Why Adjust for Enrollment Time?
In a response-adaptive trial, treatment allocation can change as participants enroll. If prognosis also changes over enrollment, treatment groups may contain different proportions of participants enrolled at different times. A crude difference in outcome means can then mix a treatment contrast with a calendar-time contrast. Linear adjustment offers a compact way to account for a gradual time trend, provided that the trend is adequately represented by the model and there is enough treatment overlap across time.
This tutorial focuses on a continuous outcome and an additive model with a treatment indicator and centered linear enrollment time. The model is not a cure for every imbalance: its adjusted treatment coefficient is meaningful only under explicit modeling and design assumptions. In particular, a fitted coefficient is not automatically a causal effect merely because treatment was randomized.
The Additive Model and Its Coefficients
The model assumes a linear time trend and a common slope in both arms: it contains no treatment-by-time interaction. Thus it represents the treatment contrast as constant over the modeled time range. Centering time makes the intercept refer to average enrollment time; it does not by itself make the model correct or create treatment overlap.
For an observed outcome, a common estimation procedure is ordinary least squares (OLS), which chooses coefficients to minimize the sum of squared residuals. The fitted treatment coefficient is the adjusted contrast under this model. Its interpretation depends on the model, the observed design, and—if a causal interpretation is intended—the assignment mechanism and relevant pre-treatment information.
Assumptions Behind an Adjusted Treatment Contrast
It is useful to separate assumptions about describing outcomes from assumptions about causal interpretation. For the model to describe conditional means well, the mean outcome should be adequately linear in centered time within each treatment arm, and the treatment contrast should be adequately constant across time. Other prognostic features may require adjustment if they are associated with assignment and outcomes. Residual patterns can reveal some failures of this model, but no diagnostic can prove it is correct.
For a causal interpretation, define \(Y_i(1)\) and \(Y_i(0)\) as participant \(i\)’s potential outcomes under treatment and control. At a minimum, assignment must be conditionally exchangeable: given the relevant pre-assignment information, treatment assignment must not select participants according to their potential outcomes. In a response-adaptive design, that information may include the assignment history or other recorded variables used by the allocation rule. If conditioning on enrollment time alone does not suffice, a model containing only treatment and time may not remove the resulting selection.
There must also be treatment overlap at relevant enrollment times: both treatment choices need a nonzero chance of assignment, and the observed data must provide enough support to estimate their contrast without relying excessively on extrapolation. The treatment and outcome definitions must be consistent, and one participant’s assignment should not change another participant’s outcome for the causal contrast being studied. Finally, a causal interpretation of this particular coefficient requires the additive mean structure to be suitable for the target contrast. Randomization supports causal inference, but neither randomization nor a fitted regression makes an incorrect time model correct.
Worked Example: A Crude Difference and a Time-Adjusted Difference
Consider a hypothetical trial with a continuous outcome measured in points. The centered enrollment times are in months. The six records below are constructed to follow the additive model exactly.
| Participant | Treatment \(A_i\) | Centered time \(t_i\) (months) | Outcome \(Y_i\) (points) |
|---|---|---|---|
| 1 | 0 | -1 | 8 |
| 2 | 0 | -1 | 8 |
| 3 | 0 | 0 | 10 |
| 4 | 1 | 0 | 13 |
| 5 | 1 | 1 | 15 |
| 6 | 1 | 1 | 15 |
The sample mean centered time is zero: \((-1-1+0+0+1+1)/6=0\). The crude treatment mean is \((13+15+15)/3=43/3\), and the control mean is \((8+8+10)/3=26/3\). Therefore the crude difference is $$ \frac{43}{3}-\frac{26}{3}=\frac{17}{3}\approx 5.667\text{ points}. $$
Now consider \(\hat{Y}_i=10+3A_i+2t_i\). For a control at \(t=-1\), the prediction is \(10+3(0)+2(-1)=8\); for a control at \(t=0\), it is \(10+3(0)+2(0)=10\). For a treated participant at \(t=0\), it is \(10+3(1)+2(0)=13\); at \(t=1\), it is \(10+3(1)+2(1)=15\). Thus this equation fits all six records exactly.
