Survival Analysis in R: The survival Package

The hazard is not the same thing as the probability of an event.

Hazard Ratio

The hazard ratio compares hazards between two groups.

For an experimental treatment compared with control:

under the proportional-hazards interpretation.

For example:

is commonly interpreted as the experimental group having an estimated hazard 70% that of the reference group, corresponding to an approximately 30% lower instantaneous event hazard under the proportional-hazards model.

Hazard Ratio Is Not Risk Ratio

This distinction is essential.

A hazard ratio is not generally equivalent to:

The hazard ratio incorporates the timing of events.

It should therefore not be casually described as a relative reduction in "risk" without appropriate qualification.

The Cox Proportional Hazards Model

The Cox proportional hazards model is one of the most widely used regression models for survival data.

The model is commonly written:

where:

• \(h_0(t)\) is the baseline hazard
• \(X_1,\ldots,X_p\) are covariates
• \(\beta_1,\ldots,\beta_p\) are regression coefficients

The baseline hazard is left unspecified, which is why the Cox model is often called a semi-parametric model.

Fitting a Cox Model in R

The coxph() function fits a Cox proportional hazards model.

cox_fit <- coxph(
  Surv(time, status) ~ treatment,
  data = trial
)

summary(cox_fit)

This estimates the coefficient for treatment on the log-hazard scale.

Interpreting the Cox Coefficient

Suppose the fitted coefficient is:

The corresponding hazard ratio is:

Thus:

exp(coef(cox_fit))

returns the hazard ratio.

Confidence Intervals for the Hazard Ratio

A common approach is:

exp(confint(cox_fit))

A complete summary can also be obtained from:

summary(cox_fit)

The resulting table typically contains the coefficient, exponentiated coefficient, standard error, test statistic, and p-value.

Example Hazard-Ratio Interpretation

Suppose a clinical trial reports:

A suitable interpretation would be:

The confidence interval provides information about the precision of the estimated treatment effect.

The Reference Group Matters

If treatment is coded as a factor, R uses one level as the reference category.

For example:

trial$treatment <-
  factor(
    trial$treatment,
    levels = c("Control", "Experimental")
  )

The resulting coefficient for Experimental is interpreted relative to Control.

Multiple Covariates

A Cox model can include several prognostic or treatment-related variables.

cox_fit <- coxph(
  Surv(time, status) ~
    treatment +
    age +
    stage +
    biomarker,
  data = trial
)

summary(cox_fit)

This produces adjusted hazard ratios for the included covariates.

Continuous Covariates

Suppose age is entered as a continuous variable.

coxph(
  Surv(time, status) ~ age,
  data = trial
)

The hazard ratio is then associated with a one-unit increase in age.

If:

the model estimates a 3% higher hazard per one-unit increase in age, assuming the model is otherwise correctly specified.

Sometimes a more interpretable contrast is a 10-year increase.

If the coefficient is \(\beta\), the hazard ratio for a 10-unit increase is:

Categorical Covariates

Categorical variables should generally be explicitly coded as factors.

trial$stage <-
  factor(
    trial$stage,
    levels = c("I", "II", "III", "IV")
  )

R then estimates each non-reference category relative to the reference.

Stratification

Sometimes a covariate does not satisfy the proportional-hazards assumption but is important for controlling the baseline hazard.

The Cox model can use stratification:

cox_fit <- coxph(
  Surv(time, status) ~
    treatment +
    age +
    strata(region),
  data = trial
)

A stratified variable receives a separate baseline hazard for each stratum, while its own coefficient is not estimated.

The Proportional Hazards Assumption

The Cox model assumes that hazard ratios remain proportional over time.

For two patients or groups:

over the relevant follow-up period.

This does not mean that the hazards themselves are constant.

It means their ratio is assumed to be constant.

Testing Proportional Hazards in R

The cox.zph() function is commonly used to assess the proportional-hazards assumption.

ph_check <- cox.zph(cox_fit)

ph_check

A graphical assessment is also useful:

plot(ph_check)

The analysis should not rely exclusively on a single p-value.

