Effect Of Allocation Ratio On Logrank Information

Why Allocation Ratio Changes Information

In a logrank analysis, information accumulates from the comparisons made at event times. At each event, the treatment and control participants who are still at risk determine how much the event can distinguish the groups. If one group is sparsely represented in the risk set, the event generally contributes less information than it would with a more balanced risk set.

This creates a design trade-off. Equal allocation often gives the greatest logrank information for a fixed number of events, but trials may use unequal allocation for clinical, recruitment, or operational reasons. The effect of a planned allocation ratio is therefore best understood through the treatment fraction actually present in the risk sets, not through the enrollment ratio alone.

Definition: At event time $j$, let $Y_{1j}$ and $Y_{0j}$ be the numbers at risk in the treatment and control groups, respectively. The risk-set treatment fraction is $q_j=Y_{1j}/(Y_{1j}+Y_{0j})$. For a single event at that time, the usual logrank variance contribution is $$ V_j=\frac{Y_{1j}Y_{0j}}{(Y_{1j}+Y_{0j})^2}=q_j(1-q_j). $$ The cumulative logrank information through a set of distinct event times is the sum of these variance contributions.

This expression is the risk-set variance for the logrank score under the null hypothesis, with the usual independent-subject and censoring assumptions for the logrank test. It is a property of the groups at risk at the event time. If there are tied events, the variance is calculated using the applicable tied-event logrank variance rather than treating every tied event as an independent single-event contribution. The examples below use distinct event times so that the displayed expression applies directly.

From Allocation Ratio to Information

Write $r=n_1/n_0$ for the planned treatment-to-control enrollment ratio, where $n_1$ and $n_0$ are the numbers assigned to treatment and control. The planned treatment fraction is $q=r/(1+r)$. If the risk-set treatment fraction remains close to this planned fraction over the event times, the approximate information per event is

q(1-q)=\frac{r}{(1+r)^2}.

This approximation assumes that the event count is fixed for the comparison and that risk-set composition is reasonably represented by the planned allocation fraction. It is not an identity for every realized trial. Risk-set membership can differ from enrollment because of event histories and censoring; consequently, the event-level risk-set fractions are the direct inputs to observed information.

Worked Example: Information From One Risk Set

In a hypothetical trial, 40 treatment participants and 60 control participants are at risk immediately before one event. Calculate the logrank variance contribution from this event.

The total risk set is $40+60=100$, so the treatment fraction is $q=40/100=0.40$. Using the single-event formula,

V=\frac{40(60)}{100^2}=\frac{2400}{10000}=0.2400.

This event contributes 0.2400 units of logrank information. The contribution is less than the maximum of 0.2500 for a risk set of the same size split equally between groups. The risk-set size alone does not determine the contribution: its division between treatment and control matters too.

Why Equal Allocation Maximizes Information Per Event

For a fixed event count and a stable risk-set treatment fraction $q$, the approximate information is the event count multiplied by $q(1-q)$. The maximum of this expression can be established directly:

q(1-q)=\frac14-\left(q-\frac12\right)^2\leq\frac14.

Equality holds only when $q=1/2$. In terms of the treatment-to-control ratio $r$, the same result follows by differentiating $f(r)=r/(1+r)^2$:

f'(r)=\frac{1-r}{(1+r)^3}.

For $r>0$, the denominator is positive. Thus $f'(r)>0$ when $r<1$, $f'(r)=0$ when $r=1$, and $f'(r)<0$ when $r>1$. Therefore $f(r)$ is maximized at $r=1$, the equal-allocation ratio. This is a statement about information per event when the risk-set fraction is represented by the planned fraction. It does not say that every trial should use equal allocation: the number of events, enrollment cost, treatment availability, and other design constraints can differ across allocation choices.

Worked Example: Equal Versus 2:1 Allocation at Fixed Event Count

Compare two hypothetical designs with the same total enrollment of 600 participants and approximately 300 observed events. In the first design, enrollment is equal; in the second, the treatment-to-control ratio is 2:1. Assume for this planning comparison that the risk-set fraction remains close to the enrollment fraction and that the event count is the same.

With equal allocation, $q=300/600=0.50$. Approximate information is

I_{\mathrm{equal}}\approx 300(0.50)(0.50)=75.00.

With 2:1 allocation, 400 participants are assigned to treatment and 200 to control, so $q=400/600=2/3$. The approximate information is

I_{2:1}\approx 300\left(\frac23\right)\left(\frac13\right) =300\left(\frac29\right)=66.67.

The 2:1 design has $66.67/75.00=0.8889$, or about 88.89%, of the approximate information of the equal-allocation design for this event count. Equivalently, it has about 11.11% less information under the stated approximation. This is not a claim that 2:1 allocation is always inferior overall: it uses the same total enrollment here, but design decisions may also depend on reasons for assigning more participants to treatment.

Planned Allocation Is Not the Risk-Set Allocation

The approximation $I\approx Dq(1-q)$, where $D$ is the number of events and $q$ is the planned treatment fraction, is useful for planning comparisons. In an observed analysis, the cumulative information is obtained by adding the variance contributions from the actual risk sets. The distinction matters because the number still at risk in each group can change over time. Even when enrollment begins at a fixed ratio, the treatment fraction among participants still at risk need not remain constant.

