Handling Tied Failures in Cause-Specific Cox Models

Why Tie Handling Matters

A Cox model uses the ordering of event times to compare the covariates of people who fail with those still at risk. If several target-cause failures share one recorded time, that ordering is unavailable. Ties can occur because time is recorded in days or weeks, because event times are genuinely discrete, or because a database rounds more precise times. The method used to handle them can affect the partial likelihood, coefficient estimates, and standard errors, especially when ties are frequent or risk sets are small.

Here the event of interest is cause 1. A cause-specific Cox model describes the instantaneous rate of a cause-1 event among people who remain at risk:

\lambda_1(t\mid X_i)=\lambda_{10}(t)\exp(\beta^\top X_i).

The coefficient vector is \(\beta\), \(X_i\) is the covariate vector for person \(i\), and \(\lambda_{10}(t)\) is an unspecified baseline cause-specific hazard. In this analysis, a competing event removes a person from the risk set after that event; it is not counted as a cause-1 failure. The focus here is the partial-likelihood contribution when two or more cause-1 failures share a recorded time.

Definition: Let \(t_j\) be a distinct recorded time at which at least one cause-1 event occurs. The risk set \(\mathcal R_j\) contains people at risk immediately before \(t_j\), under the chosen event-time convention. The tied target-cause failure set \(\mathcal D_j\) contains the people in \(\mathcal R_j\) with a cause-1 event at \(t_j\), and \(d_j=|\mathcal D_j|\). Write \(w_i=\exp(\beta^\top X_i)\), \(S_j=\sum_{i\in\mathcal R_j}w_i\), and \(S_{Dj}=\sum_{i\in\mathcal D_j}w_i\).

The risk set is specific to the cause-specific analysis. People who experienced a competing event before \(t_j\) are no longer at risk. When a competing event and a cause-1 event have the same recorded time, the risk-set convention must be applied consistently; typically both are in the risk set immediately before that time, while only cause-1 events belong to \(\mathcal D_j\). All tied methods below use the same risk sets and event sets. They differ in how they represent the tied failures within a set.

Three Partial-Likelihood Contributions

For one failure, the usual Cox partial-likelihood contribution is the failing person’s relative risk divided by the sum of relative risks in the risk set. With \(d_j>1\), the exact sequence of failures is not observed. The three standard choices are Breslow, Efron, and exact handling.

Definition: For a tied set at \(t_j\), the Breslow contribution is \(\prod_{i\in\mathcal D_j}w_i/S_j^{d_j}\). The Efron contribution is \(\prod_{i\in\mathcal D_j}w_i\) divided by \(\prod_{\ell=0}^{d_j-1}(S_j-\ell S_{Dj}/d_j)\). The exact contribution is \(\prod_{i\in\mathcal D_j}w_i\) divided by \(\sum_{A\subset\mathcal R_j,\ |A|=d_j}\prod_{i\in A}w_i\).

The Breslow method uses the same full risk-set denominator for each of the \(d_j\) failures. The Efron method reduces that denominator in equal fractions of the tied group’s total relative risk: at step \(\ell\), it subtracts \(\ell/d_j\) of \(S_{Dj}\). The exact method sums over every possible subset of \(d_j\) people from the risk set. It conditions on there being \(d_j\) failures at that time and assigns each candidate subset a weight equal to the product of its members’ relative risks.

For several distinct event times, the partial likelihood is the product of the contributions across event times. The model estimate maximizes this likelihood with respect to \(\beta\). The baseline hazard does not appear in these contributions. When \(d_j=1\), all three methods reduce to the same contribution, \(w_i/S_j\).

Worked Example: Calculating All Three Contributions

Suppose a hypothetical cause-specific analysis has four people in the risk set at a tied time. Their relative risks at a specified coefficient value are \(2,\ 1.5,\ 1,\) and \(0.5\). The first two people have cause-1 failures at that time, so \(d=2\), the numerator is \(2(1.5)=3\), and the total risk-set weight is \(S=5\). The tied-event weight is \(S_D=3.5\) (that is, \(2+1.5\)).

Breslow. Its denominator is \(S^d=5^2=25\), so the contribution is \(3/25=0.1200\).

Efron. The two denominators are \(5\) and \(5-(1/2)(3.5)=3.25\). Their product is \(16.25\), so the contribution is \(3/16.25=0.1846\).

