## # A tibble: 3 × 2 ## sx_date last_fup_date ##  ## 1 2007-06-22 2017-04-15 ## 2 2004-02-13 2018-07-04 ## 3 2010-10-27 2016-10-31

We see these are both character variables, but we need them to be formatted as dates.

We will use the package to work with dates. In this case, we need to use the ymd() function to change the format, since the dates are currently in the character format where the year comes first, followed by the month, and followed by the day.

date_ex % mutate( sx_date = ymd(sx_date), last_fup_date = ymd(last_fup_date) ) date_ex

## # A tibble: 3 × 2 ## sx_date last_fup_date ##  ## 1 2007-06-22 2017-04-15 ## 2 2004-02-13 2018-07-04 ## 3 2010-10-27 2016-10-31

Now we see that the two dates are formatted as date rather than as character. Access the help page with ?ymd to see all date format options.

Now that the dates are formatted, we need to calculate the difference between start and end dates in some units, usually months or years. Using the package, the operator %--% designates a time interval, which is then converted to the number of elapsed seconds using as.duration() and finally converted to years by dividing by dyears(1) , which gives the number of seconds in a year.

date_ex % mutate( os_yrs = as.duration(sx_date %--% last_fup_date) / dyears(1) ) date_ex

## # A tibble: 3 × 3 ## sx_date last_fup_date os_yrs ##   ## 1 2007-06-22 2017-04-15 9.82 ## 2 2004-02-13 2018-07-04 14.4 ## 3 2010-10-27 2016-10-31 6.01

Now we have our observed time for use in survival analysis.

Note: we need to load the package using a call to library in order to be able to access the special operators (similar to situation with pipes - i.e. we can’t use lubridate::ymd() and then expect to use the special operators).

Creating survival objects and curves

The Kaplan-Meier method is the most common way to estimate survival times and probabilities. It is a non-parametric approach that results in a step function, where there is a step down each time an event occurs.

The Surv() function from the package creates a survival object for use as the response in a model formula. There will be one entry for each subject that is the survival time, which is followed by a + if the subject was censored. Let’s look at the first 10 observations:

Surv(lung$time, lung$status)[1:10]

## [1] 306 455 1010+ 210 883 1022+ 310 361 218 166

We see that subject 1 had an event at time 306 days, subject 2 had an event at time 455 days, subject 3 was censored at time 1010 days, etc.

The survfit() function creates survival curves using the Kaplan-Meier method based on a formula. Let’s generate the overall survival curve for the entire cohort, assign it to object s1 , and look at the structure using str() :

## List of 16 ## $ n : int 228 ## $ time : num [1:186] 5 11 12 13 15 26 30 31 53 54 . ## $ n.risk : num [1:186] 228 227 224 223 221 220 219 218 217 215 . ## $ n.event : num [1:186] 1 3 1 2 1 1 1 1 2 1 . ## $ n.censor : num [1:186] 0 0 0 0 0 0 0 0 0 0 . ## $ surv : num [1:186] 0.996 0.982 0.978 0.969 0.965 . ## $ std.err : num [1:186] 0.0044 0.00885 0.00992 0.01179 0.01263 . ## $ cumhaz : num [1:186] 0.00439 0.0176 0.02207 0.03103 0.03556 . ## $ std.chaz : num [1:186] 0.00439 0.0088 0.00987 0.01173 0.01257 . ## $ type : chr "right" ## $ logse : logi TRUE ## $ conf.int : num 0.95 ## $ conf.type: chr "log" ## $ lower : num [1:186] 0.987 0.966 0.959 0.947 0.941 . ## $ upper : num [1:186] 1 1 0.997 0.992 0.989 . ## $ call : language survfit(formula = Surv(time, status) ~ 1, data = lung) ## - attr(*, "class")= chr "survfit"

Some key components of this survfit object that will be used to create survival curves include:

• time : the timepoints at which the curve has a step, i.e. at least one event occurred
• surv : the estimate of survival at the corresponding time

Kaplan-Meier plots

Note: alternatively, survival plots can be created using base R or the package.

