Two-Sided Logrank Monitoring For Benefit And Harm

Why Monitor Both Directions?

A group sequential logrank analysis can be designed to stop early for evidence that a treatment reduces event risk. In some trials, however, evidence in the opposite direction also matters: the treatment may increase event risk, and the monitoring plan should permit an early stop for harm. Two-sided monitoring makes both directions explicit. It does not mean that every interim result is decisive, or that the same numerical threshold must be used for benefit and harm.

The first requirement is a clear sign convention. A positive logrank Z statistic should consistently indicate fewer observed events in the treatment group than expected under the null; a negative value should indicate more. Once that convention is set, an upper boundary can represent evidence of benefit and a lower boundary evidence of harm. Reversing the statistic without reversing its boundaries reverses the clinical interpretation.

Definition: At look \(k\), let \(Z_k\) be the standardized logrank statistic, oriented so that positive values favor treatment. Let \(c_{U,k}\) be the upper boundary and \(c_{L,k}\) the lower boundary, with \(c_{L,k}<c_{U,k}\). An upper crossing occurs if \(Z_k\ge c_{U,k}\) and signals evidence in the direction of benefit. A lower crossing occurs if \(Z_k\le c_{L,k}\) and signals evidence in the direction of harm. If neither inequality holds, there is no boundary crossing at that look.

Orienting the Logrank Statistic

For a simple event-time comparison, consider event times in order. At an event time with no tied events, let \(X_i=1\) if the event is in the treatment group and \(X_i=0\) otherwise. Let \(p_i\) be the proportion of individuals in the risk set assigned to treatment under the null. A convenient score contribution is \(p_i-X_i\). Summing these contributions gives the treatment-group expected number of events minus its observed number, under the risk-set comparison. Under the null, its variance is estimated using the risk sets.

U_k=\sum_{i=1}^{D_k}(p_i-X_i),\qquad V_k\ \text{is the estimated variance of }U_k,\qquad Z_k=\frac{U_k}{\sqrt{V_k}}.

Here \(D_k\) is the number of events included by look \(k\). If the treatment group has fewer events than expected, then \(U_k\) tends to be positive; if it has more, \(U_k\) tends to be negative. Thus the chosen orientation makes positive \(Z_k\) favor treatment. With tied events or stratification, the score and variance use the corresponding standard logrank calculations; the direction convention remains the same.

A short justification makes the sign concrete. At one event time, \(X_i=1\) for a treatment event. If \(p_i=0.5\), the contribution is \(0.5-1=-0.5\): an event in treatment pushes the score toward harm. If instead the event is in control, \(X_i=0\), and the contribution is \(0.5-0=0.5\): an event in control pushes the score toward benefit. Across the analysis, the variance standardizes the accumulated score but does not change this direction.

What Two-Sided Sequential Boundaries Mean

At each planned look, compare the current statistic with both boundaries. A statistic between them is not evidence to stop under this boundary rule, even if its sign points toward benefit or harm. A crossing is assessed at the first look where it occurs; if neither boundary has been crossed, monitoring continues according to the protocol. These rules concern the prespecified test statistic and boundaries, not every fluctuation in a plotted estimate.

\text{Stop for benefit at look }k\text{ if }Z_k\ge c_{U,k}; \qquad \text{stop for harm at look }k\text{ if }Z_k\le c_{L,k}.

The boundaries must be calibrated for the planned sequence of looks. Under the null hypothesis \(H_0\), the standardized statistics are generally dependent across looks: later statistics include information already accumulated at earlier looks. If the planned total two-sided type I error is \(\alpha\), calibration aims to control the probability of crossing either boundary at any look under \(H_0\):

P_{H_0}\left(\text{at least one upper or lower crossing over all looks}\right)\le \alpha.

Consequently, a boundary at one look is not generally the ordinary critical value for a single two-sided test at level \(\alpha\). Using that value independently at every look can inflate the overall type I error. The upper and lower boundaries also need not be mirror images: their values depend on the chosen group sequential design and how error is allocated across looks and tails. The canonical joint normal approximation uses correlations determined by information fractions; the analysis must use the planned sequential calibration, not a collection of unadjusted single-look tests.

The usual approximation requires a correctly calculated logrank score and variance, appropriate handling of risk sets and censoring, and event information that is adequate for the normal approximation. Randomization supports the null comparison; censoring should be non-informative for the comparison, possibly conditional on specified strata. Proportional hazards is not required merely to state the null test's direction, but it supports a simple interpretation of a sustained treatment effect and the usual drift approximation. If effects change over time, the ordinary logrank statistic summarizes evidence across event times, and its sign alone does not describe every part of the survival experience.

Worked Example: Checking the Direction From Risk-Set Contributions

Suppose, purely hypothetically, that at four event times the treatment proportions in the risk sets are \(0.50, 0.50, 0.60,\) and \(0.40\). The events occur in control, control, treatment, and control, respectively. For this illustration, use the simple untied-event contributions \(p_i-X_i\).

The contributions are \(0.50-0=0.50\), \(0.50-0=0.50\), \(0.60-1=-0.40\), and \(0.40-0=0.40\). Therefore \(U=0.50+0.50-0.40+0.40=1.00\). Using the corresponding simple variance contributions \(p_i(1-p_i)\), \(V=0.25+0.25+0.24+0.24=0.98\), so

Z=\frac{1.00}{\sqrt{0.98}}=1.01.

The statistic is positive, so its direction favors treatment: the accumulated events were fewer in treatment than expected under the null, accounting for the risk sets. This small illustration does not cross any particular sequential boundary; a boundary decision requires the boundaries calibrated for the actual monitoring plan.