Clinical interpretation, graphical diagnostics, and knowledge of the endpoint should all contribute to the assessment.

What If Proportional Hazards Does Not Hold?

If the proportional-hazards assumption is substantially violated, several approaches may be considered.

• Include time-dependent effects
• Stratify on an appropriate variable
• Report time-specific effects
• Use RMST
• Use alternative survival models
• Describe the non-proportional pattern directly

The appropriate strategy depends on the estimand, endpoint, trial design, and statistical analysis plan.

Kaplan–Meier and Cox Models Answer Different Questions

These methods complement one another.

Kaplan–Meier vs. Cox Regression

A Kaplan–Meier curve can show:

• How survival changes over time
• Where events occur
• Where censoring occurs
• Whether curves appear to separate

A Cox model can additionally provide:

• Hazard ratios
• Confidence intervals
• Covariate adjustment
• Multivariable modeling

In a clinical study report, the two are frequently presented together.

A Typical Clinical-Trial Workflow

Clinical Trial Data Structure

A simplified survival-analysis dataset might look like:

The exact variables depend on the clinical-trial data standard and analysis dataset.

Censoring Indicators in Clinical Trials

Clinical-trial datasets often use a censoring variable rather than directly coding the event as 1.

For example:

• CNSR = 0 may indicate an event
• CNSR = 1 may indicate censoring

If that convention is used, the event indicator for Surv() may need to be derived as:

event <- 1 - CNSR

Surv(
  time = AVAL,
  event = event
)

Time Units Matter

Survival time may be expressed in:

• Days
• Weeks
• Months
• Years

The model itself does not require a particular unit.

However, the chosen unit must be consistently defined and clearly documented.

time_months <-
  (event_date - baseline_date) / 30.4375

The exact derivation should follow the analysis specification rather than an arbitrary conversion chosen during plotting.

Overall Survival

For overall survival, the event is commonly death from any cause.

The survival object might therefore be:

os <- Surv(
  time = OS_TIME,
  event = OS_EVENT
)

A Kaplan–Meier estimate can then be obtained with:

os_km <- survfit(
  os ~ TRT01P,
  data = adtte
)

The exact event and censoring rules should come from the prespecified endpoint definition.

Progression-Free Survival

Progression-free survival is more complicated because the event may consist of either:

• Disease progression
• Death

The endpoint derivation therefore typically requires integration of tumor assessment data, clinical events, treatment discontinuation rules, and censoring conventions.

Duration of Response

Duration of response is generally defined only among patients who achieve a prespecified response.

The event may be progression or death, depending on the endpoint definition.

A typical structure is:

The exact definition, however, must follow the protocol and SAP.

Competing Risks

Standard Kaplan–Meier methods can be inappropriate when another event prevents the event of interest from occurring and that competing event is handled as ordinary censoring.

For example, if the endpoint is time to relapse and death without relapse prevents relapse from occurring, death can be a competing event.

Competing-risk analyses may require methods such as cumulative incidence functions rather than treating competing events as non-informative censoring.

Left Truncation and Delayed Entry

Some survival studies include patients who enter the risk set after time zero.

This is sometimes called left truncation or delayed entry.

The survival object can represent entry and exit times:

Surv(
  time = entry,
  time2 = exit,
  event = status
)

For example:

Surv(
  entry_time,
  exit_time,
  event
)

This tells the analysis that the patient was only under observation and at risk from the entry time onward.

Time-Dependent Covariates

Some covariates change over time.

Examples include:

• Biomarker measurements
• Treatment exposure
• Time-varying disease status
• Transplant status

The Cox framework can accommodate time-dependent covariates, but the dataset must be structured appropriately.

For example:

coxph(
  Surv(start, stop, event) ~
    treatment +
    biomarker,
  data = time_varying_data
)

This is substantially more complex than a standard one-row-per-patient analysis and should be specified carefully.

