Cox Proportional Hazards Model

Definition

The Cox Proportional Hazards (Cox-PH) Model is a semiparametric regression model for survival data that relates hazard to covariates (the $x_i$s) without assuming a distribution for the baseline hazard.

Model:

$h(t,\mathbf{x})=h_0(t)\exp ({\beta }_1{x}_1+\cdots +{\beta }_P{x}_P)=h_0(t)e^{{\mathbf{\beta }}^{\prime }\mathbf{x}}$

In linear form: $\log h(t,\mathbf{x})=\log h_0(t)+{\beta }_1{x}_1+\cdots +{\beta }_P{x}_P$

where:

• $h_0(t)$ : baseline hazard — the hazard when all covariates equal 0 (or reference level). This is a function of time $t$ only.
• $\mathbf{\beta }=({\beta }_1,\dots ,{\beta }_P)$ : regression coefficients, each ${\beta }_j$ corresponds to covariate $x_j$
• $\mathbf{x}=(x_1,\dots ,x_P)$ : vector of covariates (can be categorical or numeric)

Semiparametric Nature

The model splits into two components:

Component	Nature	Description
$h_0(t)$	Non-parametric	Distribution ignored — no parametric form assumed
$e^{{\mathbf{\beta }}^{\prime }\mathbf{x}}$	Parametric	Parameters ${\beta }_j$ estimated from data

The model is semiparametric because it contains both a non-parametric part ($h_0(t)$) and a parametric part (the covariate contribution). $h_0(t)$ is treated as a nuisance parameter — inference focuses on $\mathbf{\beta }$.

Why “Proportional Hazards”?

For two subjects A and B with covariate vectors ${\mathbf{x}}_A$ and ${\mathbf{x}}_B$:

$\frac{h(t,{\mathbf{x}}_B)}{h(t,{\mathbf{x}}_A)}=\frac{h_0(t)e^{{\mathbf{\beta }}^{\prime }{\mathbf{x}}_B}}{h_0(t)e^{{\mathbf{\beta }}^{\prime }{\mathbf{x}}_A}}=e^{{\mathbf{\beta }}^{\prime }({\mathbf{x}}_B-{\mathbf{x}}_A)}$

The hazard ratio (HR) is constant over time — it depends only on the difference in covariates, not on $t$. This is the proportional hazards assumption.

Covariate Types

$h(t,\mathbf{x})=h_0(t)e^{{\beta }_1{x}_1+\cdots +{\beta }_P{x}_P}$

• Numeric: $x_j$ enters directly (e.g., age, dose). Higher-order terms (e.g., $x^2$) can be added.
• Categorical: Dummy variables required. For $L$ categories, $L-1$ dummy variables with one category as reference.

Dummy Variable Construction

For a categorical variable stage with levels 1–4 (reference = stage 4):

$\begin{array}{rl}{S}_1& =\{\begin{array}{ll}1& \text{if stage}=1\\ 0& \text{otherwise}\end{array}\\ S_2& =\{\begin{array}{ll}1& \text{if stage}=2\\ 0& \text{otherwise}\end{array}\\ S_3& =\{\begin{array}{ll}1& \text{if stage}=3\\ 0& \text{otherwise}\end{array}\end{array}$

Model: $h(t,\mathbf{x})=h_0(t)e^{{\beta }_1{S}_1+{\beta }_2{S}_2+{\beta }_3{S}_3}$

Stage	Model	Hazard relative to Stage 4
1	$h_0(t)e^{{\beta }_1}$	$e^{{\beta }_1}$
2	$h_0(t)e^{{\beta }_2}$	$e^{{\beta }_2}$
3	$h_0(t)e^{{\beta }_3}$	$e^{{\beta }_3}$
4 (ref)	$h_0(t)$	$1$

Interactions

If the effect of $x_1$ depends on $x_2$, add an interaction term: $h(t,\mathbf{x})=h_0(t)e^{{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_1{x}_2}$

For interactions between categorical (dummy) and numeric variables, multiply each dummy by the numeric variable.

Dependency Graph

graph TD
    classDef def fill:#1e3a5f,stroke:#90caf9,stroke-width:2px,color:#bbdefb
    classDef proc fill:#1b3b28,stroke:#a5d6a7,stroke-width:2px,color:#c8e6c9
    classDef sheet fill:#3d2e15,stroke:#ffcc80,stroke-width:2px,color:#ffe0b2

    subgraph Prereqs["Prerequisites"]
        ST["Survival Time — def"]:::def
        CD["Censored Data — def"]:::def
        HF["Hazard Function — def"]:::def
        LCD["Likelihood (Censored) — def"]:::def
        KM["Kaplan-Meier — def"]:::def
        LR["Log-Rank Test — def"]:::def
        STR["Stratified Test — def"]:::def
        
        ST --> HF
        CD --> LCD
        HF --> KM
        LCD --> KM
        KM --> LR
        LR --> STR
    end

    subgraph CoxPH["Cox Proportional Hazards"]
        CPH["Cox PH Model — def"]:::def
        HR["Hazard Ratio — def"]:::def
        PHC["Checking PH Assumption — proc"]:::proc
        PL["Partial Likelihood — def"]:::def
        LWT["Likelihood with Ties — def"]:::def
        CS["Cox PH Cheatsheet — sheet"]:::sheet
        
        CPH --> HR
        CPH --> PL
        HR --> PHC
        PL --> LWT
        PHC --> CS
        LWT --> CS
    end

    HF --> CPH
    STR --> CPH
    LCD --> PL

Related

• Hazard Function
• Survival Function