Review Cox Proportional Hazard Model (Biostat II)
Model diagnostics
Time-dependent covariates and time-dependent coefficients
the instantaneous rate of the occurrence of the event at time t
hazard function can be unique for each subject and very overtime
hazard function is connected to the survival function
\[ h(t) = \lim_{\delta \rightarrow 0} \frac{Pr(t<T<t+\delta \mid T>t)}{\delta} \]
\[ S(t) = \exp(-H(t)) = \exp(- \int_0^t h(s) ds) \]
\[ \lambda(t) = \lambda_0(t) \exp(\beta_1x_1 + \ldots + \beta_k x_k) \]
\[ log(\lambda(t)) = log(\lambda_0(t)) + \beta_1x_1 + \ldots + \beta_k x_k \] - log hazard function is expressed in a regression format - \(log(\lambda_0(t))\) can be viewed as intercept - models parameters \(\beta_1, \ldots, \beta_k\) are the log hazard ratios that explain the difference on the hazard of event by varying values of the corresponding covariates - covariates can be continuous, discrete, time-varying, and possible interactions
\(\lambda_0(t)\) is called baseline hazard function
it reflects the underlying hazard for individuals with all covariates set to 0 (“reference group”)
one attractive feature of the Cox model framework is its ability to estimate \(\beta\) without \(\lambda_0(t)\).
\(\lambda_0(t)\) is estimated non-parametrically (i.e., no assumption on its distribution) using Breslow estimator
Cox PH is a semiparametric model.
library(survival)
args(coxph)function (formula, data, weights, subset, na.action, init, control,
ties = c("efron", "breslow", "exact"), singular.ok = TRUE,
robust, model = FALSE, x = FALSE, y = TRUE, tt, method = ties,
id, cluster, istate, statedata, nocenter = c(-1, 0, 1), ...)
NULLNCCTG Lung Cancer Data: Survival in patients with advanced lung cancer from the North Central Cancer Treatment Group. Performance scores rate how well the patient can perform usual daily activities.
library(tidyverse)
library(gtsummary)
#checking data and variable status;
str(lung)'data.frame': 228 obs. of 10 variables:
$ inst : num 3 3 3 5 1 12 7 11 1 7 ...
$ time : num 306 455 1010 210 883 ...
$ status : num 2 2 1 2 2 1 2 2 2 2 ...
$ age : num 74 68 56 57 60 74 68 71 53 61 ...
$ sex : num 1 1 1 1 1 1 2 2 1 1 ...
$ ph.ecog : num 1 0 0 1 0 1 2 2 1 2 ...
$ ph.karno : num 90 90 90 90 100 50 70 60 70 70 ...
$ pat.karno: num 100 90 90 60 90 80 60 80 80 70 ...
$ meal.cal : num 1175 1225 NA 1150 NA ...
$ wt.loss : num NA 15 15 11 0 0 10 1 16 34 ...#simple way to check for missing column in R;
colSums(is.na(lung)) inst time status age sex ph.ecog ph.karno pat.karno
1 0 0 0 0 1 1 3
meal.cal wt.loss
47 14 #display summary table on R Markdown files;
lung %>% select(-inst) %>%
tbl_summary(
by = status,
missing_text = "(Missing)") %>%
add_overall()| Characteristic | Overall, N = 2281 | 1, N = 631 | 2, N = 1651 |
|---|---|---|---|
| time | 256 (167, 397) | 284 (222, 429) | 226 (135, 387) |
| age | 63 (56, 69) | 62 (55, 67) | 64 (57, 70) |
| sex | |||
| 1 | 138 (61%) | 26 (41%) | 112 (68%) |
| 2 | 90 (39%) | 37 (59%) | 53 (32%) |
| ph.ecog | |||
| 0 | 63 (28%) | 26 (41%) | 37 (23%) |
| 1 | 113 (50%) | 31 (49%) | 82 (50%) |
| 2 | 50 (22%) | 6 (9.5%) | 44 (27%) |
| 3 | 1 (0.4%) | 0 (0%) | 1 (0.6%) |
| (Missing) | 1 | 0 | 1 |
| ph.karno | |||
| 50 | 6 (2.6%) | 1 (1.6%) | 5 (3.0%) |
| 60 | 19 (8.4%) | 3 (4.8%) | 16 (9.8%) |
| 70 | 32 (14%) | 3 (4.8%) | 29 (18%) |
| 80 | 67 (30%) | 20 (32%) | 47 (29%) |
| 90 | 74 (33%) | 25 (40%) | 49 (30%) |
| 100 | 29 (13%) | 11 (17%) | 18 (11%) |
| (Missing) | 1 | 0 | 1 |
