Comparing Coin Rules While Holding the Target Map Fixed

Why Hold the Target and State Fixed?

A response-adaptive allocation rule has two distinct ingredients: a target map, which specifies desired arm shares at the current response-data state, and a coin correction, which uses the discrepancy between those shares and current allocation shares to produce the next assignment probabilities. Comparing rules is meaningful only if these ingredients are separated. If both the target map and correction function change at once, a difference in allocation probabilities cannot be attributed to either change alone.

Here the target shares and the current allocation state are held fixed. We compare several possible correction functions at that one state. The resulting numbers are conditional probabilities for the next participant, not target shares, accumulated allocation fractions, or estimates of treatment effects.

Definition: Suppose there are $K$ treatment arms. Let $\rho_k$ be the fixed target share for arm $k$, and let $\pi_k$ be its current allocation share. Assume $\rho_k>0$, $\pi_k>0$, $\sum_{k=1}^K\rho_k=1$, and $\sum_{k=1}^K\pi_k=1$. Define the relative allocation ratio $u_k=\rho_k/\pi_k$. Given a positive correction function $g$, the next-participant allocation probability for arm $k$ is $$ p_k=\frac{\rho_k g(u_k)}{\sum_{j=1}^K \rho_j g(u_j)}. $$

In a count-based implementation, the current share is typically $\pi_k=n_k/n$, where $n_k$ participants have been assigned to arm $k$ among $n$ assignments. The definition above assumes all current shares are positive. If an arm has not yet been assigned, its ratio is undefined; an implementation needs a specified initialization or other rule for that case. We do not silently substitute a value for a zero share.

The correction function must be positive on the ratios being used. Positivity ensures each unnormalized weight is positive, and normalization then gives valid probabilities summing to one. The examples below also use increasing correction functions with $g(1)=1$: an arm exactly at its target receives no relative correction in its weight. These features make the functions comparable, but do not make their allocation probabilities identical.

Three Correction Functions

The power correction is $g(u)=u^\gamma$, with $\gamma>0$. Larger values of $\gamma$ strengthen the relative response to differences in target-to-current ratios. A linear-blend correction is $g(u)=1-\lambda+\lambda u$, with $0<\lambda\leq 1$. It blends a constant weight with the ratio itself; $\lambda=1$ gives $g(u)=u$. A bounded rational correction is $g(u)=(1+\eta)u/(1+\eta u)$, with $\eta>0$. It is increasing, equals one at $u=1$, and is bounded above by $(1+\eta)/\eta$ as $u$ grows.

These functions are alternative specifications, not interchangeable labels for the same rule. The power function can grow without bound as $u$ grows; the linear blend also grows without bound when $\lambda>0$; the rational function limits the size of the multiplier. A bounded multiplier can temper the effect of an extreme ratio, but it does not eliminate the need to normalize across all arms.

Worked Example: Three Corrections at One Allocation State

Consider a hypothetical three-arm trial with fixed target shares $(\rho_1,\rho_2,\rho_3)=(0.50,0.30,0.20)$ and current shares $(\pi_1,\pi_2,\pi_3)=(0.40,0.40,0.20)$. Both share vectors are positive and sum to one, so the ratios are defined: $u_1=0.50/0.40=1.25$, $u_2=0.30/0.40=0.75$, and $u_3=0.20/0.20=1$.

Power correction, $\gamma=0.5$. The unnormalized weights are $0.50\sqrt{1.25}=0.5590$, $0.30\sqrt{0.75}=0.2598$, and $0.20\sqrt{1}=0.2000$. Their sum is $1.0188$. Thus $p_1=0.5590/1.0188=0.5487$, $p_2=0.2598/1.0188=0.2550$, and $p_3=0.2000/1.0188=0.1963$, to four significant figures. These are valid probabilities: they are positive and sum to one up to rounding. In this particular calculation, arm 1's probability, $0.5487$, exceeds its target share, $0.50$.

Linear blend, $\lambda=0.5$. The correction multipliers are $1.125$, $0.875$, and $1$. The weights are therefore $0.50(1.125)=0.5625$, $0.30(0.875)=0.2625$, and $0.20(1)=0.2000$, with total $1.0250$. The probabilities are $0.5625/1.0250=0.5488$, $0.2625/1.0250=0.2561$, and $0.2000/1.0250=0.1951$.

