Survival data
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/log-rank/log rank 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 Time-dependent covariates
- 1.20 Books
- 1.21 HER2-positive breast cancer
- 2 Cox proportional hazards model and the partial log-likelihood function
- 2.1 Get the partial likelihood of a Cox PH Model with new data
- 2.2 Compare the partial likelihood to the full likelihood
- 2.3 z-column (Wald statistic) from R's coxph()
- 2.4 How exactly can the Cox-model ignore exact times?
- 2.5 Partial likelihood when there are ties; hypothesis testing: Likelihood Ratio Test, Wald Test & Score Test
- 2.6 Hazard (function) and survival function
- 2.7 Check the proportional hazard (constant HR over time) assumption by cox.zph()
- 2.8 strata
- 2.9 Sample size calculators
- 2.10 Extract p-values
- 2.11 Expectation of life & expected future lifetime
- 2.12 Hazard Ratio (exp^beta) vs Relative Risk
- 2.13 Piece-wise constant baseline hazard model, Poisson model and Breslow estimate
- 2.14 Estimate baseline hazard [math]h_0(t)[/math], Breslow cumulative baseline hazard [math]H_0(t)[/math], baseline survival function [math]S_0(t)[/math] and the survival function [math]S(t)[/math]
- 2.15 Predicted survival probability in Cox model: survfit.coxph(), plot.survfit() & summary.survfit( , times)
- 2.16 Loglikelihood
- 2.17 High dimensional data
- 2.18 glmnet + Cox models
- 2.19 Prognostic index/risk scores
- 2.20 Survival risk prediction
- 2.21 Assessing the performance of prediction models
- 2.21.1 Hazard ratio
- 2.21.2 D index
- 2.21.3 AUC
- 2.21.4 Concordance index/C-index/C-statistic interpretation and R packages
- 2.21.5 Integrated brier score (≈ "mean squared error" of prediction for survival data)
- 2.21.6 Kendall's tau, Goodman-Kruskal's gamma, Somers' d
- 2.21.7 C-statistics
- 2.21.8 C-statistic limitations
- 2.21.9 C-statistic applications
- 2.21.10 C-statistic vs LRT comparing nested models
- 2.21.11 Time dependent ROC curves
- 2.21.12 Calibration
- 2.22 Prognostic markers vs predictive markers (and other biomarkers)
- 2.23 Quantifying treatment differences in confirmatory trials under non-proportional hazards
- 2.24 Computation for gene expression (microarray) data
- 2.25 Single gene vs mult-gene survival models
- 2.26 Random papers using C-index, AUC or Brier scores
- 2.27 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 [math]t_1 \lt t_2 \lt \cdots \lt t_D[/math]. At time [math]t_i[/math] there are [math]d_i[/math] events. Let [math]Y_i[/math] be the number of individuals who are at risk at time [math]t_i[/math]. The quantity [math]d_i/Y_i[/math] provides an estimate of the conditional probability that an individual who survives to just prior to time [math]t_i[/math] experiences the event at time [math]t_i[/math]. 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 ([math]t_1 \le t[/math]):
- [math] \begin{align} \hat{S}(t) &= \prod_{t_i \le t} [1 - d_i/Y_i] \\ \hat{H}(t) &= \sum_{t_i \le t} d_i/Y_i \end{align} [/math]
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 [math]t \in [t_j, t_{j+1})[/math] for a certain j, we have [math]\hat{S}(t) = \prod_{i=1}^j (1-d_i/n_i)[/math] where [math]d_i[/math] is the number people who have an event during the interval [math][t_i, t_{i+1})[/math] and [math]n_i[/math] is the number of people at risk just before the beginning of the interval [math][t_i, t_{i+1})[/math].
- The product-limit estimator can be constructed by using a reduced-sample approach. We can estimate the [math]P(T \gt t_i | T \ge t_i) = \frac{Y_i - d_i}{Y_i}[/math] for [math]i=1,2,\cdots,D[/math]. [math] S(t_i) = \frac{S(t_i)}{S(t_{i-1})} \frac{S(t_{i-1})}{S(t_{i-2})} \cdots \frac{S(t_2)}{S(t_1)} \frac{S(t_1)}{S(0)} S(0) = P(T \gt t_i | T \ge t_i) P(T \gt t_{i-1} | T \ge t_{i-1}) \cdots P(T\gt t_2|T \ge t_2) P(T\gt t_1 | T \ge t_1)[/math] because S(0)=1 and, for a discrete distribution, [math]S(t_{i-1}) = P(T \gt t_{i-1}) = P(T \ge t_i)[/math].
- 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, [math]\hat{S}(t) = \frac{1}{n}\sum I(X_i \gt t)[/math].
- 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 [math]0 \le H(t) \lt \infty[/math] and [math]H(t) \rightarrow \infty[/math] 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 [math]H(t) = -ln[S(t)][/math] or [math]S(t) = exp[-H(t)] [/math]. 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:KMannotation.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 Weighting
Robust Inference Using Inverse Probability Weighting, pdf
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
library(survminer) ggsurvplot(survo, risk.table = TRUE, pval=TRUE, pval.method = TRUE)
Paper examples
Questions:
- When I need to put two KM curves plot side by side, the function arrange_ggsurvplots() will create two graph devices and the first one is blank?
- How to remove tick mark on censored observations especially the case with a large sample size?
KMunicate
https://cran.r-project.org/web/packages/KMunicate/index.html
GGally package
ggsurv() from the Gally package.
Advantage: return object class is c("gg", "ggplot").
To combine multiple ggplot2 plots, use the ggpubr package. gridExtra is not developed after 2017.
library(GGally) library(survival) data(lung, package = "survival") sf.lung <- survfit(Surv(time, status) ~ sex, data = lung) p1 <- ggsurv(sf.lung, plot.cens = FALSE, lty.est = c(1, 3), size.est = 0.8, xlab = "Time", ylab = "Survival", main = "Lower score") p1 <- p1 + annotate("text", x=0, y=.25, hjust=0, label="zxcvb") p2 <- ggsurv(sf.lung, plot.cens = FALSE, lty.est = c(1, 3), size.est = 0.8, xlab = "Time", ylab = "Survival", main = "High score") p2 <- p2 + annotate("text", x=0, y=.25, hjust=0, label="asdfg") # gridExtra::grid.arrange(p1, p2, ncol=2, nrow =1) # no common legend option ggpubr::ggarrange(p1, p2, common.legend = TRUE, legend = "right") # return object class: "gg" "ggplot" "ggarrange"
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/log-rank/log rank 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.
- survdiff, Extract p-value from survdiff
sdf <- survdiff(Surv(time, status) ~ treatment, data=mydf) pvalue <- 1 - pchisq(sdf$chisq, length(sdf$n) - 1)
- 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
- Stratified log-rank tests
- survival package has a strata function that we can use in the survdiff() function.
- Differentiate group and strata
- The strata is useful when we suspect there is a confounding factor
- Log-rank Test: When does it Fail and how to fix it
- Survival Analysis: Logrank Test
- Comparing Survival Curves
- Survival analysis using a 5‐step stratified testing and amalgamation routine (5‐STAR) in randomized clinical trials by Mehrotra 2020
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 [math]F(\cdot)[/math] and the CDF of the censoring time C is [math]G(\cdot)[/math],
- [math] \begin{align} P(T\gt t, \delta=1) &= \int_t^\infty (1-G(s))dF(s), \\ P(T\gt t, \delta=0) &= \int_t^\infty (1-F(s))dG(s) \end{align} [/math]
- 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 [math]f(T, \delta)[/math]
- https://web.stanford.edu/~lutian/coursepdf/unit2.pdf#page=3 joint density of [math]f(T, \delta)[/math]
- http://data.princeton.edu/wws509/notes/c7.pdf#page=6
- Special case: T follows Log normal distribution and C follows [math]U(0, \xi)[/math].
R
- flexsurv package.
- Parametric survival modeling which uses the flexsurv package.
- Used in simsurv package
Parametric models and likelihood function for uncensored data
- Exponential. [math] T \sim Exp(\lambda) [/math]. [math]H(t) = \lambda t.[/math] and [math]ln(S(t)) = -H(t) = -\lambda t.[/math]
- Weibull. [math] T \sim W(\lambda,p).[/math] [math]H(t) = \lambda^p t^p.[/math] and [math]ln(-ln(S(t))) = ln(\lambda^p t^p)=const + p ln(t) [/math].
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).
[math] log T_i = x_i' \beta + \epsilon_i [/math] Therefore
- [math]T_i = exp(x_i' \beta) T_{0i} [/math]. So if there are two groups (x=1 and x=0), and [math]exp(\beta) = 2[/math], it means one group live twice as long as people in another group.
- [math]S_1(t) = S_0(t/ exp(x' \beta))[/math]. This explains the meaning of accelerated failure-time. Depending on the sign of [math]\beta' x[/math], the time is either accelerated by a constant factor or degraded by a constant factor. If [math]exp(\beta)=2[/math], 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 [math]\lambda_1(t) = \lambda_0(t/exp(x'\beta))/ exp(x'\beta) [/math]. So if [math]exp(\beta)=2[/math], 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 [math]\lambda[/math] satisfies the log linear model [math]\log \lambda_i = x_i' \beta[/math].
Proportional hazard models
Note PH models is a type of multiplicative hazard rate models [math]h(x|Z) = h_0(x)c(\beta' Z)[/math] where [math]c(\beta' Z) = \exp(\beta ' Z)[/math].
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 [math]Z[/math] and [math]Z^*[/math] is a constant function time.
- [math] \begin{align} \frac{h(t|Z)}{h(t|Z^*)} = \frac{h_0(t)\exp(\beta 'Z)}{h_0(t)\exp(\beta ' Z^*)} = \exp(\beta' (Z-Z^*)) \mbox{ independent of time} \end{align} [/math]
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) [math]h(t) = p \lambda (\lambda t)^{p-1}[/math] and [math]S(t) = exp(-\lambda t^p)[/math]. 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. [math]h(t) = \lambda^p p t^{p-1}[/math] and [math]S(t) = exp(-(\lambda t)^p)[/math]. R and wikipedia also follows this parametrization except that [math]h(t) = p t^{p-1}/\lambda^p[/math] and [math]S(t) = exp(-(t/\lambda)^p)[/math].
- Exponential distribution [math]h(t)[/math] = constant (independent of t). This is a special case of Weibull distribution (p=1).
- Weibull (and also exponential)
distributionregression model is the only case which belongs to both the proportional hazards and the accelerated life families.