| pat.karno | |||
| 30 | 2 (0.9%) | 1 (1.6%) | 1 (0.6%) |
| 40 | 2 (0.9%) | 1 (1.6%) | 1 (0.6%) |
| 50 | 4 (1.8%) | 0 (0%) | 4 (2.5%) |
| 60 | 30 (13%) | 3 (4.8%) | 27 (17%) |
| 70 | 41 (18%) | 10 (16%) | 31 (19%) |
| 80 | 51 (23%) | 12 (19%) | 39 (24%) |
| 90 | 60 (27%) | 22 (35%) | 38 (23%) |
| 100 | 35 (16%) | 14 (22%) | 21 (13%) |
| (Missing) | 3 | 0 | 3 |
| meal.cal | 975 (635, 1,150) | 975 (588, 1,075) | 1,025 (685, 1,175) |
| (Missing) | 47 | 16 | 31 |
| wt.loss | 7 (0, 16) | 4 (0, 15) | 8 (0, 16) |
| (Missing) | 14 | 1 | 13 |
| 1 Median (IQR); n (%) | |||
#removing observation with any missing data;
lung2 <- lung %>%
filter(complete.cases(.)) %>%
mutate(sex = as.factor(sex),
ph.ecog = as.factor(ph.ecog))
# Model1: full non-interaction additive model;
fit1 <- coxph(Surv(time, status) ~ age + sex + ph.ecog + ph.karno + pat.karno + meal.cal + wt.loss , data = lung2)
# Model2: excluding predictor pat.karno (nested under Model1);
fit2 <- coxph(Surv(time, status) ~ age + sex + ph.ecog + ph.karno + meal.cal + wt.loss, data = lung2)
# Model3: full non-interaction additive model with inst as cluster for robust variance estimator;
fit3 <- coxph(Surv(time, status) ~ age + sex + ph.ecog + ph.karno + meal.cal + wt.loss + cluster(inst), data = lung2)
# Model4: full non-interaction additive model with spline for continuous variable age;
fit4 <- coxph(Surv(time, status) ~ pspline(age,3) + sex + ph.ecog + ph.karno + meal.cal + wt.loss, data = lung2)
fit1Call:
coxph(formula = Surv(time, status) ~ age + sex + ph.ecog + ph.karno +
pat.karno + meal.cal + wt.loss, data = lung2)
coef exp(coef) se(coef) z p
age 9.979e-03 1.010e+00 1.173e-02 0.851 0.39503
sex2 -5.489e-01 5.776e-01 2.023e-01 -2.713 0.00667
ph.ecog1 6.470e-01 1.910e+00 2.822e-01 2.293 0.02187
ph.ecog2 1.454e+00 4.282e+00 4.621e-01 3.147 0.00165
ph.ecog3 2.762e+00 1.584e+01 1.123e+00 2.460 0.01389
ph.karno 2.207e-02 1.022e+00 1.132e-02 1.949 0.05128
pat.karno -1.150e-02 9.886e-01 8.579e-03 -1.340 0.18013
meal.cal 2.752e-05 1.000e+00 2.609e-04 0.105 0.91601
wt.loss -1.436e-02 9.857e-01 7.835e-03 -1.833 0.06681
Likelihood ratio test=28.63 on 9 df, p=0.000748
n= 167, number of events= 120 # display regression output in R markdown;
fit1 %>% tbl_regression(exp=TRUE)| Characteristic | HR1 | 95% CI1 | p-value |
|---|---|---|---|
| age | 1.01 | 0.99, 1.03 | 0.4 |
| sex | |||
| 1 | — | — | |
| 2 | 0.58 | 0.39, 0.86 | 0.007 |
| ph.ecog | |||
| 0 | — | — | |
| 1 | 1.91 | 1.10, 3.32 | 0.022 |
| 2 | 4.28 | 1.73, 10.6 | 0.002 |
| 3 | 15.8 | 1.75, 143 | 0.014 |
| ph.karno | 1.02 | 1.00, 1.05 | 0.051 |
| pat.karno | 0.99 | 0.97, 1.01 | 0.2 |
| meal.cal | 1.00 | 1.00, 1.00 | >0.9 |
| wt.loss | 0.99 | 0.97, 1.00 | 0.067 |
| 1 HR = Hazard Ratio, CI = Confidence Interval | |||
# checking goodness-of-fit;
# Akaike Information Criterion;
AIC(fit1)[1] 1005.604AIC(fit2) #the smaller the better;[1] 1005.379AIC(fit3) #likelihood don't change under robust variance;[1] 1005.379AIC(fit4) #it seems age non-linear fits slightly better;[1] 1005.127# comparing between nested models;
# using likelihood ratio test;
anova(fit1, fit2) Analysis of Deviance Table
Cox model: response is Surv(time, status)
Model 1: ~ age + sex + ph.ecog + ph.karno + pat.karno + meal.cal + wt.loss
Model 2: ~ age + sex + ph.ecog + ph.karno + meal.cal + wt.loss
loglik Chisq Df Pr(>|Chi|)
1 -493.80
2 -494.69 1.7759 1 0.1827anova(fit1, fit4) Analysis of Deviance Table
Cox model: response is Surv(time, status)
