Deriving Bias Under Linear Outcome Drift

Why enrollment time can enter an unadjusted contrast

An unadjusted treatment-minus-control mean difference compares outcomes by assigned arm without accounting for when participants enrolled. If outcomes change over enrollment time, the two arm means can reflect both the treatment effect and a difference in the times represented in each arm. Response-adaptive randomization can make such a time difference systematic: allocation probabilities may change as responses accrue, and those probabilities determine how enrollment times are distributed across arms.

The purpose here is to isolate the contribution of a common linear outcome trend. This is not a claim that every bias under adaptive randomization is caused by time drift. The earlier material on adaptive feedback establishes why allocation can be related to enrollment time; the key calculation is to translate that time imbalance into outcome units.

Definition: For participant \(i\), let \(A_i=1\) indicate assignment to treatment and \(A_i=0\) indicate assignment to control. Let \(t_i\) be the participant’s enrollment time on a consistent numerical scale. With \(n_1\) treatment participants and \(n_0\) control participants, the unadjusted contrast is \(\hat{\Delta}=\bar{Y}_1-\bar{Y}_0\). Define the arm-specific mean enrollment times as \(\bar{t}_1\) and \(\bar{t}_0\), and their difference as \(D_t=\bar{t}_1-\bar{t}_0\).

Model and finite-sample drift decomposition

Suppose the outcome has a constant treatment effect \(\tau\) and a common linear trend over enrollment time. A working model for the observed outcome is

Y_i=\mu+\tau A_i+\gamma t_i+\varepsilon_i,

where \(\mu\) is a reference mean, \(\gamma\) is the change in mean outcome per unit of enrollment time, and \(\varepsilon_i\) is the residual. The model assumes the same time slope in both arms and a treatment effect that does not vary with enrollment time. The enrollment-time scale and outcome units must be specified: for example, days and symptom-score points. To interpret the contrast as an effect plus drift, both arms must have at least one participant.

Taking the mean within each arm and subtracting gives an identity for any realized allocation schedule:

\hat{\Delta} =\tau+\gamma(\bar{t}_1-\bar{t}_0) +\left(\bar{\varepsilon}_1-\bar{\varepsilon}_0\right) =\tau+\gamma D_t+\left(\bar{\varepsilon}_1-\bar{\varepsilon}_0\right).

This identity separates the time-trend contribution, \(\gamma D_t\), from the residual contrast. If the expected residual contrast is zero under the design and outcome model, then the expected bias of the unadjusted contrast for \(\tau\) is

E(\hat{\Delta})-\tau=\gamma E(D_t).

The condition on the residual contrast matters. In a response-adaptive trial, assignments may depend on earlier outcomes. That dependence can affect the expected residual contrast, so the identity alone does not prove that the entire expected bias equals the drift contribution. It does show exactly how linear drift contributes; treating that contribution as the full bias requires the stated zero-mean condition or a design and model justification for it.

Key result: For a realized schedule, the linear-drift contribution to the treatment-minus-control contrast is exactly \(\gamma D_t\). In expectation, that contribution is \(\gamma E(D_t)\). It has outcome units because an outcome-per-time slope is multiplied by a time difference.

Worked calculations

Worked Example: Later Treatment Enrollment and Improving Outcomes

Consider a hypothetical trial with six participants enrolled in weeks 1 through 6. Suppose the first three are assigned to control and the last three to treatment. Assume a constant treatment effect of 2.0 points, a common outcome trend of \(+0.5\) points per week, and zero expected residual contrast for this schedule. Larger scores indicate better outcomes.

The mean enrollment time in treatment is

\bar{t}_1=\frac{4+5+6}{3}=5\text{ weeks},

and in control it is

\bar{t}_0=\frac{1+2+3}{3}=2\text{ weeks}.

Thus \(D_t=5-2=3\) weeks. The drift contribution is

\gamma D_t=(0.5\text{ points/week})(3\text{ weeks})=1.5\text{ points}.