- [math] \begin{align} \frac{h(x|Z_1)}{h(x|Z_2)} = \frac{h_0(x\exp(-\gamma' Z_1)) \exp(-\gamma ' Z_1)}{h_0(x\exp(-\gamma' Z_2)) \exp(-\gamma ' Z_2)} = \frac{(a/b)\left(\frac{x \exp(-\gamma ' Z_1)}{b}\right)^{a-1}\exp(-\gamma ' Z_1)}{(a/b)\left(\frac{x \exp(-\gamma ' Z_2)}{b}\right)^{a-1}\exp(-\gamma ' Z_2)} \quad \mbox{which is independent of time x} \end{align} [/math]
- 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 [math]b[/math], 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) | [math]\lambda \exp(-\lambda t)[/math] | [math]\lambda[/math] | [math]\exp(-\lambda t)[/math] | [math]1/\lambda[/math] |
Weibull (Klein, Bender, wikipedia) | [math]p\lambda t^{p-1}\exp(-\lambda t^p)[/math] | [math]p\lambda t^{p-1}[/math] | [math]exp(-\lambda t^p)[/math] | [math]\frac{\Gamma(1+1/p)}{\lambda^{1/p}}[/math] |
Exponential (R) | [math]\lambda \exp(-\lambda t)[/math], [math]\lambda[/math] is rate | [math]\lambda[/math] | [math]\exp(-\lambda t)[/math] | [math]1/\lambda[/math] |
Weibull (R, wikipedia) | [math]\frac{a}{b}\left(\frac{t}{b}\right)^{a-1} \exp(-(\frac{t}{b})^a)[/math], [math]a[/math] is shape, and [math]b[/math] is scale |
[math]\frac{a}{b}\left(\frac{t}{b}\right)^{a-1}[/math] | [math]\exp(-(\frac{t}{b})^a)[/math] | [math]b\Gamma(1+1/a)[/math] |
- Accelerated failure-time model. Let [math]Y=\log(T)=\mu + \gamma'Z + \sigma W[/math]. Then the survival function of [math]T[/math] at the covariate Z,
- [math] \begin{align} S_T(t|Z) &= P(T \gt t |Z) \\ &= P(Y \gt \ln t|Z) \\ &= P(\mu + \sigma W \gt \ln t-\gamma' Z | Z) \\ &= P(e^{\mu + \sigma W} \gt t\exp(-\gamma'Z) | Z) \\ &= S_0(t \exp(-\gamma'Z)). \end{align} [/math]
where [math]S_0(t)[/math] denote the survival function T when Z=0. Since [math]h(t) = -\partial \ln (S(t))[/math], the hazard function of T with a covariate value Z is related to a baseline hazard rate [math]h_0[/math] by (p56 Klein)
- [math] \begin{align} h(t|Z) = h_0(t\exp(-\gamma' Z)) \exp(-\gamma ' Z) \end{align} [/math]
> 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
Weibull distribution and reliability
Survival Analysis – Fitting Weibull Models for Improving Device Reliability in R (simulation)
Optimisation of a Weibull survival model using Optimx()
Optimisation of a Weibull survival model using Optimx() in R
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.
- http://www.bioconductor.org/packages/release/bioc/manuals/genefilter/man/genefilter.pdf#page=4
y <- rexp(10) cen <- runif(10) status <- ifelse(cen < .7, 1, 0)
- Inference on Selected Subgroups in Clinical Trials [math]\lambda(t) = \lambda_0(t) e^{\beta_i D}[/math] for subgroup i=1,2, respectively where D is the treatment indicator and [math]\lambda_0(t)[/math] is the baseline hazard function of Weibull(1,1). The subjects fall into one of the two subgroups with probability 0.5, and the treatment assignment is also random with equal probability. The response generated from the above model is then censored randomly from the right by a censoring variable C, where log(C) follows the uniform distribution on (-1.25, 1.00). The censoring rate is about 40% across different choices of [math]\beta_i[/math] considered in this study.
- How much power/accuracy is lost by using the Cox model instead of Weibull model when both model are correct? [math]h(t|x)=\lambda=e^{3x+1} = h_0(t)e^{\beta x}[/math] where [math]h_0(t)=e^1, \beta=3[/math].
- Note that for the exponential distribution, larger rate/[math]\lambda[/math] 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)$coef # 2.717794 # See the pdf note for the rest of code
- Intercept in survreg for the exponential distribution. http://www.stat.columbia.edu/~madigan/W2025/notes/survival.pdf#page=25.
- [math] \begin{align} \lambda = exp(-intercept) \end{align} [/math]
> futime <- rexp(1000, 5) > survreg(Surv(futime,rep(1,1000))~1,dist="exponential")$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.
- [math] \begin{align} \gamma &= 1/scale \\ \alpha &= exp(-(Intercept)*\gamma) \end{align} [/math]
> 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 [math]i[/math] with covariate [math]X_i[/math], [math]\log(scale_i)[/math] = [math]\beta ' X_i[/math]. Note that if we want to make the [math]\beta[/math] sign to be consistent with the Cox model, we want to use [math]\log(scale_i)[/math] = [math]-\beta ' X_i[/math] 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
- [math] h(t|x) = 1/scale = \frac{1}{\lambda/e^{\beta 'x}} = \frac{e^{\beta ' x}}{\lambda} = h_0(t) \exp(\beta' x) [/math]
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 [math]h_0(t)=\lambda \rho t^{\rho - 1} [/math] where shape 𝜌>0 and scale 𝜆>0. Following the inverse probability method, a realisation of 𝑇∼𝑆(⋅|𝐱) is obtained by computing [math] t = \left( - \frac{\log(v)}{\lambda \exp(x' \beta)} \right) ^ {1/\rho} [/math] 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
[math]T=H_0^{-1}[-\log(U) \exp(\beta' x)]
[/math] Bender2005.png, Bender2005table2.png
- 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 [math] \lambda = exp(\beta' x), \; S(t)= \exp(-\lambda t) = \exp(-t e^{\beta' x}) \sim Unif(0,1) [/math]
- Prediction Accuracy Measures for a Nonlinear Model and for Right-Censored Time-to-Event Data Li and Wang
- Simple example from glmnet
set.seed(10101) N = 1000 p = 30 nzc = p/3 x = matrix(rnorm(N * p), N, p) beta = rnorm(nzc) fx = x[, seq(nzc)] %*% beta/3 hx = exp(fx) ty = rexp(N, hx) tcens = rbinom(n = N, prob = 0.3, size = 1) # censoring indicator y = cbind(time = ty, status = 1 - tcens) # y=Surv(ty,1-tcens) with library(survival) fit = glmnet(x, y, family = "cox") pred = predict(fit, newx = x) Cindex(pred, y)
- A non-standard baseline hazard function [math]\lambda_0(t)=(t - .5)^2[/math] from the paper: A new nonparametric screening method for ultrahigh-dimensional survival data Liu 2018. The censoring time [math]C = \widetilde{C} \wedge \tau[/math], where [math]\widetilde{C}[/math] was generated from Unif (0, [math]\tau + 2[/math]) where [math]\tau[/math] was chosen to yield the desirable censoring rates of 20% and 40%, respectively.
- 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. See also Tian 2014 JASA p1525. X ~ standard Gaussian. True survival time exp(beta X + k · Z). Z ~ N(0,1), and k is chosen so that the signal-to-noise ratio is 3.0 or to induce a certain censoring rate. Censoring time C = exp (k · Z). The observed survival time T = min{Y, C}.
- survParamSim: Parametric Survival Simulation with Parameter Uncertainty
- vivaGen – a survival data set generator for software testing BMC Bioinformatics 2020
Warning on multiple rates
Search Vectorize() function in this page.
mean(rexp(1000, rate=2) ) # [1] 0.5258078 mean(rexp(1000, rate=1) ) # [1] 0.9712124 z = rexp(1000, rate=c(1, 2)) mean(z[seq(1, 1000, by=2)]) # [1] 1.041969 mean(z[seq(2, 1000, by=2)]) # [1] 0.5079594
Markov model
Fake Survival Data for the Disease Progression Model
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.
- Cross- Validated Cox Regression on Microarray Gene Expression Data van Houwelingen HC, Bruinsma T, Hart AAM, van’t Veer LJ, Wessels LFA (2006). Statistics in Medicine, 25, 3201–3216
- Using cross-validation to evaluate predictive accuracy of survival risk classifiers based on high-dimensional data. Simon et al, Brief Bioinform. 2011
- Testing the additional predictive value of high-dimensional molecular data. the cross-validated probabilities are not independent from each other.
- A study of pre-validation. Annals of Applied Statistics. 2008
- CVPL (cross-validated partial likelihood)
- https://www.rdocumentation.org/packages/survcomp/versions/1.22.0/topics/cvpl (lower is better)
- https://rdrr.io/cran/dynpred/man/CVPL.html. source code. 1. it does LOOCV so no need to set a random seed. 2. it seems the function does not include lasso/glmnet 3. the formula on pages 173-174 of the book Dynamic Prediction in Clinical Survival Analysis says the partial log likelihood should include the penalty term. 4. concordance measures like Harrell’s C-index are not appropriate because they only measure the discrimination and not the calibration. PS: I downloaded and looked at the chapter source code. It uses optL1() function from the penalized package to obtain cross validated log partial likelihood.
R> library(dynpred) R> data(ova) R> CVPL(Surv(tyears, d) ~ 1, data = ova) [1] NA R> CVPL(Surv(tyears, d) ~ Karn + Broders + FIGO + Ascites + Diam, data = ova) [1] -1652.169 R> coxph(Surv(tyears, d) ~ Karn + Broders + FIGO + Ascites + Diam, data = ova)$loglik[2] # No CV [1] -1374.717
- optL1() from the penalized package. It seems the penalized package has its own sequence of lambdas and these lambdas are totally different from glmnet() has created though the CV plot from each package shows a convex shape.
- Gsslasso paper. CVPL does not include the penalty term.
- https://web.stanford.edu/~hastie/Papers/v39i05.pdf#page=10 (larger is better)
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
- How is the overall survival measured?
- The length of time from either the date of diagnosis or the start of treatment for a disease, such as cancer, that patients diagnosed with the disease are still alive. In a clinical trial, measuring the overall survival is one way to see how well a new treatment works. NCI Dictionary of Cancer Terms
- Overall survival, or OS, or sometimes just “survival” is the percentage of people in a group who are alive after a length of time—usually a number of years.
- How is progression-free survival measured?
- 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. In a clinical trial, measuring the progression-free survival is one way to see how well a new treatment works. NCI Dictionary of Cancer Terms
- Progression-free survival (PFS) was measured as the interval between the initiation of treatment until either disease recurrence or last documented follow-up of the patient if he/she remains disease-free.
- OS vs PFS
- 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)
Time-dependent covariates
- Using Time Dependent Covariates and Time Dependent Coefficients in the Cox Mode
- Building an Elastic-Net Cox Model with Time-Dependent covariates
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
Class notes
- https://myweb.uiowa.edu/pbreheny/7210/f15/notes.html
- http://www.stat.columbia.edu/~madigan/W2025/notes/survival.pdf
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
[math] \lambda(t|X) = \lambda_0(t)\exp(\beta_1X_1 + \cdots + \beta_pX_p) = \lambda_0(t)\exp(X \beta^\prime). [/math]
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
[math] L(\beta) = \prod\limits_{i:C_i=1}\frac{\theta_i}{\sum_{j:Y_j\ge Y_i}\theta_j}, [/math]
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
[math] \ell(\beta) = \sum_{i:C_i=1} \left(X_i \beta^\prime - \log \sum_{j:Y_j\ge Y_i}\theta_j\right). [/math]
This function can be maximized over β to produce maximum partial likelihood estimates of the model parameters.
