Why Unequal Randomization Changes the Calculation
A logrank analysis compares observed and expected treatment-arm events as participants remain at risk. When more participants are assigned to one arm, the treatment and control risk sets are generally unequal. The calculation still uses the actual risk sets at each event time; it does not replace them with an equal-allocation formula. Unequal allocation also changes the information available per event, which affects the number of events needed for a target level of precision or power.
The focus here is a worked calculation for a two-arm trial with event-driven interim analyses. The cumulative score and its variance are updated at each look, and the resulting statistic is standardized by the square root of the accumulated information. The calculations assume the direction is defined so that a positive score means more observed events in the treatment arm than expected under the null. If the treatment is beneficial, its score will tend to be negative under this convention.
For a tied event time with \(d\) events, the usual hypergeometric variance contribution is \(d(Y_1/Y)(Y_0/Y)(Y-d)/(Y-1)\), when \(Y>1\). This form accounts for sampling without replacement from the risk set. The single-event formula is the special case \(d=1\). In actual analyses, use the prespecified handling of ties consistently at every look.
Assumptions Behind the Worked Calculations
The event-level score and variance are based on the risk sets and event counts recorded at each analysis cutoff. For the usual logrank test, participants are assigned to one of two arms, and event or censoring times and group membership are available. Under the null hypothesis of equal event hazards, and with censoring that is independent of event time within the relevant groups, the logrank score has mean approximately zero and its variance is estimated by the accumulated event-level variance contributions. The independent-increment conditions established in the earlier material on independent logrank increments support the sequential use of cumulative scores and information.
The risk-set calculation itself does not require that the realized treatment fraction among those at risk equal the randomization fraction. Risk-set composition can change through censoring, staggered entry, and differences in event experience. The approximation that information per event is \(q(1-q)\), where \(q\) is the treatment allocation proportion, is most useful for planning when the risk-set fraction remains near \(q\). In particular, it is a planning approximation under the null or under modest departures from it, not a substitute for calculating information from actual risk sets.
Worked Example: Cumulative Logrank Statistic With a 3:2 Allocation
Suppose a hypothetical trial randomizes participants in a 3:2 treatment-to-control ratio. At four successive single-event times, the risk sets and observed treatment events are as follows. The table gives \(p=Y_1/Y\), expected treatment events \(p\) for each single event, and the variance contribution \(p(1-p)\).
| Event | \(Y_1\) | \(Y_0\) | \(O_1\) | Expected treatment events | Variance contribution |
|---|---|---|---|---|---|
| 1 | 60 | 40 | 1 | 0.6000 | 0.2400 |
| 2 | 48 | 32 | 0 | 0.6000 | 0.2400 |
| 3 | 30 | 20 | 1 | 0.6000 | 0.2400 |
| 4 | 20 | 13 | 0 | 0.6061 | 0.2388 |
For the first event, \(U_1=1-60/100=0.4000\) and \(V_1=(60/100)(40/100)=0.2400\). For the second, \(U_2=0-48/80=-0.6000\), with \(V_2=(48/80)(32/80)=0.2400\). The third contribution is \(U_3=1-30/50=0.4000\), \(V_3=0.2400\). The fourth is \(U_4=0-20/33=-0.6061\), with \(V_4=(20/33)(13/33)=260/1089=0.2388\).
Adding the contributions gives \(U=0.4000-0.6000+0.4000-0.6061=-0.4061\), and \(V=0.2400+0.2400+0.2400+0.2388=0.9588\). Therefore,
The statistic is slightly negative, indicating fewer treatment-arm events than expected under the null given these risk sets. This is a descriptive direction, not evidence of a treatment benefit by itself. A decision would require the prespecified sequential boundary or other inferential procedure. Notice that the last risk-set fraction, \(20/33=0.6061\), is not exactly the original allocation fraction \(0.6000\); the actual risk sets correctly determine the event contribution.
Sequential Information at Unequal Allocation
At each look, add the new score and variance contributions to the cumulative totals. The information fraction at a look is the accumulated information divided by the planned or final information on the same scale. It need not equal the fraction of events observed: individual event contributions can differ because the risk-set allocation differs across event times.
Worked Example: Two Looks With 2:1 Randomization
Consider a hypothetical trial with treatment-to-control randomization of 2:1, so the planned treatment fraction is \(q=2/3\). Assume, for this planning illustration, that the risk-set fraction stays close to \(2/3\) throughout follow-up. Suppose the first 40 events produce a cumulative score of \(U_1=2.400\), and the next 60 events contribute an additional score of \(1.800\). Use the approximate variance contribution \(q(1-q)\) per event.
The approximate information at the first look is \(I_1=40(2/3)(1/3)=8.889\). At the final look it is \(I_2=100(2/3)(1/3)=22.22\). Thus the information fraction at the interim look is \(8.889/22.22=0.4000\), equal here to the event fraction because the assumed information contribution per event is constant.
The interim statistic is \(Z_1=2.400/\sqrt{8.889}=0.80498\). The final cumulative score is \(U_2=2.400+1.800=4.200\), so \(Z_2=4.200/\sqrt{22.22}=0.8910\). These statistics are not decisions: efficacy or futility decisions depend on the boundary rules set for the design. In an actual analysis, calculate each look's variance from its observed risk sets rather than assuming a constant contribution.