The expected unadjusted contrast under these assumptions is \(2.0+1.5=3.5\) points. The extra 1.5 points do not represent a larger treatment effect: they arise because treatment participants enrolled later, when outcomes were improving.

Worked Example: Translating Changing Allocation Probabilities into Time Imbalance

Suppose four hypothetical enrollment times are \(t=(0,1,2,3)\) days, with treatment allocation probabilities \(\pi=(0.2,0.3,0.5,0.8)\). These probabilities rise across the four times. To approximate the resulting arm-specific time difference, weight each time by its treatment probability for treatment and by its control probability for control.

The treatment-weighted mean time is

\frac{\sum_i \pi_i t_i}{\sum_i \pi_i} =\frac{0(0.2)+1(0.3)+2(0.5)+3(0.8)}{0.2+0.3+0.5+0.8} =\frac{3.7}{1.8} \approx 2.056\text{ days}.

The control-weighted mean time is

\frac{\sum_i (1-\pi_i)t_i}{\sum_i (1-\pi_i)} =\frac{0(0.8)+1(0.7)+2(0.5)+3(0.2)}{0.8+0.7+0.5+0.2} =\frac{2.3}{2.2} \approx 1.045\text{ days}.

The approximate time difference is \(2.056-1.045=1.010\) days. If the common trend is \(+0.4\) points per day, the corresponding approximate drift contribution is \((0.4)(1.010)=+0.404\) points. This is a probability-weighted approximation, not the exact expected contrast for every adaptive design: actual arm sizes are random, and the allocation probabilities can depend on previous outcomes.

Worked Example: Checking the Sign When Outcomes Worsen

In a different hypothetical trial, suppose treatment participants enroll an average of 2 weeks later than control participants, so \(D_t=+2\) weeks. Larger outcomes still mean better status, but suppose outcomes worsen by 0.3 points per week. Then the slope is \(\gamma=-0.3\) points per week.

The drift contribution is

\gamma D_t=(-0.3\text{ points/week})(2\text{ weeks})=-0.6\text{ points}.

If the treatment effect is \(\tau=1.4\) points and the expected residual contrast is zero, the expected unadjusted contrast is \(1.4-0.6=0.8\) points. Later treatment enrollment produces a negative drift contribution here because later outcomes are worse under this outcome coding. If the outcome were instead coded so lower values were better, the slope signs and interpretation would need to be reconsidered together.

Using allocation probabilities carefully

A useful summary of a changing allocation rule is the association between enrollment time and the probability of treatment assignment. For fixed enrollment times \(t_i\) and probabilities \(\pi_i\), define \(\bar{t}=n^{-1}\sum_i t_i\) and \(\bar{\pi}=n^{-1}\sum_i\pi_i\). The weighted treatment-minus-control time difference is

D_t^{(w)} =\frac{\sum_i\pi_i t_i}{\sum_i\pi_i} -\frac{\sum_i(1-\pi_i)t_i}{\sum_i(1-\pi_i)} =\frac{\operatorname{Cov}_n(\pi,t)}{\bar{\pi}(1-\bar{\pi})},

where \(\operatorname{Cov}_n(\pi,t)=n^{-1}\sum_i(\pi_i-\bar{\pi})(t_i-\bar{t})\), and \(0<\bar{\pi}<1\). To verify the equality, write each weighted mean as a deviation from \(\bar{t}\). The treatment deviation is \(\operatorname{Cov}_n(\pi,t)/\bar{\pi}\); the control deviation is \(-\operatorname{Cov}_n(\pi,t)/(1-\bar{\pi})\). Subtracting the latter from the former gives the displayed result.

Thus an increasing association between treatment probability and time gives a nonnegative weighted time difference. It is strictly positive when the covariance is positive; monotone nondecreasing probabilities alone need not imply strict positivity if the probabilities do not vary over the observed times. The weighted summary is informative about the expected direction and scale of imbalance, but should not be confused with \(E(D_t)\) without conditions supporting that approximation. In particular, a random denominator such as the realized number assigned to treatment means that a ratio of expected counts is not generally the expectation of a ratio.

