Why Plot a DBCD Sequence?
A DBCD changes its randomization probabilities as response data and allocation counts accumulate. A plot can make that feedback visible: it can show how estimated target shares move, how the realized allocation proportions develop, and what probabilities the design assigns to the next participant. These are related quantities, but they answer different questions. Reading one curve as if it were another can make a correctly operating design look wrong—or conceal a genuine problem.
A trajectory plot is a diagnostic of a sequential path, not a proof that a design is valid or that its long-run behavior is satisfactory. The plotted values depend on the target estimator, the observed sequence of outcomes and assignments, the chosen DBCD rule, and the point in the sequence at which each value is recorded. A useful plot makes those dependencies explicit.
Three Quantities, Three Interpretations
Consider \(K\) treatment arms. After \(n\) participants have been assigned, let \(N_j(n)\) be the number assigned to arm \(j\), and let \(r_{j,n}=N_j(n)/n\) be its observed allocation proportion when \(n>0\). Let \(\hat{\rho}_{j,n}\) be the estimated target share for arm \(j\), calculated from the response information available at that state. Target shares sum to one. They describe the allocation proportions the design is currently aiming toward; they are not themselves randomization probabilities.
The DBCD uses the estimated targets and current allocation proportions to determine the probability of assigning the next participant. For the standard power-weight form, with tuning parameter \(\gamma>0\), define the unnormalized weight for arm \(j\) as \(\hat{\rho}_{j,n}(\hat{\rho}_{j,n}/r_{j,n})^\gamma\). Normalizing across arms gives the next-assignment probability \(\pi_{j,n}\). This expression presumes positive target shares and positive observed allocation proportions; initialization and zero-count conventions must be specified when those conditions do not hold.
The three plotted quantities therefore have distinct meanings:
- Estimated target share \(\hat{\rho}_{j,n}\): the current estimated target for the arm, based on response data available through participant \(n\).
- Observed allocation proportion \(r_{j,n}\): the realized fraction of the first \(n\) participants assigned to the arm.
- Assignment probability \(\pi_{j,n}\): the probability, given the state after participant \(n\), of assigning participant \(n+1\) to the arm.
The timing distinction matters. A target or probability marked at state \(n\) is computed from information available after participant \(n\); \(\pi_{j,n}\) governs the next assignment. The allocation proportion \(r_{j,n}\), by contrast, summarizes the first \(n\) assignments. At \(n=0\), \(r_{j,0}\) is undefined, so a design needs an initialization rule before the usual formula can be evaluated.
Constructing an Informative Trajectory Plot
For each state \(n\) at which the design has a defined target and allocation state, record the estimated target shares and observed allocation proportions. Calculate the assignment probabilities from that same state and label them as probabilities for the next participant. For a two-arm design, one arm’s share determines the other’s, so a single arm’s curves are sufficient; for several arms, plot each arm separately or use clearly distinguished panels.
A practical plot usually puts enrollment state on the horizontal axis and share or probability, from zero to one, on the vertical axis. Use separate line styles for target, observed proportion, and next-assignment probability. Label the vertical axis as a share, proportion, or probability—not simply “allocation.” Mark the initial state or state the initialization convention. If response estimates are updated only at specified times, indicate those update times; a target that remains unchanged between updates should not be mistaken for a calculation error.
The observed proportion is a stepwise record of assignments, while the estimated target and probability can change whenever the state is updated. Do not smooth away the step structure when the purpose is to inspect the realized randomization path. If curves from multiple simulated sequences are summarized, show that they are summaries across sequences rather than a single trial’s realized path. A single trajectory cannot show how variable the design would be over repeated runs.
Worked Example: Calculating Three Curves for Two Arms
Suppose a hypothetical two-arm DBCD uses \(\gamma=1\). At enrollment state \(n=4\), there are \(N_A(4)=3\) assignments to arm A and \(N_B(4)=1\) to arm B. The estimated target shares are \(\hat{\rho}_{A,4}=0.60\) and \(\hat{\rho}_{B,4}=0.40\). Calculate the three plotted values for arm A.
The observed allocation proportion is \(r_{A,4}=3/4=0.75\). The target curve is at \(0.60\). The DBCD weight for A is \(0.60(0.60/0.75)^1=0.48\); for B it is \(0.40(0.40/0.25)^1=0.64\). Thus the probability of assigning the next participant to A is
At state \(n=4\), the plot therefore shows target \(0.60\), observed proportion \(0.75\), and next-assignment probability about \(0.4286\). Since A is currently above its estimated target, the DBCD probability for A is below that target. This is the intended direction of feedback for this state; it does not require the next assignment to be to B.
Why the Curves Do Not Coincide
A target share is a current aim, an observed proportion is a historical result, and an assignment probability governs a future random assignment. They need not match at any particular state. In particular, \(\pi_{j,n}\) is not a prediction that exactly a fraction \(\pi_{j,n}\) of participants already enrolled will be in arm \(j\). It is a conditional probability for one assignment, given the current state.
A short calculation makes the distinction precise. Given the information available after \(n\) assignments, write \(r_{j,n}=N_j(n)/n\), and let \(I_{j,n+1}\) equal one if participant \(n+1\) is assigned to arm \(j\), and zero otherwise. Its conditional mean is \(\pi_{j,n}\). Since \(N_j(n+1)=N_j(n)+I_{j,n+1}\), the conditional expected allocation proportion after the next assignment is
This is a weighted average of the current proportion and the next-assignment probability, not the next-assignment probability alone. A realized assignment can also differ from its probability: a probability of \(0.70\) permits either assignment, with arm \(j\) more likely but not certain. Over a short sequence, that randomness can leave the observed proportion visibly above or below the target.