The partial score function is [math] \ell^\prime(\beta) = \sum_{i:C_i=1} \left(X_i - \frac{\sum_{j:Y_j\ge Y_i}\theta_jX_j}{\sum_{j:Y_j\ge Y_i}\theta_j}\right), [/math]
and the Hessian matrix of the partial log likelihood is
[math] \ell^{\prime\prime}(\beta) = -\sum_{i:C_i=1} \left(\frac{\sum_{j:Y_j\ge Y_i}\theta_jX_jX_j^\prime}{\sum_{j:Y_j\ge Y_i}\theta_j} - \frac{\sum_{j:Y_j\ge Y_i}\theta_jX_j\times \sum_{j:Y_j\ge Y_i}\theta_jX_j^\prime}{[\sum_{j:Y_j\ge Y_i}\theta_j]^2}\right). [/math]
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.
Get the partial likelihood of a Cox PH Model with new data
offset was used. See https://stackoverflow.com/questions/26721551/is-there-a-way-to-get-the-partial-likelihood-of-a-cox-ph-model-with-new-data-and
How to compute partial log-likelihood function in Cox proportional hazards model?
set.seed(1) n <- 1000 t <- rexp(100) c <- rbinom(100, 1, .2) ## censoring indicator (independent process) x <- rbinom(100, 1, exp(-t)) ## some arbitrary relationship btn x and t betamax <- coxph(Surv(t, c) ~ x) beta1 <- coxph(Surv(t, c) ~ x, init = c(1), control=coxph.control(iter.max=0)) betamax$loglik[2] # [1]=initial, [2]=final # [1] -52.81476 beta1$loglik[2] # [1] -52.85067
Compare the partial likelihood to the full likelihood
http://math.ucsd.edu/~rxu/math284/slect5.pdf#page=10
z-column (Wald statistic) from R's coxph()
- https://socialsciences.mcmaster.ca/jfox/Books/Companion/appendix/Appendix-Cox-Regression.pdf#page=6 The ratio of each regression coefficient to its standard error, a Wald statistic which is asymptotically standard normal under the hypothesis that the corresponding β is 0.
- http://dni-institute.in/blogs/cox-regression-interpret-result-and-predict/
How exactly can the Cox-model ignore exact times?
The Cox model does not depend on the times itself, instead it only needs an ordering of the events.
library(survival) survfit(Surv(time, status) ~ x, data = aml) fit <- coxph(Surv(time, status) ~ x, data = aml) coef(fit) # 0.9155326 min(diff(sort(unique(aml$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.
[math] h(t) = \lim_{\Delta t \to 0} \frac{P(t \leq T \lt t+\Delta t|T \geq t)}{\Delta t} = \frac{f(t)}{S(t)} = -\partial{ln[S(t)]} [/math]
Therefore
[math] H(x) = \int_0^x h(u) d(u) = -ln[S(x)]. [/math]
or
[math] S(x) = e^{-H(x)} [/math]
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 [math]\log h(t) = c + \rho t[/math], S(t) is Gompertz distribution.
- If [math]\log h(t)=c + \rho \log (t)[/math], S(t) is Weibull distribution.
- For Cox regression, the survival function can be shown to be [math]S(t|X) = S_0(t) ^ {\exp(X\beta)}[/math].
- [math] \begin{align} S(t|X) &= e^{-H(t)} = e^{-\int_0^t h(u|X)du} \\ &= e^{-\int_0^t h_0(u) exp(X\beta) du} \\ &= e^{-\int_0^t h_0(u) du \cdot exp(X \beta)} \\ &= S_0(t)^{exp(X \beta)} \end{align} [/math]
Alternatively,
- [math] \begin{align} S(t|X) &= e^{-H(t)} = e^{-\int_0^t h(u|X)du} \\ &= e^{-\int_0^t h_0(u) exp(X\beta) du} \\ &= e^{-H_0(t) \cdot exp(X \beta)} \end{align} [/math]
where the cumulative baseline hazard at time t, [math]H_0(t)[/math], is commonly estimated through the non-parametric Breslow estimator.
Check the proportional hazard (constant HR over time) assumption by cox.zph()
library(survival) res.cox <- coxph(Surv(time, status) ~ age + sex + wt.loss, data = lung) test.ph <- cox.zph(res.cox) test.ph
- survival package
- 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
- https://www.rdocumentation.org/packages/gof/versions/0.9.1/topics/cumres.coxph
- http://rstudio-pubs-static.s3.amazonaws.com/5043_145684af0d364175bf5e5e6bb792ca28.html
- Residuals and model diagnostics from the lecture notes of Patrick Breheny
- Cumulative martingale residuals (Lin et al Biometrika 1993)
strata
stratification in cox model. In a Cox model, stratification allows for as many different hazard functions as there are strata. Beta coefficients (hazard ratios) optimized for all strata are then fitted.
bladder1 <- bladder[bladder$enum < 5, ] o <- coxph(Surv(stop, event) ~ rx + size + number + strata(enum) , bladder1) # the strata will not be a term in covariate in the model fitting anova(o)
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.
- [math] \begin{align} \mu &= \int_0^\infty t f(t) dt \\ &= \int_0^\infty S(t) dt \end{align} [/math]
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 [math]S(\infty)=0[/math]. The average lifetime may not be bounded if you have censored data, there's censored observations that last beyond your last recorded death.
- [math]\frac{1}{S(t_0)} \int_0^{\infty} t\,f(t_0+t)\,dt = \frac{1}{S(t_0)} \int_{t_0}^{\infty} S(t)\,dt,[/math]
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. See an example here.
- Hazard ratio is a measure of an effect of an intervention of an outcome of interest over time. See an example here.
- 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)$conf.int; confint(fit)
- The coefficient beta represents the expected change in log hazard if X changes by one unit and all other variables are held constant in Cox models. See Variable selection – A review and recommendations for the practicing statistician by Heinze et al 2018.
- Understanding the endpoints in oncology: overall survival, progression free survival, hazard ratio, censored value
Another example (John Fox, Cox Proportional-Hazards Regression for Survival Data) is assuming Y ~ age + prio + others.
- If exp(beta_age) = 0.944. It means an additional year of age reduces the hazard by a factor of .944 on average, or (1-.944)*100 = 5.6 percent.
- If exp(beta_prio) = 1.096, it means each prior conviction increases the hazard by a factor of 1.096, or 9.6 percent.
How do you explain the difference between hazard ratio and relative risk to a layman? from Quora.
Odds Ratio, Hazard Ratio and Relative Risk by Janez Stare
For two groups that differ only in treatment condition, the ratio of the hazard functions is given by [math]e^\beta[/math], where [math]\beta[/math] is the estimate of treatment effect derived from the regression model. See here.
Compute ratio ratios from coxph() in R (Hint: exp(beta)).
Prognostic index is defined on http://www.math.ucsd.edu/~rxu/math284/slect6.pdf#page=2.
Basics of the Cox proportional hazards model. Good prognostic factor (b<0 or HR<1) and bad prognostic factor (b>0 or HR>1).
Variable selection: variables were retained in the prediction models if they had a hazard ratio of <0.85 or >1.15 (for binary variables) and were statistically significant at the 0.01 level. see Development and validation of risk prediction equations to estimate survival in patients with colorectal cancer: cohort study.
library(KMsurv) # No ties. Section 8.2 data(btrial) coxph(Surv(time, death) ~ im, data = btrial) summary(coxph(Surv(time, death) ~ im, data = btrial))$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
- Poisson regression
- 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.
- Glm family functions in glmnet 4.0 also mentions Cox model (and multinomial regression, and multi-response Gaussian) is a special case of the Poisson GLM.
- 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
- Optimisation of a Poisson survival model using Optimx in R
Estimate baseline hazard [math]h_0(t)[/math], Breslow cumulative baseline hazard [math]H_0(t)[/math], baseline survival function [math]S_0(t)[/math] and the survival function [math]S(t)[/math]
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.
- 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 [math]\sum_{t_i \le t} \frac{d_i}{Y_i}[/math] ([math]Y_i[/math] is the number at risk at time [math]t_i[/math]) when there are no covariates present; see p283 of Klein 2003.
- [math] \begin{align} \hat{H}_0(t) &= \sum_{t_i \le t} \frac{d_i}{W(t_i;b)}, \\ W(t_i;b) &= \sum_{j \in R(t_i)} \exp(b' z_j) \end{align} [/math]
where [math] t_1 \lt t_2 \lt \cdots \lt t_D[/math] denotes the distinct death times and [math]d_i[/math] be the number of deaths at time [math]t_i[/math]. The estimator of the baseline survival function [math]S_0(t) = \exp [-H_0(t)][/math] is given by [math]\hat{S}_0(t) = \exp [-\hat{H}_0(t)][/math]. 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: [math]S_0(t)^{\exp(\hat{\beta}_{age}*60)}[/math], [math]S_0(t)^{\exp(\hat{\beta}_{age}*60+\hat{\beta}_{stageII})}[/math], [math]S_0(t)^{\exp(\hat{\beta}_{age}*60+\hat{\beta}_{stageIII})}[/math], [math]S_0(t)^{\exp(\hat{\beta}_{age}*60+\hat{\beta}_{stageIV})}[/math].
library(KMsurv) # Data package for Klein & Moeschberge data(larynx) larynx$stage <- factor(larynx$stage) coxobj <- coxph(Surv(time, delta) ~ age + stage, data = larynx) # Figure 8.3 from Section 8.6 plot(survfit(coxobj, newdata = data.frame(age=rep(60, 4), stage=factor(1:4))), lty = 1:4) # Estimated probability for a 60-year old for different stage patients out <- summary(survfit(coxobj, data.frame(age = rep(60, 4), stage=factor(1:4))), times = 5) out$surv # time n.risk n.event survival1 survival2 survival3 survival4 # 5 34 40 0.702 0.665 0.51 0.142 sum(larynx$time >=5) # n.risk # [1] 34 sum(larynx$delta[larynx$time <=5]) # n.event # [1] 40 sum(larynx$time >5) # Wrong # [1] 31 sum(larynx$delta[larynx$time <5]) # Wrong # [1] 39 # 95% confidence interval out$lower # 0.8629482 0.9102532 0.7352413 0.548579 out$upper # 0.5707952 0.4864903 0.3539527 0.03691768
We need to pay attention when the number of covariates is large (and we don't want to specify each covariates in the formula). The key is to create a data frame and use dot (.) in the formula. This is to fix a warning message: 'newdata' had XXX rows but variables found have YYY rows from running survfit(, newdata).
Another way is to use as.formula() if we don't want to create a new object.
xsub <- data.frame(xtrain) colnames(xsub) <- paste0("x", 1:ncol(xsub)) coxobj <- coxph(Surv(ytrain[, "time"], ytrain[, "status"]) ~ ., data = xsub) newdata <- data.frame(xtest) colnames(newdata) <- paste0("x", 1:ncol(newdata)) survprob <- summary(survfit(coxobj, newdata=newdata), times = 5)$surv[1, ] # since there is only 1 time point, we select the first row in surv (surv is a matrix with one row).
The predictSurvProb() function from the pec package can also be used to extract survival probability predictions from various modeling approaches.
