Treatment-by-Period Interaction Models

Why Model Treatment Effects by Period?

In a platform trial, enrollment and available treatment arms can change over calendar time. Even when treatment and control are compared concurrently, the estimated treatment effect in one period need not equal the effect in another. The underlying treatment response may change, or differences in patients, care, or measurement may alter the observed comparison. A treatment-by-period interaction makes this variation explicit: rather than assuming one common treatment effect, it allows the effect to differ by period.

The earlier material on period-stratified analysis compares treatment and concurrent control within each period. An interaction model expresses those period-specific comparisons in one regression model. It provides a direct estimate and test of how much the treatment contrast changes between periods. It does not, by itself, establish why effects differ or remove bias from nonrandomized comparisons.

Model and Interpretation

First consider one experimental treatment versus its concurrent control, a continuous outcome, and two prespecified calendar periods. Let \(Y_i\) be the outcome for participant \(i\), \(A_i=1\) indicate assignment to the experimental treatment and \(A_i=0\) indicate control, and \(P_i=1\) indicate the later period, with \(P_i=0\) for the reference period. A linear model with an interaction is

E(Y_i\mid A_i,P_i)=\beta_0+\beta_A A_i+\beta_P P_i+\beta_{AP}A_iP_i.

The coefficients describe the four treatment-period cell means. The reference-period control mean is \(\beta_0\); the later-period control mean is \(\beta_0+\beta_P\). The treatment effect in the reference period is \(\beta_A\). In the later period it is \(\beta_A+\beta_{AP}\). Therefore, \(\beta_{AP}\) is the difference between those two treatment effects, not the treatment effect in the later period itself.

Definition: In a model with a treatment-by-period interaction, the period-specific treatment effect is the model-based treatment-versus-control contrast within that period. The interaction coefficient for a nonreference period is the difference between that period's treatment effect and the reference period's effect, on the model's outcome scale.

For more than two periods, use indicator variables for the categorical periods. With period 1 as the reference and \(K\) periods in total, one form is

E(Y_i\mid A_i,P_i) = \beta_0+\beta_A A_i +\sum_{k=2}^{K}\beta_k I(P_i=k) +\sum_{k=2}^{K}\delta_k A_i I(P_i=k).

Here \(I(P_i=k)\) equals 1 when participant \(i\) is in period \(k\) and 0 otherwise. The treatment effect in period 1 is \(\beta_A\); in period \(k\) it is \(\beta_A+\delta_k\). The null hypothesis of no treatment-effect variation across periods is \(H_0:\delta_2=\cdots=\delta_K=0\). With covariate adjustment, additional covariate terms may be included; the treatment effects are then conditional contrasts from that specified model.

Assumptions and a Contrast Derivation

Interpretation depends on the design and model, not just on including an interaction term. For a causal treatment comparison, treatment assignment should be randomized within the relevant period, or a defensible set of adjustment assumptions must support an observational comparison. In every period whose treatment effect is to be estimated, both treatment and control participants must be represented. The outcome and treatment definitions should be consistent across periods, and the model must represent the intended mean contrasts. For the usual regression standard errors, observations must also be independent, with an appropriate variance model; robust standard errors may be used when the variance is not constant.

The interaction interpretation follows directly from subtracting cell means. In the two-period model, the control mean changes by \(\beta_P\) from reference to later period. The treatment mean changes by \(\beta_P+\beta_{AP}\). Thus the change in the treatment mean minus the change in the control mean is

\big[(\beta_0+\beta_A+\beta_P+\beta_{AP})-(\beta_0+\beta_A)\big] - \big[(\beta_0+\beta_P)-\beta_0\big] = \beta_{AP}.

This is a difference between period-specific treatment contrasts. For a continuous outcome in a linear model, it is measured in the outcome's units. In a logistic model, the corresponding interaction is on the log-odds scale; it does not directly represent a difference in risk differences. Interpretation must therefore name the model and scale.

Worked Example: Estimating Effects in Two Periods

Suppose a hypothetical randomized platform comparison has 100 participants in each treatment-period cell. Mean symptom scores, in points with lower scores better, are 10 for control and 7 for treatment in period 1, and 14 for control and 12 for treatment in period 2. Fit the two-period model using period 1 as the reference.

The reference-period control mean gives \(\hat{\beta}_0=10\). The control mean increases by \(14-10=4\) points, so \(\hat{\beta}_P=4\). The period 1 treatment effect is \(7-10=-3\) points, so \(\hat{\beta}_A=-3\). In period 2 the treatment effect is \(12-14=-2\) points. The interaction is the change in that effect: \(\hat{\beta}_{AP}=(-2)-(-3)=1\) point.

\widehat{E}(Y_i\mid A_i,P_i)=10-3A_i+4P_i+1(A_iP_i).

Substitution recovers the observed cell means: period 2 treatment is \(10-3+4+1=12\), while period 2 control is \(10+4=14\). The estimated treatment effect is a 3-point reduction in period 1 and a 2-point reduction in period 2. The interaction of 1 point means the treatment contrast is 1 point less favorable in period 2. These means alone do not provide a standard error or establish whether the interaction differs from zero.

Inference About Effect Variation

For two periods, a test of \(H_0:\beta_{AP}=0\) asks whether the modeled treatment effects differ. If the estimated interaction is \(\hat{\beta}_{AP}\) with standard error \(SE(\hat{\beta}_{AP})\), a large-sample Wald statistic is \(z=\hat{\beta}_{AP}/SE(\hat{\beta}_{AP})\), compared with a standard normal distribution under the null. A confidence interval is \(\hat{\beta}_{AP}\pm1.96SE(\hat{\beta}_{AP})\) for an approximate 95% interval. The interval concerns the difference in effects, in the model's units and on its scale.