One Row Per Patient Is Not Always Enough

For a basic right-censored endpoint, a dataset may contain one row per patient.

For time-dependent analyses, a patient may contribute multiple observation intervals.

The interval structure allows the covariate to change over time.

Stratified Kaplan–Meier Curves

Sometimes the analyst wants curves within another grouping variable.

survfit(
  Surv(time, status) ~ treatment + stage,
  data = trial
)

This can produce separate curves for combinations of treatment and stage.

For reporting, however, the display should be designed deliberately so that the number of curves does not become excessive.

Confidence Bands in a Kaplan–Meier Plot

Confidence bands can communicate uncertainty around the survival estimate.

A basic base-R display can include confidence intervals:

plot(
  km_fit,
  conf.int = TRUE,
  xlab = "Time",
  ylab = "Survival Probability",
  lwd = 2
)

For publication or CSR graphics, analysts often use a more customizable plotting workflow.

Extracting Kaplan–Meier Results

The fitted object contains the estimated survival information.

summary(km_fit)

Useful components include:

km_fit$time
km_fit$surv
km_fit$n.risk
km_fit$n.event
km_fit$n.censor

These components can be useful when creating custom tables or figures.

Extracting the Cox Model Results

summary(cox_fit)

Useful quantities include:

coef(cox_fit)
exp(coef(cox_fit))
confint(cox_fit)
exp(confint(cox_fit))

This allows a programmer to construct a formatted hazard-ratio table.

A Simple Hazard-Ratio Table

A clinical analysis table might contain:

Forest Plots

Hazard ratios are often presented graphically in a forest plot.

A forest plot makes it easy to see:

• The direction of effects
• The magnitude of hazard ratios
• Confidence intervals
• The reference value of HR = 1

The survival package supplies the model; the actual forest-plot visualization can be constructed using additional R packages or custom plotting code.

Adjusted vs Unadjusted Analyses

A treatment-only Cox model:

coxph(
  Surv(time, status) ~ treatment,
  data = trial
)

produces an unadjusted treatment effect.

A model containing baseline covariates:

coxph(
  Surv(time, status) ~
    treatment +
    age +
    stage +
    biomarker,
  data = trial
)

produces adjusted estimates.

The appropriate primary analysis depends on the prespecified statistical methodology.

Do Not Select Covariates Solely by p-Value

A common mistake is to fit many variables and retain only those with small p-values.

Clinical-trial models should instead follow the prespecified analysis strategy, including decisions about:

• Baseline covariates
• Stratification factors
• Treatment effects
• Interactions
• Missing data
• Model assumptions

Exploratory model selection and confirmatory analysis should not be confused.

Interactions in Cox Models

An interaction can be modeled with:

coxph(
  Surv(time, status) ~
    treatment * biomarker,
  data = trial
)

This evaluates whether the treatment effect varies according to the biomarker.

Interaction analyses require careful interpretation and are especially sensitive to sample size and multiplicity.

Clinical Interpretation of a Hazard Ratio

Suppose:

The estimated hazard is lower in the treatment group.

However, the statement:

"Treatment increases survival by 35%"

is generally too simplistic.

A more accurate description is that the estimated hazard ratio is 0.65, corresponding to an approximately 35% lower hazard under the proportional hazards interpretation.

Survival Probability Is Often Easier to Explain

Suppose the Kaplan–Meier estimates are:

These values can provide an intuitive description of the observed survival experience.

They complement rather than replace the hazard ratio.

Crossing Kaplan–Meier Curves

A particularly important situation occurs when survival curves cross.

For example:

• Treatment A may initially have worse survival.
• The curves may cross after 12 months.
• Treatment A may subsequently have better survival.

This pattern can indicate that a single proportional hazard ratio does not adequately summarize the treatment effect.

Non-Proportional Hazards

Non-proportional hazards can arise from:

• Delayed treatment effects
• Early treatment toxicity
• Immunotherapy-related delayed effects
• Treatment switching
• Changing disease risk over time

In these settings, the hazard ratio may be difficult to interpret as a single summary of the entire follow-up period.

