Why Adjust for Calendar Time?
In a platform trial, treatment arms may enter or leave at different calendar dates. During that time, eligibility rules, clinical practice, outcome measurement, or the mix of enrolled patients may change. A treatment group enrolled mostly later can therefore differ from a control group for reasons besides treatment. Regression adjustment represents calendar time in an outcome model so that the treatment comparison is made conditional on the modelled time pattern.
Adjustment is not a substitute for a credible comparison group. In particular, including calendar time in a regression does not by itself make nonconcurrent controls exchangeable with patients assigned to a newer arm. The model needs adequate overlap in enrollment time and a defensible assumption that relevant differences in outcomes are captured by the included variables and their specified forms.
Two Common Ways to Represent Time
With \(K\) calendar periods, a categorical-period model includes an indicator for each period except a reference period. For a continuous outcome, an additive model can be written as:
Here, \(\beta_A\) is the modelled treatment-minus-control difference at the same period. The period indicators allow each period to have its own mean level, without requiring the changes between periods to follow a straight line. The additive form assumes the treatment contrast is the same in every period; it does not include a treatment-by-period interaction.
If time is represented by a numeric index, a simple smooth trend is a linear term:
This model assumes that the expected outcome changes by the same amount for each one-unit increase in \(t_i\), and again assumes a constant treatment contrast across time. A more flexible smooth specification could use a prespecified polynomial or spline basis, replacing \(\beta_t t_i\) with a sum of smooth basis functions. Greater flexibility can represent curvature, but requires enough data across time to estimate the additional parameters reliably.
The choice is substantive, not merely computational. Categorical periods make fewer assumptions about the shape of the time pattern but can be unstable when some periods contain few participants or no comparison group. A linear trend is parsimonious, but can leave important temporal structure unexplained if the outcome changes nonlinearly. The introductory material on calendar time as a confounder motivates why time matters; here, the focus is how the outcome model encodes it.
Assumptions Behind the Adjusted Coefficient
An adjusted coefficient has the interpretation implied by its model only when several conditions are plausible. First, the time variable or period definition must correspond to the calendar-time differences relevant to the outcome. Enrollment date is not the same as follow-up time: an outcome measured six months after randomization should not be adjusted using follow-up duration as though it were calendar time.
- Conditional comparability: after conditioning on treatment, calendar time, and any other included covariates, there are no unmeasured causes of both treatment-group membership and the outcome that remain relevant to the comparison. This is especially demanding for nonconcurrent controls.
- Overlap: there must be sufficient treatment and control information at comparable calendar times to support the adjusted comparison. If treatment appears only in late periods and control only in early periods, a model may produce a coefficient by extrapolating its assumed time trend rather than by comparing like periods.
- Correct outcome-model form: the chosen period indicators or smooth function must adequately represent the outcome's temporal pattern, conditional on the other model terms.
- Constant contrast for an additive model: without a treatment-by-time term, the model assumes the treatment effect on the model's scale does not vary over calendar time.
- Appropriate outcome and error model: ordinary least squares is suitable for estimating a conditional mean for a continuous outcome when its mean structure is correctly specified. Inference also requires an appropriate variance calculation; for example, heteroskedasticity or clustering may require robust or cluster-aware standard errors.
For a binary outcome, an analogous model can use a link function, such as the logit link. Then the treatment coefficient is on the link scale (log odds for logistic regression), not automatically an adjusted risk difference. The scale must be stated, and the assumptions of that model still apply.
Worked Example: Categorical Period Adjustment
Suppose a hypothetical platform comparison has two calendar periods, with 10 participants in each treatment-by-period cell. The table shows the observed mean outcome, in points.
| Calendar period | Control mean | Treatment mean | Treatment minus control |
|---|---|---|---|
| Period 1 | 10 | 8 | -2 |
| Period 2 | 16 | 11 | -5 |
Assume for this calculation that there are 10 observations in every cell. The overall control mean is \((10+16)/2=13\), the treatment mean is \((8+11)/2=9.5\), and the grand mean is \((10+16+8+11)/4=11.25\). In the balanced design, the additive treatment-plus-period model's fitted cell means equal the row mean plus the column mean minus the grand mean.