Bounded rational correction, $\eta=1$. Here $g(u)=2u/(1+u)$. The multipliers are $2(1.25)/(2.25)=1.1111$, $2(0.75)/(1.75)=0.8571$, and $1$. The weights are $0.5556$, $0.2571$, and $0.2000$; their sum is $1.0127$. The resulting probabilities are approximately $0.5486$, $0.2539$, and $0.1975$.

All three calculated probabilities for arm 1 exceed $0.50$ in this example. That is a result of these full sets of weights and their denominators, not a general consequence of arm 1 being below target. The other arms' weights affect the normalization.

What the Normalization Does

A useful way to understand the comparison is to divide the normalized probability by the target share. For any arm $k$,

\frac{p_k}{\rho_k} = \frac{g(u_k)}{\sum_{j=1}^K \rho_j g(u_j)}.

The denominator is the target-share-weighted average of all correction multipliers. Therefore, $p_k>\rho_k$ exactly when arm $k$'s multiplier exceeds this weighted average. The direction of the correction function matters, but the other arms and their target weights matter too.

The comparison also has a simple odds form. For any two arms $k$ and $\ell$, provided their target shares are positive,

\frac{p_k}{p_\ell} = \frac{\rho_k}{\rho_\ell} \frac{g(u_k)}{g(u_\ell)}.

This identity shows how the coin correction modifies the target odds. Under the power rule, for example, the odds multiplier is $(u_k/u_\ell)^\gamma$. Under the rational rule, it is the ratio of two bounded-function values. The identity is about relative probabilities; the normalization is still needed to obtain each arm's probability.

Justification: Let $W=\sum_{j=1}^K\rho_jg(u_j)$. Since every $\rho_j$ and $g(u_j)$ is positive, $W>0$. The probabilities $p_k=\rho_kg(u_k)/W$ are positive, and $$ \sum_{k=1}^Kp_k =\frac{\sum_{k=1}^K\rho_kg(u_k)}{W} =\frac{W}{W}=1. $$ Also, dividing $p_k$ by $\rho_k>0$ gives the weighted-average criterion above. Thus comparing an arm's probability with its target requires comparing its multiplier with the entire denominator, not merely checking whether $u_k$ exceeds one.

Worked Example: An Under-Target Arm Can Still Fall Below Its Target Probability

Take the hypothetical target shares $(0.10,0.80,0.10)$, current shares $(0.09,0.10,0.81)$, and power correction with $\gamma=0.5$. The ratios are $u_1=0.10/0.09=1.1111$, $u_2=8$, and $u_3=0.10/0.81=0.12346$. Arm 1 is below its target in the current allocation share.

Its weight is $0.10\sqrt{1.1111}=0.1054$. The other weights are $0.80\sqrt{8}=2.2627$ and $0.10\sqrt{0.12346}=0.03514$. The total is approximately $2.4033$, so $p_1=0.1054/2.4033=0.0439$. This is less than the target share $0.10$. Arm 1's multiplier is above one, but arm 2's large target-weighted correction makes the denominator much larger than that multiplier. The example demonstrates why “below target” alone cannot establish that $p_k>\rho_k$.

How Correction Strength Changes the Result

Worked Example: A Two-Arm Sensitivity Comparison

In a separate hypothetical state, let target shares be $(0.60,0.40)$ and current shares be $(0.50,0.50)$. The ratios are $1.20$ and $0.80$. Compare the power rule at $\gamma=0.5$ and $\gamma=1$, and the linear blend at $\lambda=0.25$.

For $\gamma=0.5$, the weights are $0.60\sqrt{1.20}=0.6573$ and $0.40\sqrt{0.80}=0.3578$, with total $1.0150$. The probabilities are approximately $0.6475$ and $0.3525$.

For $\gamma=1$, the weights are $0.60(1.20)=0.7200$ and $0.40(0.80)=0.3200$, with total $1.0400$. The probabilities are $0.6923$ and $0.3077$. For the linear blend, the multipliers are $1-0.25+0.25(1.20)=1.05$ and $1-0.25+0.25(0.80)=0.95$. The weights are $0.6300$ and $0.3800$, giving probabilities $0.6300/1.0100=0.6238$ and $0.3800/1.0100=0.3762$.

At this fixed state, the stronger power correction produces the largest probability for arm 1 among these three specifications. This is a numerical comparison at one state, not a general ranking for all targets, allocation states, or parameter values.

