The Tuning Parameter Gamma

Why Gamma Matters

In a two-arm doubly adaptive biased coin design (DBCD), a target allocation proportion and a next-patient assignment probability are different quantities. The target describes the allocation the design is tracking; the assignment probability governs one random assignment, conditional on the trial history so far. The tuning parameter $\gamma$ determines how strongly the current allocation imbalance modifies that next-patient probability.

This distinction matters in practice. A target may favor one treatment for ethical or statistical reasons, while the design still uses randomization to correct departures from that target. Gamma controls the correction, not the target itself. A larger gamma increases the response of the allocation probability to a given departure, but it does not change the target map or guarantee that the realized allocation immediately reaches the target.

The Two-Arm Rule and Its Conditions

Definition: At an assignment step, let $N_1$ and $N_2$ be the numbers already assigned to arms 1 and 2, with $n=N_1+N_2>0$. Let $x=N_1/n$ be the current proportion on arm 1, and let $\rho$ be the design's current target proportion for arm 1. The target for arm 2 is $1-\rho$. For $0<\rho<1$, $0<x<1$, and $\gamma\geq 0$, the two-arm DBCD assigns the next patient to arm 1 with probability $$ \pi_1 = \frac{\rho(\rho/x)^\gamma} {\rho(\rho/x)^\gamma+(1-\rho)((1-\rho)/(1-x))^\gamma}. $$ The probability of assignment to arm 2 is $\pi_2=1-\pi_1$.

The formula presumes both arms have already received at least one patient, so $x$ is defined and lies strictly between 0 and 1. It also presumes an interior target. These are conditions for this form of the rule; a design's separate initialization or boundary-handling provisions must be used when they do not hold. The target $\rho$ may be updated from data in a response-adaptive design, but at this assignment step the rule uses the target value currently supplied by the design.

At $\gamma=0$, both ratio terms are raised to zero, so the probability reduces to the target, $\pi_1=\rho$. For positive gamma, the ratio terms adjust the target-based weights according to how current allocation compares with the target. This tutorial focuses on that adjustment at a given state; the limiting behavior at gamma zero and at large gamma is treated separately.

Deriving Gamma's Role

The odds form makes the correction especially clear. Divide the arm 1 weight by the arm 2 weight. Taking logarithms then separates the target contribution from the contribution controlled by gamma.

\begin{aligned} \frac{\pi_1}{1-\pi_1} &= \frac{\rho(\rho/x)^\gamma} {(1-\rho)((1-\rho)/(1-x))^\gamma}\\ &= \frac{\rho}{1-\rho} \left(\frac{\rho(1-x)}{x(1-\rho)}\right)^\gamma,\\ \log\left(\frac{\pi_1}{1-\pi_1}\right) &= \operatorname{logit}(\rho) +\gamma\left\{\operatorname{logit}(\rho)-\operatorname{logit}(x)\right\}. \end{aligned}

Here $\operatorname{logit}(z)=\log\{z/(1-z)\}$ for $0<z<1$. To verify the last line, expand the logarithm of the powered ratio: $\log\{\rho(1-x)/(x(1-\rho))\}=\operatorname{logit}(\rho)-\operatorname{logit}(x)$. Thus gamma multiplies the log-odds discrepancy between the target and current arm 1 proportions. It is not a multiplier on $\rho$ itself.

Interpretation: If $x<\rho$, arm 1 is below its target share, and the log-odds discrepancy is positive. Increasing gamma therefore increases $\pi_1$ at that fixed state. If $x>\rho$, the discrepancy is negative, and increasing gamma decreases $\pi_1$. If $x=\rho$, the discrepancy is zero, so the assignment probability equals $\rho$ for every gamma.

This is a fixed-state statement: $\rho$ and $x$ are held constant while gamma varies. Across actual patients, allocations change and the target may also be updated, so the effect of changing gamma on an entire trial cannot be read from a single-step derivative alone.

Worked Probabilities

Worked Example: Correcting an Underrepresented Arm

Hypothetical setup. Suppose 100 patients have already been assigned, with 40 on arm 1 and 60 on arm 2. The current target for arm 1 is $\rho=0.60$, and the design uses $\gamma=1$. Then $x=40/100=0.40$, so arm 1 is below its target share. The counts are positive and the target is interior, satisfying the rule's conditions.

The arm-specific weights are $$ w_1=0.60(0.60/0.40)^1=0.60(1.5)=0.90, \qquad w_2=0.40(0.40/0.60)^1=0.40(0.6667)=0.2667. $$ Therefore, $$ \pi_1=\frac{0.90}{0.90+0.2667} =\frac{0.90}{1.1667} \approx 0.7714. $$

The next patient has about a 77.14% probability of assignment to arm 1 under this hypothetical rule. This exceeds the 60% target because arm 1 is currently underrepresented; it does not mean that the realized proportion after one assignment will become 77.14%.

