uses the
**flexsurv**package. - Used in simsurv package

## Parametric models and likelihood function for uncensored data

- Exponential. . and
- Weibull. and .

http://www.math.ucsd.edu/~rxu/math284/slect4.pdf

See also accelerated life models where a set of covariates were used to model survival time.

## Survival modeling

### Accelerated life models - a direct extension of the classical linear model

http://data.princeton.edu/wws509/notes/c7.pdf and also Kalbfleish and Prentice (1980).

Therefore

- . So if there are two groups (x=1 and x=0), and , it means one group live twice as long as people in another group.
- . This explains the meaning of
**accelerated failure-time**.**Depending on the sign of , the time is either accelerated by a constant factor or degraded by a constant factor**. If , the probability that a member in group one (eg treatment) will be alive at age t is exactly the same as the probability that a member in group zero (eg control group) will be alive at age t/2. - The hazard function . So if , at any given age people in group one would be exposed to half the risk of people in group zero half their age.

In applications,

- If the errors are normally distributed, then we obtain a log-normal model for the T. Estimation of this model for censored data by maximum likelihood is known in the econometric literature as a Tobit model.
- If the errors have an extreme value distribution, then T has an exponential distribution. The hazard satisfies the log linear model .

### Proportional hazard models

Note PH models is a type of multiplicative hazard rate models where .

Assumption: Survival curves for two strata (determined by the particular choices of values for covariates) must have **hazard functions that are proportional over time** (i.e. **constant relative hazard over time**). Proportional hazards assumption meaning. The ratio of the hazard rates from two individuals with covariate value and is a constant function time.

Test the assumption

- Survival Analysis Tutorial by Jacob Lindell and Joe Berry.
- cox.zph() can be used to test the proportional hazards assumption for a Cox regression model fit.
- Log-log Kaplan-Meier curves and other methods.
- https://stats.idre.ucla.edu/other/examples/asa2/testing-the-proportional-hazard-assumption-in-cox-models/. If the predictor satisfy the proportional hazard assumption then the graph of the survival function versus the survival time should results in a graph with parallel curves, similarly the graph of the log(-log(survival)) versus log of survival time graph should result in parallel lines if the predictor is proportional. This method does not work well for continuous predictor or categorical predictors that have many levels because the graph becomes to “cluttered”.

## Weibull and Exponential model to Cox model

- https://socserv.socsci.mcmaster.ca/jfox/Books/Companion/appendix/Appendix-Cox-Regression.pdf. It also includes model diagnostic and all stuff is illustrated in R.
- http://stat.ethz.ch/education/semesters/ss2011/seminar/contents/handout_9.pdf

In summary:

- Weibull distribution (Klein) and . If p >1, then the risk increases over time. If p<1, then the risk decreases over time.
- Note that Weibull distribution has a different parametrization. See http://data.princeton.edu/pop509/ParametricSurvival.pdf#page=2. and . R and wikipedia also follows this parametrization except that and .

- Exponential distribution = constant (independent of t). This is a special case of Weibull distribution (p=1).
- Weibull (and also exponential)
~~distribution~~regression model is the only case which belongs to both the proportional hazards and the accelerated life families.

- Using the Weibull baseline hazard is the only circumstance under which the model satisfies both the proportional hazards, and accelerated failure time models
- If X is exponential distribution with mean , then X^(1/a) follows Weibull(a, b). See Exponential distribution and Weibull distribution.
- Derivation of mean and variance of Weibull distribution.

f(t)=h(t)*S(t) | h(t) | S(t) | Mean | |
---|---|---|---|---|

Exponential (Klein p37) | ||||

Weibull (Klein, Bender, wikipedia) | ||||

Exponential (R) | , is rate | |||

Weibull (R, wikipedia) | , is shape, and is scale |

- Accelerated failure-time model. Let . Then the survival function of at the covariate Z,

where denote the survival function T when Z=0. Since , the hazard function of T with a covariate value Z is related to a baseline hazard rate by (p56 Klein)

> mean(rexp(1000)^(1/2)) [1] 0.8902948 > mean(rweibull(1000, 2, 1)) [1] 0.8856265 > mean((rweibull(1000, 2, scale=4)/4)^2) [1] 1.008923

### Graphical way to check Weibull, AFT, PH

http://stat.ethz.ch/education/semesters/ss2011/seminar/contents/handout_9.pdf#page=40

- Log(Weibull) = Extreme value
- Extreme Value Distributions from mathwave.com
- Generalized extreme value distribution from wikipedia
- Density, distribution function, quantile function, and random generation for the (largest) extreme value distribution from EnvStats R package
- Lesson 60 – Extreme value distributions in R

### Weibull distribution and bathtub

- https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1740-9713.2018.01177.x by John Crocker
- https://www.sciencedirect.com/topics/materials-science/weibull-distribution
- https://en.wikipedia.org/wiki/Bathtub_curve

## CDF follows Unif(0,1)

https://stats.stackexchange.com/questions/161635/why-is-the-cdf-of-a-sample-uniformly-distributed

Take the Exponential distribution for example

stem(pexp(rexp(1000))) stem(pexp(rexp(10000)))

Another example is from simulating survival time. Note that this is exactly Bender et al 2005 approach. See also the simsurv (newer) and survsim (older) packages.

set.seed(100) #Define the following parameters outlined in the step: n = 1000 beta_0 = 0.5 beta_1 = -1 beta_2 = 1 b = 1.6 #This will be changed later as mentioned in Step 5 of documentation #Step 1 x_1<-rbinom(n, 1, 0.25) x_2<-rbinom(n, 1, 0.7) #Step 2 U<-runif(n, 0,1) T<-(-log(U)*exp(-(beta_0+beta_1*x_1+beta_2*x_2))) #Eqn (5) Fn <- ecdf(T) # https://stat.ethz.ch/R-manual/R-devel/library/stats/html/ecdf.html # verify F(T) or 1-F(T) ~ U(0, 1) hist(Fn(T)) # look at the plot of survival probability vs time plot(T, 1 - Fn(T))

