Introduction and Overview

From description to mathematics. Section 1 used Kaplan–Meier curves and life tables to describe survival in a sample. Those descriptions are useful, but to compare groups, build regression models, or simulate survival processes we need a more formal language. This section introduces the four interlocking functions that make up that language — the survivor S(t), failure F(t), hazard h(t), and cumulative hazard H(t) functions. Each describes the same underlying time-to-event distribution from a different angle, and each becomes the foundation for a different modelling approach in Sections 3 and 4.

The Survivor Function

The survivor function (also called the survival function) is the cornerstone quantity in survival analysis. It gives the probability that an individual survives beyond a specified time t.

At time t = 0, S(0) = 1 (everyone is alive). As time increases, S(t) decreases. If we follow subjects long enough, S(t) will approach zero. A steep decline in the survivor function indicates rapid event occurrence, while a gradual decline indicates slow event occurrence.

The Failure Function

The failure function (also called the cumulative distribution function, CDF) is simply the complement of the survivor function. It gives the probability that the event has already occurred by time t.

The Probability Density Function and Hazard Function

The probability density function f(t) = dF(t)/dt describes the instantaneous rate of failure events at time t. However, in survival analysis, the most important function is the hazard function, which conditions on survival up to time t.

The hazard function gives the instantaneous rate of the event at time t, conditional on having survived to that point. It is not a probability (it can exceed 1) but rather a rate per unit time. The hazard is the quantity most directly related to the biological or clinical mechanism driving events.

The Cumulative Hazard Function

The cumulative hazard integrates the hazard over time and has a direct relationship with the survivor function:

Hazard Function Shapes

The shape of the hazard function tells us about the underlying process generating the events. Different parametric distributions correspond to different hazard shapes.

Comparing Survival Curves

When comparing the survival experience of two or more groups, several statistical tests are available. Each test weights the contributions of different time points differently.

Introduction and Overview

From description to regression. Sections 1 and 2 gave us the language and the descriptive tools for time-to-event data. We can now describe a survival distribution — but the central epidemiologic question is comparative: does the hazard differ between exposure groups, and by how much, after adjusting for covariates? The Cox proportional hazards model is the regression workhorse for that question. It plugs predictors into the hazard function we just defined while sidestepping the need to specify the baseline hazard’s shape — a feature that makes it the default first model in most applied survival analyses.

The Cox Model

The Cox proportional hazards model is the most widely used regression model for survival data. Introduced by Sir David Cox (1972), it is a semi-parametric model—it models the effect of predictors on the hazard without making any assumptions about the shape of the baseline hazard function.

In this model, h₀(t) is the baseline hazard (the hazard when all predictors are zero), and e^βX is the multiplicative effect of the predictor(s). The baseline hazard h₀(t) is left completely unspecified—it can take any shape. This is the key advantage of the Cox model: no distributional assumption about the underlying survival times is needed.

The Hazard Ratio

The hazard ratio (HR) is the primary measure of effect in the Cox model. For a one-unit increase in a predictor X, the hazard is multiplied by e^β. For example, if β = 0.693 for a treatment variable, then HR = e^0.693 = 2.0, meaning the treated group has twice the hazard (rate of events) compared to the reference group. An HR > 1 indicates increased hazard (shorter survival), while HR < 1 indicates decreased hazard (longer survival). Caution: under non-proportional hazards or selection effects induced by conditioning on survival, the HR can be a misleading effect measure (Hernán, 2010).

Estimation: Partial Likelihood

Because the baseline hazard is left unspecified, the Cox model cannot use standard maximum likelihood estimation. Instead, it uses partial likelihood (also called conditional likelihood), which estimates the β coefficients without needing to estimate h₀(t). The partial likelihood considers only the ordering of events—at each failure time, it asks: given the current risk set, what is the probability that this particular individual was the one to fail?

Handling Ties

When two or more events occur at exactly the same time (ties), the exact partial likelihood becomes computationally expensive. Several approximations are available:

• Breslow method—the simplest approximation, adequate when ties are few
• Efron method—a better approximation that is the default in many software packages
• Exact methods—computationally intensive but most accurate when ties are common

Example: Cox Model Results

In this example, patients receiving the new drug have 37% lower hazard of death (HR = 0.63) compared to the control group, after adjusting for age and disease stage. Each 10-year increase in age is associated with a 42% increase in the hazard. Stage III patients have 3 times the hazard compared to Stage I patients.

Stratified Cox Models

When the proportional hazards assumption is violated for a particular variable, one solution is to stratify on that variable. In the stratified Cox model, each stratum j has its own baseline hazard h_0j(t), but the regression coefficients (β) are assumed to be the same across all strata. This allows the stratifying variable to have a completely flexible effect on survival without needing to estimate or specify its functional form.