Worked Example: Reducing Assignment to an Overrepresented Arm

Hypothetical setup. Suppose 50 patients have been assigned, 25 to arm 1 and 25 to arm 2, and the current target is $\rho=0.40$. Thus $x=25/50=0.50$, so arm 1 is above its target share. Take $\gamma=2$. Both current counts are positive, and the target is between zero and one.

The weights are $$ w_1=0.40(0.40/0.50)^2=0.40(0.64)=0.256, \qquad w_2=0.60(0.60/0.50)^2=0.60(1.44)=0.864. $$ Hence $$ \pi_1=\frac{0.256}{0.256+0.864} =\frac{0.256}{1.120} \approx 0.2286. $$

The next patient is assigned to arm 1 with probability about 22.86%, below its 40% target. The rule does not assign nobody to arm 1; it makes assignment to the overrepresented arm less likely, while preserving a positive probability for each arm in this interior example.

Worked Example: Local Sensitivity to Gamma

Hypothetical setup. Consider a state with $\rho=0.50$ and $x=0.40$, based, for example, on 40 of 100 previous patients assigned to arm 1. Let $\gamma=1$. The target is interior and both arms have positive counts, as required.

The odds formula gives $$ \frac{\pi_1}{1-\pi_1} = \frac{0.50}{0.50} \left(\frac{0.50(1-0.40)}{0.40(1-0.50)}\right)^1 = 1\left(\frac{0.30}{0.20}\right) =1.5. $$ Thus $\pi_1=1.5/(1+1.5)=0.60$.

For a local sensitivity calculation, write $\pi_1=\{1+\exp(-\eta)\}^{-1}$, where $\eta=\operatorname{logit}(\rho)+\gamma\{\operatorname{logit}(\rho)-\operatorname{logit}(x)\}$. Differentiating with respect to gamma while holding the state fixed gives $$ \frac{\partial\pi_1}{\partial\gamma} = \pi_1(1-\pi_1) \{\operatorname{logit}(\rho)-\operatorname{logit}(x)\}. $$

At the stated values, the discrepancy is $\log(0.50/0.50)-\log(0.40/0.60)=\log(1.5)\approx0.4055$. Therefore, $$ \frac{\partial\pi_1}{\partial\gamma} = (0.60)(0.40)(0.4055) \approx 0.0973. $$

At this state, a small increase of 0.1 in gamma would increase the arm 1 probability by approximately $0.0973(0.1)=0.00973$, or 0.973 percentage points, as a first-order approximation. This is a local approximation, not an exact change for a finite increase, and it applies only while the target and current allocation are held fixed.

Justification of the Restoring Direction

The odds representation also justifies the direction of correction without relying on numerical examples. Define $\delta=\operatorname{logit}(\rho)-\operatorname{logit}(x)$. Since the logit function is strictly increasing on $(0,1)$, $x<\rho$ implies $\delta>0$, while $x>\rho$ implies $\delta<0$. At fixed $\rho$ and $x$, the log odds are $\operatorname{logit}(\rho)+\gamma\delta$. The logistic function is strictly increasing, so the assignment probability moves upward with gamma when $\delta>0$, and downward with gamma when $\delta<0$. This is the restoring direction: the correction favors the arm that is below its target share.

More precisely, differentiation gives $\partial\pi_1/\partial\gamma=\pi_1(1-\pi_1)\delta$. Because $0<\pi_1<1$, the factor $\pi_1(1-\pi_1)$ is positive. The derivative's sign is therefore exactly the sign of the target-versus-current discrepancy. When the proportions match, $\delta=0$, the derivative is zero and the probability remains at the target.

Choosing and Checking Gamma

Gamma is a tuning parameter, not an estimated treatment effect or a measure of uncertainty. Choosing it expresses how strongly the design should react to the current allocation discrepancy under the specified rule. The same gamma can produce different probabilities at different states because the size and direction of the log-odds discrepancy also matter.

A useful check before implementation is to compute probabilities at representative interior states, including states below and above the target. Verify that the underrepresented arm receives a probability above its target share when gamma is positive, and that the overrepresented arm receives a probability below its target share. Also check that the probabilities sum to one and that both are between zero and one. These checks can expose swapped arm labels, an inverted allocation ratio, or an incorrect sign on the correction.

