Comparing Survival Curves With the Log-Rank Test

Why Compare Survival Curves?

Kaplan–Meier curves describe survival over time, but a visual difference between curves does not by itself quantify the evidence against equal survival distributions. The log-rank test provides a formal comparison by asking, at each event time, how many events occurred in each group and how many would be expected if the groups had the same event-time distribution. It combines those comparisons across event times while accounting for the changing risk sets.

The test is a global comparison: it summarizes evidence over the observed follow-up rather than testing whether the curves differ at one selected time. Its calculation uses the event and censoring information that determines the Kaplan–Meier estimator, but its purpose is inferential rather than descriptive.

Risk Sets, Expected Counts, and Variance

An individual is at risk immediately before a time if they have entered observation and have not yet experienced the event or been censored. Someone censored at an event time is included in the risk set for that time; their censoring removes them from later risk sets. With delayed entry, only individuals who have entered by that time belong in the risk set. The earlier material on event times, censoring, and risk sets establishes this bookkeeping.

At an event time, the expected number of events in group \(g\) under equal survival distributions is the total number of events multiplied by the group’s share of the risk set. This accounts for unequal numbers at risk: if one group has twice as many people at risk, it is expected to contribute twice as many events at that time under the null.

When multiple events occur at the same time, the usual log-rank calculation treats the \(d_j\) events as a tied set. Conditional on \(d_j\) events among \(n_j\) people at risk, the number assigned to a group follows a hypergeometric allocation under the null. For two groups, the variance contribution for group 1 is therefore:

The factor \((n_j-d_j)/(n_j-1)\) is the finite-population correction for sampling without replacement. If there is one event, the expression reduces to the variance of a single event’s group assignment. If all at-risk individuals experience the event at that time, \(d_j=n_j\), the allocation is fixed and the variance is zero. Event times with zero variance contribute no information to the test.

Forming the Test Statistic

For two groups, let \(U=O_1-E_1\) and let \(V_1=\sum_j V_{1j}\). Since the total observed and expected counts agree across groups, \(O_2-E_2=-U\); thus one group’s difference contains all the information. Under the null hypothesis and with enough information for a large-sample approximation, the standardized difference is approximately standard normal, and its square is approximately chi-square with one degree of freedom.

The statistic is nonnegative and tests for a difference in either direction. The sign of \(O_1-E_1\), considered separately, indicates whether group 1 had more or fewer events than expected under the null; more events than expected generally point toward poorer survival in that group. The p-value is the probability, assuming the null hypothesis and test assumptions, of obtaining a statistic at least as large as the observed one. It is not the probability that the null hypothesis is true.

For \(G\) groups, define \(U_g=O_g-E_g\). The vector \(U\) sums to zero, so only \(G-1\) components are independent. At each event time the covariance matrix has diagonal and off-diagonal entries:

Here \(\mathbf{1}(g=h)\) is 1 when \(g=h\) and 0 otherwise. Because the components of \(U\) sum to zero, \(V\) is singular if all \(G\) components are retained. Use any \(G-1\) independent components and their corresponding covariance submatrix \(V^*\); the statistic \(U^{*T}(V^*)^{-1}U^*\) is approximately chi-square with \(G-1\) degrees of freedom when the large-sample approximation is adequate.

Why the Calculation Has This Form

The expected count follows from the null allocation of events among the risk set. Given \(d_j\) events, each of the \(n_j\) at-risk individuals has the same event probability in the tied-set calculation. A group containing \(n_{gj}\) of those individuals therefore has expected event count \(d_jn_{gj}/n_j\). This expectation adds across event times.

For two groups, let \(Y_j\) be the number of the \(d_j\) events assigned to group 1. Under the tied-set null, \(Y_j\) is hypergeometric with population size \(n_j\), \(n_{1j}\) group-1 individuals, and \(d_j\) event selections. Its mean is \(d_jn_{1j}/n_j\), and its variance is:

When event times are distinct, the event-time contributions are accumulated across successive risk sets, giving \(U=O_1-E_1\) and \(V_1=\sum_j\operatorname{Var}(Y_j)\). Standardization by \(\sqrt{V_1}\) gives an approximately standard normal statistic under the null; squaring it gives the one-degree-of-freedom test. The same reasoning across groups yields the covariance matrix and the \(G-1\) degree-of-freedom test.

Assumptions, Diagnostics, and Alternatives

The usual interpretation of the log-rank p-value relies on several conditions. Individuals’ event outcomes should be independent across subjects, and censoring should be independent of the event time within groups, at least conditional on factors used to define the analysis. If people with a higher near-term event risk are more likely to be censored, the observed risk sets may no longer represent those still capable of experiencing the event. Groups should also be defined consistently from a common time origin and eligibility rule. If delayed entry occurs, risk sets must include only individuals who have entered observation and remain event-free.

The chi-square reference distribution is an approximation. It can be unreliable when there are very few events, sparse group risk sets, or expected event counts that are too small for a normal approximation. In such settings, consider an exact or simulation-based procedure appropriate to the design, and report the limited information rather than treating the asymptotic p-value as definitive. Tied event times should be handled with a stated convention; the hypergeometric variance above is the standard tied-set calculation.

A practical check is to inspect the Kaplan–Meier curves and the risk-set counts alongside the test. Look for implausible sudden changes, unexpectedly empty risk sets, and discrepancies between the event records and the reported numbers at risk. Such checks can reveal data or bookkeeping errors, but a visual inspection cannot verify independent censoring. Also inspect whether the curves cross: the log-rank test remains a test of equality under its assumptions, but crossing patterns can cause early and late differences to offset, reducing power.

The log-rank test is closely related to, but distinct from, several alternatives. Kaplan–Meier curves describe survival and can show when differences occur; they do not replace a test of the global null. Weighted log-rank tests give different emphasis to event times and may be more sensitive to particular patterns, but the choice of weights should be justified rather than selected after viewing results. A Cox proportional hazards model can estimate covariate associations and is a different modeling approach; its treatment is separate from this test.

Common Pitfall

A frequent mistake is to count everyone originally assigned to a group in every risk set, or to count censored observations as events. Neither is correct. The expected count at time \(t_j\) uses only those at risk immediately before that time, and only actual events contribute to \(d_j\). For example, in the third worked example the censored person in B is included in the first risk set, then excluded from the second; that person contributes no event. Using the original group sizes at the second time would give the wrong expected count and variance.