Predicted survival probabilities from glmnet: c060/peperr, biospear packages and manual computation
- Terry Therneau: The answer is that you cannot predict survival time, in general
- https://rdrr.io/cran/c060/man/predictProb.glmnet.html
## S3 method for class 'glmnet' predictProb(object, response, x, times, complexity, ...) set.seed(1234) junk <- biospear::simdata(n=500, p=500, q.main = 10, q.inter = 0, prob.tt = .5, m0=1, alpha.tt=0, beta.main= -.5, b.corr = .7, b.corr.by=25, wei.shape = 1, recr=3, fu=2, timefactor=1) summary(junk$time) library(glmnet) library(c060) # Error: object 'predictProb' not found library(peperr) y <- cbind(time=junk$time, status=junk$status) x <- cbind(1, junk[, "treat", drop = FALSE]) names(x) <- c("inter", "treat") x <- as.matrix(x) cvfit <- cv.glmnet(x, y, family = "cox") obj <- glmnet(x, y, family = "cox") xnew <- matrix(c(0,0), nr=1) predictProb(obj, y, xnew, times=1, complexity = cvfit$lambda.min) # Error in exp(lp[response[, 1] >= t.unique[i]]) : # non-numeric argument to mathematical function # In addition: Warning message: # In is.na(x) : is.na() applied to non-(list or vector) of type 'NULL'
- https://www.rdocumentation.org/packages/biospear/versions/1.0.1/topics/expSurv and manual computation (search bhaz)
expSurv(res, traindata, method, ci.level = .95, boot = FALSE, nboot, smooth = TRUE, pct.group = 4, time, trace = TRUE, ncores = 1) # S3 method for resexpSurv predict(object, newdata, ...)
# continue the example # BMsel() takes a little while resBM <- biospear::BMsel( data = junk, method = "lasso", inter = FALSE, folds = 5) # Note: if we specify time =5 in expsurv(), we will get an error message # 'time' is out of the range of the observed survival time. # Note: if we try to specify more than 1 time point, we will get the following msg # 'time' must be an unique value; no two values are allowed. esurv <- biospear::expSurv( res = resBM, traindata = junk, boot = FALSE, time = median(junk$time), trace = TRUE) # debug(biospear:::plot.resexpSurv) plot(esurv, method = "lasso") # This is equivalent to doing the following xx <- attributes(esurv)$m.score[, "lasso"] wc <- order(xx); wgr <- 1:nrow(esurv$surv) p1 <- plot(x = xx[wc], y = esurv$surv[wgr, "lasso"][wc], xlab='prognostic score', ylab='survival prob') # prognostic score beta*x in this cases. # ignore treatment effect and interactions xxmy <- as.vector(as.matrix(junk[, names(resBM$lasso)]) %*% resBM$lasso) # See page4 of the paper. Scaled scores were used in the plot range(abs(xx - (xxmy-quantile(xxmy, .025)) / (quantile(xxmy, .975) - quantile(xxmy, .025)))) # [1] 1.500431e-09 1.465241e-06 h0 <- bhaz(resBM$lasso, junk$time, junk$status, junk[, names(resBM$lasso)]) newtime <- median(junk$time) H0 <- sapply(newtime, function(tt) sum(h0$h[h0$dt <= tt])) # newx <- junk[ , names(resBM$lasso)] # Compute the estimate of the survival probability at training x and time = median(junk$time) # using Breslow method S2 <- outer(exp(-H0), exp(xxmy), "^") # row = newtime, col = newx range(abs(esurv$surv[wgr, "lasso"] - S2)) # [1] 6.455479e-18 2.459136e-06 # My implementation of the prognostic plot # Note that the x-axis on the plot is based on prognostic scores beta*x, # not on treatment modifying scores gamma*x as described in the paper. # Maybe it is because inter = FALSE in BMsel() we have used. p2 <- plot(xxmy[wc], S2[wc], xlab='prognostic score', ylab='survival prob') # cf p1 > names(esurv) [1] "surv" "lower" "upper" > str(esurv$surv) num [1:500, 1:2] 0.772 0.886 0.961 0.731 0.749 ... - attr(*, "dimnames")=List of 2 ..$ : NULL ..$ : chr [1:2] "lasso" "oracle" esurv2 <- predict(esurv, newdata = junk) esurv2$surv # All zeros?
Bug from the sample data (interaction was considered here; inter = TRUE) ?
set.seed(123456) resBM <- BMsel( data = Breast, x = 4:ncol(Breast), y = 2:1, tt = 3, inter = TRUE, std.x = TRUE, folds = 5, method = c("lasso", "lasso-pcvl")) esurv <- expSurv( res = resBM, traindata = Breast, boot = FALSE, smooth = TRUE, time = 4, trace = TRUE ) Computation of the expected survival Computation of analytical confidence intervals Computation of smoothed B-splines Error in cobs(x = x, y = y, print.mesg = F, print.warn = F, method = "uniform", : There is at least one pair of adjacent knots that contains no observation.
Loglikelihood
- fit$loglik is a vector of length 2 (initial model, fitted model). So deviance can be calculated by -2*fit$loglik[2]; see here for an example from BhGLM package.
- Use survival::anova() command to do a likelihood ratio test. Note this function does not work on glmnet object.
- residuals.coxph Calculates martingale, deviance, score or Schoenfeld residuals for a Cox proportional hazards model.
- No deviance() on coxph object!
- logLik() function will return fit$loglik[2]
- Gradient descent for the elastic net Cox-PH model
glmnet
- It seems AIC does not require the assumption of nested models.
- https://en.wikipedia.org/wiki/Akaike_information_criterion, (akaike pronunciation in Japanese)
- [math] \begin{align} \mathrm{AIC} &= 2k - 2\ln(\hat L) \\ \mathrm{AICc} &= \mathrm{AIC} + \frac{2k^2 + 2k}{n - k - 1} \end{align} [/math]
- Is it possible to calculate AIC and BIC for lasso regression models?. See the references about the degrees of freedom in Lasso regressions.
fit <- glmnet(x, y, family = "multinomial") tLL <- fit$nulldev - deviance(fit) # ln(L) k <- fit$df n <- fit$nobs AICc <- -tLL+2*k+2*k*(k+1)/(n-k-1) AICc
f <- glmnet(x = x, y = y, family = family) f$aic <- deviance(f) + 2 * f$df
- For glmnet object, see ?deviance.glmnet, ?coxnet.deviance and R: Getting AIC/BIC/Likelihood from GLMNet. An example with all continuous variables
set.seed(10101) N=1000;p=6 nzc=p/3 x=matrix(rnorm(N*p),N,p) beta=rnorm(nzc) fx=x[,seq(nzc)]%*%beta/3 hx=exp(fx) ty=rexp(N,hx) tcens=rbinom(n=N,prob=.3,size=1)# censoring indicator y=cbind(time=ty,status=1-tcens) # y=Surv(ty,1-tcens) with library(survival) coxobj <- coxph(Surv(ty, 1-tcens) ~ x) coxobj_small <- coxph(Surv(ty, 1-tcens) ~1) anova(coxobj, coxobj_small) # Analysis of Deviance Table # Cox model: response is Surv(ty, 1 - tcens) # Model 1: ~ x # Model 2: ~ 1 # loglik Chisq Df P(>|Chi|) # 1 -4095.2 # 2 -4102.7 15.151 6 0.01911 * fit2 <- glmnet(x,y,family="cox", lambda=0) # ridge regression deviance(fit2) # 2*(loglike_sat - loglike) # [1] 8190.313 coxnet.deviance(x=x, y=y, beta=coef(fit2)) # 2*(loglike_sat - loglike) # [1] 8190.313 # https://github.com/cran/glmnet/blob/master/R/coxnet.deviance.R#L79 assess.glmnet(fit2, x=x, y=y) # returns deviance and c-index fit2$df # [1] 6 fit2$nulldev - deviance(fit2) # Log-Likelihood ratio statistic # [1] 15.15097 1-pchisq(fit2$nulldev - deviance(fit2), fit2$df) # [1] 0.01911446
Here is another example including a factor covariate:
library(KMsurv) # Data package for Klein & Moeschberge data(larynx) larynx$stage <- factor(larynx$stage) coxobj <- coxph(Surv(time, delta) ~ age + stage, data = larynx) coef(coxobj) # age stage2 stage3 stage4 # 0.0190311 0.1400402 0.6423817 1.7059796 coxobj_small <- coxph(Surv(time, delta)~age, data = larynx) anova(coxobj, coxobj_small) # Analysis of Deviance Table # Cox model: response is Surv(time, delta) # Model 1: ~ age + stage # Model 2: ~ age # loglik Chisq Df P(>|Chi|) # 1 -187.71 # 2 -195.55 15.681 3 0.001318 ** # --- # Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 # Now let's look at the glmnet() function. # It seems glmnet does not recognize factor covariates. coxobj2 <- with(larynx, glmnet(cbind(age, stage), Surv(time, delta), family = "cox", lambda=0)) coxobj2$nulldev - deviance(coxobj2) # Log-Likelihood ratio statistic # [1] 15.72596 coxobj1 <- with(larynx, glmnet(cbind(1, age), Surv(time, delta), family = "cox", lambda=0)) deviance(coxobj1) - deviance(coxobj2) # [1] 13.08457 1-pchisq(deviance(coxobj1) - deviance(coxobj2) , coxobj2$df-coxobj1$df) # [1] 0.0002977376
High dimensional data
https://cran.r-project.org/web/views/Survival.html
glmnet + Cox models
- Robust estimation of the expected survival probabilities from high-dimensional Cox models with biomarker-by-treatment interactions in randomized clinical trials by Nils Ternès, Federico Rotolo and Stefan Michiels, BMC Medical Research Methodology, 2017 (open review available). The corresponding software biospear on cran and rdocumentation.org.
- Accounting for grouped predictor variables or pathways in high-dimensional penalized Cox regression models Belhechmi1 2020
- Expected time of survival in glmnet for cox family
Error in glmnet: x should be a matrix with 2 or more columns
Error in coxnet: (list) object cannot be coerced to type 'double'
Fix: do not use data.frame in X. Use cbind() instead.
Prognostic index/risk scores
- International Prognostic Index
- In R,
- Low scores correspond to the lowest predicted risk and high scores correspond to the greatest predicted risk.
- The test data were first segregated into high-risk and low-risk groups by the median of training risk scores. Assessment of performance of survival prediction models for cancer prognosis
- On the paper "The C-index is not proper for the evaluation of t-year predicted risk" Blanche et al 2018 defined the true t-year predicted risk by [math]P(T \le t | Z) = 1 - Survival[/math]
linear.predictors component in coxph object
The $linear.predictors component is not [math]\beta' x[/math]. It is [math]\beta' (x-\mu_x)[/math]. See this post.
predict.coxph, prognostic index & risk score
- predict.coxph() Compute fitted values and regression terms for a model fitted by coxph. The Cox model is a relative risk model; predictions of type "linear predictor", "risk", and "terms" are all relative to the sample from which they came. By default, the reference value for each of these is the mean covariate within strata. The primary underlying reason is statistical: a Cox model only predicts relative risks between pairs of subjects within the same strata, and hence the addition of a constant to any covariate, either overall or only within a particular stratum, has no effect on the fitted results. Returned value: a vector or matrix of predictions, or a list containing the predictions (element "fit") and their standard errors (element "se.fit") if the se.fit option is TRUE.
predict(object, newdata, type=c("lp", "risk", "expected", "terms", "survival"), se.fit=FALSE, na.action=na.pass, terms=names(object$assign), collapse, reference=c("strata", "sample"), ...)
type:
- "lp": linear predictor
- "risk": risk score exp(lp)
- "expected": the expected number of events given the covariates and follow-up time. The survival probability for a subject is equal to exp(-expected).