In this fitted model, \(\hat{\beta}_A=3\) points and \(\hat{\beta}_T=2\) points per month. The adjusted treatment contrast is 3 points at any fixed time under the common-slope model. The crude difference is larger because the treated participants are enrolled later in these data and later time has a higher modeled outcome. Notice the limited overlap: both arms are observed at \(t=0\), but only control is observed at \(t=-1\), and only treatment at \(t=1\). The numerical fit is exact, yet extending a constant treatment contrast across the full range depends strongly on the model.
Why the Treatment Coefficient Has This Meaning
The model’s interpretation follows by comparing its predicted mean for treatment with its predicted mean for control at the same time. Let \(m(a,t)=\beta_0+\beta_A a+\beta_Tt\). Then
The time term cancels, so the treatment coefficient is a same-time contrast. This is a model-based comparison, not necessarily a comparison directly observed at every time. If the true treatment contrast changes with time, or if the linear time trend is wrong and treatment groups have different time distributions, the coefficient need not represent the desired time-specific or population-average causal effect.
Centering changes the reference point for the intercept, not the treatment contrast. If time is recentered as \(t_i^*=t_i-c\), then
The intercept changes to reflect the new reference time, while the treatment coefficient and slope remain the same in this additive model. Choosing the sample’s mean enrollment time as the reference is convenient, but another prespecified reference time can be used if it is meaningful and lies in a region with adequate data support.
Worked Example: Interpreting a Fitted Model in Context
Suppose a hypothetical study measures a symptom score in points, with lower scores indicating fewer symptoms. Enrollment time is centered at the study’s mean and measured in weeks. The fitted equation is \(\hat{Y}_i=18.4-2.7A_i-0.15t_i\).
At average enrollment time, \(t=0\), the predicted control score is \(18.4-2.7(0)-0.15(0)=18.4\) points. The predicted treatment score is \(18.4-2.7(1)-0.15(0)=15.7\) points. Their adjusted difference is \(15.7-18.4=-2.7\) points. Under the model, treatment is associated with a 2.7-point lower mean score than control at the same enrollment time.
The time slope is \(-0.15\) points per week: for either arm, the model predicts a mean score 0.15 points lower for each additional enrollment week. At \(t=8\), the predicted means are \(18.4-0.15(8)=17.2\) points for control and \(18.4-2.7-0.15(8)=14.5\) points for treatment. Their difference remains \(-2.7\) points. These are model-based interpretations; calling the contrast causal additionally requires the exchangeability, overlap, and model assumptions described above.
When Time and Treatment Cannot Be Separated
Linear adjustment cannot recover a treatment effect when assignment is perfectly determined by enrollment time. The issue is not solved by centering. If, for example, everyone at \(t=-1\) receives control and everyone at \(t=1\) receives treatment, then \(A_i=(t_i+1)/2\) for every participant. Substitution into the linear predictor gives
The data identify only the combined intercept and slope shown on the right, not \(\beta_A\) and \(\beta_T\) separately. For example, hypothetical mean outcomes of 10 points for control at \(t=-1\) and 14 points for treatment at \(t=1\) give an observed difference of 4 points. That difference could arise from a 4-point treatment effect with no time trend, or from no treatment effect and a 2-point-per-unit time trend, among many other combinations. Without overlap, the data alone cannot distinguish these explanations.
Worked Example: Diagnosing a Linear-Trend Failure
Suppose a hypothetical trial has equal numbers in both arms at each centered time \(t=-2,-1,0,1,2\). The control mean outcomes are 16, 11, 10, 11, and 16 points, respectively. At each time, the treatment mean is exactly 3 points higher.
The control mean across times is \((16+11+10+11+16)/5=64/5=12.8\) points. By symmetry, the linear slope estimated from these five control means is zero: the paired deviations at \(-2\) and \(2\), and at \(-1\) and \(1\), cancel. Because allocation is balanced at each time and the treatment mean is always 3 points higher, the additive regression estimates an adjusted treatment contrast of 3 points in this example.