## Simulate survival data

Note that status = 1 means an event (e.g. death) happened; Ti <= Ci. That is, the status variable used in R/Splus means the death indicator.

y <- rexp(10) cen <- runif(10) status <- ifelse(cen < .7, 1, 0)

**Note that**for the**exponential**distribution, larger rate/ corresponds to a smaller mean. This relation matches with the Cox regression where a large covariate corresponds to a smaller survival time. So the coefficient 3 in myrates in the below example has the same sign as the coefficient (2.457466 for censored data) in the output of the Cox model fitting.

n <- 30 x <- scale(1:n, TRUE, TRUE) # create covariates (standardized) # the original example does not work on large 'n' myrates <- exp(3*x+1) set.seed(1234) y <- rexp(n, rate = myrates) # generates the r.v. cen <- rexp(n, rate = 0.5 ) # E(cen)=1/rate ycen <- pmin(y, cen) di <- as.numeric(y <= cen) survreg(Surv(ycen, di)~x, dist="weibull")$coef[2] # -3.080125 coxph(Surv(ycen, di)~x)$coef # 2.457466 # no censor survreg(Surv(y,rep(1, n))~x,dist="weibull")$coef[2] # -3.137603 survreg(Surv(y,rep(1, n))~x,dist="exponential")$coef[2] # -3.143095 coxph(Surv(y,rep(1, n))~x)$coef # 2.717794 # See the pdf note for the rest of code

- Intercept in survreg for the exponential distribution. http://www.stat.columbia.edu/~madigan/W2025/notes/survival.pdf#page=25.

> futime <- rexp(1000, 5) > survreg(Surv(futime,rep(1,1000))~1,dist="exponential")$coef (Intercept) -1.618263 > exp(1.618263) [1] 5.044321

- Intercept and scale in survreg for a Weibull distribution. http://www.stat.columbia.edu/~madigan/W2025/notes/survival.pdf#page=28.

> survreg(Surv(futime,rep(1,1000))~1,dist="weibull") Call: survreg(formula = Surv(futime, rep(1, 1000)) ~ 1, dist = "weibull") Coefficients: (Intercept) -1.639469 Scale= 1.048049 Loglik(model)= 620.1 Loglik(intercept only)= 620.1 n= 1000

- rsurv() function from the ipred package
- Use Weibull distribution to model survival data. We assume the shape is constant across subjects. We then allow the scale to vary across subjects. For subject with covariate , = . Note that if we want to make the sign to be consistent with the Cox model, we want to use = instead.
- http://sas-and-r.blogspot.com/2010/03/example-730-simulate-censored-survival.html. Assuming shape=1 in the Weibull distribution, then the hazard function can be expressed as a proportional hazard model

n = 10000 beta1 = 2; beta2 = -1 lambdaT = .002 # baseline hazard lambdaC = .004 # hazard of censoring set.seed(1234) x1 = rnorm(n,0) x2 = rnorm(n,0) # true event time T = Vectorize(rweibull)(n=1, shape=1, scale=lambdaT*exp(-beta1*x1-beta2*x2)) # No censoring event2 <- rep(1, length(T)) coxph(Surv(T, event2)~ x1 + x2) # coef exp(coef) se(coef) z p # x1 1.99825 7.37613 0.01884 106.07 <2e-16 # x2 -1.00200 0.36715 0.01267 -79.08 <2e-16 # # Likelihood ratio test=15556 on 2 df, p=< 2.2e-16 # n= 10000, number of events= 10000 # Censoring C = rweibull(n, shape=1, scale=lambdaC) #censoring time time = pmin(T,C) #observed time is min of censored and true event = time==T # set to 1 if event is observed coxph(Surv(time, event)~ x1 + x2) # coef exp(coef) se(coef) z p # x1 2.01039 7.46622 0.02250 89.33 <2e-16 # x2 -0.99210 0.37080 0.01552 -63.95 <2e-16 # # Likelihood ratio test=11321 on 2 df, p=< 2.2e-16 # n= 10000, number of events= 6002 mean(event) # [1] 0.6002

- https://stats.stackexchange.com/a/135129 (Bender's inverse probability method). Let where shape 𝜌>0 and scale 𝜆>0. Following the inverse probability method, a realisation of 𝑇∼𝑆(⋅|𝐱) is obtained by computing with 𝑣 a uniform variate on (0,1). Using results on transformations of random variables, one may notice that 𝑇 has a conditional Weibull distribution (given 𝐱) with shape 𝜌 and scale 𝜆exp(𝐱′𝛽).

# N = sample size # lambda = scale parameter in h0() # rho = shape parameter in h0() # beta = fixed effect parameter # rateC = rate parameter of the exponential distribution of censoring variable C simulWeib <- function(N, lambda, rho, beta, rateC) { # covariate --> N Bernoulli trials x <- sample(x=c(0, 1), size=N, replace=TRUE, prob=c(0.5, 0.5)) # Weibull latent event times v <- runif(n=N) Tlat <- (- log(v) / (lambda * exp(x * beta)))^(1 / rho) # censoring times C <- rexp(n=N, rate=rateC) # follow-up times and event indicators time <- pmin(Tlat, C) status <- as.numeric(Tlat <= C) # data set data.frame(id=1:N, time=time, status=status, x=x) } # Test set.seed(1234) betaHat <- rate <- rep(NA, 1e3) for(k in 1:1e3) { dat <- simulWeib(N=100, lambda=0.01, rho=1, beta=-0.6, rateC=0.001) fit <- coxph(Surv(time, status) ~ x, data=dat) rate[k] <- mean(dat$status == 0) betaHat[k] <- fit$coef } mean(rate) # [1] 0.12287 mean(betaHat) # [1] -0.6085473