Survival Analysis in Immuno-Oncology

Immunotherapy trials can provide examples where the proportional-hazards assumption deserves particular attention.

Some trials may show:

• Delayed separation of curves
• Early similar event rates
• Later separation
• Long-term survival plateaus

The analysis should therefore consider whether the prespecified estimand and statistical model appropriately capture the treatment effect.

Censoring Mechanisms

Standard survival methods generally rely on assumptions about censoring.

In simplified terms, censoring should not systematically provide information about the future event time after accounting for relevant observed information.

Examples of potentially informative censoring include:

• Patients withdrawing because they are deteriorating
• Patients discontinuing because they are doing particularly well
• Loss to follow-up associated with prognosis

Clinical-trial censoring rules and sensitivity analyses should therefore be prespecified.

Administrative Censoring

Administrative censoring occurs because the study observation period ends before a patient experiences the event.

For example, a patient might remain alive when the database cutoff is reached.

That patient contributes all of their available follow-up without being classified as having experienced the event.

Informative vs Non-Informative Censoring

The distinction is important.

The statistical method does not automatically solve an informative-censoring problem.

Clinical Study Report Considerations

Survival analyses in a CSR typically require clear documentation of:

• Endpoint definition
• Event definition
• Censoring definition
• Analysis population
• Analysis cutoff date
• Time origin
• Time units
• Kaplan–Meier methodology
• Confidence-interval methodology
• Comparison method
• Cox model specification
• Covariate handling
• Missing-data rules

Time Origin Is Critical

The starting point of the survival clock must be clearly defined.

Possible time origins include:

• Randomization
• First dose
• Diagnosis
• Surgery
• Response date
• Transplantation

The appropriate time origin depends on the endpoint.

Analysis Cutoff Dates

Clinical-trial survival analyses are often based on a prespecified data cutoff.

Patients who are still event-free at the cutoff are generally censored according to the endpoint-specific rules.

This means that survival analysis is inherently linked to the timing of the database snapshot.

Example: Building a Kaplan–Meier Analysis

library(survival)

km_fit <- survfit(
  Surv(
    OS_TIME,
    OS_EVENT
  ) ~ TRT01P,
  data = adtte
)

summary(km_fit)

plot(
  km_fit,
  xlab = "Months from Randomization",
  ylab = "Overall Survival Probability",
  lwd = 2,
  conf.int = TRUE
)

This basic workflow can be expanded into a fully specified clinical-trial analysis.

Example: Cox Analysis

cox_fit <- coxph(
  Surv(
    OS_TIME,
    OS_EVENT
  ) ~ TRT01P + AGE + SEX + STAGE,
  data = adtte
)

summary(cox_fit)

The treatment coefficient provides the adjusted treatment hazard ratio, assuming the model is appropriately specified.

Checking the Cox Model

ph_test <- cox.zph(cox_fit)

print(ph_test)

plot(ph_test)

The output should be reviewed together with the Kaplan–Meier curves and the scientific context.

Creating a Survival Analysis Dataset

A robust programming workflow separates endpoint derivation from statistical modeling.

Validation of Event Dates

For each patient, the programmer should be able to answer:

• What was the time origin?
• What was the event date?
• If there was no event, what was the censoring date?
• Why was the patient censored?
• Was the event date after the time origin?
• Does the analysis time agree with the source dates?

A survival curve can be mathematically correct while still being clinically wrong if the endpoint derivation is incorrect.