The package works best if you create the survfit object using the included ggsurvfit::survfit2() function, which uses the same syntax to what we saw previously with survival::survfit() . The ggsurvfit::survfit2() tracks the environment from the function call, which allows the plot to have better default values for labeling and p-value reporting.

survfit2(Surv(time, status) ~ 1, data = lung) %>% ggsurvfit() + labs( x = "Days", y = "Overall survival probability" )

The default plot in ggsurvfit() shows the step function only. We can add the confidence interval using add_confidence_interval() :

survfit2(Surv(time, status) ~ 1, data = lung) %>% ggsurvfit() + labs( x = "Days", y = "Overall survival probability" ) + add_confidence_interval()

Typically we will also want to see the numbers at risk in a table below the x-axis. We can add this using add_risktable() :

survfit2(Surv(time, status) ~ 1, data = lung) %>% ggsurvfit() + labs( x = "Days", y = "Overall survival probability" ) + add_confidence_interval() + add_risktable()

Plots can be customized using many standard options.

Estimating \(x\) -year survival

One quantity often of interest in a survival analysis is the probability of surviving beyond a certain number of years, \(x\) .

For example, to estimate the probability of surviving to \(1\) year, use summary with the times argument (Note: the time variable in the lung data is actually in days, so we need to use times = 365.25 )

summary(survfit(Surv(time, status) ~ 1, data = lung), times = 365.25)

## Call: survfit(formula = Surv(time, status) ~ 1, data = lung) ## ## time n.risk n.event survival std.err lower 95% CI upper 95% CI ## 365 65 121 0.409 0.0358 0.345 0.486

We find that the \(1\) -year probability of survival in this study is 41%.

The associated lower and upper bounds of the 95% confidence interval are also displayed.

The \(1\) -year survival probability is the point on the y-axis that corresponds to \(1\) year on the x-axis for the survival curve.

What happens if you use a “naive” estimate? Here “naive” means that the patients who were censored prior to 1-year are considered event-free and included in the denominator.

121 of the 228 patients in the lung data died by \(1\) year so the “naive” estimate is calculated as:

\[\Big(1 - \frac\Big) \times 100 = 47\%\] You get an incorrect estimate of the \(1\) -year probability of survival when you ignore the fact that 42 patients were censored before \(1\) year.

Recall the correct estimate of the \(1\) -year probability of survival, accounting for censoring using the Kaplan-Meier method, was 41%.

Ignoring censoring leads to an overestimate of the overall survival probability. Imagine two studies, each with 228 subjects. There are 165 deaths in each study. Censoring is ignored in one (blue line), censoring is accounted for in the other (yellow line). The censored subjects only contribute information for a portion of the follow-up time, and then fall out of the risk set, thus pulling down the cumulative probability of survival. Ignoring censoring erroneously treats patients who are censored as part of the risk set for the entire follow-up period.

We can produce nice tables of \(x\) -time survival probability estimates using the tbl_survfit() function from the package:

survfit(Surv(time, status) ~ 1, data = lung) %>% tbl_survfit( times = 365.25, label_header = "**1-year survival (95% CI)**" )

Estimating median survival time

Another quantity often of interest in a survival analysis is the average survival time, which we quantify using the median. Survival times are not expected to be normally distributed so the mean is not an appropriate summary.

We can obtain the median survival directly from the survfit object:

survfit(Surv(time, status) ~ 1, data = lung)

## Call: survfit(formula = Surv(time, status) ~ 1, data = lung) ## ## n events median 0.95LCL 0.95UCL ## [1,] 228 165 310 285 363

We see the median survival time is 310 days The lower and upper bounds of the 95% confidence interval are also displayed.

Median survival is the time corresponding to a survival probability of \(0.5\) :

What happens if you use a “naive” estimate? Here “naive” means that you exclude the censored patients from the calculation entirely to estimate median survival time among the patients who have had the event.

Summarize the median survival time among the 165 patients who died:

lung %>% filter(status == 1) %>% summarize(median_surv = median(time))

## median_surv ## 1 226

You get an incorrect estimate of median survival time of 226 days when you ignore the fact that censored patients also contribute follow-up time.

Recall the correct estimate of median survival time is 310 days.

Ignoring censoring will lead to an underestimate of median survival time because the follow-up time that censored patients contribute is excluded (blue line). The true survival curve accounting for censoring in the lung data is shown in yellow for comparison.

We can produce nice tables of median survival time estimates using the tbl_survfit() function from the package:

survfit(Surv(time, status) ~ 1, data = lung) %>% tbl_survfit( probs = 0.5, label_header = "**Median survival (95% CI)**" )

Comparing survival times between groups

We can conduct between-group significance tests using a log-rank test. The log-rank test equally weights observations over the entire follow-up time and is the most common way to compare survival times between groups. There are versions that more heavily weight the early or late follow-up that could be more appropriate depending on the research question (see ?survdiff for different test options).