- Generating survival times to simulate Cox proportional hazards models Bender et al 2005
- survsim package and the paper on JSS. See this post.
- simsurv package (new, 2 vignettes).
- Get a desired percentage of censored observations in a simulation of Cox PH Model. The answer is based on Bender et al 2005. Generating survival times to simulate Cox proportional hazards models. Statistics in Medicine 24: 1713–1723. The censoring time is fixed and the distribution of the censoring indicator is following the binomial. In fact, when we simulate survival data with a predefined censoring rate, we can pretend the survival time is already censored and only care about the censoring/status variable to make sure the censoring rate is controlled.
- (Search github) Using inverse CDF
- Prediction Accuracy Measures for a Nonlinear Model and for Right-Censored Time-to-Event Data Li and Wang

- Regularization paths for Cox's proportional hazards model via coordinate descent. J Stat Software Simon et al 2011. Gsslasso Cox: a Bayesian hierarchical model for predicting survival and detecting associated genes by incorporating pathway information by Tang 2019.

## Standardize covariates

coxph() does not have an option to standardize covariates but glmnet() does.

library(glmnet) library(survival) N=1000;p=30 nzc=p/3 beta <- c(rep(1, 5), rep(-1, 5)) set.seed(1234) x=matrix(rnorm(N*p),N,p) x[, 1:5] <- x[, 1:5]*2 x[, 6:10] <- x[, 6:10] + 2 fx=x[,seq(nzc)] %*% beta hx=exp(fx) ty=rexp(N,hx) tcens <- rep(0,N) y=cbind(time=ty,status=1-tcens) # y=Surv(ty,1-tcens) with library(survival) coxph(Surv(ty, 1-tcens) ~ x) %>% coef %>% head(10) # x1 x2 x3 x4 x5 x6 x7 # 0.6076146 0.6359927 0.6346022 0.6469274 0.6152082 -0.6614930 -0.5946101 # x8 x9 x10 # -0.6726081 -0.6275205 -0.7073704 xscale <- scale(x, TRUE, TRUE) # halve the covariate values coxph(Surv(ty, 1-tcens) ~ xscale) %>% coef %>% head(10) # double the coef # xscale1 xscale2 xscale3 xscale4 xscale5 xscale6 xscale7 # 1.2119940 1.2480628 1.2848646 1.2857796 1.1959619 -0.6431946 -0.5941309 # xscale8 xscale9 xscale10 # -0.6723137 -0.6188384 -0.6793313 set.seed(1) fit=cv.glmnet(x,y,family="cox", nfolds=10, standardize = TRUE) as.vector(coef(fit, s = "lambda.min"))[seq(nzc)] # [1] 0.9351341 0.9394696 0.9187242 0.9418540 0.9111623 -0.9303783 # [7] -0.9271438 -0.9597583 -0.9493759 -0.9386065 set.seed(1) fit=cv.glmnet(x,y,family="cox", nfolds=10, standardize = FALSE) as.vector(coef(fit, s = "lambda.min"))[seq(nzc)] # [1] 0.9357171 0.9396877 0.9200247 0.9420215 0.9118803 -0.9257406 # [7] -0.9232813 -0.9554017 -0.9448827 -0.9356009 set.seed(1) fit=cv.glmnet(xscale,y,family="cox", nfolds=10, standardize = TRUE) as.vector(coef(fit, s = "lambda.min"))[seq(nzc)] # [1] 1.8652889 1.8436015 1.8601198 1.8719515 1.7712951 -0.9046420 # [7] -0.9263966 -0.9593383 -0.9362407 -0.9014015 set.seed(1) fit=cv.glmnet(xscale,y,family="cox", nfolds=10, standardize = FALSE) as.vector(coef(fit, s = "lambda.min"))[seq(nzc)] # [1] 1.8652889 1.8436015 1.8601198 1.8719515 1.7712951 -0.9046420 # [7] -0.9263966 -0.9593383 -0.9362407 -0.9014015

## Predefined censoring rates

Simulating survival data with predefined censoring rates for proportional hazards models

## Cross validation

- Cross validation in survival analysis by Verweij & van Houwelingen, Stat in medicine 1993.
- Using cross-validation to evaluate predictive accuracy of survival risk classifiers based on high-dimensional data. Simon et al, Brief Bioinform. 2011

## Competing risk

- https://www.mailman.columbia.edu/research/population-health-methods/competing-risk-analysis
- Page 61 of Klein and Moeschberger "Survival Analysis"

## Survival rate terminology

- Disease-free survival (DFS): the period after curative treatment [
**disease eliminated**] when no disease can be detected - Progression-free survival (PFS), overall survival (OS). PFS is the length of time during and after the treatment of a disease, such as cancer, that a patient lives with the
**disease but it does not get worse**. See an use at NCI-MATCH trial. - Time to progression: The length of time from the date of diagnosis or the start of treatment for a disease until the disease starts to get worse or spread to other parts of the body. In a clinical trial, measuring the time to progression is one way to see how well a new treatment works. Also called TTP.
- Metastasis-free survival (MFS) time: the period until metastasis is detected
- Understanding Statistics Used to Guide Prognosis and Evaluate Treatment (DFS & PFS rate)

## Books

- Survival Analysis, A Self-Learning Text by Kleinbaum, David G., Klein, Mitchel
- Applied Survival Analysis Using R by Moore, Dirk F.
- Regression Modeling Strategies by Harrell, Frank
- Regression Methods in Biostatistics by Vittinghoff, E., Glidden, D.V., Shiboski, S.C., McCulloch, C.E.
- https://tbrieder.org/epidata/course_reading/e_tableman.pdf
- Survival Analysis: Models and Applications by Xian Liu