Model 1: ~ age + sex + ph.ecog + ph.karno + pat.karno + meal.cal + wt.loss
Model 2: ~ pspline(age, 3) + sex + ph.ecog + ph.karno + meal.cal + wt.loss
loglik Chisq Df Pr(>|Chi|)
1 -493.80
2 -492.54 2.5303 1.0268 0.1157anova(fit2, fit4) Analysis of Deviance Table
Cox model: response is Surv(time, status)
Model 1: ~ age + sex + ph.ecog + ph.karno + meal.cal + wt.loss
Model 2: ~ pspline(age, 3) + sex + ph.ecog + ph.karno + meal.cal + wt.loss
loglik Chisq Df Pr(>|Chi|)
1 -494.69
2 -492.54 4.3062 2.0268 0.1188#LRT test revealed that model without pat.karno and spline does not change the fit significantly, we can keep the simpler model;
# checking c-statistics;
#you can compute Wald confidence interval of C-index;
# pointest +/- 1.96*se; For example, for model 1, we have
paste(0.654-1.96*0.029, 0.654+1.96*0.029, sep = ",")[1] "0.59716,0.71084"# we can visulize the spline fit for variable age;
# terms=, specify the index order of the variable we wish to plot;
termplot(fit4, se=T, terms=1, ylabs="Log hazard")
we cannot use anova() to conduct likelihood ratio test with clustered data, however, adding a cluster term using cluster() does not change the loglikelihood of the model (hence the observed identical value of AIC between fit2 and fit3)
Note that, \(k\) represents number of parameters in the model and \(l(\cdot)\) is the partial loglikelihood of a specified model.
\[ AIC = -2 \times l(\hat{\beta}) + 2 \times k \]
\[ M_i(t) = N_i(t) - \int_0^t Y_i(u)exp\{\beta'x_i\} \lambda_0 (u)du \] - can be view as the difference between the observed event process and the model estimated event process
The \(\hat{M}_i\) for \(i=1, \ldots, n\) can be plotted against continuous covariates to check for non-linearity of the covariate effects.
This is because any systematic deviation from linearity (on the modelled log hazard ratio) will show as systematic difference in the residuals.
If there is evidence of linearity violation - that is an additive linear combination of the covariates a not suitable, we can consider including interactions, higher order terms, splines and other function forms etc.
Deviance residuals are re-scaled versions of martingale residuals, to make them more symmetric around zero.
Some subjects may have an especially large influence on the parameter estimates. It is helpful for the data analyst to have tools that can identify those subjects.
Score residual and dfbeta measure the influence of the observation \(i\) on the parameter estimates when \(i\) is removed from the model fit. These are calculated separately for each regression coefficient.
We can conduct sensitivity analysis by comparing the estimated log Hazard ratios (the Cox regression coefficients) with and without the identified influential observations.
Schoenfeld residuals are defined from the score function as \[ \hat{r}_i(t) = x_i(t) - \frac{\sum_{l=1}^n Y_l(t)exp(\hat{\beta}'x_l(t)) x_l(t) }{\sum_{l=1}^n Y_l(t)exp(\hat{\beta}'x_l(t)) }, \]
can be seen as the difference between the observed covariate and the expected covariate at each failure time.
the expected covariate is a weighted-average of the covariate, weighted by each individual’s likelihood of event at \(t\).
Schoenfeld residuals are calculated for each covariates and are defined only for non-censored observations
Schoenfeld (1982) showed that these residuals are asymptotically uncorrelated and have expectation valued at 0. Thus a plot of Schoenfeld residuals against time should be centered around zero.