However, the fitted control mean is 12.8 points at each time, leaving residual patterns \(3.2,-1.8,-2.8,-1.8,3.2\) points. The residuals show a pronounced curved pattern rather than random scatter around a linear time trend. Here the balanced comparisons preserve the 3-point contrast, but the model’s time component does not describe the outcome trend. With unequal treatment allocation over time, a misspecified trend can also affect the estimated treatment coefficient. Residual plots against enrollment time, arm-by-time summaries, and an allocation-probability timeline can help identify such concerns; they cannot establish that a linear adjustment is adequate.
Choosing Among Practical Approaches
| Approach | Strength | Main limitation |
|---|---|---|
| Crude difference in means | Simple and does not impose a linear outcome-time relationship. | Can be sensitive to chance or systematic differences in enrollment times when outcomes drift. |
| Linear time adjustment | Uses a compact model and adjusts for a shared linear trend across observed enrollment times. | Can mislead if the trend is not adequately linear, treatment effects vary over time, or overlap is weak. |
| Time-stratified standardization | Compares arms within time strata and combines contrasts using a stated target distribution. | Requires adequate observations in the relevant arm-by-time strata; sparse strata can make estimates unstable or unavailable. |
| Temporal blocking | Designs allocation to improve balance across enrollment periods. | Prevents neither outcome drift nor every realized imbalance; it is a design strategy, not a substitute for checking the outcome model. |
The linear model is most attractive when a gradual, approximately linear trend is plausible and there is meaningful treatment overlap through time. Time-stratified treatment effect standardization avoids imposing one linear slope but depends on support within the chosen strata and on the target distribution used to combine contrasts. Temporal blocking addresses allocation balance by design, whereas outcome-model adjustment addresses the relationship between outcomes, treatment, and time. These approaches answer related but distinct problems.
Common Mistakes and Practical Checks
- Calling \(\hat{\beta}_A\) the causal effect automatically. It is first a model-adjusted contrast. A causal claim also needs a defensible assignment and outcome model, relevant confounding control, and overlap.
- Assuming that centering fixes collinearity. Centering makes the intercept easier to interpret; it does not separate treatment from time when assignment is determined by time.
- Assuming the treatment contrast is constant because the fitted model says so. A common coefficient imposes a common contrast. It does not demonstrate that treatment effects are actually constant across enrollment.
- Checking only the fitted treatment coefficient. Inspect arm-by-time counts and outcome residuals against time. Sparse overlap and structured residual patterns are warnings that the coefficient may rely on extrapolation or an inadequate trend.
A fitted linear adjustment also does not settle every question about uncertainty. Adaptive assignment and dependence across participants can matter for uncertainty calculations, so the validity of a conventional model-based standard error should not be inferred from the coefficient’s interpretation alone.
Check Your Understanding
Answer each question, then compare your reasoning with the concise solutions.
- A fitted model is \(\hat{Y}=30+1.5A-0.4t\), where \(Y\) is measured in points and \(t\) in days from average enrollment. What is the adjusted treatment contrast, and what is the time slope?
- Why does centering enrollment time change the intercept but not the treatment coefficient in an additive model?
- At \(t=-1\), all participants are control; at \(t=1\), all are treated. Can a treatment coefficient be separated from a linear time trend? Explain.
- Name two diagnostics that could flag problems with a linear time adjustment, and state one limitation of each.
Solutions. (1) The adjusted treatment-minus-control contrast is \(1.5\) points at the same enrollment time. The modeled outcome changes by \(-0.4\) points per additional day for either arm. (2) Re-expressing time as \(t^*=t-c\) transfers the constant \(\beta_Tc\) into the intercept; the treatment term \(\beta_A A\) is unchanged. (3) No. Treatment is determined by time, so their effects are aliased; the observed difference combines treatment and time, and the model cannot identify them separately. (4) Arm-by-time counts can reveal sparse overlap but do not establish outcome-model adequacy. Residual plots against time can reveal systematic departures from a linear mean trend but cannot prove the absence of unobserved drift or other model failures.