## HER2-positive breast cancer

- https://www.mayoclinic.org/breast-cancer/expert-answers/FAQ-20058066
- https://en.wikipedia.org/wiki/Trastuzumab (antibody, injection into a vein or under the skin)

# Cox proportional hazards model and the partial log-likelihood function

Let *Y*_{i} denote the observed time (either censoring time or event time) for subject *i*, and let *C*_{i} be the indicator that the time corresponds to an event (i.e. if *C*_{i} = 1 the event occurred and if *C*_{i} = 0 the time is a censoring time). The hazard function for the Cox proportional hazard model has the form

This expression gives the hazard at time *t* for an individual with covariate vector (explanatory variables) *X*. Based on this hazard function, a **partial likelihood** (defined on hazard function) can be constructed from the datasets as

where *θ*_{j} = exp(*X*_{j }*β*^{′}) and *X*_{1}, ..., *X*_{n} are the covariate vectors for the *n* independently sampled individuals in the dataset (treated here as column vectors). This pdf or this note give a toy example

The corresponding log partial likelihood is

This function can be maximized over *β* to produce maximum partial likelihood estimates of the model parameters.

The partial score function is

and the Hessian matrix of the partial log likelihood is

Using this score function and Hessian matrix, the partial likelihood can be maximized using the Newton-Raphson algorithm. The inverse of the Hessian matrix, evaluated at the estimate of *β*, can be used as an approximate variance-covariance matrix for the estimate, and used to produce approximate standard errors for the regression coefficients.

If X is age, then the coefficient is likely >0. If X is some treatment, then the coefficient is likely <0.

## Compare the partial likelihood to the full likelihood

http://math.ucsd.edu/~rxu/math284/slect5.pdf#page=10

## z-column (Wald statistic) from R's coxph()

- https://socialsciences.mcmaster.ca/jfox/Books/Companion/appendix/Appendix-Cox-Regression.pdf#page=6 The ratio of each regression coefficient to its standard error, a Wald statistic which is asymptotically standard normal under the hypothesis that the corresponding β is 0.
- http://dni-institute.in/blogs/cox-regression-interpret-result-and-predict/

## How exactly can the Cox-model ignore exact times?

The Cox model does not depend on the times itself, instead it only needs an ordering of the events.

library(survival) survfit(Surv(time, status) ~ x, data = aml) fit <- coxph(Surv(time, status) ~ x, data = aml) coef(fit) # 0.9155326 min(diff(sort(unique(aml$time)))) # 1 # Shift survival time for some obs but keeps the same order # make sure we choose obs (n=20 not works but n=21 works) with twins rbind(order(aml$time), sort(aml$time), aml$time[order(aml$time)]) # [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15] [,16] # [1,] 12 13 14 15 1 16 2 3 17 4 5 18 19 6 20 7 # [2,] 5 5 8 8 9 12 13 13 16 18 23 23 27 28 30 31 # [3,] 5 5 8 8 9 12 13 13 16 18 23 23 27 28 30 31 # [,17] [,18] [,19] [,20] [,21] [,22] [,23] # [1,] 21 8 22 9 23 10 11 # [2,] 33 34 43 45 45 48 161 # [3,] 33 34 43 45 45 48 161 aml$time2 <- aml$time aml$time2[order(aml$time)[1:21]] <- aml$time[order(aml$time)[1:21]] - .9 fit2 <- coxph(Surv(time2, status) ~ x, data = aml); fit2 coef(fit2) # 0.9155326 coef(fit) == coef(fit2) # TRUE aml$time3 <- aml$time aml$time3[order(aml$time)[1:20]] <- aml$time[order(aml$time)[1:20]] - .9 fit3 <- coxph(Surv(time3, status) ~ x, data = aml); fit3 coef(fit3) # 0.8891567 coef(fit) == coef(fit3) # FALSE

## Partial likelihood when there are ties; hypothesis testing: Likelihood Ratio Test, Wald Test & Score Test

http://math.ucsd.edu/~rxu/math284/slect5.pdf#page=29

In R's coxph(): Nearly all Cox regression programs use the *Breslow* method by default, but not this one. The * Efron approximation * is used as the default here, it is more accurate when dealing with tied death times, and is as efficient computationally.

http://sfb649.wiwi.hu-berlin.de/fedc_homepage/xplore/tutorials/xaghtmlnode28.html (include the case when there is a partition of parameters). The formulas for 3 tests are also available on Appendix B of Klein book.

The following code does not test for models. But since there is only one coefficient, the results are the same. If there is more than one variable, we can use anova(model1, model2) to run LRT.