In principle, a Schoenfeld residuals plot that shows a non-random pattern against time is evidence of violation of the PH assumption.
cox.zph() function to check for PH assumption of a fitted Cox model.# calculating residuals from fox model;
fit <- coxph(Surv(time, status) ~ age + trt + celltype + karno + prior, data = veteran)
fit %>% tbl_regression(exp=FALSE)| Characteristic | log(HR)1 | 95% CI1 | p-value |
|---|---|---|---|
| age | -0.01 | -0.03, 0.01 | 0.3 |
| trt | 0.29 | -0.11, 0.70 | 0.2 |
| celltype | |||
| squamous | — | — | |
| smallcell | 0.86 | 0.33, 1.4 | 0.002 |
| adeno | 1.2 | 0.61, 1.8 | <0.001 |
| large | 0.40 | -0.15, 0.96 | 0.2 |
| karno | -0.03 | -0.04, -0.02 | <0.001 |
| prior | 0.01 | -0.03, 0.05 | 0.7 |
| 1 HR = Hazard Ratio, CI = Confidence Interval | |||
martingaleres <- residuals(fit, type=c('martingale'))
dfbeta <- residuals(fit, type=c('dfbeta'))
# marginal residual for continuous variables;
par(mfrow=c(1,2))
# 1. age;
plot(veteran$age, martingaleres, xlab='Age', ylab='Martingale residual')
lines(lowess(veteran$age, martingaleres), lwd=2, col='blue')
abline(h=0, lty='dotted')
# 2. karno;
plot(veteran$karno, martingaleres, xlab='Karnofsky performance score', ylab='Martingale residual')
lines(lowess(veteran$karno, martingaleres), lwd=2, col='blue')
abline(h=0, lty='dotted')
# dfbeta
index.obs <- 1:length(veteran$time)
par(mfrow=c(4,2))
for (i in 1:length(fit$coefficients)){
plot(dfbeta[,i] ~ index.obs, type="h",
xlab="Observation", ylab=paste("Change in coefficient", names(fit$coefficients)[i], sep = " "))
abline(h=0)
}
#return observations are influential
#where the log(HR) on age changes more than 15% of the model estimated value at 0.01;
which(abs(dfbeta[,1])>0.01*0.15) 11 17 18 36 44 85 91 118
11 17 18 36 44 85 91 118 
# Schoenfeld Residuals;
par(mfrow=c(3,2))
cox.zph(fit, transform="km") chisq df p
age 1.826 1 0.17655
trt 0.262 1 0.60896
celltype 15.118 3 0.00172
karno 12.913 1 0.00033
prior 2.131 1 0.14437
GLOBAL 31.716 7 4.6e-05plot(cox.zph(fit, transform="km"))
coxph(Surv(time, status) ~ age + trt*strata(celltype) + karno + prior, data = veteran)Call:
coxph(formula = Surv(time, status) ~ age + trt * strata(celltype) +
karno + prior, data = veteran)
coef exp(coef) se(coef) z p
age -0.011831 0.988239 0.009735 -1.215 0.224
trt -0.173251 0.840927 0.400176 -0.433 0.665
karno -0.037634 0.963065 0.005895 -6.384 1.73e-10
prior 0.010154 1.010206 0.021324 0.476 0.634
trt:strata(celltype)smallcell 0.720645 2.055759 0.522560 1.379 0.168
trt:strata(celltype)adeno 0.144155 1.155063 0.610662 0.236 0.813
trt:strata(celltype)large 0.804962 2.236612 0.581084 1.385 0.166
Likelihood ratio test=47.2 on 7 df, p=5.092e-08
n= 137, number of events= 128 coxph(Surv(time, status) ~ age + trt*strata(celltype) + karno + tt(karno) + prior, tt=function(x,t, ...) x*log(t), data=veteran)Call:
coxph(formula = Surv(time, status) ~ age + trt * strata(celltype) +
karno + tt(karno) + prior, data = veteran, tt = function(x,
t, ...) x * log(t))
coef exp(coef) se(coef) z p
age -0.011801 0.988268 0.009734 -1.212 0.2254
trt -0.311686 0.732212 0.408472 -0.763 0.4454