- "terms": the terms of the linear predictor.
- http://stats.stackexchange.com/questions/44896/how-to-interpret-the-output-of-predict-coxph. The $linear.predictors component represents [math]\beta (x - \bar{x})[/math]. The risk score (type='risk') corresponds to [math]\exp(\beta (x-\bar{x}))[/math]. Factors are converted to dummy predictors as usual; see model.matrix.
- http://www.togaware.com/datamining/survivor/Lung1.html
library(coxph) fit <- coxph(Surv(time, status) ~ age , lung) fit # Call: # coxph(formula = Surv(time, status) ~ age, data = lung) # coef exp(coef) se(coef) z p # age 0.0187 1.02 0.0092 2.03 0.042 # # Likelihood ratio test=4.24 on 1 df, p=0.0395 n= 228, number of events= 165 fit$means # age # 62.44737 # type = "lr" (Linear predictor) as.numeric(predict(fit,type="lp"))[1:10] # [1] 0.21626733 0.10394626 -0.12069589 -0.10197571 -0.04581518 0.21626733 # [7] 0.10394626 0.16010680 -0.17685643 -0.02709500 0.0187 * (lung$age[1:10] - fit$means) # [1] 0.21603421 0.10383421 -0.12056579 -0.10186579 -0.04576579 0.21603421 # [7] 0.10383421 0.15993421 -0.17666579 -0.02706579 fit$linear.predictors[1:10] # [1] 0.21626733 0.10394626 -0.12069589 -0.10197571 -0.04581518 # [6] 0.21626733 0.10394626 0.16010680 -0.17685643 -0.02709500 # type = "risk" (Risk score) > as.numeric(predict(fit,type="risk"))[1:10] [1] 1.2414342 1.1095408 0.8863035 0.9030515 0.9552185 1.2414342 1.1095408 [8] 1.1736362 0.8379001 0.9732688 > exp((lung$age-mean(lung$age)) * 0.0187)[1:10] [1] 1.2411448 1.1094165 0.8864188 0.9031508 0.9552657 1.2411448 [7] 1.1094165 1.1734337 0.8380598 0.9732972 > exp(fit$linear.predictors)[1:10] [1] 1.2414342 1.1095408 0.8863035 0.9030515 0.9552185 1.2414342 [7] 1.1095408 1.1736362 0.8379001 0.9732688
Survival risk prediction
- Using cross-validation to evaluate predictive accuracy of survival risk classifiers based on high-dimensional data Simon 2011. The authors have noted the CV has been used for optimization of tuning parameters but the data available are too limited for effective into training & test sets.
- The CV Kaplan-Meier curves are essentially unbiased and the separation between the curves gives a fair representation of the value of the expression profiles for predicting survival risk.
- The log-rank statistic does not have the usual chi-squared distribution under the null hypothesis. This is because the data was used to create the risk groups.
- Survival ROC curve can be used as a measure of predictive accuracy for the survival risk group model at a certain landmark time.
- The ROC curve can be misleading. For example if re-substitution is used, the AUC can be very large.
- The p-value for the significance of the test that AUC=.5 for the cross-validated survival ROC curve can be computed by permutations.
- Measure of assessment for prognostic prediction
0/1 Survival Sensitivity [math]P(Pred=1|True=1)[/math] [math]P(\beta' X \gt c | T \lt t)[/math] Specificity [math]P(Pred=0|True=0)[/math] [math]P(\beta' X \le c | T \ge t)[/math]
- An evaluation of resampling methods for assessment of survival risk prediction in high-dimensional settings Subramanian, et al 2010.
- The conditional probabilities can be estimated by Heagerty et al 2000 (R package survivalROC). The AUC(t) can be used for comparing and assessing prognostic models (a measure of accuracy) for future samples. In the absence of an independent large dataset, an estimate for AUC(t) is obtained through resampling from the original sample [math]S_n[/math].
- Resubstitution estimate of AUC(t) (i.e. all observations were used for feature selection, model building as well as the estimation of accuracy) is too optimistic. So k-fold CV method is considered.
- There are two ways to compute k-fold CV estimate of AUC(t): the pooling strategy (used in the paper) and average strategy (AUC(t)s are first computed for each test set and are then averaged). In the pooling strategy, all the test set risk-score predictions are first collected and AUC(t) is calculated on this combined set.
- Conclusions: sample splitting and LOOCV have a higher mean square error than other methods. 5-fold or 10-fold CV provide a good balance between bias and variability for a wide range of data settings.
- Gene Expression–Based Prognostic Signatures in Lung Cancer: Ready for Clinical Use? Subramanian, et al 2010.
- Assessment of survival prediction models based on microarray data Martin Schumacher, et al. 2007
- Semi-Supervised Methods to Predict Patient Survival from Gene Expression Data Eric Bair , Robert Tibshirani, 2004
- Time dependent ROC curves for censored survival data and a diagnostic marker. Heagerty et al, Biometrics 2000
- An introduction to survivalROC by Saha, Heagerty. If the AUCs are computed at several time points, we can plot the AUCs vs time for different models (eg different covariates) and compare them to see which model performs better.
- The survivalROC package does not draw an ROC curve. It outputs FP (x-axis) and TP (y-axis). We can use basic R or ggplot to draw the curve.
- survivalROC() calculates AUC at specified time by using NNE method (default). We can use the prognostic index as marker when there are more than one markers is used. Note that survAUC::AUC.uno() uses Uno (2007) to calculate FP and TP.
- Assessment of Discrimination in Survival Analysis (C-statistics, etc)
- plotROC package by Sachs for showing ROC curves from multiple time points on the same plot.
- Time-dependent ROC for Survival Prediction Models in R
- survivalROC怎么看最佳cut-off值？/ HOW to use the survivalROC to get optimal cut-off values?
- Survival prediction from clinico-genomic models - a comparative study Hege M Bøvelstad, 2009
- Assessment and comparison of prognostic classification schemes for survival data. E. Graf, C. Schmoor, W. Sauerbrei, et al. 1999
- What do we mean by validating a prognostic model? Douglas G. Altman, Patrick Royston, 2000
- On the prognostic value of survival models with application to gene expression signatures T. Hielscher, M. Zucknick, W. Werft, A. Benner, 2000
- Accuracy of point predictions in survival analysis, Henderson et al, Statist Med, 2001.
- Assessment of performance of survival prediction models for cancer prognosis Hung-Chia Chen et al 2012
- Accuracy of predictive ability measures for survival models Flandre, Detsch and O'Quigley, 2017.
- Association between expression of random gene sets and survival is evident in multiple cancer types and may be explained by sub-classification Yishai Shimoni, PLOS 2018
- Development and validation of risk prediction equations to estimate survival in patients with colorectal cancer: cohort study
- Cox-nnet: An artificial neural network method for prognosis prediction of high-throughput omics data Ching et al 2018.
Assessing the performance of prediction models
- Investigating the prediction ability of survival models based on both clinical and omics data: two case studies by Riccardo De Bin, Statistics in Medicine 2014. (not useful)
- Assessment of performance of survival prediction models for cancer prognosis Chen et al , BMC Medical Research Methodology 2012
- A simulation study of predictive ability measures in a survival model I: Explained variation measures Choodari‐Oskooei et al, Stat in Medicine 2011
- Assessing the performance of prediction models: a framework for some traditional and novel measures by Ewout W. Steyerberg, Andrew J. Vickers, [...], and Michael W. Kattan, 2010.
- survcomp: an R/Bioconductor package for performance assessment and comparison of survival models paper in 2011 and Introduction to R and Bioconductor Survival analysis where the survcomp package can be used. The summary here is based on this paper.
- How to compare predictive power of survival models?
- How to compare Harrell C-index from different models in survival analysis? and Frank Harrell's comment: Doing model comparison with LR statistics is more powerful than using methods that depend on an asymptotic distribution of the C-index.
Hazard ratio
hazard.ratio(x, surv.time, surv.event, weights, strat, alpha = 0.05, method.test = c("logrank", "likelihood.ratio", "wald"), na.rm = FALSE, ...)
D index
D.index(x, surv.time, surv.event, weights, strat, alpha = 0.05, method.test = c("logrank", "likelihood.ratio", "wald"), na.rm = FALSE, ...)
AUC
See ROC curve.
Comparison:
Definition | Interpretation | |
---|---|---|
Two class | [math] P(Z_{case} \gt 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 \gt \beta' Z_2|T_1 \lt T_2) [/math] | (Roughly speaking) 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). (Precisely) fraction of pairs in your data, where the observation with the higher survival time has the higher probability of survival predicted by your model. |
p95 of Heagerty and Zheng (2005) gave a relationship of C-statistic:
[math] C = P(M_j \gt M_k | T_j \lt T_k) = \int_t \mbox{AUC(t) w(t)} \; dt [/math]
where M is the marker value and [math]w(t) = 2 \cdot f(t) \cdot S(t) [/math]. When the interest is in the accuracy of a regression model we will use [math]M_i = Z_i^T \beta[/math].
The time-dependent AUC is also related to time-dependent C-index. [math] C_\tau = P(M_j \gt M_k | T_j \lt T_k, T_j \lt \tau) = \int_t \mbox{AUC(t)} \mbox{w}_{\tau}(t) \; dt [/math] where [math] w_\tau(t) = 2 \cdot f(t) \cdot S(t)/(1-S^2(\tau))[/math].
Concordance index/C-index/C-statistic interpretation and R packages
- The area under ROC curve (plot of sensitivity of 1-specificity) is also called C-statistic. It is a measure of discrimination generalized for survival data (Harrell 1982 & 2001). The ROC are functions of the sensitivity and specificity for each value of the measure of model. (Nancy Cook, 2007)
- The sensitivity of a test is the probability of a positive test result, or of a value above a threshold, among those with disease (cases).
- The specificity of a test is the probability of a negative test result, or of a value below a threshold, among those without disease (noncases).
- Perfect discrimination corresponds to a c-statistic of 1 & is achieved if the scores for all the cases are higher than those for all the non-cases.
- The c-statistic is the probability that the measure or predicted risk/risk score is higher for a case than for a noncase.
- The c-statistic is not the probability that individuals are classified correctly or that a person with a high test score will eventually become a case.
- C-statistic is a rank-based measure. The c-statistic describes how well models can rank order cases and noncases, but not a function of the actual predicted probabilities.
- How to interpret the output for calculating concordance index (c-index)? [math]
P(\beta' Z_1 \gt \beta' Z_2|T_1 \lt T_2)
[/math] where T is the survival time and Z is the covariates.
- It is the fraction of pairs in your data, where the observation with the higher survival time has the higher probability of survival predicted by your model.
- High values mean that your model predicts higher probabilities of survival for higher observed survival times.