For control in Period 1, the fitted mean is \(13+9-11.25=10.75\). For control in Period 2, it is \(13+13.5-11.25=15.25\). For treatment in Period 1, it is \(9.5+9-11.25=7.25\), and for treatment in Period 2 it is \(9.5+13.5-11.25=11.75\).
The fitted treatment contrast in each period is therefore \(7.25-10.75=-3.5\) points and \(11.75-15.25=-3.5\) points. Thus, the additive model estimates a common adjusted contrast of \(-3.5\) points, even though the observed contrasts are \(-2\) and \(-5\). The discrepancy occurs because the additive model imposes a common contrast; its fitted means need not reproduce each observed cell mean when those contrasts differ. This example demonstrates the model's implication, not evidence that a constant treatment effect is true.
What a Smooth-Time Adjustment Estimates
For a linear regression with an intercept, treatment indicator \(A_i\), and time \(t_i\), the estimated coefficient of \(A_i\) can be understood by first removing the linear association with time from both treatment and outcome. Regress \(A_i\) on an intercept and \(t_i\), and separately regress \(Y_i\) on an intercept and \(t_i\). The coefficient of treatment in the full regression is the cross-product of those two sets of residuals divided by the sum of squared residuals from the regression of treatment on time. This is the partial-regression identity.
This identity explains the role of overlap. If treatment assignment is nearly determined by time, the residual variation in treatment after removing its time trend is small. The adjusted coefficient then relies on little information about treatment comparisons at similar times and may be highly uncertain. If the residual variation is exactly zero, the treatment and time terms are collinear and their separate coefficients cannot be estimated.
Worked Example: Linear Calendar-Time Adjustment
Consider nine hypothetical participants with a continuous outcome measured in points. Let \(t=0,1,2,3\) index successive calendar periods.
| Participant | \(A_i\) | \(t_i\) | \(Y_i\) |
|---|---|---|---|
| 1 | 1 | 0 | 25 |
| 2 | 1 | 1 | 4 |
| 3 | 1 | 1 | 6 |
| 4 | 1 | 2 | 2 |
| 5 | 1 | 2 | 3 |
| 6 | 1 | 3 | 12 |
| 7 | 0 | 1 | 10 |
| 8 | 0 | 1 | 11 |
| 9 | 0 | 1 | 13 |
The treatment mean is \(52/6=8.667\) points and the control mean is \(34/3=11.33\) points. The crude difference is \(8.667-11.33=-2.667\) points. For the adjusted coefficient, the required totals are \(\sum A_i=6\), \(\sum t_i=12\), \(\sum t_i^2=22\), \(\sum A_it_i=9\), \(\sum Y_i=86\), \(\sum A_iY_i=52\), and \(\sum t_iY_i=90\). In particular, the products \(t_iY_i\) sum to \(90\).
After centering, the treatment sum of squares is \(6-6^2/9=2\), the time sum of squares is \(22-12^2/9=6\), and the treatment-time cross-product is \(9-(6)(12)/9=1\). The residualized treatment sum of squares is therefore:
The centered treatment-outcome cross-product is \(52-(6)(86)/9=-16/3\), and the centered time-outcome cross-product is \(90-(12)(86)/9=-74/3\). Removing the linear time association from their cross-product gives:
Consequently, the treatment coefficient is \((-11/9)/(11/6)=-2/3\), or approximately \(-0.667\) points. Under the assumed additive linear-time model, treatment is associated with a \(0.667\)-point lower mean outcome than control at the same modelled calendar time. The crude difference is not the adjusted estimate. This tiny dataset is for arithmetic illustration only; it provides no basis for reliable statistical inference, and its timing pattern makes the overlap concern especially important.
Checking Whether the Time Specification Is Plausible
A smooth trend should not be selected just because it gives a convenient coefficient. Plot-based diagnostics can reveal temporal imbalance and outcome patterns; model diagnostics should then check whether the chosen regression leaves structure unexplained. Examine residuals against calendar time, compare observed and fitted outcomes across time, and inspect whether treatment and control observations occupy overlapping periods. A residual pattern that rises and falls systematically suggests that a straight-line trend may be inadequate. Very sparse periods, highly influential observations, and a treatment indicator that is almost predictable from time are warnings that the adjusted comparison may be driven by modelling assumptions.