Common Data-Programming Mistakes

1. Reversing the event indicator. This can turn events into censored observations and vice versa.
2. Using the wrong time origin. All subsequent survival times become misaligned.
3. Ignoring endpoint-specific censoring rules. Different endpoints can require different handling of events and follow-up.
4. Using the database cutoff incorrectly. This can produce incorrect censoring dates.
5. Mixing time units. Days, months, and years should not be mixed without explicit conversion.
6. Ignoring competing events. Some endpoints require competing-risk methods rather than ordinary Kaplan–Meier estimation.
7. Reporting an unstable tail. Very few patients at risk can make late estimates highly uncertain.
8. Ignoring proportional-hazards diagnostics. A Cox model should not automatically be interpreted as a constant treatment effect.

Kaplan–Meier Curve vs. Cumulative Incidence

These displays answer different questions.

The distinction becomes important in competing-risk settings.

Survival Probability vs. Cumulative Event Probability

For a simple single-event setting:

Thus, if:

then the estimated probability of experiencing the event by 12 months is approximately:

or 20%, under the corresponding single-event framework.

Why the Kaplan–Meier Curve Is a Step Function

The Kaplan–Meier estimator changes only when an event occurs.

Censoring changes the number at risk but does not itself reduce the estimated survival probability.

Therefore, the curve:

• Steps downward at events
• Remains flat between events
• Does not drop at censoring times

An Example of the Risk Set

Suppose there are 100 patients at risk immediately before an event time, and two events occur.

The survival estimate is multiplied by:

If the survival estimate immediately before the event was 0.90, then after the event:

The curve therefore falls to approximately 88.2%.

Multiple Events at the Same Time

If multiple events occur at the same time, the Kaplan–Meier estimator accounts for all of them at that event time.

This is one reason why understanding the risk set is important when validating survival estimates.

Survival Analysis and Ties

In clinical-trial data, multiple patients may experience events on the same day.

The Cox model therefore needs a method for handling tied event times.

The coxph() function supports established approaches for ties, and the analysis should use the method specified by the statistical methodology.

Reproducible R Code

A clean survival-analysis program should separate:

• Data preparation
• Endpoint derivation
• Statistical modeling
• Figure generation
• Output formatting
• Quality control

For example:

# 1. Prepare analysis data

adtte <- analysis_data


# 2. Create survival object

os_surv <- with(
  adtte,
  Surv(OS_TIME, OS_EVENT)
)


# 3. Kaplan-Meier model

km_fit <- survfit(
  os_surv ~ TRT01P,
  data = adtte
)


# 4. Cox model

cox_fit <- coxph(
  os_surv ~ TRT01P + AGE + SEX,
  data = adtte
)


# 5. Diagnostics

ph_check <- cox.zph(cox_fit)


# 6. Review outputs

summary(km_fit)
summary(cox_fit)
print(ph_check)

Using Formula Syntax Correctly

R survival models use formula notation.

For example:

Surv(time, status) ~ treatment

can be read as:

The left side contains the survival outcome. The right side contains predictors or grouping variables.

Multiple Groups

Kaplan–Meier estimation can accommodate more than two groups.

km_fit <- survfit(
  Surv(time, status) ~ dose_group,
  data = trial
)

The resulting object contains one survival curve per group.

If there are many groups, the figure can quickly become difficult to read.

Stratified Randomized Trials

Some clinical trials use stratified randomization.

The Cox model may incorporate those stratification factors according to the prespecified analysis.

cox_fit <- coxph(
  Surv(time, status) ~
    treatment +
    strata(randomization_factor),
  data = trial
)

The exact model should follow the SAP rather than being chosen after inspecting the results.

Missing Baseline Covariates

Missing covariates can cause patients to be excluded from a complete-case Cox model unless missingness is explicitly handled.

For example:

coxph(
  Surv(time, status) ~
    treatment + biomarker,
  data = trial
)

may use only observations with complete values for the variables included in the model.

Landmark Analyses

Sometimes investigators want to examine survival after a defined landmark time.

For example, patients might be classified according to response status at Week 12 and then followed thereafter.

Such analyses require careful handling of selection and guarantee-time bias.

They should be prespecified or clearly labeled as exploratory.

Immortal Time Bias

A related issue is immortal time bias.