Assumptions, diagnostics, and alternatives

Before interpreting \(\gamma D_t\) as the full expected bias, check the assumptions that give it that role:

Common linear drift: mean outcomes change linearly with the stated time scale, at a common slope in both arms.
Constant treatment effect: \(\tau\) is the target at every enrollment time. A treatment effect that changes over time is outside this calculation.
Nonempty arms and a meaningful time scale: both arm means exist, and the slope and time difference use compatible units.
Residual condition: the expected residual mean difference is zero if \(\gamma E(D_t)\) is to represent the full expected bias. Adaptive feedback can make this condition nontrivial.
Allocation summary is appropriate: a probability-weighted time difference is an approximation unless the design justifies replacing the random realized time difference by that summary.

A practical diagnostic is to calculate \(\bar{t}_1-\bar{t}_0\) from the realized enrollment times and arm assignments, then multiply by a plausible or estimated slope. Compare the magnitude with the treatment contrast and report both the time difference and the slope in their units. Check that allocation probabilities actually varied over time and that their pattern supports the proposed direction. Also examine whether a single linear slope is plausible over the enrollment period; a changing or nonmonotone trend can make one slope misleading. These checks diagnose the drift calculation, but do not establish that the residual contrast is unbiased.

Approach	What it tells you	Important qualification
Unadjusted arm-mean difference	The observed difference in average outcomes	Combines treatment, time imbalance, and residual contrast.
Drift decomposition	The part attributable to a common linear trend, \(\gamma D_t\)	Needs a slope and time difference; it is not automatically the total bias.
Probability-weighted time summary	The direction and approximate scale of time imbalance implied by allocation probabilities	Does not generally equal the exact expected realized time difference.

Randomization does not guarantee that the realized arm means have identical enrollment times. Nor does a time difference alone prove bias: its contribution is zero when the slope is zero, and its sign depends on both the slope and the time difference. The calculation is most useful as a transparent decomposition, not as a substitute for checking the assignment mechanism, residual behavior, and plausibility of the trend model.

Common mistake

A frequent error is to infer the sign of the drift contribution from “improvement” without checking how the outcome is coded. If larger values mean better outcomes, improvement over time means \(\gamma>0\), not \(\gamma<0\). With later average treatment enrollment, \(D_t>0\), so \(\gamma D_t>0\). Conversely, if larger values mean better outcomes and outcomes worsen over time, then \(\gamma<0\), and the same positive time difference gives a negative contribution. Always write the units and multiply the signed slope by the signed time difference.

Key takeaway: Under a common linear trend, the drift component of an unadjusted treatment-minus-control contrast is the outcome change per time unit multiplied by the difference in arm-specific mean enrollment times. This equals the full expected bias only when the expected residual contrast is also zero.

Check Your Understanding

Answer each question, then compare with the solutions.

Outcomes increase by 0.25 points per day, and the treatment group enrolls an average of 3 days later. What is the drift contribution?
With larger values indicating better outcomes, what sign does \(\gamma\) have if outcomes worsen over time? If \(D_t>0\), what is the sign of the drift contribution?
Why is the probability-weighted time difference not automatically the exact expected realized time difference in an adaptive trial?
If the time difference is zero, does that prove the unadjusted contrast is unbiased for the treatment effect? Explain.

Solutions. (1) \((0.25\text{ points/day})(3\text{ days})=+0.75\) points. (2) Worsening means \(\gamma<0\); with \(D_t>0\), the product \(\gamma D_t\) is negative. (3) Realized arm sizes and assignments are random, so expectations of ratios need not equal ratios formed from expected counts; response-adaptive probabilities may also depend on earlier outcomes. (4) No. It removes the linear-drift contribution for that schedule, but the residual contrast can remain nonzero, and model assumptions such as a common linear trend may fail.