library(KMsurv) # No ties. Section 8.2 data(btrial) str(btrial) # 'data.frame': 45 obs. of 3 variables: # $ time : int 19 25 30 34 37 46 47 51 56 57 ... # $ death: int 1 1 1 1 1 1 1 1 1 1 ... # $ im : int 1 1 1 1 1 1 1 1 1 1 ... table(subset(btrial, death == 1)$time) # death time is unique coxph(Surv(time, death) ~ im, data = btrial) # coef exp(coef) se(coef) z p # im 0.980 2.665 0.435 2.25 0.024 # Likelihood ratio test=4.45 on 1 df, p=0.03 # n= 45, number of events= 24 # Ties, Section 8.3 data(kidney) str(kidney) # 'data.frame': 119 obs. of 3 variables: # $ time : num 1.5 3.5 4.5 4.5 5.5 8.5 8.5 9.5 10.5 11.5 ... # $ delta: int 1 1 1 1 1 1 1 1 1 1 ... # $ type : int 1 1 1 1 1 1 1 1 1 1 ... table(subset(kidney, delta == 1)$time) # 0.5 1.5 2.5 3.5 4.5 5.5 6.5 8.5 9.5 10.5 11.5 15.5 16.5 18.5 23.5 26.5 # 6 1 2 2 2 1 1 2 1 1 1 2 1 1 1 1 # Default: Efron method coxph(Surv(time, delta) ~ type, data = kidney) # coef exp(coef) se(coef) z p # type -0.613 0.542 0.398 -1.54 0.12 # Likelihood ratio test=2.41 on 1 df, p=0.1 # n= 119, number of events= 26 summary(coxph(Surv(time, delta) ~ type, data = kidney)) # n= 119, number of events= 26 # coef exp(coef) se(coef) z Pr(>|z|) # type -0.6126 0.5420 0.3979 -1.539 0.124 # # exp(coef) exp(-coef) lower .95 upper .95 # type 0.542 1.845 0.2485 1.182 # # Concordance= 0.497 (se = 0.056 ) # Rsquare= 0.02 (max possible= 0.827 ) # Likelihood ratio test= 2.41 on 1 df, p=0.1 # Wald test = 2.37 on 1 df, p=0.1 # Score (logrank) test = 2.44 on 1 df, p=0.1 # Breslow method summary(coxph(Surv(time, delta) ~ type, data = kidney, ties = "breslow")) # n= 119, number of events= 26 # coef exp(coef) se(coef) z Pr(>|z|) # type -0.6182 0.5389 0.3981 -1.553 0.12 # # exp(coef) exp(-coef) lower .95 upper .95 # type 0.5389 1.856 0.247 1.176 # # Concordance= 0.497 (se = 0.056 ) # Rsquare= 0.02 (max possible= 0.827 ) # Likelihood ratio test= 2.45 on 1 df, p=0.1 # Wald test = 2.41 on 1 df, p=0.1 # Score (logrank) test = 2.49 on 1 df, p=0.1 # Discrete/exact method summary(coxph(Surv(time, delta) ~ type, data = kidney, ties = "exact")) # coef exp(coef) se(coef) z Pr(>|z|) # type -0.6294 0.5329 0.4019 -1.566 0.117 # # exp(coef) exp(-coef) lower .95 upper .95 # type 0.5329 1.877 0.2424 1.171 # # Rsquare= 0.021 (max possible= 0.795 ) # Likelihood ratio test= 2.49 on 1 df, p=0.1 # Wald test = 2.45 on 1 df, p=0.1 # Score (logrank) test = 2.53 on 1 df, p=0.1

## Hazard (function) and survival function

A hazard is the rate at which events happen, so that the probability of an event happening in a short time interval is the length of time multiplied by the hazard.

Therefore

or

Hazards (or probability of hazards) may vary with time, while the assumption in proportional hazard models for survival is that the hazard is a constant proportion.

Examples:

- If h(t)=c, S(t) is exponential. f(t) = c exp(-ct). The mean is 1/c.
- If , S(t) is Gompertz distribution.
- If , S(t) is Weibull distribution.
- For Cox regression, the survival function can be shown to be .

Alternatively,

where the cumulative baseline hazard at time t, , is commonly estimated through the non-parametric Breslow estimator.

## Check the proportional hazard (constant HR over time) assumption by cox.zph()

- Methods to evaluate the validity of the Cox model assumptions
- https://www.rdocumentation.org/packages/survival/versions/2.41-2/topics/cox.zph
- Draw a hazard rate plot. How to calculate predicted hazard rates from a Cox PH model?. The hazard ratio should be constant; h(t|Z) / h(t|Z*) is independent of t.
- An online updating approach for testing the proportional hazards assumption with streams of survival data Xue 2019

## Sample size calculators

- Calculate Sample Size Needed to Test Time-To-Event Data: Cox PH, Equivalence including a reference
- http://www.sample-size.net/sample-size-survival-analysis/ including a reference
- Evolution of survival sample size methods demonstrated by nQuery software.
**Sample size refers the number of events; status=1 (not the number of observations)** - http://www.icssc.org/Documents/AdvBiosGoa/Tab%2026.00_SurvSS.pdf no reference
- powerSurvEpi R package
- NPHMC R package (based on the Proportional Hazards Mixture Cure Model) and the paper
- Hmisc::cpower() function.

### How many events are required to fit the Cox regression reliably?

If we have only 1 covariate and the covariate is continuous, we need at least 2 events (one for the baseline hazard and one for beta).

If the covariate is discrete, we need at least one event from (each of) two groups in order to fit the Cox regression reliably. For example, if status=(0,0,0,1,0,1) and x=(0,0,1,1,2,2) works fine.

library(survival) head(ovarian) # futime fustat age resid.ds rx ecog.ps # 1 59 1 72.3315 2 1 1 # 2 115 1 74.4932 2 1 1 # 3 156 1 66.4658 2 1 2 # 4 421 0 53.3644 2 2 1 # 5 431 1 50.3397 2 1 1 # 6 448 0 56.4301 1 1 2 ova <- ovarian # n=26 ova$time <- ova$futime ova$status <- 0 ova$status[1:4] <- 1 coxph(Surv(time, status) ~ rx, data = ova) # OK summary(survfit(Surv(time, status) ~ rx, data =ova)) # rx=1 # time n.risk n.event survival std.err lower 95% CI upper 95% CI # 59 13 1 0.923 0.0739 0.789 1 # 115 12 1 0.846 0.1001 0.671 1 # 156 11 1 0.769 0.1169 0.571 1 # rx=2 # time n.risk n.event survival std.err lower 95% CI upper 95% CI # 421.0000 10.0000 1.0000 0.9000 0.0949 0.7320 1.0000 # Suspicious Cox regression result due to 0 sample size in one group ova$status <- 0 ova$status[1:3] <- 1 coxph(Surv(time, status) ~ rx, data = ova) # coef exp(coef) se(coef) z p # rx -2.13e+01 5.67e-10 2.32e+04 0 1 # # Likelihood ratio test=4.41 on 1 df, p=0.04 # n= 26, number of events= 3 # Warning message: # In fitter(X, Y, strats, offset, init, control, weights = weights, : # Loglik converged before variable 1 ; beta may be infinite. summary(survfit(Surv(time, status) ~ rx, data = ova)) # rx=1 # time n.risk n.event survival std.err lower 95% CI upper 95% CI # 59 13 1 0.923 0.0739 0.789 1 # 115 12 1 0.846 0.1001 0.671 1 # 156 11 1 0.769 0.1169 0.571 1 # rx=2 # time n.risk n.event survival std.err lower 95% CI upper 95% CI