A larger gamma can make probabilities more responsive and, in some states, close to zero or one. That may reduce random assignment to an arm for a period and can make the design sensitive to a temporary imbalance or a noisy, changing target. A numerical implementation should evaluate the rule stably, especially if weights become very different in magnitude. The interior formula alone does not specify what to do when an arm has no prior assignments, when a target reaches a boundary, or when the design imposes minimum assignment probabilities; those require explicit design provisions.

Gamma also does not by itself establish that a trial has acceptable inferential properties. Allocation behavior must be evaluated in the context of the target definition, response-estimation procedure, and trial objectives. In particular, a target that changes with estimated responses can move at the same time that the allocation correction acts. The fixed-state interpretation remains valid for each step, but it does not describe the whole adaptive trajectory.

How This Rule Differs From Alternatives

Approach	Role of allocation	How it differs from positive-gamma DBCD
Fixed-probability randomization at the target	Uses probability $\rho$ for arm 1 at each assignment, regardless of current counts.	Does not modify that probability to correct the current allocation discrepancy.
Two-arm DBCD	Uses the target and current allocation proportions to form adjusted weights.	Gamma sets the strength of the current-state correction in the assignment probability.
Deterministic allocation to the underrepresented arm	Would assign the next patient based on which arm needs more allocation.	DBCD remains randomized for interior inputs; correction is probabilistic rather than a guaranteed next assignment.

These approaches answer different design needs. The target expresses the desired allocation objective; gamma controls the DBCD response to current imbalance. Changing gamma is not a substitute for choosing or justifying the target, and a more aggressive correction is not automatically preferable. The appropriate choice depends on the desired balance between tracking the target and retaining assignment randomness, as assessed for the intended design.

Common Mistakes and Limitations

Confusing the target with the assignment probability. The target is $\rho$; the next-patient probability is $\pi_1$, which generally differs from $\rho$ when gamma is positive and $x\neq\rho$.
Claiming gamma changes the target. Gamma changes the correction applied at the current state. It does not redefine the target proportion.
Reversing the correction. If arm 1 is below target, $x<\rho$, positive gamma should increase $\pi_1$ above $\rho$, not decrease it.
Interpreting a single probability as a realized allocation. A probability of 0.77 is a chance for the next assignment, not a promise about the next outcome or the final trial proportion.
Extrapolating a fixed-state result to the entire adaptive trial. After assignment, $x$ changes; a response-adaptive target may change as well. Whole-trial behavior requires considering that sequence of updates.

Key takeaway: In the two-arm DBCD rule, gamma multiplies the log-odds discrepancy between the target arm 1 proportion and the current arm 1 proportion. At a fixed interior state, increasing gamma strengthens correction in the restoring direction; it does not alter the target itself.

Check Your Understanding

Answer each question, then compare with the solutions below.

For $\rho=0.55$, $x=0.45$, and positive gamma, is $\pi_1$ above or below the target? Why?
If $x=\rho=0.30$, what is $\pi_1$ for any allowable gamma in the interior rule?
Suppose $\rho=0.40$, $x=0.50$, and $\gamma=1$. Calculate the arm 1 assignment probability from the odds form.
What does a positive value of $\partial\pi_1/\partial\gamma$ mean when the state is held fixed?
Name one condition under which the stated interior formula cannot be applied directly.

Solutions. (1) Since $x<\rho$, the discrepancy $\operatorname{logit}(\rho)-\operatorname{logit}(x)$ is positive. For positive gamma, the arm 1 log odds exceed the target log odds, so $\pi_1>\rho=0.55$. (2) The discrepancy is zero, so $\pi_1=\rho=0.30$. (3) The odds are $\frac{0.40}{0.60}\left\{\frac{0.40(1-0.50)}{0.50(1-0.40)}\right\} =\frac{2}{3}\left(\frac{0.20}{0.30}\right)=\frac{4}{9}$. Thus $\pi_1=(4/9)/(1+4/9)=4/13\approx0.3077$. (4) It means a small increase in gamma increases the arm 1 assignment probability at that particular fixed target and allocation state. (5) Examples include $x=0$ or $x=1$, where an arm has no prior assignments, or a target $\rho=0$ or $\rho=1$; these require separate design provisions.

Approach	Role of allocation	How it differs from positive-gamma DBCD
Fixed-probability randomization at the target	Uses probability \(\rho\) for arm 1 at each assignment, regardless of current counts.	Does not modify that probability to correct the current allocation discrepancy.
Two-arm DBCD	Uses the target and current allocation proportions to form adjusted weights.	Gamma sets the strength of the current-state correction in the assignment probability.
Deterministic allocation to the underrepresented arm	Would assign the next patient based on which arm needs more allocation.	DBCD remains randomized for interior inputs; correction is probabilistic rather than a guaranteed next assignment.

What You'll Learn