- The c index estimates the probability of concordance between predicted and observed responses. A value of 0.5 indicates no predictive discrimination and a value of 1.0 indicates perfect separation of patients with different outcomes. (p371 Harrell 1996)
- Drawback of C-statistics:
- Even though rank indexes such as c are widely applicable and easily interpretable, they are not sensitive for detecting small differences in discrimination ability between two models. This is due to the fact that a rank method considers the (prediction, outcome) pairs (0.01,0), (0.9, 1) as no more concordant than the pairs (0.05,0), (0.8, 1). A more sensitive likelihood-ratio Chi-square-based statistic that reduces to R2 in the linear regression case may be substituted. (p371 Harrell 1996)
- If the model is correct, the likelihood based measures may be more sensitive in detecting differences in prediction ability, compared to rank-based measures such as C-indexes. (Uno 2011 p 1113)
- http://dmkd.cs.vt.edu/TUTORIAL/Survival/Slides.pdf
- Concordance vignette from the survival package. It has a good summary of different ways (such as Kendall's tau and Somers' d) to calculate the concordance statistic. The concordance function in the survival package can be used with various types of models including logistic and linear regression.
- Assessment of Discrimination in Survival Analysis (C-statistics, etc) webpage
- 5 Ways to Estimate Concordance Index for Cox Models in R, Why Results Aren't Identical?, C-index/C-statistic 计算的5种不同方法及比较. The 5 functions are rcorrcens() from Hmisc, summary()$concordance from survival, survConcordance() from survival, concordance.index() from survcomp and cph() from rms.
- Summary of R packages to compute C-statistic
Package Function New data? survival summary(coxph(formula, data))$concordance["C"], Cindex() no, yes survC1 Est.Cval() no survAUC UnoC() yes survcomp concordance.index() ? Hmisc rcorr.cens() no pec cindex() yes
Integrated brier score (≈ "mean squared error" of prediction for survival data)
Assessment and comparison of prognostic classification schemes for survival data Graf et al Stat. Med. 1999 2529-45, Consistent Estimation of the Expected Brier Score in General Survival Models with Right‐Censored Event Times Gerds et al 2006.
- Because the point predictions of event-free times will almost inevitably given inaccurate and unsatisfactory result, the mean square error of prediction [math]\frac{1}{n}\sum_1^n (T_i - \hat{T}(X_i))^2[/math] method will not be considered. See Parkes 1972 or Henderson 2001.
- Another approach is to predict the survival or event status [math]Y=I(T \gt \tau)[/math] at a fixed time point [math]\tau[/math] for a patient with X=x. This leads to the expected Brier score [math]E[(Y - \hat{S}(\tau|X))^2][/math] where [math]\hat{S}(\tau|X)[/math] is the estimated event-free probabilities (survival probability) at time [math]\tau[/math] for subject with predictor variable [math]X[/math].
- The time-dependent Brier score (without censoring)
- [math] \begin{align} \mbox{Brier}(\tau) &= \frac{1}{n}\sum_1^n (I(T_i\gt \tau) - \hat{S}(\tau|X_i))^2 \end{align} [/math]
- The time-dependent Brier score (with censoring, C is the censoring variable)
- [math] \begin{align} \mbox{Brier}(\tau) = \frac{1}{n}\sum_i^n\bigg[\frac{(\hat{S}_C(t_i))^2I(t_i \leq \tau, \delta_i=1)}{\hat{S}_C(t_i)} + \frac{(1 - \hat{S}_C(t_i))^2 I(t_i \gt \tau)}{\hat{S}_C(\tau)}\bigg] \end{align} [/math]
where [math]\hat{S}_C(t_i) = P(C \gt t_i)[/math], the Kaplan-Meier estimate of the censoring distribution with [math]t_i[/math] the survival time of patient i. The integration of the Brier score can be done by over time [math]t \in [0, \tau][/math] with respect to some weight function W(t) for which a natual choice is [math](1 - \hat{S}(t))/(1-\hat{S}(\tau))[/math]. The lower the iBrier score, the larger the prediction accuracy is.
- Useful benchmark values for the Brier score are 33%, which corresponds to predicting the risk by a random number drawn from U[0, 1], and 25% which corresponds to predicting 50% risk for everyone. See Evaluating Random Forests for Survival Analysis using Prediction Error Curves by Mogensen et al J. Stat Software 2012 (pec package). The paper has a good summary of different R package implementing Brier scores.
R function
- pec by Thomas A. Gerds. The plot.pec() can plot prediction error curves (defined by Brier score). See an example from this paper. The .632+ bootstrap prediction error curves is from the paper Boosting for high-dimensional time-to-event data with competing risks 2009
- peperr package. The package peperr is an early branch of pec.
- survcomp::sbrier.score2proba().
- ipred::sbrier()
Papers on high dimensional covariates
- Assessment of survival prediction models based on microarray data, Bioinformatics , 2007, vol. 23 (pg. 1768-74)
- Allowing for mandatory covariates in boosting estimation of sparse high-dimensional survival models, BMC Bioinformatics , 2008, vol. 9 pg. 14
Kendall's tau, Goodman-Kruskal's gamma, Somers' d
- https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient
- https://en.wikipedia.org/wiki/Goodman_and_Kruskal%27s_gamma
- https://en.wikipedia.org/wiki/Somers%27_D
- Survival package has a good summary. Especially concordance = (d+1)/2.
C-statistics
- For two groups data (one with event, one without), C-statistic has an intuitive interpretation: if two individuals are selected at random, one with the event and one without, then the C-statistic is the probability that the model predicts a higher risk for the individual with the event. Analysis of Biomarker Data: logs, odds ratios and ROC curves by Grund 2010
- C-statistics is the probability of concordance between predicted and observed survival.
- Comparing two correlated C indices with right‐censored survival outcome: a one‐shot nonparametric approach Kang et al, Stat in Med, 2014. compareC package for comparing two correlated C-indices with right censored outcomes. Harrell’s Concordance. The s.e. of the Harrell's C-statistics can be estimated by the delta method. [math] \begin{align} C_H = \frac{\sum_{i,j}I(t_i \lt t_{j}) I(\hat{\beta} Z_i \gt \hat{\beta} Z_j) \delta_i}{\sum_{i,j} I(t_i \lt t_j) \delta_i} \end{align} [/math] converges to a censoring-dependent quantity [math] P(\beta'Z_1 \gt \beta' Z_2|T_1 \lt T_2, T_1 \lt \text{min}(D_1,D_2)).[/math] Here D is the censoring variable.
- On the C-statistics for Evaluating Overall Adequacy of Risk Prediction Procedures with Censored Survival Data by Uno et al 2011. Let [math]\tau[/math] be a specified time point within the support of the censoring variable. [math]
\begin{align}
C(\tau) = \text{UnoC}(\hat{\pi}, \tau)
= \frac{\sum_{i,i'}(\hat{S}_C(t_i))^{-2}I(t_i \lt t_{i'}, t_i \lt \tau) I(\hat{\beta}'Z_i \gt \hat{\beta}'Z_{i'}) \delta_i}{\sum_{i,i'}(\hat{S}_C(t_i))^{-2}I(t_i \lt t_{i'}, t_i \lt \tau) \delta_i}
\end{align}
[/math], a measure of the concordance between [math]\hat{\beta} Z_i[/math] (the linear predictor) and the survival time. [math]\hat{S}_C(t)[/math] is the Kaplan-Meier estimator for the censoring distribution/variable/time (cf event time); flipping the definition of [math]\delta_i[/math]/considering failure events as "censored" observations and censored observations as "failures" and computing the KM as usual; see p207 of Satten 2001 and the source code from the kmcens() in survC1. Note that [math]C_\tau[/math] converges to [math] P(\beta'Z_1 \gt \beta' Z_2|T_1 \lt T_2, T_1 \lt \tau).[/math]
- Uno's estimator does not require the fitted model to be correct . See also table V in the simulation study where the true model is log-normal regression.
- Uno's estimator is consistent for a population concordance measure that is free of censoring. See the coverage result in table IV and V from his simulation study. Other forms of C-statistic estimate population parameters that may depend on the current study-specific censoring distribution.
- To accommodate discrete risk scores, in survC1::Est.Cval(), it is using the formula [math]. \begin{align} \frac{\sum_{i,i'}[ (\hat{S}_C(t_i))^{-2}I(t_i \lt t_{i'}, t_i \lt \tau) I(\hat{\beta}'Z_i \gt \hat{\beta}'Z_{i'}) \delta_i + 0.5 * (\hat{S}_C(t_i))^{-2}I(t_i \lt t_{i'}, t_i \lt \tau) I(\hat{\beta}'Z_i = \hat{\beta}'Z_{i'}) \delta_i ]}{\sum_{i,i'}(\hat{S}_C(t_i))^{-2}I(t_i \lt t_{i'}, t_i \lt \tau) \delta_i} \end{align} [/math]. Note that pec::cindex() is using the same formula but survAUC::UnoC() does not.
- If the specified [math]\tau[/math] (tau) is 'too' large such that very few events were observed or very few subjects were followed beyond this time point, the standard error estimate for [math]\hat{C}_\tau[/math] can be quite large.
- Uno mentioned from (page 95) Heagerty and Zheng 2005 that when T is right censoring, one would typically consider [math]C_\tau[/math] with a fixed, prespecified follow-up period [math](0, \tau)[/math].
- Uno also mentioned that when the data is right censored, the censoring variable D is usually shorter than that of the failure time T, the tail part of the estimated survival function of T is rather unstable. Thus we consider a truncated version of C.
- Heagerty and Zheng (2005) p95 said [math]C_\tau[/math] is the probability that the predictions for a random pair of subjects are concordant with their outcomes, given that the smaller event time occurs in [math](0, \tau)[/math].
- real data 1: fit a Cox model. Get risk scores [math]\hat{\beta}'Z[/math]. Compute the point and confidence interval estimates (M=500 indep. random samples with the same sample size as the observation data) of [math]C_\tau[/math] for different [math]\tau[/math]. Compare them with the conventional C-index procedure (Korn).
- real data 1: compute [math]C_\tau[/math] for a full model and a reduce model. Compute the difference of them ([math]C_\tau^{(A)} - C_\tau^{(B)} = .01[/math]) and the 95% confidence interval (-0.00, .02) of the difference for testing the importance of some variable (HDL in this case). Though HDL is quite significant (p=0) with respect to the risk of CV disease but its incremental value evaluated via C-statistics is quite modest.
- real data 2: goal - evaluate the prognostic value of a new gene signature in predicting the time to death or metastasis for breast cancer patients. Two models were fitted; one with age+ER and the other is gene+age+ER. For each model we can calculate the point and interval estimates of [math]C_\tau[/math] for different [math]\tau[/math]s.
- simulation: T is from Weibull regression for case 1 and log-normal regression for case 2. Covariates = (age, ER, gene). 3 kinds of censoring were considered. Sample size is 100, 150, 200 and 300. 1000 iterations. Compute coverage probabilities and average length of 95% confidence intervals, bias and root mean square error for [math]\tau[/math] equals to 10 and 15. Compared with the conventional approach, the new method has higher coverage probabilities and less bias in 6 scenarios.
- Statistical methods for the assessment of prognostic biomarkers (Part I): Discrimination by Tripep et al 2010
- Gonen and Heller 2005 concordance index for Cox models
- [math]P(T_2\gt T_1|g(Z_1)\gt g(Z_2))[/math]. Gonen and Heller's c statistic which is independent of censoring.
- GHCI() from survAUC package. Strangely only one parameter is needed. survAUC allows for testing data but CPE package does not have an option for testing data.