karno -0.074643 0.928075 0.018392 -4.059 4.94e-05
tt(karno) 0.010638 1.010695 0.004907 2.168 0.0302
prior 0.007112 1.007137 0.021352 0.333 0.7391
trt:strata(celltype)smallcell 0.798075 2.221261 0.525909 1.518 0.1291
trt:strata(celltype)adeno 0.310392 1.363960 0.617398 0.503 0.6151
trt:strata(celltype)large 0.812943 2.254533 0.583375 1.394 0.1635
Likelihood ratio test=52.17 on 8 df, p=1.558e-08
n= 137, number of events= 128 vet2 <- survSplit(Surv(time, status) ~ ., data= veteran, cut=c(90, 180), episode= "tgroup", id="id")
head(vet2) trt celltype karno diagtime age prior id tstart time status tgroup
1 1 squamous 60 7 69 0 1 0 72 1 1
2 1 squamous 70 5 64 10 2 0 90 0 1
3 1 squamous 70 5 64 10 2 90 180 0 2
4 1 squamous 70 5 64 10 2 180 411 1 3
5 1 squamous 60 3 38 0 3 0 90 0 1
6 1 squamous 60 3 38 0 3 90 180 0 2fit2<-coxph(Surv(tstart, time, status) ~ age + trt*strata(celltype) + karno:strata(tgroup) + prior, data=vet2)
fit2Call:
coxph(formula = Surv(tstart, time, status) ~ age + trt * strata(celltype) +
karno:strata(tgroup) + prior, data = vet2)
coef exp(coef) se(coef) z p
age -0.012305 0.987770 0.009765 -1.260 0.2076
trt -0.404353 0.667409 0.414695 -0.975 0.3295
prior 0.006850 1.006874 0.021471 0.319 0.7497
trt:strata(celltype)smallcell 0.897756 2.454089 0.534178 1.681 0.0928
trt:strata(celltype)adeno 0.242645 1.274616 0.621954 0.390 0.6964
trt:strata(celltype)large 0.848816 2.336879 0.589146 1.441 0.1497
karno:strata(tgroup)tgroup=1 -0.048355 0.952796 0.006758 -7.155 8.35e-13
karno:strata(tgroup)tgroup=2 0.011038 1.011100 0.015801 0.699 0.4848
karno:strata(tgroup)tgroup=3 -0.016906 0.983236 0.020528 -0.824 0.4102
Likelihood ratio test=60.89 on 9 df, p=9.03e-10
n= 225, number of events= 128 cox.zph(fit2) chisq df p
age 4.122 1 0.042
trt 0.186 1 0.667
prior 2.260 1 0.133
trt:strata(celltype) 3.774 3 0.287
karno:strata(tgroup) 2.207 3 0.531
GLOBAL 14.382 9 0.109We might be interested to know if the ones who received transplant have better survival outcome than those who don’t.
We can check the Kaplan Meier curve of Survival between the transplant receivers and the non receivers.
The plot shows that receivers have clear better survival than those who received transplant.
library(survminer)
fit<- survfit(Surv(futime, fustat) ~ transplant, data = jasa)
ggsurvplot(fit = fit, data = jasa, conf.int = TRUE, pval = TRUE)
This result may appear to indicate (as it did to Clark et al. in 1971) that transplants are extremely effective in increasing survival of the recipients.
The problem here is that receipt of a transplant is a time dependent covariate;
Oscar winning actors and actresses live longer than their less successful peers.
If we study the survival of winners versus nominees, we are looking at survival conditioning on something happened in the future.
Those winners are guaranteed to survive until their winning, resulting a immortal time window.
Survival of patients on the waiting list for the Stanford heart transplant program.
J Crowley and M Hu (1977), Covariance analysis of heart transplant survival data. Journal of the American Statistical Association, 72, 27-36.