Worked Example: Reading a Three-Arm State
In a hypothetical three-arm design at \(n=20\), suppose the counts are \(10,6,4\), the target shares are \(0.40,0.35,0.25\), and \(\gamma=1\). The observed proportions are \(0.50,0.30,0.20\). Calculate the next-assignment probabilities.
The three unnormalized weights are \(0.40(0.40/0.50)=0.3200\), \(0.35(0.35/0.30)\approx0.4083\), and \(0.25(0.25/0.20)=0.3125\). Their sum is approximately \(1.0408\). Normalizing yields
A suitable plot at state 20 would show the target vector \((0.40,0.35,0.25)\), observed proportions \((0.50,0.30,0.20)\), and next-assignment probabilities approximately \((0.3074,0.3923,0.3002)\). Arm 1 is overrepresented relative to its target and has a lower next-assignment probability than its target share; arm 2 is underrepresented and has a higher probability than its target share. The probabilities sum to one, up to rounding.
Diagnostics, Failure Modes, and Alternatives
A trajectory plot is most useful when it prompts a check, not when it is treated as a pass/fail test by visual impression alone. Examine whether each probability vector is in the interval from zero to one and sums to one, whether target shares sum to one, and whether the target and probability values are aligned with the state that generated them. Check abrupt changes against changes in response information or allocation counts. Unexpected jumps may reflect a genuine target update, sparse-data instability, a coding or logging error, or an inconsistent time index.
The plot also helps reveal the direction of feedback. Under the stated DBCD rule with positive shares and \(\gamma>0\), an arm whose current allocation proportion exceeds its target receives a weight adjustment below one, while an underrepresented arm receives an adjustment above one. The normalized probability depends on every arm’s weight, so this direction should be checked in the full probability vector rather than inferred from one curve in isolation. Large oscillations can be a sign of volatile target estimates or an aggressive response to allocation imbalance; they are not, by themselves, proof of an error.
Three presentation choices commonly mislead:
- Putting a next-assignment probability at the wrong time. Record \(\pi_{j,n}\) as the probability used for participant \(n+1\), or explicitly shift its plotted index. Do not silently compare a probability computed before an assignment with a proportion computed after it.
- Calling the target the assignment probability. The target share and DBCD probability are different quantities; display and label them separately.
- Expecting every realized allocation proportion to track the target closely. Random assignments and early imbalances can produce visible gaps. A gap is a reason to inspect the states and rule, not automatic evidence that randomization failed.
Trajectory plots complement, rather than replace, other summaries. A final allocation table reports end-of-study counts but hides how the design got there. A probability plot isolates the coin rule’s behavior but does not show whether the target estimator itself is moving. A target plot shows estimator movement but cannot reveal whether realized assignments followed the probabilities. A table of selected states can make arithmetic easy to audit, but can miss short-lived changes between the selected states.
| Display | What it shows well | What it can hide |
|---|---|---|
| Target trajectory | How estimated desired shares change | How assignments actually accumulated |
| Observed-proportion trajectory | The realized allocation history | The probability used at each step |
| Assignment-probability trajectory | The design’s next-assignment behavior | The realized random outcomes and target estimation process |
| Final allocation summary | End-of-sequence allocation counts | Intermediate feedback and volatility |
Plots have important limits. They do not establish that the target estimator is appropriate, that assumptions behind the target are met, or that a long-run theoretical property holds. A smooth-looking path from one hypothetical or observed sequence does not guarantee stable behavior in other sequences. For simulation studies, compare multiple trajectories or summarize across repeated runs when the question concerns variability; keep those summaries distinct from the path of any one run.
Worked Example: Diagnosing a Timing Misinterpretation
At state \(n=8\) in a hypothetical two-arm run, suppose \(N_A(8)=5\), so \(r_{A,8}=5/8=0.625\), and the design calculates \(\pi_{A,8}=0.674\). Participant 9 is assigned to B. A reviewer sees the post-assignment proportion \(r_{A,9}=5/9\approx0.5556\) and says the \(0.674\) probability “failed” because the observed proportion fell.
The comparison is mistaken. The value \(0.674\) was the conditional probability of assigning participant 9 to A, not a target value for the proportion after that participant. Since participant 9 went to B, the count for A stays at 5, and its observed share becomes \(5/9\approx0.5556\). That realized outcome is possible under a probability of \(0.674\); it does not contradict the randomization rule. The plot should label \(0.674\) as the probability at state 8 for participant 9, and \(0.5556\) as the realized proportion at state 9.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- At state \(n\), what does \(\pi_{j,n}\) refer to, and how does it differ from \(r_{j,n}\)?
- A two-arm design has \(r_{A,n}=0.40\) and \(\hat{\rho}_{A,n}=0.60\). For \(\gamma>0\), what qualitative direction should the DBCD feedback take for arm A?
- Why can the observed allocation proportion move away from a target even when the plotted assignment probability is calculated correctly?
- What should be checked if the plotted assignment probabilities for three arms sum to \(1.08\)?
Solutions. (1) \(\pi_{j,n}\) is the conditional probability of assigning the next participant, \(n+1\), to arm \(j\); \(r_{j,n}\) is the realized fraction of the first \(n\) participants assigned to that arm. (2) Arm A is below its target, so the DBCD adjustment favors it relative to its current share; its assignment probability should be interpreted after normalization with the other arm’s weight. (3) The probability applies to one random assignment, not a guaranteed fractional allocation. A realized assignment can be to another arm, changing the observed proportion accordingly. (4) Check the unnormalized weights, normalization denominator, and whether the values are probabilities from the same state. Small discrepancies may be rounding, but a sum of \(1.08\) is too large to attribute to ordinary rounding of displayed values alone.