Time-Varying Predictors and Effects

The standard Cox model assumes that predictor values are measured at baseline and remain constant. However, some predictors change over the course of follow-up:

• Time-varying predictors: The actual value of the predictor changes during follow-up (e.g., a patient is discharged from hospital and their treatment status changes from “inpatient” to “outpatient”). This requires splitting the follow-up time into intervals and updating predictor values.
• Time-varying effects: The predictor itself may be fixed at baseline, but its effect on the hazard changes over time. This is modelled by including a predictor × time interaction term and indicates a violation of the proportional hazards assumption.

library(survival);  library(survminer)
phaa <- read.csv("phaa_followup.csv", stringsAsFactors = FALSE)
phaa$smoker <- factor(phaa$smoker, levels = c("No","Yes"))

# 1. Build the Surv object: time + event indicator
y <- Surv(time = phaa$fu_years, event = phaa$cv_event)

# 2. Kaplan-Meier: event-free probability over time, by smoking status
km <- survfit(y ~ smoker, data = phaa)
ggsurvplot(km, data = phaa, conf.int = TRUE,
           pval = TRUE, risk.table = TRUE,
           xlab = "Years of follow-up")

# 3. Log-rank test: do the curves differ overall?
survdiff(y ~ smoker, data = phaa)

# 4. Cox model with multiple covariates -> hazard ratios
cox <- coxph(y ~ smoker + age + gender + bmi + hypertension,
             data = phaa)
summary(cox)

# 5. Test the proportional-hazards assumption
cox.zph(cox)

Validating the Cox Model

Thorough model validation involves checking several aspects of model performance. Different types of residuals serve different diagnostic purposes.

Introduction and Overview

From semi-parametric to fully parametric — and beyond. The Cox model bought flexibility by leaving the baseline hazard unspecified, but that flexibility comes at a cost: less efficiency, no direct prediction of absolute survival times, and limited ability to extrapolate. When we have a defensible reason to specify the shape of the hazard — or when our research question requires absolute predictions — we shift to fully parametric models (exponential, Weibull, Gompertz). This section also introduces accelerated failure time (AFT) models, which reframe covariate effects as stretching or shrinking survival time rather than scaling the hazard, and frailty models, which extend survival regression to handle unmeasured heterogeneity — a natural bridge to the clustered-data lessons that follow.

Why Parametric Models?

While the Cox model’s flexibility is a strength, parametric models offer important advantages when the distributional assumption is correct. By specifying the form of the baseline hazard h₀(t), parametric models are more statistically efficient—they produce narrower confidence intervals and more powerful tests. They also allow direct estimation of the baseline hazard, prediction of survival times, and extrapolation beyond the observed data.

Common Parametric Models

Accelerated Failure Time (AFT) Models

Parametric survival models can also be formulated as accelerated failure time models, which model the effect of predictors on the log of survival time directly, rather than on the hazard.

In the AFT framework, the measure of effect is the time ratio (TR = e^β) rather than the hazard ratio. A time ratio of 2 means the expected survival time is doubled; a TR of 0.5 means it is halved. The AFT model essentially “accelerates” or “decelerates” time for different covariate patterns.

Choosing a Parametric Model

Several strategies help guide the choice of parametric distribution:

• Fit the generalised gamma model and test whether simpler models are adequate (κ = 1 for Weibull, κ = 0 for log-normal, etc.)
• Compare models using AIC (Akaike Information Criterion)—lower values indicate better fit after penalising for complexity
• Examine diagnostic plots (e.g., ln H(t) vs ln(t) for Weibull, ln H(t) vs t for Gompertz)
• Consider the biological plausibility of the hazard shape implied by each distribution

Frailty Models

Frailty models address the problem of unmeasured heterogeneity in survival data. Even after including all known predictors, individuals may differ in their underlying susceptibility to the event due to unmeasured factors. Frailty models introduce a random effect α that represents this unobserved heterogeneity.

Bringing It All Together

This lesson built up the survival-analysis toolkit from the ground up. We started with the defining feature of time-to-event data — censoring — and the non-parametric estimators (life tables, Kaplan–Meier, Nelson–Aalen) that describe survival without making distributional assumptions. Section 2 turned those descriptions into the formal language of survivor, failure, hazard, and cumulative hazard functions, the four interlocking views that every later model rests on.

Section 3 introduced the Cox proportional hazards model — the workhorse semi-parametric regression that scales the baseline hazard by exp(βX) and produces hazard ratios as its primary effect measure. Section 4 added fully parametric and accelerated failure time alternatives, which trade flexibility for efficiency and the ability to predict absolute survival times, plus frailty extensions that bridge into the clustered-data lessons coming next. Together these methods cover descriptive, comparative, and predictive questions about time-to-event outcomes.

The final assessment asks you to recognise censoring patterns on sight, choose between non-parametric, semi-parametric, and parametric tools, and interpret hazard ratios on the rate scale rather than the probability scale.