Worked Boundary Decisions

Worked Example: Positive Statistic Below the Benefit Boundary

In a hypothetical trial, the prespecified upper boundary at the current look is \(c_{U,k}=2.45\), and the lower boundary is \(c_{L,k}=-2.45\). The observed logrank score is \(U_k=12.0\) with estimated variance \(V_k=25.0\). Assume the analysis uses the stated positive-favors-treatment convention and the specified event information is available.

The standardized statistic is

Z_k=\frac{U_k}{\sqrt{V_k}}=\frac{12.0}{\sqrt{25.0}}=\frac{12.0}{5.00}=2.40.

Since \(2.40<2.45\), there is no upper crossing; since \(2.40>-2.45\), there is no lower crossing either. The result points toward benefit, but does not meet the prespecified stopping criterion. It would be incorrect to stop for benefit just because \(Z_k\) is positive or because it is close to the boundary.

Worked Example: Crossing the Harm Boundary

In a second hypothetical trial, the current calibrated boundaries are \(c_{U,k}=2.60\) and \(c_{L,k}=-2.40\). At this look, the logrank score is \(U_k=-15.0\) and its estimated variance is \(V_k=36.0\). Assume the data and risk sets have been updated to the planned cutoff and the logrank approximation is appropriate.

The statistic is

Z_k=\frac{-15.0}{\sqrt{36.0}}=\frac{-15.0}{6.00}=-2.50.

Because \(-2.50\le -2.40\), the lower boundary is crossed. Under this sign convention the crossing is in the direction of harm: the score indicates more treatment-group events than expected under the null. The protocol's statistical stopping rule is met. This is evidence against \(H_0\) in the harmful direction under the specified monitoring plan; the boundary crossing by itself does not prescribe clinical management or replace a safety review.

Worked Example: Following Both Tails Across Looks

Consider a hypothetical two-look plan with calibrated boundaries \((c_{L,1},c_{U,1})=(-2.80,2.80)\) at the first look and \((c_{L,2},c_{U,2})=(-2.10,2.10)\) at the second. Suppose the first-look statistic is \(Z_1=-1.70\). Since \(-2.80<-1.70<2.80\), neither boundary is crossed, so monitoring continues.

At the second look, suppose the updated cumulative score and variance are \(U_2=-18.0\) and \(V_2=64.0\). Then

Z_2=\frac{-18.0}{\sqrt{64.0}}=\frac{-18.0}{8.00}=-2.25.

Now \(-2.25\le -2.10\), so the lower boundary is crossed and the stopping signal is for harm. The first-look statistic is not added to the second-look statistic: \(Z_2\) is already a standardized cumulative statistic through look 2. The planned joint calibration accounts for the dependence between these cumulative statistics.

Diagnostics, Alternatives, and Limits

Before interpreting a crossing, verify the items that determine its direction and validity:

Check the sign convention. Confirm from the analysis definition that positive values favor treatment. If the software reports the opposite contrast, reverse the interpretation and use boundaries defined for that orientation.
Check the analysis time and information. The statistic must use the risk sets and events available at the planned cutoff. A mismatch between realized and planned information can make a planned boundary inappropriate.
Check both tails at every look. A positive statistic is not automatically a benefit decision, and a negative statistic is not automatically a harm decision. Compare with the correct look-specific boundary.
Check data handling and assumptions. Errors in event status, treatment assignment, censoring, stratification, or variance calculation can change the statistic or its calibration. Unusually sparse information may also weaken the normal approximation.
Compare the score with clinical context. Review the survival curves, event patterns, and treatment effect estimates alongside the sequential test. A single logrank summary can conceal time variation in the treatment effect.

Two-sided monitoring is appropriate when a formally calibrated decision in either direction is part of the design. A one-sided efficacy design paired with a separate futility rule is an alternative when the formal stopping objective is benefit and a lower efficacy boundary is not intended. Those designs answer different questions: a futility rule is not automatically a calibrated harm test. Nor should an ordinary, unadjusted confidence interval or nominal p-value at an interim look be substituted for the prespecified sequential rule.

The main limitation is that the boundary governs statistical evidence for the logrank comparison, not the full clinical judgment. In particular, crossing the lower boundary does not establish why harm arose, and failure to cross it does not establish safety. When hazards vary over time, the logrank score can combine event-time contributions with different signs; therefore an overall positive or negative statistic should not be read as proof that the treatment helped or harmed every subgroup or at every time.

Key takeaway: With the positive-favors-treatment convention, the upper sequential boundary is for benefit and the lower boundary is for harm. Interpret a result only after checking its sign, its look-specific boundaries, and the joint two-sided calibration across all planned looks.

Check Your Understanding

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

With positive \(Z_k\) defined to favor treatment, what direction does a lower-boundary crossing represent?
At a look, \(U_k=8.0\), \(V_k=16.0\), and the upper boundary is \(2.20\). Is there a benefit crossing?
Why is it generally wrong to use the ordinary two-sided level-\(\alpha\) critical value at every interim look?
A statistic is negative but lies between the lower and upper boundaries. What can be concluded under the boundary rule?

Solutions. (1) It represents evidence in the direction of harm: with this orientation, a negative statistic indicates more treatment-group events than expected under the null. (2) \(Z_k=8.0/\sqrt{16.0}=2.00\), and \(2.00<2.20\), so there is no benefit crossing, although the direction favors treatment. (3) The cumulative statistics across looks are dependent, and repeated unadjusted testing can make the probability of any false crossing exceed \(\alpha\); the two tails and all looks must be jointly calibrated. (4) There is no boundary crossing at that look. The statistic points toward harm, but the prespecified stopping criterion has not been met.