This can occur when patients must survive event-free for a period before they can be classified into an exposure group, but that guaranteed survival period is not handled correctly in the analysis.

Time-dependent methods or landmark approaches may sometimes be more appropriate.

Clinical Interpretation of Kaplan–Meier Curves

When reviewing a Kaplan–Meier figure, ask:

• What is the time origin?
• What is the event?
• How many events occurred?
• How much censoring occurred?
• How many patients remain at risk?
• Do the curves separate early or late?
• Do the curves cross?
• Is the median estimable?
• Is the tail based on many or few patients?

Clinical Interpretation of Cox Results

When reviewing a hazard ratio, ask:

• Which group is the reference?
• What is the hazard ratio?
• What is the confidence interval?
• Does the confidence interval include 1?
• Was the model adjusted?
• Were the proportional-hazards assumptions evaluated?
• Is the estimated effect clinically meaningful?
• Does the HR adequately summarize the observed curve pattern?

Statistical Significance vs. Clinical Significance

A small p-value does not necessarily imply a clinically important treatment effect.

For example:

could be statistically significant in a very large trial while representing a relatively modest treatment effect.

Conversely, a clinically important hazard ratio may have a wide confidence interval in a small study.

Effect size, precision, and clinical context should therefore be considered together.

A Practical Reporting Framework

Survival Analysis Quality Control

Survival-analysis programming should undergo independent validation or other appropriate quality-control procedures.

At minimum, verify:

• Correct analysis population
• Correct treatment assignment
• Correct time origin
• Correct event date
• Correct censoring date
• Correct event indicator
• Correct analysis time
• Correct time units
• Correct Kaplan–Meier estimates
• Correct number-at-risk counts
• Correct median survival estimates
• Correct hazard ratios
• Correct confidence intervals
• Correct p-values
• Correct reference category
• Correct model covariates
• Appropriate assessment of proportional hazards

Patient-Level Validation

A useful QC strategy is to trace individual patients through the survival derivation.

For selected patients, verify:

This type of patient-level review can detect derivation errors that may not be obvious from the final Kaplan–Meier curve.

Cross-Checking Kaplan–Meier Output

For important outputs, independently verify selected survival estimates.

For example, confirm that:

• The survival probability changes only at event times.
• Censoring does not directly cause a drop.
• The number at risk decreases appropriately.
• The reported median corresponds to the curve crossing 0.50.

Common Mistakes With Surv()

1. Using the wrong event coding. Always confirm what 0 and 1 mean.
2. Passing dates directly without defining analysis time. The survival model should receive a meaningful time scale.
3. Using event date as time without subtracting the time origin. The time variable must represent elapsed follow-up.
4. Ignoring delayed entry. Some datasets require start-stop survival objects.
5. Assuming all endpoints use the same censoring rule. OS, PFS, DoR, and other endpoints can have different definitions.

Common Mistakes With Kaplan–Meier Curves

1. Ignoring the number at risk. Late curves can be based on very few patients.
2. Overinterpreting crossing curves. A single hazard ratio may not adequately summarize non-proportional effects.
3. Reporting a median when it was not reached. The median is not automatically the final follow-up time.
4. Confusing censoring with an event. Censored patients remain part of the risk-set calculation up to their censoring time.
5. Using the wrong time origin. This changes the endpoint itself.

Common Mistakes With Cox Models

1. Interpreting HR as a risk ratio. The hazard ratio describes relative hazard, not simply cumulative event probability.
2. Ignoring the reference group. The direction of the HR depends on which group is the reference.
3. Assuming proportional hazards automatically holds. It should be assessed.
4. Using post hoc variable selection for a confirmatory model. The model should follow the prespecified analysis strategy.
5. Reporting only the p-value. The effect estimate and confidence interval are essential.

Survival Analysis in a Statistical Analysis Plan

An SAP should clearly describe the planned survival analysis.

