Cox Proportional Hazards Model

While the Kaplan–Meier estimator tells us how survival probability changes over time, it doesn’t explain why — it cannot handle predictors or covariates.

The Cox model answers that question by linking covariates (e.g., age, treatment, income) to the hazard rate, i.e., the instantaneous risk of the event occurring.

The Model Structure

The hazard for individual $i$ with covariates $X_i = (X_{i1}, X_{i2}, …, X_{im})$

\[h(t|X_i) = h_0(t)\exp(\beta_1 X_{i1} + \beta_2 X_{i2} + ... + \beta_n X_{im})\]

where:

• $h_0(t)$ is the baseline hazard function — unspecified (non-parametric).
• $\exp(\beta_j)$ is the multiplicative effect of covariate $X_j$ on the hazard.
• $\beta_j$ is the log-hazard ratio coefficient.

Each covariate $X_k$ multiplies the hazard by a fixed proportion $e^{\beta_k}$.

So, for example if $\beta_k = 0.7$, then one-unit increase in $X_k$ increases the hazard by $e^{0.7} \approx 2.01$ times. And if $\beta_k = -0.7$, for example, then the hazard is multiplied by $e^{-0.7} \approx 0.5$, so it’s effectively halved.

Crucially, this increase/decrease is proportionate, not absolute — it scales the whole hazard curve up or down, keeping its shape over time identical.

The hazard ratio between two individuals $i$ and $j$ with covariate vectors $X_i$ and $X_j$ is:

\[\frac{h(t|X_i)}{h(t|X_j)} = \exp(\beta^{T} (X_i - X_j))\]

Notice how the baseline hazard function disappeared. Importantly, this ratio is independent of time, which is the assumption of proportional hazards model.

Partial Likelihood Estimation

Unlike parametric models, the Cox model doesn’t specify $h_0(t)$. Yet, we still want to estimate the regression coefficients $\beta$ the effect of covariates on the hazard. The partial likelihood cleverly eliminates $h_0(t)$ by conditioning on the set of individuals who are at risk when an event occurs.

If the ordered event times are $t_{(1)}, t_{(2)}, …, t_{(n)}$, and $R(t_{(k)})$ is the risk set (subjects still under observation just before $t_{(k)}$), the partial likelihood is:

\[L(\beta) = \prod_{k=1}^n \frac{\exp(\beta^T X_{(k)})}{\sum_{j \in R(t_{(k)})} \exp(\beta^T X_{j})}\]

Taking logs gives the partial log-likelihood, which is maximized to estimate $\hat \beta$.

This method automatically accounts for right-censoring, because censored individuals remain in the risk sets up to their censoring time.

Interpretation of Hazard Ratio

Each coefficient $\beta_j$ corresponds to a log hazard ratio:

\[HR_j = \exp(\beta_j)\] \[\ln HR_j = \beta_j\]

• $HR_j > 1$: higher risk of the event.
• $HR_j < 1$: protective (lower risk).
• $HR_j = 1$: no effect.

Confidence intervals for $HR_j$ are constructed as:

\[\exp(\beta_j \pm z_{\alpha/2 \text{SE}(\beta_j)})\]

Extensions of the Cox Model

Stratified Cox Model

Used when the baseline hazard differs across subgroups (e.g., men vs. women), but covariate effects are assumed common.

\[h(t|X, \text{stratum}=s) = h_{0s}(t)\exp(\beta^T X)\]

Each stratum has its own $h_{0s}(t)$, but a shared $\beta$. This allows the shape and level of the hazard over time to differ across groups — while keeping the same multiplicative effect of covariates on the hazard within each stratum.

Partial likelihood is constructed is constructed per stratum, and then multiplied together:

\[L(\beta) = \prod_{s=1}^S \prod_{k=1}^n \frac{\exp(\beta^T X_{(k)})}{\sum_{j \in R(t_{(k)})} \exp(\beta^T X_{j})}\]

Time-Dependent Covariates

Some covariates change over time — e.g., blood pressure, medication status. These are modeled as $X_i (t)$:

\[h(t|X_i (t)) = h_{0}(t)\exp(\beta^T X)\]

This allows modeling dynamic effects such as the impact of treatment after initiation.

Non-Proportional Hazards and Interactions

If the proportionality assumption fails, interactions with time can be added:

\[h(t|X_i (t)) = h_{0}(t)\exp(\beta_1 X + \beta_2 X \cdot g(t))\]

where $g(t)$ might be $\log t$, $t$, or a spline term, allowing covariate effects to vary over time.