jasa: original data
birth.dt: birth date
accept.dt: acceptance into program
tx.date: transplant date
fu.date: end of followup
fustat: dead or alive
surgery: prior bypass surgery
age: age (in years)
futime: followup time
wait.time: time before transplant
transplant: transplant indicator
mismatch: mismatch score
hla.a2: particular type of mismatch
mscore: another mismatch score
reject: rejection occurred
library(DT)
datatable(jasa[,c("accept.dt", "tx.date","fu.date","fustat","transplant","wait.time","futime")],
options = list(dom = 't'))tmerge() function in the survival package to create multiple time-based intervals per subject# step 1: adding subject identifier to the data;
jasa <- jasa %>% mutate(id = 1:nrow(jasa),
age = round(age,2))
# step 2: check if there is any event time that is 0;
# we need to update this to 0.5, representing event happened mid-day;
# the analysis cannot handle an event time at the start of the study;
which(jasa$futime==0)[1] 15jasa[which(jasa$futime==0), "futime"] <- 0.5
# Step 3: create multiple entries for each subject by transplant status and time;
jasa_new <- tmerge(data1 = jasa[,c("id", "age","accept.dt","surgery")],
#data1 is the baseline data to be retained in the analysis (time-independent);
data2 = jasa[,c("id","transplant","wait.time","futime","fustat")],
#data2 is source for new data including events and time-dependent var;
id = id, #subject identifier used to merge the data together;
death = event(futime, fustat),
transplant=tdc(wait.time))
datatable(jasa_new,
rownames = FALSE,
options = list(dom = 't'))#creating a new variable called year for cohort entry year;
jasa_new$year = ifelse(format(as.Date(jasa_new$accept.dt, format="%d/%m/%Y"),"%Y")<1970,0,1)
# correct time-dependent analysis;
fit_correct <- coxph(Surv(tstart, tstop, death) ~ age + transplant + surgery, data = jasa_new)
fit_correctCall:
coxph(formula = Surv(tstart, tstop, death) ~ age + transplant +
surgery, data = jasa_new)
coef exp(coef) se(coef) z p
age 0.03139 1.03188 0.01392 2.254 0.0242
transplant -0.07898 0.92406 0.30608 -0.258 0.7964
surgery -0.77035 0.46285 0.35959 -2.142 0.0322
Likelihood ratio test=10.78 on 3 df, p=0.01295
n= 169, number of events= 75 # incorrect analysis using the original data;
fit_incorrect <- coxph(Surv(futime, fustat) ~ age + transplant + surgery, data = jasa)
fit_incorrectCall:
coxph(formula = Surv(futime, fustat) ~ age + transplant + surgery,
data = jasa)
coef exp(coef) se(coef) z p
age 0.05890 1.06067 0.01505 3.914 9.10e-05
transplant -1.71722 0.17956 0.27854 -6.165 7.04e-10
surgery -0.41899 0.65771 0.37118 -1.129 0.259
Likelihood ratio test=45.85 on 3 df, p=6.089e-10
n= 103, number of events= 75 This data is a continuation of the PBC data set, and contains the follow-up laboratory data for each study patient. An analysis based on the data can be found in Murtagh, et. al. (1994)
Murtaugh PA. Dickson ER. Van Dam GM. Malinchoc M. Grambsch PM. Langworthy AL. Gips CH. “Primary biliary cirrhosis: prediction of short-term survival based on repeated patient visits.” Hepatology. 20(1.1):126-34, 1994.
id: case number
age: in years
sex: m/f
trt: 1/2/NA for D-penicillmain, placebo, not randomised
time: number of days between registration and the earlier of death,
transplantion, or study analysis in July, 1986
status: status at endpoint, 0/1/2 for censored, transplant, dead
day: number of days between enrollment and this visit date
all measurements below refer to this date
datatable(pbcseq,
rownames = FALSE,
options = list(dom = 't')) %>%
formatRound(columns=c('age'), digits=1)There are three event status, we can treat the data as a multi-state problem (if we are interested in estimating the hazard rate between each status change).
In the original paper, the primary analysis focus on death, patients were censored at the time of transplantation!
The analysis below is to demonstrate how you would fit Cox model with follow-up repeatedly measured lab values where we censore patient if they received transplant;
pbcseq2 <- pbcseq %>% group_by(id) %>%
mutate(tstart = day, tstop = lead(day))
# impute the last missing lead value with the end of follow up time;
pbcseq2$tstop[is.na(pbcseq2$tstop)] <- pbcseq2$futime[is.na(pbcseq2$tstop)]
fit_example <- coxph(Surv(tstart, tstop, status==2) ~ age + sex + trt + albumin + bili, data = pbcseq2)
fit_example %>% tbl_regression(exp=TRUE)| Characteristic | HR1 | 95% CI1 | p-value |
|---|---|---|---|
| age | 1.03 | 1.02, 1.04 | <0.001 |
| sex | |||
| m | — | — | |
| f | 0.56 | 0.46, 0.68 | <0.001 |
| trt | 0.92 | 0.80, 1.07 | 0.3 |
| albumin | 0.45 | 0.39, 0.53 | <0.001 |
| bili | 1.09 | 1.08, 1.10 | <0.001 |
| 1 HR = Hazard Ratio, CI = Confidence Interval | |||