Why Tied Event Times Need Special Treatment
The Cox partial likelihood compares the covariates of a person who fails at a given time with those of people still at risk at that time. Its usual single-failure contribution has one observed failure in the numerator and a sum over the risk set in the denominator. If two or more failures are recorded at the same time, the data do not identify a unique ordering of those failures. Simply applying the single-failure formula repeatedly would require choosing an order and deciding which individuals remain in each successive denominator.
Ties can arise because event times are recorded only to the nearest day or month, even when underlying event times differ. They can also represent genuinely grouped event times. The treatment of ties can affect coefficient estimates and standard errors, particularly when tied failures make up a substantial share of the events or when risk sets are small. The methods below differ in how they form the denominator for a tied block.
Here \(\mathbf{x}_i\) is the covariate vector and \(\boldsymbol{\beta}\) the regression coefficient vector. The tied-event numerator is the product of the weights of the observed failures, \(\prod_{i\in D(t)}w_i\), in all three methods. Their distinction is the denominator. Contributions across distinct event times are multiplied to form the partial likelihood. The formulas below concern one tied block, holding the risk set immediately before that time fixed.
Three Denominators
The Breslow approximation uses the full risk-set sum for each of the \(d\) failures. The Efron approximation reduces that sum in fractional steps as if tied failures were removed successively, without specifying their order. The exact partial likelihood sums over all possible subsets of \(d\) failures from the risk set, with each subset weighted by the product of its members’ relative risks.
In the exact denominator, \(A\) ranges over every subset of \(d\) distinct members of the risk set. The exact calculation is often described as conditional: given that \(d\) failures occur at that time, it evaluates the relative weight of the observed failure set among all possible sets of \(d\) failures. It does not assume that a particular ordering of the observed tied failures is known.
The Efron denominator subtracts \(0, S_D(t)/d, 2S_D(t)/d,\ldots,(d-1)S_D(t)/d\) from the risk-set sum. These fractional reductions approximate removal of the tied failures through the block. When \(d=1\), the denominator in each method is simply \(S_R(t)\), so the methods agree.
Worked Example: Two Tied Failures in a Small Risk Set
Suppose four individuals are at risk at an observed event time. Their relative risk weights are 2, 1.5, 1, and 0.5. The first two individuals fail at that time, so \(d=2\), \(S_R=5\), and \(S_D=3.5\). Assume these weights are the fitted model weights for this calculation.
The numerator is the product of the two observed failures’ weights: \(2(1.5)=3\). Breslow uses \(S_R^2=5^2=25\), giving \(L_{\mathrm{B}}=3/25=0.1200\).
For Efron, the two denominator factors are \(5\) and \(5-(1/2)(3.5)=3.25\). Their product is \(16.25\), so \(L_{\mathrm{E}}=3/16.25=0.1846\).
For the exact denominator, enumerate the six pairs from the risk set. Their weight products are \(3, 2, 1, 1.5, 0.75,\) and \(0.5\), which sum to \(8.75\). Thus \(L_{\mathrm{exact}}=3/8.75=0.3429\).
This calculation illustrates how strongly the denominator can depend on the method when the risk set is small and half its members fail together. The numerical contributions are not a basis for comparing the overall fit of the different methods: their likelihood constructions have different normalizing factors. The relevant practical question is whether their coefficient estimates or inferences differ meaningfully when each method is applied consistently to the full data.
Why Efron Approximates the Exact Treatment
The exact denominator counts distinct failure sets and uses their products of risk weights. Breslow instead raises the total risk weight to the \(d\)th power, which includes repeated selections of the same individual when interpreted as an expansion. Efron makes a correction by reducing the risk-set sum across the tied block. This is a useful compromise: it avoids enumerating every subset while representing the depletion of risk weight caused by the observed tied failures.
A short algebraic check shows why the methods coincide for one failure. If \(d=1\), the exact denominator sums the weights over all one-member subsets, which is \(S_R\). The Breslow denominator is \(S_R^1\), and the only Efron factor, at \(\ell=0\), is also \(S_R\). For multiple failures, the denominators generally differ because they account differently for selecting distinct failures.
The exact denominator can be computed as an elementary symmetric sum of order \(d\) in the risk-set weights. Directly enumerating all subsets may be costly when both \(d\) and the risk-set size are large; efficient algorithms can calculate the same sum without listing every subset. Breslow and Efron are generally less computationally demanding, which helps explain their use in large data sets.
Worked Example: Comparing Methods in a Larger Risk Set
Suppose 100 individuals are at risk. Three fail at the same recorded time, with weights \(1.2\), \(0.9\), and \(1.1\); each of the other 97 has weight \(1\). Then \(S_R=100.2\), \(S_D=3.2\), and the tied numerator is \(1.2(0.9)(1.1)=1.188\).
Breslow gives a denominator of \(100.2^3=1{,}006{,}012.008\), so its contribution is approximately \(1.180\times10^{-6}\).
For Efron, the denominator is \(100.2(100.2-3.2/3)(100.2-2(3.2/3))\), which is approximately \(974{,}112\). Its contribution is approximately \(1.220\times10^{-6}\).