We get the log-rank p-value using the survdiff function. For example, we can test whether there was a difference in survival time according to sex in the lung data:

survdiff(Surv(time, status) ~ sex, data = lung)

## Call: ## survdiff(formula = Surv(time, status) ~ sex, data = lung) ## ## N Observed Expected (O-E)^2/E (O-E)^2/V ## sex=1 138 112 91.6 4.55 10.3 ## sex=2 90 53 73.4 5.68 10.3 ## ## Chisq= 10.3 on 1 degrees of freedom, p= 0.001

We see that there was a significant difference in overall survival according to sex in the lung data, with a p-value of p = 0.001.

The Cox regression model

We may want to quantify an effect size for a single variable, or include more than one variable into a regression model to account for the effects of multiple variables.

The Cox regression model is a semi-parametric model that can be used to fit univariable and multivariable regression models that have survival outcomes.

\[h(t|X_i) = h_0(t) \exp(\beta_1 X_ + \cdots + \beta_p X_)\]

\(h(t)\) : hazard, or the instantaneous rate at which events occur \(h_0(t)\) : underlying baseline hazard

Some key assumptions of the model:

• non-informative censoring
• proportional hazards

Note: parametric regression models for survival outcomes are also available, but they won’t be addressed in this tutorial

We can fit regression models for survival data using the coxph() function from the package, which takes a Surv() object on the left hand side and has standard syntax for regression formulas in R on the right hand side.

coxph(Surv(time, status) ~ sex, data = lung)

## Call: ## coxph(formula = Surv(time, status) ~ sex, data = lung) ## ## coef exp(coef) se(coef) z p ## sex -0.5310 0.5880 0.1672 -3.176 0.00149 ## ## Likelihood ratio test=10.63 on 1 df, p=0.001111 ## n= 228, number of events= 165

We can obtain tables of results using the tbl_regression() function from the package, with the option to exponentiate set to TRUE to return the hazard ratio rather than the log hazard ratio:

coxph(Surv(time, status) ~ sex, data = lung) %>% tbl_regression(exp = TRUE) 

The quantity of interest from a Cox regression model is a hazard ratio (HR). The HR represents the ratio of hazards between two groups at any particular point in time. The HR is interpreted as the instantaneous rate of occurrence of the event of interest in those who are still at risk for the event. It is not a risk, though it is commonly mis-interpreted as such. If you have a regression parameter \(\beta\) , then HR = \(\exp(\beta)\) .

A HR < 1 indicates reduced hazard of death whereas a HR >1 indicates an increased hazard of death.

So the HR = 0.59 implies that 0.59 times as many females are dying as males, at any given time. Stated differently, females have a significantly lower hazard of death than males in these data.

Part 2: Landmark Analysis and Time Dependent Covariates

In Part 1 we covered using log-rank tests and Cox regression to examine associations between covariates of interest and survival outcomes. But these analyses rely on the covariate being measured at baseline, that is, before follow-up time for the event begins. What happens if you are interested in a covariate that is measured after follow-up time begins?

Example: Overall survival is measured from treatment start, and interest is in the association between complete response to treatment and survival.

Anderson et al (JCO, 1983) described why traditional methods such as log-rank tests or Cox regression are biased in favor of responders in this scenario, and proposed the landmark approach. The null hypothesis in the landmark approach is that survival from landmark does not depend on response status at landmark.

Anderson, J., Cain, K., & Gelber, R. (1983). Analysis of survival by tumor response. Journal of Clinical Oncology : Official Journal of the American Society of Clinical Oncology, 1(11), 710-9.

Some other possible covariates of interest in cancer research that may not be measured at baseline include:

• transplant failure
• graft versus host disease
• second resection
• adjuvant therapy
• compliance
• adverse events

The BMT dataset

Throughout this section we will use the BMT dataset from package as an example dataset. The data consist of 137 bone marrow transplant patients. Variables of interest include:

• T1 time (in days) to death or last follow-up
• delta1 death indicator; 1=Dead, 0=Alive
• TA time (in days) to acute graft-versus-host disease
• deltaA acute graft-versus-host disease indicator; 1-Developed acute graft-versus-host disease, 0-Never developed acute graft-versus-host disease

First, load the data for use in examples throughout:

# install.packages("SemiCompRisks") data(BMT, package = "SemiCompRisks")

Here are the first 6 observations:

head(BMT[, c("T1", "delta1", "TA", "deltaA")])

## T1 delta1 TA deltaA ## 1 2081 0 67 1 ## 2 1602 0 1602 0 ## 3 1496 0 1496 0 ## 4 1462 0 70 1 ## 5 1433 0 1433 0 ## 6 1377 0 1377 0

Landmark approach

1. Select a fixed time after baseline as your landmark time. Note: this should be done based on clinical information, prior to data inspection
2. Subset population for those followed at least until landmark time. Note: always report the number excluded due to the event of interest or censoring prior to the landmark time.
3. Calculate follow-up from landmark time and apply traditional log-rank tests or Cox regression.