Important components may include:

• Endpoint definition
• Time origin
• Event definition
• Censoring rules
• Analysis population
• Kaplan–Meier methodology
• Confidence intervals
• Median survival estimation
• Treatment comparison
• Cox model specification
• Covariates
• Stratification
• Proportional-hazards assessment
• Sensitivity analyses

Example SAP Language

The actual wording should reflect the trial-specific estimand and SAP.

Survival Curves Are Not Just "Lines"

A Kaplan–Meier figure contains substantial statistical information.

The curve reflects:

• Event timing
• Risk-set sizes
• Censoring
• Accumulated survival probability

This is why simply connecting observed event proportions with ordinary lines would not produce a valid Kaplan–Meier estimate.

Why Censoring Does Not Cause a Drop

Suppose a patient is censored at 10 months.

That patient was event-free through month 10.

Therefore, the patient's information contributes to the risk set before 10 months.

After censoring, the patient is removed from the risk set.

But because no event occurred, the survival probability does not drop at that time.

An Intuitive Example

Suppose five patients are followed:

At month 3, five patients are initially at risk and one event occurs.

The survival estimate becomes:

At month 5, Patient B is censored. There is no survival drop.

At month 7, another event occurs among the remaining patients, producing the next drop.

Survival Analysis and Reproducibility

The survival package makes the statistical analysis reproducible because the same analysis can be regenerated directly from the analysis dataset and R code.

A reproducible workflow should retain:

• Source data version
• Analysis dataset
• Endpoint derivation code
• Statistical model code
• Figure code
• Output version
• Software/package versions where required

Minimal End-to-End Example

library(survival)

# Kaplan-Meier analysis

km <- survfit(
  Surv(time, status) ~ treatment,
  data = trial
)

summary(km)

# Treatment comparison

logrank <- survdiff(
  Surv(time, status) ~ treatment,
  data = trial
)

logrank

# Cox proportional hazards model

cox <- coxph(
  Surv(time, status) ~ treatment,
  data = trial
)

summary(cox)

# Hazard ratio

exp(coef(cox))

# 95% CI

exp(confint(cox))

# Proportional hazards diagnostic

cox.zph(cox)

A More Complete Clinical-Trial Workflow

library(survival)

# -------------------------------
# Analysis endpoint
# -------------------------------

adtte$event <- 1 - adtte$CNSR

adtte$time <-
  adtte$AVAL / 30.4375


# -------------------------------
# Kaplan-Meier
# -------------------------------

km_fit <- survfit(
  Surv(time, event) ~ TRT01P,
  data = adtte
)


# -------------------------------
# Log-rank comparison
# -------------------------------

logrank <- survdiff(
  Surv(time, event) ~ TRT01P,
  data = adtte
)


# -------------------------------
# Cox model
# -------------------------------

cox_fit <- coxph(
  Surv(time, event) ~
    TRT01P +
    AGE +
    SEX +
    STAGE,
  data = adtte
)


# -------------------------------
# Hazard ratios
# -------------------------------

hr <- exp(coef(cox_fit))

ci <- exp(
  confint(cox_fit)
)


# -------------------------------
# PH assessment
# -------------------------------

ph <- cox.zph(cox_fit)

print(ph)

What the survival Package Does Not Do Automatically

The package provides statistical methods, but it does not know your clinical endpoint definition.

It does not automatically determine:

• What constitutes progression
• Which death dates belong to the endpoint
• Which assessments qualify as valid
• When censoring should occur
• What the analysis cutoff should be
• Which population belongs in the analysis

Statistical Programming vs. Clinical Endpoint Derivation

This distinction is particularly important in clinical trials.

A programmer may write:

survfit(
  Surv(AVAL, EVENT) ~ TRT01P,
  data = adtte
)

perfectly.

If EVENT or AVAL is incorrectly derived, however, the final result remains incorrect.

Therefore, survival-analysis quality depends on both:

• Correct endpoint derivation
• Correct statistical implementation

Interactive Figure: What the Curve Is Telling You

The following simplified figure emphasizes the relationship between survival probability and follow-up.