Exact. The denominator sums the weights of all six pairs from the four-person risk set: \(2(1.5)+2(1)+2(0.5)+1.5(1)+1.5(0.5)+1(0.5)=8.75\). The contribution is \(3/8.75=0.3429\).

These are likelihood contributions at the specified coefficient value, not three estimated effects. They differ because each method handles the unobserved within-time ordering differently.

Why the Denominators Differ

The exact denominator follows directly from conditioning on the size of the failure set: every subset \(A\) of size \(d_j\) is a possible tied set, with relative weight \(\prod_{i\in A}w_i\). Normalizing those weights gives a conditional probability, and the observed set’s contribution is its weight divided by the sum of all candidate weights.

The Breslow and Efron denominators can be understood as approximations to the sequential removal of failures from the risk set. If the tied failures had a known order, each successive failure would be compared with a risk set reduced by the people who had already failed. But the order is unknown. Breslow leaves the risk-set denominator unchanged at every step; Efron approximates the removal by subtracting a fraction of the tied group’s total weight. This gives the Efron method a more refined denominator when several failures have substantial weight relative to the full risk set.

For \(d_j=2\), the Efron denominator is \(S_j(S_j-S_{Dj}/2)\). The first factor uses the full risk set; the second subtracts half of the total tied-event weight. This construction is an approximation, not a claim that half of each person literally fails first.

Worked Example: Comparing Score Contributions

Consider a hypothetical risk set of five people: three have a binary exposure \(x=1\), and two have \(x=0\). At \(\beta=\log(2)\), exposed people have weight \(2\) and unexposed people weight \(1\). Suppose two people fail from cause 1 at the tied time, one exposed and one unexposed. Thus \(S=3(2)+2(1)=8\), \(S_D=2+1=3\), and the observed event-set weight is \(2(1)=2\).

The Breslow contribution is \(2/8^2=0.03125\). The Efron contribution is \(2/[8(8-3/2)]=2/52=0.03846\). For the exact denominator, the possible pairs contribute \(12\) for two exposed people, \(12\) for one exposed and one unexposed, and \(1\) for two unexposed people. The denominator is \(25\), giving an exact contribution of \(2/25=0.08000\).

The score is the derivative of the log contribution with respect to \(\beta\). For Breslow, the observed exposed count is \(1\), while the risk-set weighted mean exposed count per denominator is \(6/8=0.75\), repeated twice. Its score contribution is \(1-2(0.75)=-0.5000\).

For Efron, the two denominator terms have weighted exposed means \(6/8=0.75\) and \([6-(1/2)(2)]/6.5=5/6.5=0.7692\). The score contribution is \(1-0.75-0.7692=-0.5192\). For the exact method, the weighted mean number exposed in a candidate pair is \([2(12)+1(12)+0(1)]/25=36/25=1.44\); the score contribution is \(1-1.44=-0.4400\). These are local score contributions at \(\beta=\log(2)\), not final fitted coefficients. The arithmetic illustrates how tie handling can change the direction and size of the likelihood’s local push.

Assumptions, Diagnostics, and Practical Choice

The tie methods rely on the cause-specific Cox model and its risk sets being defined appropriately. In particular, the model specifies a common coefficient vector \(\beta\) over the modeled time range unless the model has been extended; the relative risk at a given time is \(\exp(\beta^\top X_i)\). The usual assumptions needed for Cox regression, including an appropriate censoring mechanism, remain relevant. Tie handling does not repair a misspecified covariate effect, an incorrect event classification, or an incorrectly constructed risk set.

The continuous-time Cox model assigns probability zero to exact simultaneous failures under ideal continuous event times. In practice, recorded ties may be induced by time rounding, or may reflect genuinely discrete observation times. The tie method does not determine which situation produced the ties. If recording precision is coarse, finer time data may change the observed tie groups and the analysis.

Useful checks include counting the number and sizes of tied cause-1 event sets, confirming the time unit and recording precision, and verifying how competing events at the same recorded time enter the risk set. Refit using Efron and Breslow, or exact when feasible, and compare coefficients, standard errors, and substantive conclusions. Large changes indicate that the tie approximation matters for these data; they are a reason to investigate time resolution and the model, not an automatic test that one method is correct.