In the BMT data, interest is in the association between acute graft versus host disease (aGVHD) and survival. But aGVHD is assessed after the transplant, which is our baseline, or start of follow-up, time.

Step 1 Select landmark time

Typically aGVHD occurs within the first 90 days following transplant, so we use a 90-day landmark.

Step 2 Subset population for those followed at least until landmark time

lm_dat % filter(T1 >= 90) 

We exclude 15 patients who were not followed until the landmark time of 90 days.

Note: All 15 excluded patients died before the 90 day landmark.

Step 3 Calculate follow-up time from landmark and apply traditional methods.

lm_dat % mutate( lm_T1 = T1 - 90 )

survfit2(Surv(lm_T1, delta1) ~ deltaA, data = lm_dat) %>% ggsurvfit() + labs( x = "Days from 90-day landmark", y = "Overall survival probability" ) + add_risktable()

In Cox regression you can use the subset option in coxph to exclude those patients who were not followed through the landmark time, and we can view the results using the tbl_regression() function from the package:

coxph( Surv(T1, delta1) ~ deltaA, subset = T1 >= 90, data = BMT ) %>% tbl_regression(exp = TRUE)

Time-dependent covariate approach

An alternative to a landmark analysis is incorporation of a time-dependent covariate. This may be more appropriate than landmark analysis when:

• the value of a covariate is changing over time
• there is not an obvious landmark time
• use of a landmark would lead to many exclusions

There was no ID variable in the BMT data, which is needed to create the special dataset, so create an ID variable called my_id :

Use the tmerge function with the event and tdc function options to create the special dataset.

• tmerge() creates a long dataset with multiple time intervals for the different covariate values for each patient
• event() creates the new event indicator to go with the newly created time intervals
• tdc() creates the time-dependent covariate indicator to go with the newly created time intervals

td_dat % select(my_id, T1, delta1), data2 = BMT %>% select(my_id, T1, delta1, TA, deltaA), death = event(T1, delta1), agvhd = tdc(TA) )

To see what this does, let’s look at the data for the first 5 individual patients. The variables of interest in the original data looked like:

## my_id T1 delta1 TA deltaA ## 1 1 2081 0 67 1 ## 2 2 1602 0 1602 0 ## 3 3 1496 0 1496 0 ## 4 4 1462 0 70 1 ## 5 5 1433 0 1433 0

The new dataset for these same patients looks like:

## my_id T1 delta1 tstart tstop death agvhd ## 1 1 2081 0 0 67 0 0 ## 2 1 2081 0 67 2081 0 1 ## 3 2 1602 0 0 1602 0 0 ## 4 3 1496 0 0 1496 0 0 ## 5 4 1462 0 0 70 0 0 ## 6 4 1462 0 70 1462 0 1 ## 7 5 1433 0 0 1433 0 0

We see that patients 1 and 4 now have multiple rows of data because they both developed aGVHD at some point in time after baseline.

Now we can analyze this time-dependent covariate as usual using Cox regression with coxph and an alteration to our use of Surv to include arguments to both time and time2 :

coxph( Surv(time = tstart, time2 = tstop, event = death) ~ agvhd, data = td_dat ) %>% tbl_regression(exp = TRUE)

We find that acute graft versus host disease is not significantly associated with death using either landmark analysis or a time-dependent covariate.

Often one will want to use landmark analysis for visualization of a single covariate, and Cox regression with a time-dependent covariate for univariable and multivariable modeling.

Part 3: Competing Risks

Competing risks analyses may be used when subjects have multiple possible events in a time-to-event setting.

• recurrence
• death from disease
• death from other causes
• treatment response

All or some of these (among others) may be possible events in any given study.

The fundamental problem that may lead to the need for specialized statistical methods is unobserved dependence among the various event times. For example, one can imagine that patients who recur are more likely to die, and therefore times to recurrence and times to death would not be independent events.