The reason that allocation enters information can be seen directly from a single-event variance contribution. If a fraction \(q\) of the risk set is in treatment, then \(Y_1/Y\) is approximately \(q\), and the contribution is approximately \(q(1-q)\). This product is largest at equal allocation and decreases as the allocation becomes more unbalanced.
Given treatment risk-set fraction \(q\), the single-event logrank variance is \(q(1-q)\).
If the risk-set fraction is approximately stable over \(D\) events, the total information is approximately \(I=Dq(1-q)\).
For the same target information, an allocation with smaller \(q(1-q)\) requires more events, approximately in inverse proportion to that product.
How Allocation Ratio Changes Event Requirements
For a proportional-hazards planning calculation, a common large-sample approximation for the required number of events is
where \(D\) is the total event count, \(HR\) is the target treatment-to-control hazard ratio, \(1-\beta\) is the target power, and \(z_a\) is the \(a\)-quantile of the standard normal distribution. This is a conventional fixed-design approximation for a two-sided test; a sequential design must also account for its monitoring boundaries and planned power. The equation isolates the allocation effect: holding the other quantities fixed, event requirements are proportional to \(1/[q(1-q)]\).
Worked Example: Comparing Equal and 2:1 Allocation
Suppose a hypothetical proportional-hazards design targets \(HR=0.70\), two-sided \(\alpha=0.05\), and power \(0.80\). Use \(z_{0.975}=1.960\), \(z_{0.80}=0.8416\), and \(\log(0.70)=-0.3567\). The common numerator is \((1.960+0.8416)^2=7.849\), and \((\log(0.70))^2=0.1272\).
With equal allocation, \(q=0.5\), so \(q(1-q)=0.25\). The approximate event count is
or about 247 events. With 2:1 allocation to treatment, \(q=2/3\) and \(q(1-q)=2/9=0.2222\). Thus
or about 278 events. The event-count ratio is approximately \(0.25/(2/9)=1.125\), so this approximation requires 12.5% more events under 2:1 allocation than under equal allocation for the same target information. These are not complete sequential sample-size results: sequential boundaries, accrual, censoring, and the conversion from events to participants must be incorporated in a full design.
Diagnostics, Alternatives, and Limitations
Unequal allocation is not a reason to abandon the logrank method, but it makes careful risk-set bookkeeping especially important. At every look, check that treatment and control risk-set counts sum to the total risk set, that observed event counts sum to the total events, and that expected treatment events use the treatment fraction in the risk set at that event time. Reconstruct cumulative scores and variances by summing event-level contributions. A discrepancy between the event count and the total number of events in the data cutoff, a sudden change in risk-set composition, or a variance inconsistent with the risk sets can signal a data or calculation error.
A frequent mistake is to use the original randomization fraction for every event even when actual risk sets are available. That can produce incorrect expected counts and information. Use actual \(Y_1/Y\) and \(Y_0/Y\) for the analysis; use \(q\) only when making a clearly labeled planning approximation. Another mistake is to assume that 40% of the events always means 40% of the information. This is true in the constant-\(q(1-q)\) illustration, but not necessarily when risk-set composition and event contributions change over time.
Equal allocation is generally more efficient in terms of event information per participant when treatment and control participants have comparable event rates and follow-up. Unequal allocation may nevertheless be chosen for clinical, operational, or ethical reasons. A stratified logrank analysis, as covered in the earlier material on stratified sequential logrank analysis, calculates contributions within strata and then combines them; the same principle applies within each stratum, where its actual risk sets determine information. The simple \(Dq(1-q)\) approximation may then be inadequate if allocation fractions or event rates vary materially across strata.
Finally, the event-count formula assumes a proportional-hazards effect summarized by a single \(HR\). If hazards are not proportional, the standard logrank score may not have the intended power for the anticipated pattern of effects. That issue is distinct from unequal allocation: changing \(q\) changes the information efficiency, while a changing treatment effect can change what the ordinary logrank comparison detects. The calculations here describe allocation and ordinary logrank information, not a remedy for non-proportional hazards.
Check Your Understanding
Try each problem, then compare your work with the solutions below.
- At a single-event time, \(Y_1=45\), \(Y_0=55\), and the event is in the treatment arm. Calculate the score and variance contribution.
- A design has treatment allocation fraction \(q=0.75\). What is the approximate information per event, and how does it compare with equal allocation?
- Why can the information fraction differ from the event fraction in an actual sequential analysis?
- For a target information of 20, compare the approximate event counts under equal allocation and 2:1 treatment-to-control allocation.
Solutions. 1. The expected treatment events are \(45/100=0.4500\), so \(U=1-0.4500=0.5500\); the variance is \((0.45)(0.55)=0.2475\). 2. The information per event is \(0.75(0.25)=0.1875\), compared with \(0.25\) under equal allocation; it is 75% as large. 3. Event contributions depend on the treatment and control risk-set fractions, which can change over time, so accumulating events does not necessarily accumulate information at a constant rate. 4. Under equal allocation, \(D\approx20/0.25=80\) events. Under 2:1 allocation, \(D\approx20/(2/9)=90\) events. These are approximate counts assuming stable risk-set allocation.