Median Survival From the Figure

The median is the time at which the estimated survival curve first reaches or falls below 50%.

If the curve never reaches 50%:

then the median is not estimable from the observed follow-up.

A Second Important Survival Concept: The Tail

The right-hand tail of a Kaplan–Meier curve deserves special attention.

Imagine a study beginning with 500 patients. At 36 months, perhaps only 20 patients remain at risk.

The estimated curve may still be mathematically valid, but the amount of information supporting the estimate is much smaller.

This is why number-at-risk tables are essential.

Why a Large Hazard Ratio CI Matters

Suppose:

The point estimate suggests lower hazard, but the confidence interval is wide and includes 1.

This indicates substantial uncertainty about the treatment effect.

The point estimate should not be interpreted in isolation.

Sample Size and Number of Events

Survival-analysis precision is often strongly influenced by the number of observed events.

A trial can have many enrolled patients but relatively few events.

In that situation:

• Confidence intervals may be wide.
• Median survival may not be reached.
• Late Kaplan–Meier estimates may be unstable.
• Cox-model estimates may be imprecise.

The number of events should therefore be considered alongside the total sample size.

Survival Analysis and Interim Analyses

In event-driven trials, interim analyses may occur after a specified number of events rather than at a fixed calendar date.

This creates additional considerations for:

• Database cutoff
• Event counts
• Information fraction
• Statistical boundaries
• Multiplicity

The basic survival functions can calculate the survival estimates, but interim-analysis methodology requires additional statistical planning.

Survival Analysis and Treatment Switching

Treatment switching can complicate interpretation of overall survival.

If patients assigned to control later receive experimental treatment, the observed survival comparison may no longer represent the treatment effect under simple intention-to-treat interpretation.

Potential approaches include prespecified sensitivity analyses and specialized methods for treatment-switching adjustment.

These analyses should not be improvised solely during programming.

Survival Analysis and Multiplicity

A trial may evaluate several time-to-event endpoints:

• Overall survival
• Progression-free survival
• Event-free survival
• Time to response
• Duration of response

If multiple confirmatory hypotheses are tested, multiplicity control may be required.

The survival package does not decide the multiplicity strategy.

How Survival Analysis Fits Into a Clinical Reporting Package

Recommended Survival-Analysis Checklist

Key R Functions to Remember

The Most Important Concept

The most important idea in survival analysis is that time and event status must be analyzed together.

A patient who experiences an event after 24 months and a patient who is censored after 24 months are not equivalent observations.

Both contribute 24 months of follow-up, but only the first contributes an observed event at 24 months.

Survival-analysis methods are designed specifically to preserve this distinction.

Summary

Survival analysis provides a framework for analyzing time-to-event endpoints in the presence of censoring.

The survival package makes the core methodology accessible through a small number of powerful functions.

The essential workflow is:

For clinical-trial statisticians, the key concepts to master are:

• Right censoring
• Risk sets
• Kaplan–Meier estimation
• Median survival
• Number at risk
• Log-rank testing
• Hazard functions
• Hazard ratios
• Cox proportional hazards regression
• Proportional-hazards diagnostics
• Non-proportional hazards
• Endpoint-specific censoring
• Clinical-trial validation and traceability

Once these concepts are understood, the R syntax becomes comparatively straightforward.

References

Therneau, T.M. (2024). A Package for Survival Analysis in R. R package documentation for survival.
Therneau, T.M. and Grambsch, P.M. (2000). Modeling Survival Data: Extending the Cox Model. Springer.
Klein, J.P. and Moeschberger, M.L. (2003). Survival Analysis: Techniques for Censored and Truncated Data. Springer.
Collett, D. (2015). Modelling Survival Data in Medical Research. CRC Press.
Cox, D.R. (1972). Regression Models and Life-Tables. Journal of the Royal Statistical Society: Series B, 34(2), 187–220.
Kaplan, E.L. and Meier, P. (1958). Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53(282), 457–481.