There are two approaches to analysis in the presence of multiple potential outcomes:

1. Cause-specific hazard of a given event: this represents the rate per unit of time of the event among those not having failed from other events
2. Subdistribution hazard of a given event: this represents the rate per unit of time of the event as well as the influence of competing events

Each of these approaches may only illuminate one important aspect of the data while possibly obscuring others, and the chosen approach should depend on the question of interest.

When the events are independent (almost never true), cause-specific hazards is unbiased. When the events are dependent, a variety of results can be obtained depending on the setting.

Cumulative incidence using 1 minus the Kaplan-Meier estimate is always >= cumulative incidence using competing risks methods, so can only lead to an overestimate of the cumulative incidence, though the amount of overestimation depends on event rates and dependence among events.

To establish that a covariate is indeed acting on the event of interest, cause-specific hazards may be preferred for treatment or prognostic marker effect testing. To establish overall benefit, subdistribution hazards may be preferred for building prognostic nomograms or considering health economic effects to get a better sense of the influence of treatment and other covariates on an absolute scale.

Dignam JJ, Zhang Q, Kocherginsky M. The use and interpretation of competing risks regression models. Clin Cancer Res. 2012;18(8):2301-8.

Kim HT. Cumulative incidence in competing risks data and competing risks regression analysis. Clin Cancer Res. 2007 Jan 15;13(2 Pt 1):559-65.

Satagopan JM, Ben-Porat L, Berwick M, Robson M, Kutler D, Auerbach AD. A note on competing risks in survival data analysis. Br J Cancer. 2004;91(7):1229-35.

Austin, P., & Fine, J. (2017). Practical recommendations for reporting Fine‐Gray model analyses for competing risk data. Statistics in Medicine, 36(27), 4391-4400.

We can obtain the cause-specific hazard of a given event using the methods introduced in the previous section, where the event of interest counts as an event and any competing events are censored at the date of the competing even.

The rest of this section will focus on methods for subdistribtuion hazards.

The primary package we will use for competing risks analysis is the package.

The Melanoma dataset

We will use the Melanoma data from the package to illustrate these concepts. It contains variables:

• time survival time in days, possibly censored.
• status 1 died from melanoma, 2 alive, 3 dead from other causes.
• sex 1 = male, 0 = female.
• age age in years.
• year of operation.
• thickness tumor thickness in mm.
• ulcer 1 = presence, 0 = absence.

# install.packages("MASS") data(Melanoma, package = "MASS")

The status variable in these data are coded in a non-standard way. Let’s recode to avoid confusion:

Melanoma % mutate( status = as.factor(recode(status, `2` = 0, `1` = 1, `3` = 2)) )

• status 0=alive, 1=died from melanoma, 2=dead from other causes.

Here are the first 6 patients:

head(Melanoma)

## time status sex age year thickness ulcer ## 1 10 2 1 76 1972 6.76 1 ## 2 30 2 1 56 1968 0.65 0 ## 3 35 0 1 41 1977 1.34 0 ## 4 99 2 0 71 1968 2.90 0 ## 5 185 1 1 52 1965 12.08 1 ## 6 204 1 1 28 1971 4.84 1

Cumulative incidence for competing risks

A non-parametric estimate of the cumulative incidence of the event of interest. At any point in time, the sum of the cumulative incidence of each event is equal to the total cumulative incidence of any event (not true in the cause-specific setting). Gray’s test is a modified Chi-squared test used to compare 2 or more groups.

Estimate the cumulative incidence in the context of competing risks using the cuminc function from the package. By default this requires the status to be a factor variable with censored patients coded as 0.

cuminc(Surv(time, status) ~ 1, data = Melanoma)

## ## time n.risk estimate std.error 95% CI ## 1,000 171 0.127 0.023 0.086, 0.177 ## 2,000 103 0.230 0.030 0.174, 0.291 ## 3,000 54 0.310 0.037 0.239, 0.383 ## 4,000 13 0.339 0.041 0.260, 0.419 ## 5,000 1 0.339 0.041 0.260, 0.419 ## ## time n.risk estimate std.error 95% CI ## 1,000 171 0.034 0.013 0.015, 0.066 ## 2,000 103 0.050 0.016 0.026, 0.087 ## 3,000 54 0.058 0.017 0.030, 0.099 ## 4,000 13 0.106 0.032 0.053, 0.179 ## 5,000 1 0.106 0.032 0.053, 0.179