## Extract p-values

fit <- coxph(Surv(futime, fustat) ~ age, data = ovarian) # method 1: beta <- coef(fit) se <- sqrt(diag(vcov(fit))) 1 - pchisq((beta/se)^2, 1) # method 2: https://www.biostars.org/p/65315/ coef(summary(fit))[, "Pr(>|z|)"]

## Expectation of life & expected future lifetime

- The average lifetime is the same as the area under the survival curve.

by integrating by parts making use of the fact that -f(t) is the derivative of S(t), which has limits S(0)=1 and . The average lifetime may not be bounded if you have censored data, there's censored observations that last beyond your last recorded death.

## Hazard Ratio (exp^beta) vs Relative Risk

- https://en.wikipedia.org/wiki/Hazard_ratio
**Hazard**represents the**instantaneous event rate**, which means the probability that an individual would experience an event (e.g. death/relapse) at a particular given point in time after the intervention, assuming that this individual has survived to that particular point of time without experiencing any event.**Hazard ratio**is a measure of**an effect**of**an intervention**of**an outcome**of interest over time.- Hazard ratio = hazard in the intervention group / Hazard in the control group
- A hazard ratio is often reported as a “reduction in risk of death or progression” – This
**risk reduction**is calculated as**1 minus the Hazard Ratio (exp^beta)**, e.g., HR of 0.84 is equal to a 16% reduction in risk. See www.time4epi.com and stackexchange.com. - Hazard ratio and its confidence can be obtained in R by using the
**summary()**method; e.g.**fit <- coxph(Surv(time, status) ~ x); summary(fit)$conf.int; confint(fit)** - The coefficient beta represents the expected change in
**log hazard**if X changes by one unit and all other variables are held constant in Cox models. See Variable selection – A review and recommendations for the practicing statistician by Heinze et al 2018.

Another example (John Fox, Cox Proportional-Hazards Regression for Survival Data) is assuming Y ~ age + prio + others.

- If exp(beta_age) = 0.944. It means an additional year of age
**reduces the hazard by a factor**of .944 on average, or (1-.944)*100 = 5.6**percent**. - If exp(beta_prio) = 1.096, it means each prior conviction
**increases the hazard by a factor**of 1.096, or 9.6**percent**.

How do you explain the difference between hazard ratio and **relative risk** to a layman? from Quora.

Odds Ratio, Hazard Ratio and Relative Risk by Janez Stare

For two groups that differ only in treatment condition, the ratio of the hazard functions is given by , where is the estimate of treatment effect derived from the regression model. See here.

Compute ratio ratios from coxph() in R (Hint: exp(beta)).

**Prognostic index** is defined on http://www.math.ucsd.edu/~rxu/math284/slect6.pdf#page=2.

Basics of the Cox proportional hazards model. Good prognostic factor (b<0 or HR<1) and bad prognostic factor (b>0 or HR>1).

Variable selection: variables were retained in the prediction models if they had a hazard ratio of <0.85 or >1.15 (for binary variables) and were statistically significant at the 0.01 level. see Development and validation of risk prediction equations to estimate survival in patients with colorectal cancer: cohort study.

library(KMsurv) # No ties. Section 8.2 data(btrial) coxph(Surv(time, death) ~ im, data = btrial) summary(coxph(Surv(time, death) ~ im, data = btrial))$conf.int # exp(coef) exp(-coef) lower .95 upper .95 # im 2.664988 0.3752362 1.136362 6.249912

So the hazard ratio and its 95% ci can be obtained from the 1st, 3rd and 4th element in *conf.int*.

### Hazard Ratio and death probability

https://en.wikipedia.org/wiki/Hazard_ratio#The_hazard_ratio_and_survival

Suppose *S*_{0}(t)=.2 (20% survived at time t) and the hazard ratio (hr) is 2 (a group has twice the chance of dying than a comparison group), then (Cox model is assumed)

*S*_{1}(t)=*S*_{0}(t)^{hr}= .2^{2}= .04 (4% survived at t)- The corresponding death probabilities are 0.8 and 0.96.
- If a subject is exposed to twice the risk of a reference subject at every age, then the probability that the subject will be alive at any given age is the square of the probability that the reference subject (covariates = 0) would be alive at the same age. See p10 of this lecture notes.
- exp(x*beta) is the relative risk associated with covariate value x.

### Hazard Ratio Forest Plot

The forest plot quickly summarizes the hazard ratio data across multiple variables –If the line crosses the 1.0 value, the hazard ratio is not significant and there is no clear advantage for either arm.

Hazard ratio forest plot: ggforest() from survminer

### Restricted mean survival time

- Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome Royston 2013
- The Use of Restricted Mean Survival Time (RMST) Method When Proportional Hazards Assumption is in Doubt
- To estimate treatment effect for time to event, Hazard Ratio (HR) is commonly used.
- HR is often assumed to be constant over time (i.e., proportional hazard assumption).
- Recently, we have some doubt about this assumption.
- If the PH assumption does not hold, the interpretation of HR can be difficult.