Worked Example: Detecting Curvature in a Time Trend
Suppose hypothetical control-group mean outcomes in four successive periods are 10, 11, 14, and 19 points, for \(t=0,1,2,3\). A linear regression of these four means on time has mean outcome \(13.5\), mean time \(1.5\), centered time sum of squares \(5\), and centered time-outcome cross-product \(15\). Thus the fitted slope is \(15/5=3\) points per period and the intercept is \(13.5-(3)(1.5)=9\). The linear fitted means are 9, 12, 15, and 18 points.
The observed-minus-fitted residuals are \(1,-1,-1,1\) points. Their positive values at the endpoints and negative values in the middle form a curved pattern rather than random scatter around zero. In this example, the outcomes follow \(10+t^2\) exactly: a quadratic trend reproduces all four means. That does not establish that a quadratic is generally appropriate; it shows why residual patterns matter. With sparse observations, a more flexible curve may overfit, so the chosen smoothness should be justified and supported by data across the relevant time range.
Choosing a Specification and Recognizing Its Limits
Categorical periods and smooth trends are alternatives with different costs. Period indicators can capture abrupt changes between defined intervals and avoid imposing a linear shape, but require enough data and treatment-control support within periods. A linear trend uses fewer parameters and can be more stable, but assumes equal changes per time unit. Polynomial or spline terms offer intermediate flexibility; their shape and complexity should be prespecified where possible and assessed with residual and overlap diagnostics.
| Specification | Main advantage | Key assumption or risk |
|---|---|---|
| Categorical periods | Allows an unrestricted mean level for each defined period | Can be imprecise in sparse periods; still imposes a common treatment contrast if additive |
| Linear time | Parsimonious and easy to interpret as a per-period trend | Misses curvature or abrupt shifts when the true pattern is not linear |
| Flexible smooth trend | Can represent gradual nonlinear change | Needs enough data and overlap; excessive flexibility can fit noise |
These models do not resolve every platform-trial complication. If outcomes change for reasons not represented by calendar time, if measurement or eligibility changes are omitted, or if the groups have no temporal overlap, a fitted coefficient can remain biased. Nor does an additive model establish that the treatment effect is constant over time: that constancy is an imposed model restriction. The adjusted coefficient is a useful summary only when its time specification, comparison structure, and target contrast are appropriate to the scientific question.
Common Mistakes
- Assuming adjustment guarantees comparability. Adding time to a model does not remove unmeasured temporal confounding or manufacture overlap where none exists.
- Calling a linear trend assumption-free. A linear term requires the expected outcome to change at a constant rate per time unit, conditional on the other terms.
- Reading an additive coefficient as period-specific evidence. The additive model imposes one common treatment contrast; it does not estimate a separate contrast for every period.
- Confusing calendar time with follow-up time. These measure different processes and should not be interchanged in model interpretation.
- Treating a fitted trend as proof of adequate fit. Check residual patterns and the distribution of treatment and control observations across time.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- In an additive categorical-period model, what does the treatment coefficient represent, and what restriction does the model impose?
- Why may a linear time trend be unstable as a basis for adjustment when treatment assignment is almost determined by calendar time?
- In a hypothetical study, treatment and control mean outcomes differ by \(-1\) point in one period and \(-4\) points in another. Does an additive period model reproduce both contrasts? What does it impose instead?
- What would a repeated curved pattern in residuals against calendar time suggest about a linear time specification?
- Does including calendar time in a regression make nonconcurrent controls exchangeable with current participants? Give one reason.
Solutions. (1) It is the modelled treatment-minus-control contrast at the same period; the additive model imposes the same contrast in every period. (2) Little treatment variation remains after time is accounted for, so the coefficient relies on weak overlap and may be imprecise or driven by extrapolation. (3) No. Unless the contrasts are equal, the additive model cannot reproduce both exactly; it fits one common contrast. (4) The linear trend may be inadequate, and a suitably justified nonlinear time representation may be considered. (5) No. Unmeasured differences between historical and current participants or changes in care may remain after adjustment.