We can use the ggcuminc() function from the package to plot the cumulative incidence. By default it plots the first event type only. So the following plot shows the cumulative incidence of death from melanoma:

cuminc(Surv(time, status) ~ 1, data = Melanoma) %>% ggcuminc() + labs( x = "Days" ) + add_confidence_interval() + add_risktable()

If we want to include both event types, specify the outcomes in the ggcuminc(outcome=) argument:

cuminc(Surv(time, status) ~ 1, data = Melanoma) %>% ggcuminc(outcome = c("1", "2")) + ylim(c(0, 1)) + labs( x = "Days" )

Now let’s say we wanted to examine death from melanoma or other causes in the Melanoma data, according to ulcer , the presence or absence of ulceration. We can estimate the cumulative incidence at various times by group and display that in a table using the tbl_cuminc() function from the package, and add Gray’s test to test for a difference between groups over the entire follow-up period using the add_p() function.

cuminc(Surv(time, status) ~ ulcer, data = Melanoma) %>% tbl_cuminc( times = 1826.25, label_header = "**-year cuminc**") %>% add_p()

Then we can see the plot of death due to melanoma, according to ulceration status, as before using ggcuminc() from the package:

cuminc(Surv(time, status) ~ ulcer, data = Melanoma) %>% ggcuminc() + labs( x = "Days" ) + add_confidence_interval() + add_risktable()

Competing risks regression

There are two approaches to competing risks regression:

1. Cause-specific hazards
   • instantaneous rate of occurrence of the given type of event in subjects who are currently event‐free
   • estimated using Cox regression ( coxph function)
2. Subdistribution hazards
   • instantaneous rate of occurrence of the given type of event in subjects who have not yet experienced an event of that type
   • estimated using Fine-Gray regression ( crr function)

Let’s say we’re interested in looking at the effect of age and sex on death from melanoma, with death from other causes as a competing event.

The crr() function from the package will estimate the subdistribution hazards.

crr(Surv(time, status) ~ sex + age, data = Melanoma)

## ## Variable Coef SE HR 95% CI p-value ## sex 0.588 0.272 1.80 1.06, 3.07 0.030 ## age 0.013 0.009 1.01 0.99, 1.03 0.18

And we can generate tables of formatted results using the tbl_regression() function from the package, with the option exp = TRUE to obtain the hazard ratio estimates:

crr(Surv(time, status) ~ sex + age, data = Melanoma) %>% tbl_regression(exp = TRUE)

We see that male sex (recall that 1=male, 0=female in these data) is significantly associated with increased hazard of death due to melanoma, whereas age was not significantly associated with death due to melanoma.

Alternatively, if we wanted to use the cause-specific hazards regression approach, we first need to censor all subjects who didn’t have the event of interest, in this case death from melanoma, and then use coxph as before. So patients who died from other causes are now censored for the cause-specific hazard approach to competing risks. Again we generate a table of formatted results using the tbl_regression() function from the package:

coxph( Surv(time, ifelse(status == 1, 1, 0)) ~ sex + age, data = Melanoma ) %>% tbl_regression(exp = TRUE)

And in this case we obtain similar results using the two approaches to competing risks regression.

Part 4: Advanced Topics

So far we’ve covered:

• The basics of survival analysis including the Kaplan-Meier survival function and Cox regression
• Landmark analysis and time-dependent covariates
• Cumulative incidence and regression for competing risks analyses

In this section I’ll include a variety of bits and pieces of things that may come up and be handy to know:

1. Assessing the proportional hazards assumption
2. Making a smooth survival plot based on \(x\) -year survival according to a continuous covariate
3. Conditional survival

Assessing proportional hazards

One assumption of the Cox proportional hazards regression model is that the hazards are proportional at each point in time throughout follow-up. The cox.zph() function from the package allows us to check this assumption. It results in two main things:

1. A hypothesis test of whether the effect of each covariate differs according to time, and a global test of all covariates at once.
   • This is done by testing for an interaction effect between the covariate and log(time)
   • A significant p-value indicates that the proportional hazards assumption is violated
2. Plots of the Schoenfeld residuals
   • Deviation from a zero-slope line is evidence that the proportional hazards assumption is violated

mv_fit 

## chisq df p ## sex 2.608 1 0.11 ## age 0.209 1 0.65 ## GLOBAL 2.771 2 0.25

plot(cz)

# install.packages("sm") library(sm) sm.options( list( xlab = "Age (years)", ylab = "Median time to death (years)") ) sm.survival( x = lung$age, y = lung$time, status = lung$status, h = sd(lung$age) / nrow(lung)^(-1/4) )