For the exact denominator, partition possible triples into those with zero, one, two, or three of the three observed failures. The total is \(\binom{97}{3}+(3.2)\binom{97}{2}+(3.39)(97)+1.188\). Here \(3.2\) is the sum of the three failure weights, and \(3.39=1.2(0.9)+1.2(1.1)+0.9(1.1)\) is the sum of their pairwise products. Substitution gives \(147{,}440+14{,}899.2+328.83+1.188=162{,}669.218\). Therefore \(L_{\mathrm{exact}}=1.188/162{,}669.218\approx7.300\times10^{-6}\).
The denominators differ substantially in raw scale here, partly because the exact denominator sums distinct unordered triples, whereas the approximations use other normalizations. One must not interpret the differences in these single-block likelihood values as evidence that one method fits better. In practice, compare the resulting coefficient estimates and standard errors from the full analysis if sensitivity to the tie method is a concern.
Choosing a Method and Checking Its Suitability
A sound choice starts with how event times were measured and with the structure of the tied risk sets. Exact partial likelihood is attractive when ties are numerous and risk sets are small enough for the calculation to be feasible. Efron is widely used as a practical approximation and often performs well when there are tied failures. Breslow is simple and computationally economical, but can be less accurate when many individuals fail together relative to the risk-set size. These are general guides, not guarantees: the impact depends on the covariates, weights, and pattern of ties.
| Method | Denominator for a tied block | Practical consideration |
|---|---|---|
| Breslow | \(S_R^d\) | Simple; may be less accurate with substantial tied failure counts. |
| Efron | Product of \(d\) fractional-step risk sums | A common compromise that accounts approximately for depletion across the tied block. |
| Exact | Sum of products over all distinct size-\(d\) subsets | Direct conditional treatment; can be computationally demanding for large risk sets or tie counts. |
Check the time unit and recording precision. Several events recorded on one day may have occurred at different underlying times, while genuinely grouped event times call for a grouped-time interpretation.
Count failures at each tied time and compare that count with the number at risk. A small tied count in a large risk set is a different approximation problem from many failures in a small risk set.
Use an exact method when its computational cost is acceptable and a direct conditional treatment is desired; otherwise Efron is a common choice. Record the convention so results are reproducible.
Refit with a plausible alternative tie method and compare coefficient estimates and standard errors. Material changes indicate that the tie convention matters for the analysis.
The method presumes a correctly defined risk set at each event time and correct identification of the failures in the tied set. Delayed entry, censoring, and the event-time recording rule must be represented consistently; a tie method does not repair errors in those data or definitions. Tied-time calculations also do not address other model issues, such as whether the Cox proportional hazards specification is appropriate.
When event times are intrinsically discrete, a grouped-time survival model may be a more natural alternative than treating ties only as a computational complication in a continuous-time Cox model. The choice depends on the scientific time scale and data-generating process. The exact partial likelihood offers a direct tied-set calculation within the Cox framework; it does not by itself establish that this framework is the best representation of discrete event timing.
Common Mistakes and Diagnostics
- Applying the single-event formula repeatedly in an arbitrary order. This makes the answer depend on which tied failure is put first. Use a stated tied-time method rather than inventing an order.
- Assuming the methods are identical whenever there are ties. They agree for \(d=1\), but multiple failures generally produce different denominators.
- Comparing raw likelihood contributions across methods. Their normalizations differ. Assess sensitivity through fitted coefficients and standard errors, not by comparing isolated contribution magnitudes.
- Ignoring the time-recording resolution. Ties caused by rounding and true grouped event times have different interpretations. Verify the time scale before deciding how to model ties.
- Assuming exact is always preferable. Exact treatment can be computationally expensive, and a more expensive calculation does not compensate for an inappropriate model or incorrect risk sets.
Useful diagnostics include tabulating the number of tied failures at each event time and the corresponding risk-set sizes, checking whether ties cluster at the measurement resolution, and comparing coefficient estimates under reasonable tie methods when tie blocks are consequential. Large discrepancies warrant review of the event-time coding and modeling assumptions, not merely selection of whichever result is preferred.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For a single failure at a time, why do Breslow, Efron, and exact partial likelihood have the same denominator?
- A risk set has weights \(2\), \(1\), and \(1\), and the individual with weight \(2\) fails alone. Calculate the single-event partial-likelihood contribution.
- When would the exact method be especially appealing, and what practical limitation might affect its use?
- Why is it incorrect to decide that exact partial likelihood fits better just because its contribution in the small-risk-set example is numerically larger?
- A large proportion of failures occur at the same recorded monthly times. Name two checks to make before settling on a tie method.
Solutions. (1) With \(d=1\), each denominator is the sum of all risk-set weights, \(S_R\). (2) The numerator is \(2\), and \(S_R=2+1+1=4\), so the contribution is \(2/4=0.5\). (3) Exact treatment is appealing when ties are substantial and risk sets are small enough to make its calculation feasible; enumeration or computation can be demanding for large risk sets or tie counts. (4) The methods use different normalizations, so isolated contribution values are not comparable measures of fit; compare estimates from the full analysis instead. (5) Check the event-time recording resolution and whether times are genuinely grouped, and tabulate tied counts relative to their risk-set sizes. These checks inform both the interpretation of the ties and the method’s likely sensitivity.