Worked Example: Summing Information Across Changing Risk Sets

Suppose a hypothetical trial has three distinct event times. Immediately before the events, the risk sets are as follows: 30 treatment and 70 control participants, then 40 treatment and 60 control participants, then 60 treatment and 40 control participants. Calculate the total logrank information from these three event times.

For the first event, $q_1=30/100=0.30$, giving $V_1=(0.30)(0.70)=0.2100$. For the second, $q_2=40/100=0.40$, giving $V_2=(0.40)(0.60)=0.2400$. For the third, $q_3=60/100=0.60$, giving $V_3=(0.60)(0.40)=0.2400$. Therefore,

$$ I=V_1+V_2+V_3=0.2100+0.2400+0.2400=0.6900. $$

An analysis that substituted a single assumed fraction of 0.50 for all three risk sets would give $3(0.50)(0.50)=0.7500$, which is not the information calculated from these risk sets. This example illustrates why observed information should be based on the risk sets at the event times rather than inferred solely from the original enrollment ratio.

Allocation Choices Under Different Constraints

The comparison changes when the constraint is not fixed total enrollment. For example, a design may hold the control-group size fixed and add treatment participants. The total event count may then rise, so total information can increase even though information per event falls. These are different comparisons and should not be conflated.

Worked Example: Unequal Allocation With a Fixed Control Group

Consider two hypothetical designs with 200 control participants each. The first assigns 200 treatment participants, giving equal allocation and 400 participants total; the second assigns 400 treatment participants, giving a 2:1 ratio and 600 participants total. Assume, for illustration, that each design has events equal to one half of its enrollment and that risk-set fractions track enrollment fractions.

The equal-allocation design has 200 events and $q=0.50$, so its approximate information is

I_{\mathrm{equal}}\approx 200(0.50)(0.50)=50.00.

The 2:1 design has 300 events and $q=2/3$, so its approximate information is

I_{2:1}\approx 300\left(\frac23\right)\left(\frac13\right) =66.67.

Here the unequal-allocation design has more total information because it has more assumed events, despite having less information per event. It also requires 200 additional treatment participants. The example does not establish that this design is more efficient per participant; it separates the effect of allocation on each event's information from the effect of changing the number of events.

Diagnostics, Comparisons, and Limitations

For a group sequential logrank design, allocation affects the information accumulated by each analysis look because information is built from the risk-set variance contributions observed up to that look. A planned ratio can provide a useful anticipated scale, but the realized information is the sum of the contributions actually available. Thus, a planned event count by itself does not guarantee a particular amount of information.

Useful checks include the following:

At each event time, verify that both group-specific risk-set counts are nonnegative and that their sum matches the total risk set used in the variance calculation.
Check that calculated contributions are nonnegative and no greater than 0.25 for a single distinct event. The bound follows from $q(1-q)\leq 0.25$.
Compare the planned allocation fraction with the treatment fractions in the observed risk sets. Large or systematic differences mean the simple $Dq(1-q)$ approximation may be poor.
Use the variance calculation appropriate to the handling of tied event times. Applying a single-event formula to tied events without adjustment can misstate information.

The main alternative to equal allocation is not one universal ratio but a design chosen to meet constraints while accepting the resulting information per event. Equal allocation maximizes that quantity under the fixed-event approximation. Unequal allocation can be reasonable when it serves a separate design objective, but its efficiency should be evaluated using both the expected event count and the risk-set variance contributions. Comparisons are limited when the designs differ in event rates, follow-up patterns, or risk-set composition: those differences can change the total information, and the simple allocation formula alone cannot account for them.

Common pitfall: “A 2:1 enrollment ratio means every event contributes the 2:1 variance, so total information is always 8/9 of equal allocation.” The correction is that $8/9$ compares information per event under a stable 2:1 risk-set fraction. Actual information is calculated from the treatment and control counts at risk at each event time, and total information also depends on how many events occur.

Check Your Understanding

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

At a distinct event time, 25 treatment and 75 control participants are at risk. What is the logrank variance contribution?
Under the fixed-event, stable-risk-set approximation, what fraction of equal-allocation information per event is expected with a 3:1 treatment-to-control ratio?
Why can the enrollment ratio fail to equal the treatment fraction in the risk sets at later event times?
A design has fewer information units per event under unequal allocation but a larger expected event count. Can its total information exceed that of an equal-allocation design? Explain.

Solutions. (1) The treatment fraction is $25/100=0.25$, so the contribution is $(0.25)(0.75)=0.1875$. (2) For a 3:1 ratio, $q=3/4$, so per-event information is $(3/4)(1/4)=3/16=0.1875$. Equal allocation gives $0.25$, so the fraction is $0.1875/0.25=0.7500$, or 75%. (3) Participants leave risk sets through events and censoring, so the group composition among those still at risk can change; the relevant fraction is calculated at each event time. (4) Yes. Total information combines information per event with the number and risk-set composition of events, so a sufficiently larger event count can offset a smaller contribution per event.