Method	How it handles a tied set	Practical consideration
Breslow	Uses the full risk-set sum for each failure	Computationally simple; can be less accurate when ties are substantial
Efron	Fractionally reduces tied-event weight across denominator steps	Often a useful approximation when ties are present
Exact	Sums weights over every candidate subset of the tied size	Most direct conditional calculation here, but can be computationally expensive for large risk sets or tie groups

No method is universally best for every data structure. Efron is commonly used for many tied event times in continuous-time applications; Breslow is simple and can be adequate when ties are sparse relative to risk sets. Exact handling can be attractive when tied groups are important and the computation is manageable, particularly when discrete event times are central to the data structure. The exact calculation may become costly because the denominator involves \(\binom{|\mathcal R_j|}{d_j}\) candidate subsets, although software can use efficient algorithms rather than enumerate them explicitly.

These methods address the same tied-time problem within a cause-specific Cox model. They are not substitutes for choosing between cause-specific and subdistribution-hazard estimands, nor do they change the interpretation of the cause-specific coefficient as a relative hazard among those still at risk. For the broader comparison of estimands, use the earlier material on choosing between cause-specific and Fine-Gray regression.

Common Mistakes and Limits

Treating a competing event as a target-cause failure. A competing event may be in the risk set immediately before a tied time, but it is not part of \(\mathcal D_j\) unless it is a cause-1 event.
Assuming Efron exactly reconstructs the failure order. It does not. Its fractional reduction is an approximation to the unavailable ordering.
Calling exact handling automatically correct. The exact calculation is exact for its conditional tied-set likelihood, given the specified risk set and model. It cannot correct erroneous event times, risk sets, or covariate assumptions.
Ignoring the time scale. Ties produced by rounding can be sensitive to the recording unit. The recorded time scale should be reported and checked.
Interpreting a likelihood contribution as a hazard ratio. A contribution is a probability-like term in the partial likelihood at a specified coefficient value; it is not itself an effect estimate.

A practical analysis should report the chosen tie method and examine whether reasonable alternatives materially change the coefficient estimates or uncertainty. If they do, explain the scale of the ties and why the selected method is appropriate. All tie methods still depend on the proportional-hazards and other modeling assumptions; agreement across tie methods does not validate those assumptions.

Worked Example: When the Methods Agree

Suppose only one person in a hypothetical risk set has a cause-1 event at a given time. If the risk-set weights sum to \(S=4\) and the failing person’s weight is \(2\), then \(d=1\). Breslow gives \(2/4^1=0.5\); Efron has only the \(\ell=0\) denominator, so it gives \(2/4=0.5\); and the exact denominator is the sum of the weights of all one-person subsets, also \(4\), giving \(2/4=0.5\). Thus, differences among the methods arise from tied failures, not from event times with a single failure.

Key takeaway: For a tied cause-1 failure set, Breslow repeats the full risk-set denominator, Efron reduces it fractionally, and exact handling sums over all candidate subsets of the tied size. The methods share the same cause-specific risk sets but can produce different partial likelihoods and estimates.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

At a tied time, three people fail from cause 1 and one person experiences a competing event. Which people belong to the target-cause failure set?
For a tied set with \(d=2\), risk-set sum \(S=10\), and tied-event weight sum \(S_D=4\), what denominator does Efron use?
Why does the exact denominator sum over subsets of size \(d\), rather than over all subsets?
If there is only one cause-1 failure at a recorded time, how do Breslow, Efron, and exact contributions compare?
Why might comparing Efron and Breslow estimates be useful, and what does a large difference not establish by itself?

Solutions. (1) The three cause-1 failures belong to the target-cause failure set; the person with a competing event does not. That person may be in the risk set immediately before the tied time, depending on the stated event-time convention. (2) The Efron denominator is \(10(10-4/2)=10(8)=80\). (3) The observed tied set contains \(d\) failures, so the conditional candidates are precisely the possible risk-set groups of that same size. (4) They are identical: each uses the failing person’s weight divided by the full risk-set sum. (5) The comparison shows whether conclusions are sensitive to the approximation used for ties. A large difference signals sensitivity and warrants checking tie frequency, time precision, and modeling choices; it does not alone prove that either method is wrong.