- RMST is defined as the area under the survival curve up to t*, which should be pre-specified for a randomized trial. Uno 2014
- survRM2 package

## Piece-wise constant baseline hazard model, Poisson model and Breslow estimate

- https://en.wikipedia.org/wiki/Proportional_hazards_model#Relationship_to_Poisson_models
- http://data.princeton.edu/wws509/notes/c7s4.html
- It has been implemented in the biospear package (poissonize.R) with the 'grplasso' package for group-lasso method.
*We implemented a Poisson model over two-month intervals, corresponding to a piecewise constant hazard model which approximates rather well the Breslow estimator in the Cox model*. - http://r.789695.n4.nabble.com/exponential-proportional-hazard-model-td805536.html
- https://www.demogr.mpg.de/papers/technicalreports/tr-2010-003.pdf
- Does Cox Regression have an underlying Poisson distribution?
- Calculate incidence rates using poisson model: relation to hazard ratio from Cox PH model R code verification is included.

- https://rdrr.io/cran/JM/man/piecewiseExp.ph.html
- https://rdrr.io/cran/pch/man/pchreg.html
- Survival Analysis via Hazard Based Modeling and Generalized Linear Models
- https://www.rdocumentation.org/packages/mgcv/versions/1.8-23/topics/cox.pht

## Estimate baseline hazard , Breslow cumulative baseline hazard , baseline survival function and the survival function

Google: how to estimate baseline hazard rate

- survfit.object has print(), plot(), lines(), and points() methods. It returns a list with components
- n
- time
- n.risk
- n.event
- n.censor
- surv [S_0(t)]
- cumhaz [ same as -log(surv)]
- upper
- lower
- n.all

- Terry Therneau: The
*baseline survival*, which is the survival for a hypothetical subject with all covariates=0, may be useful mathematical shorthand when writing a book but I cannot think of a single case where the resulting curve would be of any practical interest in medical data. - http://www.math.ucsd.edu/~rxu/math284/slect6.pdf#page=4
**Breslow**Estimator for**cumulative**baseline hazard at a time t and**Kalbfleisch/Prentice**Estimator - When there are no covariates, the Breslow’s estimate reduces to the Fleming-Harrington (Nelson-Aalen) estimate, and K/P reduces to KM.
- stackexchange.com and
**cumulative**and non-cumulative baseline hazard - (newbie) Cox Baseline Hazard
*There are two methods of calculating the baseline survival, the default one gives the baseline hazard estimator you want. It is attributed to Aalen, Breslow, or Peto (see the next item).*An example: https://stats.idre.ucla.edu/r/examples/asa/r-applied-survival-analysis-ch-2/. - survfit.coxph(formula, newdata, type, ...)
- newdata:
**Default is the mean of the covariates used in the coxph fit** - type = "aalen", "efron", or "kalbfleisch-prentice". The default is to match the computation used in the Cox model. The Nelson-Aalen-Breslow estimate for ties='breslow', the Efron estimate for ties='efron' and the Kalbfleisch-Prentice estimate for a discrete time model ties='exact'. Variance estimates are the Aalen-Link-Tsiatis, Efron, and Greenwood. The default will be the Efron estimate for ties='efron' and the
**Aalen estimate**otherwise.

- newdata:
- Nelson-Aalen estimator in R. The easiest way to get the Nelson-Aalen estimator is

basehaz(coxph(Surv(time,status)~1,data=aml))

because the (Breslow) hazard estimator for a Cox model reduces to the Nelson-Aalen estimator when there are no covariates. You can also compute it from information returned by survfit().

fit <- survfit(Surv(time, status) ~ 1, data = aml) cumsum(fit$n.event/fit$n.risk) # the Nelson-Aalen estimator for the times given by fit$times -log(fit$surv) # cumulative hazard

### Manually compute

**Breslow estimator of the baseline cumulative hazard rate** reduces to the **Nelson-Aalen** estimator ( is the number at risk at time ) when there are no covariates present; see p283 of Klein 2003.

where denotes the distinct death times and be the number of deaths at time . The estimator of the baseline survival function is given by . Below we use the formula to compute the cumulative hazard (and survival function) and compare them with the result obtained using R's built-in functions. The following code is a modification of the snippet from the post Breslow cumulative hazard and basehaz().

bhaz <- function(beta, time, status, x) { # time can be duplicated # x (covariate) must be continuous data <- data.frame(time,status,x) data <- data[order(data$time), ] dt <- unique(data$time) k <- length(dt) risk <- exp(data.matrix(data[,-c(1:2)]) %*% beta) h <- rep(0,k) for(i in 1:k) { h[i] <- sum(data$status[data$time==dt[i]]) / sum(risk[data$time>=dt[i]]) } return(data.frame(h, dt)) } # Example 1 'ovarian' which has unique survival time all(table(ovarian$futime) == 1) # TRUE fit <- coxph(Surv(futime, fustat) ~ age, data = ovarian) # 1. compute the cumulative baseline hazard # 1.1 manually using Breslow estimator at the observed time h0 <- bhaz(fit$coef, ovarian$futime, ovarian$fustat, ovarian$age) H0 <- cumsum(h0$h) # 1.2 use basehaz (always compute at the observed time) # since we consider the baseline, we need to use centered=FALSE H0.bh <- basehaz(fit, centered = FALSE) cbind(H0, h0$dt, H0.bh) range(abs(H0 - H0.bh$hazard)) # [1] 6.352747e-22 5.421011e-20 # 2. compute the estimation of the survival function # 2.1 manually using Breslow estimator at t = observed time (one dim, sorted) # and observed age (another dim): # S(t) = S0(t) ^ exp(bx) = exp(-H0(t)) ^ exp(bx) S1 <- outer(exp(-H0), exp(coef(fit) * ovarian$age), "^") dim(S1) # row = times, col = age # 2.2 How about considering times not at observed (e.g. h0$dt - 10)? # Let's look at the hazard rate newtime <- h0$dt - 10 H0 <- sapply(newtime, function(tt) sum(h0$h[h0$dt <= tt])) S2 <- outer(exp(-H0), exp(coef(fit) * ovarian$age), "^") dim(S2) # row = newtime, col = age # 2.3 use summary() and survfit() to obtain the estimation of the survival S3 <- summary(survfit(fit, data.frame(age = ovarian$age)), times = h0$dt)$surv dim(S3) # row = times, col = age range(abs(S1 - S3)) # [1] 2.117244e-17 6.544321e-12 # 2.4 How about considering times not at observed (e.g. h0$dt - 10)? # Note that we cannot put times larger than the observed S4 <- summary(survfit(fit, data.frame(age = ovarian$age)), times = newtime)$surv range(abs(S2 - S4)) # [1] 0.000000e+00 6.544321e-12