Comparison, Diagnostics, and Limits

Correction	Multiplier	Characteristic	Check when comparing
Power	$u^\gamma$, $\gamma>0$	Increasing; grows without bound for large $u$	Record $\gamma$; larger values can make extreme ratios influential
Linear blend	$1-\lambda+\lambda u$, $0<\lambda\leq1$	Increasing and positive; grows linearly for large $u$	Check $\lambda$ and calculate each weight before normalizing
Bounded rational	$(1+\eta)u/(1+\eta u)$, $\eta>0$	Increasing and bounded above	Check $\eta$; bounded weights do not remove denominator effects

A hand calculation or implementation audit should verify the target shares and current shares separately, confirm that each sums to one, and check that no current share used in a ratio is zero. Then calculate every $u_k$, every multiplier, every unnormalized weight, the common denominator, and the final $p_k$. Finally check positivity and that the probabilities sum to one, allowing only for small rounding error. A probability outside $[0,1]$, a zero denominator, or a substantial failure to sum to one signals an input, formula, or arithmetic error.

Do not compare only one arm's unnormalized weight, and do not mistake a correction multiplier for an allocation probability. The actual probability depends on all arms through the denominator. In particular, an arm with $\pi_k<\rho_k$ has $u_k>1$, and an increasing function with $g(1)=1$ gives it a multiplier above one. That fact alone does not imply $p_k>\rho_k$.

A fixed-state comparison also has a deliberate limitation: it does not say how the alternative rules will lead to different future allocation states. After an assignment, the current shares change, and in a response-adaptive design the target map may also be updated as response information arrives. Thus the probabilities calculated here isolate the coin-rule contribution at a specified input; they are not a comparison of complete trial trajectories, operating characteristics, or clinical outcomes. The introductory material on target maps and on allocation odds provides the broader design context, but neither replaces this fixed-input calculation.

Common Mistake

Wrong reasoning: “Arm 1 is below target, so its next-assignment probability must be above its target share.” The ratio $u_1>1$ may make its correction multiplier exceed one, but the probability compares that multiplier with the target-weighted average multiplier across all arms. If other arms have sufficiently large multipliers and target weights, the normalized probability can remain below $\rho_1$. The correction is to compute the complete denominator, or equivalently compare $g(u_1)$ with $\sum_j\rho_jg(u_j)$, before making that claim.

Key takeaway: With target map and allocation state held fixed, changing the correction function changes the arm weights and can change the next-assignment probabilities. The weights must be normalized across all arms; no single arm's under- or over-allocation determines its probability relative to target by itself.

Check Your Understanding

Answer each question using the fixed-state allocation rule. Solutions follow the questions.

For two arms with target shares $(0.40,0.60)$ and current shares $(0.50,0.50)$, calculate the ratios and the next-arm probabilities under the power rule with $\gamma=1$.
State the exact condition for $p_k>\rho_k$ in terms of the correction multiplier and the other arms.
Why is it invalid to calculate $u_k=\rho_k/\pi_k$ when $\pi_k=0$ without an additional design convention?
At one fixed state, what can be concluded from finding that one correction rule gives a larger $p_1$ than another? Name one conclusion that cannot be drawn from that calculation alone.

Solutions. For question 1, the ratios are $0.40/0.50=0.80$ and $0.60/0.50=1.20$. The power-rule weights at $\gamma=1$ are $0.40(0.80)=0.32$ and $0.60(1.20)=0.72$; their sum is $1.04$. Hence $p_1=0.32/1.04=0.3077$ and $p_2=0.72/1.04=0.6923$. For question 2, $p_k>\rho_k$ if and only if $g(u_k)>\sum_j\rho_jg(u_j)$. For question 3, division by zero is undefined, so a specified initialization or other rule is needed before the ratio-based formula can be applied. For question 4, one may conclude that the first rule assigns a larger conditional probability to arm 1 at that exact target and allocation state. One cannot conclude from that single-state comparison which rule produces a preferred full-trial trajectory or better operating characteristics.

Correction	Multiplier	Characteristic	Check when comparing
Power	\(u^\gamma\), \(\gamma>0\)	Increasing; grows without bound for large \(u\)	Record \(\gamma\); larger values can make extreme ratios influential
Linear blend	\(1-\lambda+\lambda u\), \(0<\lambda\leq1\)	Increasing and positive; grows linearly for large \(u\)	Check \(\lambda\) and calculate each weight before normalizing
Bounded rational	\((1+\eta)u/(1+\eta u)\), \(\eta>0\)	Increasing and bounded above	Check \(\eta\); bounded weights do not remove denominator effects