TR <- ovarian[1:16,] TE <- ovarian[17:26,] train.fit <- coxph(Surv(futime, fustat) ~ age, x=TRUE, y=TRUE, method="breslow", data=TR) lpnew <- predict(train.fit, newdata=TE) survAUC::GHCI(lpnew) # .8515 lpnew2 <- predict(train.fit, newdata = TR) survAUC::GHCI(lpnew2) # 0.8079495 CPE::phcpe(train.fit, CPE.SE = TRUE) # $CPE # [1] 0.8079495 # $CPE.SE # [1] 0.0670646 Hmisc::rcorr.cens(-TR$age, Surv(TR$futime, TR$fustat))["C Index"] # 0.7654321 Hmisc::rcorr.cens(TR$age, Surv(TR$futime, TR$fustat))["C Index"] # 0.2345679
- Used by simulatorZ package
- Uno's C-statistics (2011) and some examples using different packages
- C-statistic may or may not be a decreasing function of tau. However, AUC(t) may not be decreasing; see Fig 1 of Blanche et al 2018.
library(survAUC); library(pec) set.seed(1234) dat <- simulWeib(N=100, lambda=0.01, rho=1, beta=-0.6, rateC=0.001) # simulWebib was defined above # coef exp(coef) se(coef) z p # x -0.744 0.475 0.269 -2.76 0.0057 TR <- dat[1:80,] TE <- dat[81:100,] train.fit <- coxph(Surv(time, status) ~ x, data=TR) plot(survfit(Surv(time, status) ~ 1, data =TR)) lpnew <- predict(train.fit, newdata=TE) Surv.rsp <- Surv(TR$time, TR$status) Surv.rsp.new <- Surv(TE$time, TE$status) sapply(c(.25, .5, .75), function(qtl) UnoC(Surv.rsp, Surv.rsp.new, lpnew, time=quantile(TR$time, qtl))) # [1] 0.2580193 0.2735142 0.2658271 sapply(c(.25, .5, .75), function(qtl) cindex( list(matrix( -lpnew, nrow = nrow(TE))), formula = Surv(time, status) ~ x, data = TE, eval.times = quantile(TR$time, qtl))$AppC$matrix) # [1] 0.5041490 0.5186850 0.5106746
- Four elements are needed for computing truncated C-statistic using survAUC::UnoC. But it seems pec::cindex does not need the training data.
- training data including covariates,
- testing data including covariates,
- predictor from new data,
- truncation time/evaluation time/prediction horizon.
- (From ?UnoC) Uno's estimator is based on inverse-probability-of-censoring weights and does not assume a specific working model for deriving the predictor lpnew. It is assumed, however, that there is a one-to-one relationship between the predictor and the expected survival times conditional on the predictor. Note that the estimator implemented in UnoC is restricted to situations where the random censoring assumption holds.
- survAUC::UnoC(). The tau parameter: Truncation time. The resulting C tells how well the given prediction model works in predicting events that occur in the time range from 0 to tau. [math] P(\beta'Z_1 \gt \beta' Z_2|T_1 \lt T_2, T_1 \lt \tau).[/math] Con: no confidence interval estimate for [math]C_\tau[/math] nor [math]C_\tau^{(A)} - C_\tau^{(B)}[/math]
- pec::cindex(). At each timepoint of eval.times the c-index is computed using only those pairs where one of the event times is known to be earlier than this timepoint. If eval.times is missing or Inf then the largest uncensored event time is used. See a more general example from here
- Est.Cval() from the survC1 package (the only package gives confidence intervals of C-statistic or deltaC, authored by H. Uno). It doesn't take new data nor the vector of predictors obtained from the test data. Pro: Inf.Cval() can compute the confidence interval (perturbation-resampling based) of [math]C_\tau[/math] & Inf.Cval.Delta() for the difference [math]C_\tau^{(A)} - C_\tau^{(B)}[/math]. File:C stat scatter.svg
library(survAUC) # require training and predict sets TR <- ovarian[1:16,] TE <- ovarian[17:26,] train.fit <- coxph(Surv(futime, fustat) ~ age, data=TR) lpnew <- predict(train.fit, newdata=TE) Surv.rsp <- Surv(TR$futime, TR$fustat) Surv.rsp.new <- Surv(TE$futime, TE$fustat) UnoC(Surv.rsp, Surv.rsp, train.fit$linear.predictors, time=365.25*1) # [1] 0.9761905 UnoC(Surv.rsp, Surv.rsp, train.fit$linear.predictors, time=365.25*2) # [1] 0.7308979 UnoC(Surv.rsp, Surv.rsp, train.fit$linear.predictors, time=365.25*3) # [1] 0.7308979 UnoC(Surv.rsp, Surv.rsp, train.fit$linear.predictors, time=365.25*4) # [1] 0.7308979 UnoC(Surv.rsp, Surv.rsp, train.fit$linear.predictors, time=365.25*5) # [1] 0.7308979 UnoC(Surv.rsp, Surv.rsp, train.fit$linear.predictors) # [1] 0.7308979 # So the function UnoC() can obtain the exact result as Est.Cval(). # Now try on a new data set. Question: why do we need Surv.rsp? UnoC(Surv.rsp, Surv.rsp.new, lpnew) # [1] 0.7333333 UnoC(Surv.rsp, Surv.rsp.new, lpnew, time=365.25*2) # [1] 0.7333333 library(pec) cindex( list(matrix( -lpnew, nrow = nrow(TE))), formula = Surv(futime, fustat) ~ age, data = TE, eval.times = 365.25*2)$AppC # $matrix # [1] 0.7333333 library(survC1) Est.Cval(cbind(TE, lpnew), tau = 365.25*2, nofit = TRUE)$Dhat # [1] 0.7333333 # tau is mandatory (>0), no need to have training and predict sets Est.Cval(ovarian[1:16, c(1,2, 3)], tau=365.25*1)$Dhat # [1] 0.9761905 Est.Cval(ovarian[1:16, c(1,2, 3)], tau=365.25*2)$Dhat # [1] 0.7308979 Est.Cval(ovarian[1:16, c(1,2, 3)], tau=365.25*3)$Dhat # [1] 0.7308979 Est.Cval(ovarian[1:16, c(1,2, 3)], tau=365.25*4)$Dhat # [1] 0.7308979 Est.Cval(ovarian[1:16, c(1,2, 3)], tau=365.25*5)$Dhat # [1] 0.7308979 svg("~/Downloads/c_stat_scatter.svg", width=8, height=5) par(mfrow=c(1,2)) plot(TR$futime, train.fit$linear.predictors, main="training data", xlab="time", ylab="predictor") mtext("C=.731 at t=2", 3) plot(TE$futime, lpnew, main="testing data", xlab="time", ylab="predictor") mtext("C=.733 at t=2", 3) dev.off()
- C-statistic may or may not be a decreasing function of tau. However, AUC(t) may not be decreasing; see Fig 1 of Blanche et al 2018.
- Assessing the prediction accuracy of a cure model for censored survival data with long-term survivors: Application to breast cancer data
- The use of ROC for defining the validity of the prognostic index in censored data
- Use and Misuse of the Receiver Operating Characteristic Curve in Risk Prediction Cook 2007
- Evaluating Discrimination of Risk Prediction Models: The C Statistic by Pencina et al, JAMA 2015
- Blanche et al(2018) The c-index is not proper for the evaluation of t-year predicted risks
- There is a bug on script line 154.
- With a fixed prediction horizon, the concordance index can be higher for a misspecified model than for a correctly specified model. The time-dependent AUC does not have this problem.
- (page 8) We now show that when a misspecified prediction model satisfies the ranking condition but the true distribution does not, then it is possible that the misspecified model achieves a misleadingly high c-index.
- The traditional C‐statistic used for the survival models is not guaranteed to identify the “best” model for estimating the risk of t-year survival. In contrast, measures of predicted error do not suffer from these limitations. See this paper The relationship between the C‐statistic and the accuracy of program‐specific evaluations by Wey et al 2018
- Unfortunately, a drawback of Harrell’s c-index for the time to event and competing risk settings is that the measure does not provide a value specific to the time horizon of prediction (e.g., a 3-year risk). See this paper The index of prediction accuracy: an intuitive measure useful for evaluating risk prediction models by Kattan and Gerds 2018.
- In Fig 1 Y-axis is concordance (AUC/C) and X-axis is time, the caption said The ability of (some variable) to discriminate patients who will either die or be transplanted within the next t-years from those who will be event-free at time t.
- The [math]\tau[/math] considered here is the maximal end of follow-up time
- AUC (riskRegression::Score()), Uno-C (pec::cindex()), Harrell's C (Hmisc::rcorr.cens() for censored and summary(fit)$concordance for uncensored) are considered.
- The C_IPCW(t) or C_Harrell(t) is obtained by artificially censoring the outcome at time t. So C_IPCW(t) is different from Uno's version.
C-statistic limitations
See the discussion section of The relationship between the C‐statistic and the accuracy of program‐specific evaluations by Wey 2018
- Correctly specified models can have low or high C‐statistics. Thus, the C‐statistic cannot identify a correctly specified model.
- the traditional C‐statistic used for the survival models is not guaranteed to identify the “best” model for estimating the risk of, for example, 1‐year survival
Importantly, there exists no measure of risk discrimination or predicted error that can identify a correctly specified model, because they all depend on unknown characteristics of the data. For example, the C‐statistic depends on the variability in recipient‐level risk, while measures of squared error such as the Brier Score depend on residual variability.
Analysis of Biomarker Data: logs, odds ratios and ROC curves. This paper does not consider the survival time data. It has some summary about C-statistic (interpretation, warnings).
- The C-statistic is relatively insensitive to the added contribution of a new marker when the two models, with and without biomarker, estimate risk on a continuous scale. In fact, many new biomarkers provide only minimal increase in the C-statistic when added to the Framingham model for CHD risk.
- The classical C-statistic assumes that high sensitivity and high specificity are equally desirable. This is not always the case – for example, when screening the general population for a low-prevalence outcome requiring invasive follow-up, high specificity is important, while cancer screening in a high-risk group would emphasize high sensitivity.
- To achieve a noticeable increase in the C-statistic, a biomarker must have a very strong independent association with the event risk (say ORs of 10 or higher per 1 SD increase).