# Example 2 'kidney' which has duplicated time fit <- coxph(Surv(time, status) ~ age, data = kidney) # manually compute the breslow cumulative baseline hazard # at the observed time h0 <- with(kidney, bhaz(fit$coef, time, status, age)) H0 <- cumsum(h0$h) # use basehaz (always compute at the observed time) # since we consider the baseline, we need to use centered=FALSE H0.bh <- basehaz(fit, centered = FALSE) head(cbind(H0, h0$dt, H0.bh)) range(abs(H0 - H0.bh$hazard)) # [1] 0.000000000 0.005775414 # manually compute the estimation of the survival function # at t = observed time (one dim, sorted) and observed age (another dim): # S(t) = S0(t) ^ exp(bx) = exp(-H0(t)) ^ exp(bx) S1 <- outer(exp(-H0), exp(coef(fit) * kidney$age), "^") dim(S1) # row = times, col = age # How about considering times not at observed (h0$dt - 1)? # Let's look at the hazard rate newtime <- h0$dt - 1 H0 <- sapply(newtime, function(tt) sum(h0$h[h0$dt <= tt])) S2 <- outer(exp(-H0), exp(coef(fit) * kidney$age), "^") dim(S2) # row = newtime, col = age # use summary() and survfit() to obtain the estimation of the survival S3 <- summary(survfit(fit, data.frame(age = kidney$age)), times = h0$dt)$surv dim(S3) # row = times, col = age range(abs(S1 - S3)) # [1] 0.000000000 0.002783715 # How about considering times not at observed (h0$dt - 1)? # We cannot put times larger than the observed S4 <- summary(survfit(fit, data.frame(age = kidney$age)), times = newtime)$surv range(abs(S2 - S4)) # [1] 0.000000000 0.002783715

- basehaz() (an alias for survfit) from stackexchange.com and here. basehaz() has a parameter
*centered*which by default is TRUE. Actually basehaz() gives**cumulative hazard H(t)**. See here. That is, exp(-basehaz(fit)$hazard) is the same as summary(survfit(fit))$surv. basehaz() function is not useful.

fit <- coxph(Surv(futime, fustat) ~ age, data = ovarian) > fit Call: coxph(formula = Surv(futime, fustat) ~ age, data = ovarian) coef exp(coef) se(coef) z p age 0.1616 1.1754 0.0497 3.25 0.0012 Likelihood ratio test=14.3 on 1 df, p=0.000156 n= 26, number of events= 12 # Note the default 'centered = TRUE' for basehaz() > exp(-basehaz(fit)$hazard) # exp(-cumulative hazard) [1] 0.9880206 0.9738738 0.9545899 0.9334790 0.8973620 0.8624781 0.8243117 [8] 0.8243117 0.8243117 0.7750981 0.7750981 0.7244924 0.6734146 0.6734146 [15] 0.5962187 0.5204807 0.5204807 0.5204807 0.5204807 0.5204807 0.5204807 [22] 0.5204807 0.5204807 0.5204807 0.5204807 0.5204807 > dim(ovarian) [1] 26 6 > exp(-basehaz(fit)$hazard)[ovarian$fustat == 1] [1] 0.9880206 0.9738738 0.9545899 0.8973620 0.8243117 0.8243117 0.7750981 [8] 0.7750981 0.5204807 0.5204807 0.5204807 0.5204807 > summary(survfit(fit))$surv [1] 0.9880206 0.9738738 0.9545899 0.9334790 0.8973620 0.8624781 0.8243117 [8] 0.7750981 0.7244924 0.6734146 0.5962187 0.5204807 > summary(survfit(fit, data.frame(age=mean(ovarian$age))), time=ovarian$futime[ovarian$fustat == 1])$surv # Same result as above > summary(survfit(fit, data.frame(age=mean(ovarian$age))), time=ovarian$futime)$surv [1] 0.9880206 0.9738738 0.9545899 0.9334790 0.8973620 0.8624781 0.8243117 [8] 0.8243117 0.8243117 0.7750981 0.7750981 0.7244924 0.6734146 0.6734146 [15] 0.5962187 0.5204807 0.5204807 0.5204807 0.5204807 0.5204807 0.5204807 [22] 0.5204807 0.5204807 0.5204807 0.5204807 0.5204807

## Predicted survival probability in Cox model: survfit.coxph(), plot.survfit() & summary.survfit( , times)

For theory, see section 8.6 Estimation of the survival function in Klein & Moeschberger.

For R, see Extract survival probabilities in Survfit by groups

plot.survfit(). fun="log" to plot log survival curve, fun="event" plot cumulative events, fun="cumhaz" plots cumulative hazard (f(y) = -log(y)).

The plot function below will draw 4 curves: , ,