Partial Likelihood Cox PH

Component of the full likelihood used to estimate $\mathbf{\beta }$ in the Cox PH model, treating the baseline hazard $h_0(t)$ as a nuisance parameter.

Partial Likelihood (No Ties)

Let there be $D$ distinct event times: $t_1<t_2<\dots <t_D$. At each $t_j$, let $R_j$ be the risk set — all subjects still at risk just before $t_j$.

$\boxed{L_1(\mathbf{\beta })=\prod _{j=1}^D\frac{e^{{\mathbf{\beta }}^{\prime }{\mathbf{x}}_{(j)}}}{{\sum }_{l\in R_j}{e}^{{\mathbf{\beta }}^{\prime }{\mathbf{x}}_l}}}$

where:

• ${\mathbf{x}}_{(j)}$ : covariate vector of the subject who experiences the event at $t_j$
• $R_j$ : set of indices of subjects at risk at $t_j$

Full Likelihood Factorization

For $n$ independent observations $(t_i,{\delta }_i,{\mathbf{x}}_i)$:

$L(\mathbf{\beta },h_0(\cdot ))={\prod }_{i=1}^{n}f(t_i|{\mathbf{x}}_i{)}^{{\delta }_i}S(t_i|{\mathbf{x}}_i{)}^{1-{\delta }_i}$

Using $f(t|\mathbf{x})=h(t|\mathbf{x})S(t|\mathbf{x})$ and the Cox PH model:

$L(\mathbf{\beta },h_0(\cdot ))={\prod }_{i=1}^n{\left[h_0(t_i)e^{{\mathbf{\beta }}^{\prime }{\mathbf{x}}_i}\right]}^{{\delta }_i}{\left[e^{-{\int }_0^{t_i}{h}_0(u)du}\right]}^{e^{{\mathbf{\beta }}^{\prime }{\mathbf{x}}_i}}$

This factorizes into:

$L(\mathbf{\beta },h_0(\cdot ))=L_1(\mathbf{\beta })\times L_2(\mathbf{\beta },h_0)$

where $L_1(\mathbf{\beta })$ is the partial likelihood — it depends only on $\mathbf{\beta }$ and the event ordering, not on $h_0(t)$.

Intuition

At each event time $t_j$, the contribution to the likelihood is the ratio:

$L(t_j)=\frac{\text{hazard of the subject who actually had the event}}{\text{sum of hazards of all subjects at risk}}=\frac{h_0(t_j)e^{{\mathbf{\beta }}^{\prime }{\mathbf{x}}_{(j)}}}{{\sum }_{l\in R_j}{h}_0(t_j)e^{{\mathbf{\beta }}^{\prime }{\mathbf{x}}_l}}=\frac{e^{{\mathbf{\beta }}^{\prime }{\mathbf{x}}_{(j)}}}{{\sum }_{l\in R_j}{e}^{{\mathbf{\beta }}^{\prime }{\mathbf{x}}_l}}$

The $h_0(t_j)$ cancels because it appears in both numerator and denominator — this is why $h_0(t)$ does not need to be specified.

Since events are assumed independent, multiply across all event times.

Estimation

Maximize $\log L_1(\mathbf{\beta })$ to obtain $\hat{\mathbf{\beta }}$ (MLE). The score function and information matrix give standard errors. This is done numerically (e.g., Newton-Raphson) in practice.

Example: Burn Data

Setup: 154 burn patients. Model: $h(t,\text{gender})=h_0(t)e^{{\beta }_1\cdot \text{gender}}$, with gender = 1 (female), 0 (male).

At $t=4$: $Y=145$ at risk (31 female, 114 male). $d=5$ events (1 female, 4 male).

Breslow contribution: $L_{\text{Breslow}}(t=4)=\frac{e^{{\beta }_1}}{[114+31e^{{\beta }_1}{]}^5}$

Efron contribution — total hazard of 5 event subjects: $4+e^{{\beta }_1}$: $L_{\text{Efron}}(t=4)=\frac{e^{{\beta }_1}}{{\prod }_{j=1}^5\left[31e^{{\beta }_1}+114-\frac{j-1}{5}(4+e^{{\beta }_1})\right]}$

Discrete contribution — 6 possible gender combinations for 5 events: ${\sum }_{l=0}^5\left(\genfrac{}{}{0ex}{}{114}{l}\right)\left(\genfrac{}{}{0ex}{}{31}{5-l}\right)e^{(5-l){\beta }_1}$

Method	Accuracy	Speed	R Default?
Breslow	Good for few ties	Fastest	No
Efron	Better	Moderate	Yes
Discrete	Exact	Slow (combinatorial)	No — use method = "exact"

Related

• Cox Proportional Hazards Model
• Likelihood for Censored Data