Likelihood with Ties Cox PH

When multiple events occur at the same time $t_i$ (ties), the partial likelihood must account for the possible event orderings. Three methods exist: breslow, efron and discrete

Notation

• $D$ : total distinct event times
• $d_i$ : number of events at time $t_i$
• ${\mathcal{D}}_i$ : set of $d_i$ subjects experiencing event at $t_i$
• $R_i$ : risk set at $t_i$
• ${\mathbf{Z}}_i={\sum }_{k\in {\mathcal{D}}_i}{\mathbf{x}}_k$ : sum of covariate vectors of event subjects
• ${\mathbf{x}}_{qj}$ : covariate vector of a possible selection of $d_i$ subjects from $R_i$

Method 1: Breslow (Approximation)

Treats ties as sequential events, but does not update the risk set between them.

$L_{\text{Breslow}}(\mathbf{\beta })={\prod }_{i=1}^D\frac{\exp ({\mathbf{\beta }}^{\prime }{\mathbf{Z}}_i)}{{\left[{\sum }_{l\in R_i}\exp ({\mathbf{\beta }}^{\prime }{\mathbf{x}}_l)\right]}^{d_i}}$

The denominator uses the full risk set $d_i$ times.

Example (Burn data, $t=4$, $Y=145$ with 31 women [W] and 114 men [M], 5 events: 1W + 4M):

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

Method 2: Efron (Improved Approximation)

Similar to Breslow but adds a correction factor to the denominator as each successive event occurs.

$L_{\text{Efron}}(\mathbf{\beta })={\prod }_{i=1}^D\frac{\exp ({\mathbf{\beta }}^{\prime }{\mathbf{Z}}_i)}{{\prod }_{j=1}^{d_i}\left[{\sum }_{l\in R_i}\exp ({\mathbf{\beta }}^{\prime }{\mathbf{x}}_l)-\frac{j-1}{d_i}{\sum }_{k\in {\mathcal{D}}_i}\exp ({\mathbf{\beta }}^{\prime }{\mathbf{x}}_k)\right]}$

At the $j$-th event, the denominator is reduced by $\frac{j-1}{d_i}$ of the total hazard from the $d_i$ event subjects.

Example (Burn data, $t=4$):

The total hazard of 5 event subjects: $h_0(4)[4+e^{{\beta }_1}]$

$L_{\text{Efron}}(t=4)=\frac{e^{{\beta }_1}}{{\prod }_{j=1}^5\left[31e^{{\beta }_1}+114-\frac{j-1}{5}(4+e^{{\beta }_1})\right]}$

Method 3: Discrete (Exact)

Enumerates all possible ways $d_i$ subjects could be selected from $R_i$ as the event set.

$L_{\text{Discrete}}(\mathbf{\beta })={\prod }_{i=1}^D\frac{\exp ({\mathbf{\beta }}^{\prime }{\mathbf{Z}}_i)}{{\sum }_{q\in Q_i}\exp ({\mathbf{\beta }}^{\prime }{\mathbf{Z}}_q^{\ast })}$

where $Q_i$ enumerates all $\left(\genfrac{}{}{0ex}{}{Y_i}{d_i}\right)$ possible selections of $d_i$ subjects from $R_i$.

Example (Burn data, $t=4$): 6 possibilities for 5 events by gender (5W, 4W+1M, 3W+2M, …, 5M):

${\sum }_{l=0}^5\left(\genfrac{}{}{0ex}{}{114}{l}\right)\left(\genfrac{}{}{0ex}{}{31}{5-l}\right)e^{(5-l){\beta }_1}$

Comparison

Method	Accuracy	Computation	When to Use
Breslow	Good for few ties	Fastest	Default in most software
Efron	Better than Breslow	Moderate	Moderate ties; R default
Discrete	Exact	Slow (combinatorial)	Many ties, small $Y_i$

NOTE

R’s coxph() uses Efron by default. The method argument accepts "efron", "breslow", or "exact".

Related

• Partial Likelihood Cox PH
• Cox Proportional Hazards Model