C-statistic applications
- Semiparametric Regression Analysis of Multiple Right- and Interval-Censored Events by Gao et al, JASA 2018
- A c statistic of 0.7–0.8 is considered good, while >0.8 is considered excellent. See this paper. 2018
- The C statistic, also termed concordance statistic or c-index, is analogous to the area under the curve and is a global measure of model discrimination. Discrimination refers to the ability of a risk prediction model to separate patients who develop a health outcome from patients who do not develop a health outcome. Effectively, the C statistic is the probability that a model will result in a higher-risk score for a patient who develops the outcomes of interest compared with a patient who does not develop the outcomes of interest. See the paper JAMA 2018
C-statistic vs LRT comparing nested models
1. Binary data
# https://stats.stackexchange.com/questions/46523/how-to-simulate-artificial-data-for-logistic-regression set.seed(666) x1 = rnorm(1000) # some continuous variables x2 = rnorm(1000) z = 1 + 2*x1 + 3*x2 # linear combination with a bias pr = 1/(1+exp(-z)) # pass through an inv-logit function y = rbinom(1000,1,pr) # bernoulli response variable df = data.frame(y=y,x1=x1,x2=x2) fit <- glm( y~x1+x2,data=df,family="binomial") summary(fit) # Estimate Std. Error z value Pr(>|z|) # (Intercept) 0.9915 0.1185 8.367 <2e-16 *** # x1 2.2731 0.1789 12.709 <2e-16 *** # x2 3.1853 0.2157 14.768 <2e-16 *** # --- # Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 # # (Dispersion parameter for binomial family taken to be 1) # # Null deviance: 1355.16 on 999 degrees of freedom # Residual deviance: 582.93 on 997 degrees of freedom # AIC: 588.93 confint.default(fit) # 2.5 % 97.5 % # (Intercept) 0.7592637 1.223790 # x1 1.9225261 2.623659 # x2 2.7625861 3.608069 # LRT - likelihood ratio test fit2 <- glm( y~x1,data=df,family="binomial") anova.res <- anova(fit2, fit) # Analysis of Deviance Table # # Model 1: y ~ x1 # Model 2: y ~ x1 + x2 # Resid. Df Resid. Dev Df Deviance # 1 998 1186.16 # 2 997 582.93 1 603.23 1-pchisq( abs(anova.res$Deviance[2]), abs(anova.res$Df[2])) # [1] 0 # Method 1: use ROC package to compute AUC library(ROC) set.seed(123) markers <- predict(fit, newdata = data.frame(x1, x2), type = "response") roc1 <- rocdemo.sca( truth=y, data=markers, rule=dxrule.sca ) auc <- AUC(roc1); print(auc) # [1] 0.9459085 markers2 <- predict(fit2, newdata = data.frame(x1), type = "response") roc2 <- rocdemo.sca( truth=y, data=markers2, rule=dxrule.sca ) auc2 <- AUC(roc2); print(auc2) # [1] 0.7259098 auc - auc2 # [1] 0.2199987 # Method 2: use pROC package to compute AUC roc_obj <- pROC::roc(y, markers) pROC::auc(roc_obj) # Area under the curve: 0.9459 # Method 3: Compute AUC by hand # https://www.r-bloggers.com/calculating-auc-the-area-under-a-roc-curve/ auc_probability <- function(labels, scores, N=1e7){ pos <- sample(scores[labels], N, replace=TRUE) neg <- sample(scores[!labels], N, replace=TRUE) # sum( (1 + sign(pos - neg))/2)/N # does the same thing (sum(pos > neg) + sum(pos == neg)/2) / N # give partial credit for ties } auc_probability(as.logical(y), markers) # [1] 0.945964
2. Survival data
library(survival) data(ovarian) head(ovarian) range(ovarian$futime) # [1] 59 1227 plot(survfit(Surv(futime, fustat) ~ 1, data = ovarian)) coxph(Surv(futime, fustat) ~ rx + age, data = ovarian) # coef exp(coef) se(coef) z p # rx -0.8040 0.4475 0.6320 -1.27 0.2034 # age 0.1473 1.1587 0.0461 3.19 0.0014 # # Likelihood ratio test=15.9 on 2 df, p=0.000355 # n= 26, number of events= 12 require(survC1) covs0 <- as.matrix(ovarian[, c("rx")]) covs1 <- as.matrix(ovarian[, c("rx", "age")]) tau=365.25*1 Delta=Inf.Cval.Delta(ovarian[, 1:2], covs0, covs1, tau, itr=200) round(Delta, digits=3) # Est SE Lower95 Upper95 # Model1 0.844 0.119 0.611 1.077 # Model0 0.659 0.148 0.369 0.949 # Delta 0.185 0.197 -0.201 0.572
Time dependent ROC curves
Calibration
Graphical calibration curves and the integrated calibration index (ICI) for survival models
Prognostic markers vs predictive markers (and other biomarkers)
- Prognostic marker (某種疾病的危險因子) are biomarkers used to measure the progress of a disease in the patient sample. Prognostic markers are useful to stratify the patients into groups, guiding towards precise medicine discovery. Prognostic markers inform about likely disease outcome independent of the treatment received. See Statistical and practical considerations for clinical evaluation of predictive biomarkers by Mei-Yin Polley et al 2013.
- Predictive marker/treatment selection markers provide information about likely outcomes with application of specific interventions. See Measuring the performance of markers for guiding treatment decisions by Janes, et al 2011.
- Statistical controversies in clinical research: prognostic gene signatures are not (yet) useful in clinical practice by Michiels 2016.
- Diagnostic biomarker, prognostic biomarker and predictive biomarkers. Disease-related biomarkers and drug-related biomarkers. https://en.wikipedia.org/wiki/Biomarker_(medicine)
- Diagnostic biomarker, prognostic biomarker and predictive biomarkers. https://en.wikipedia.org/wiki/Cancer_biomarker
- Diagnostic (確定是某種疾病): diagnose conditions, as in the case of identifying early stage cancers
- Statistical methods for building better biomarkers of chronic kidney disease by Pencina et al 2019.
- Qualitative and quantitative interactions from Interaction (statistics)
- A plot of Interactions from STAT 509 Design and Analysis of Clinical Trials.
- Understanding Prognostic versus Predictive Biomarkers 2016. No author. x-axis is time? File:Progpreg.png
- Example of a biomarker that is prognostic but not predictive (fig1)
- A biomarker that is both prognostic (negatively) and predictive (fig2)
- quantitative treatment-by-biomarker statistical interaction (fig3). Treatment effect 方向不變 但是程度不同 in two subsets.
- qualitative treatment-by-biomarker interaction (fig4). There is a change in direction of treatment effects in two subsets (biomarker-positive/biomarker-negative).
- Biomarker: Predictive or Prognostic? by Karla V. Ballman, 2015. File:Zlj9991056240002.jpeg
- Both groups derived benefit from the treatment, this is a quantitative interaction.
- A qualitative interaction occurs when one biomarker group obtains benefit from treatment and the other group obtains no benefit (or is harmed) from treatment. The definition also appeared at Testing for qualitative interactions between treatment effects and patient subsets.
- Fig2 A. A purely prognostic biomarker. X-axis is time. Left is biomarker-negative patients and RHS is biomarker-positive patients. Biomarker-positive patients have a better survival than Biomarker-negative patients independent of treatment group.
- Fig2 B. A purely predictive marker.
- Fig2 C. A biomarker is both predictive and prognostic. This is also an example of a quantitative interaction.
- Prognostic vs. Predictive Biomarkers from Clinical OMICs.
- Distinguishing prognostic and predictive biomarkers: an information theoretic approach Sechidis 2018, Bioinformatics.
- This uses a more mathematical way to discuss several issues.
- Prognostic factors versus predictive factors: Examples from a clinical trial of erlotinib Clark 2008.
- Construction and optimization of gene expression signatures for prediction of survival in two-arm clinical trials by Joachim Theilhaber et al 2020
- Evidence for Treatment-by-Biomarker interaction for FDA-approved Oncology Drugs with Required Pharmacogenomic Biomarker Testing 2017
Treatment Effect
- Tian 2014: [math]P(T^1 \geq t_0|z) - P(T^{-1} \geq t_0|z) [/math]
- Bonetti 2000: Hazard ratio
- Janes 2014: [math] \Delta(Y) = \rho_0(Y) - \rho_1(Y) = P(D=1|T=0, Y) - P(D=1|T=1, Y) [/math]
- Subjects with [math]\Delta(Y)\lt 0[/math] are called marker-negative
- Subjects with [math]\Delta(Y)\gt 0[/math] are called marker-positive. The rule is applying treatment onlyto marker-positive patients. And for this portion of patients, the average benefit of treatment is calculated by [math]B_{pos} = E(\Delta(Y) | \Delta(Y) \gt 0)[/math]. See p103 on the paper.
Quantifying treatment differences in confirmatory trials under non-proportional hazards
Quantifying treatment differences in confirmatory trials under non-proportional hazards
The source code in Github.
Computation for gene expression (microarray) data
- survival package (basic package, not designed for gene expression)
- gsa package
- samr package
- pamr package
- (Bioconductor) genefilter, source. genefilter() & coxfilter(). apply() was used.
- logpl() from survcomp package
n <- 500 g <- 10000 y <- rexp(n) status <- ifelse(runif(n) < .7, 1, 0) x <- matrix(rnorm(n*g), nr=g) treat <- rbinom(n, 1, .5) # Method 1 system.time(for(i in 1:g) coxph(Surv(y, status) ~ x[i, ] + treat + treat:x[i, ])) # 28 seconds # Method 2 system.time(apply(x, 1, function(z) coxph(Surv(y, status) ~ z + treat + treat:z))) # 29 seconds # Method 3 (Windows) dyn.load("C:/Program Files (x86)/ArrayTools/Fortran/surv64.dll") tme <- y sorted <- order(tme) stime <- as.double(tme[sorted]) sstat <- as.integer(status[sorted]) x1 <- x[,sorted] imodel <- 1 # imodel=1, fit univariate gene expression. Return p-values vector. nvar <- 1 system.time(outx1 <- .Fortran("coxfitc", as.integer(n), as.integer(g), as.integer(0), stime, sstat, t(x1), as.double(0), as.integer(imodel), double(2*n+2*nvar*nvar+3*nvar), logdiff = double(g))) # 1.69 seconds on R i386 # 0.79 seconds on R x64 # method 4: GSA genenames=paste("g", 1:g, sep="") #create some random gene sets genesets=vector("list", 50) for(i in 1:50){ genesetsi=paste("g", sample(1:g,size=30), sep="") } geneset.names=paste("set",as.character(1:50),sep="") debug(GSA.func) GSA.obj<-GSA(x,y, genenames=genenames, genesets=genesets, censoring.status=status, resp.type="Survival", nperms=1) Browse[3]> str(catalog.unique) int [1:1401] 7943 227 4069 3011 8402 1586 2443 2777 673 9021 ... Browse[3]> system.time(cox.func(x[catalog.unique,], y, censoring.status, s0=0)) # 1.3 seconds Browse[2]> system.time(cox.func(x, y, censoring.status, s0=0)) # 7.259 seconds
Single gene vs mult-gene survival models
A comparative study of survival models for breast cancer prognostication revisited: the benefits of multi-gene models by Grzadkowski et al 2018. To concordance of biomarker performance, the authors use the Concordance Correlation Coefficient (CCC) as introduced by Lin (1989) and further amended in Lin (2000).
Random papers using C-index, AUC or Brier scores
- Predicting the Survival Time for Bladder Cancer Using an Additive Hazards Model in Microarray Data 2016. AUC, Brier scores and C-index were used
More, Web tools
- This pdf file from data.princeton.edu contains estimation, hypothesis testing, time varying covariates and baseline survival estimation.
- Survival analysis: basic terms, the exponential model, censoring, examples in R and JAGS
- Survival analysis is not commonly used to predict future times to an event. Cox model would require specification of the baseline hazard function.
- SurvExpress: An Online Biomarker Validation Tool and Database for Cancer Gene Expression Data Using Survival Analysis 2013. JSP, Javascript, MySQL, Ajax, R & Apache. Public datasets were obtained from GEO, TCGA, ArrayExpress.