With \(K\) periods, test all \(K-1\) interaction coefficients jointly. Let \(\hat{\boldsymbol{\delta}}\) be their vector of estimates and \(\widehat{V}\) its estimated covariance matrix. The Wald statistic is

W=\hat{\boldsymbol{\delta}}^{\,T}\widehat{V}^{-1}\hat{\boldsymbol{\delta}}.

Under the joint null and standard large-sample conditions, \(W\) is approximately distributed as a chi-square random variable with \(K-1\) degrees of freedom. This joint test addresses whether there is evidence of any difference across periods; it does not identify which periods differ without examining contrasts and their uncertainty. Failure to reject the null is not proof that effects are identical.

Worked Example: Testing a Two-Period Interaction

In a hypothetical linear model for a functional score measured in points, the estimated interaction is \(\hat{\beta}_{AP}=2.0\) points with \(SE(\hat{\beta}_{AP})=0.9\) points. Calculate a two-sided large-sample test and an approximate 95% confidence interval.

The test statistic is \(z=2.0/0.9=2.22\). The two-sided standard normal p-value is approximately 0.026. The 95% interval is

2.0\pm1.96(0.9)=2.0\pm1.764=(0.236,\ 3.764)\text{ points}.

At the 5% level, these results provide evidence that the treatment effect differs between the two periods. The interval estimates the later-minus-reference change in the treatment contrast: from about 0.24 to 3.76 points on the score scale. It is not the treatment effect in either period; those require the treatment coefficient and the relevant coefficient sum.

Worked Example: A Joint Test Across Three Periods

Suppose a hypothetical three-period trial uses period 1 as the reference. The fitted interaction estimates are \(\hat{\delta}_2=1.2\) points and \(\hat{\delta}_3=-0.8\) points. Assume their estimated covariance is zero, with standard errors 0.6 and 0.5 points, respectively. Test the joint null that treatment effects are equal across all three periods.

With zero covariance, the Wald statistic is the sum of the squared estimate-to-standard-error ratios:

W= \left(\frac{1.2}{0.6}\right)^2+ \left(\frac{-0.8}{0.5}\right)^2 =4+2.56=6.56.

There are \(K-1=2\) interaction coefficients, so the reference distribution is chi-square with 2 degrees of freedom. The p-value is \(P(\chi^2_2\ge6.56)=e^{-6.56/2}\approx0.0376\). This gives evidence against equal treatment effects across the three periods at the 5% level. If the reference-period treatment effect were \(-2.0\) points, the model's point estimates would imply effects of \(-2.0\), \(-2.0+1.2=-0.8\), and \(-2.0-0.8=-2.8\) points. The joint test does not establish that any particular pair differs; pairwise contrasts require their own estimates and uncertainty.

Diagnostics, Alternatives, and Limits

Before interpreting an interaction, inspect the treatment and control counts in every period and plot or tabulate their outcome summaries. Sparse or empty treatment-period cells make estimates unstable or unidentified. Check whether outcome definitions, follow-up, and measurement procedures remained comparable. For a continuous-outcome model, assess residual patterns and influential observations; if the mean relationship over time is not plausibly constant within periods, use a prespecified finer period categorization or another justified time specification. Data-driven period cut points can produce misleading apparent heterogeneity.

A period-stratified analysis estimates a separate treatment contrast in each period and is transparent about within-period comparisons. The interaction model is an equivalent way to parameterize these contrasts in a saturated treatment-by-categorical-period model, while also making differences between contrasts explicit. A model with treatment and period main effects but no interaction imposes a common treatment effect; it can be more precise if that restriction is credible, but it cannot reveal effect variation. A continuous-time interaction can estimate a changing effect more parsimoniously, but imposes a functional form and may obscure abrupt changes. Choose the representation before examining results where feasible.

Most importantly, interaction modeling does not repair lack of overlap or temporal confounding. If treatment is offered only in a later period and there are no concurrent controls for that period, its treatment effect cannot be separated from period changes using this model alone. Likewise, a statistically significant interaction may reflect changing patient mix, care, or measurement rather than a biological change in efficacy. Randomization within period supports causal comparisons within that period, but differences across periods still require care in interpretation.

Common pitfall: Treating the treatment main-effect coefficient as the effect in every period is incorrect when an interaction is present. With reference-period coding, that coefficient is the effect only in the reference period; add the relevant interaction coefficient to obtain another period's effect. Also, a nonsignificant interaction is not evidence that effects are exactly equal.

Check Your Understanding

Answer each question, then compare with the solutions below.

In a two-period linear model, \(\hat{\beta}_A=-4\) points and \(\hat{\beta}_{AP}=1.5\) points. What are the estimated treatment effects in the reference and later periods?
What does \(\beta_{AP}=0\) mean in a two-period model, and what does it not establish?
For three periods, how many interaction coefficients are tested in the joint test of equal treatment effects?
Why can a treatment-by-period model not identify a later-period treatment effect if that period has no concurrent control observations?

Solutions. (1) The reference-period effect is \(-4\) points. The later-period effect is \(-4+1.5=-2.5\) points, so the modeled benefit is 1.5 points smaller in the later period. (2) It means the two modeled treatment contrasts are equal on the model's outcome scale. It does not prove equality, explain the cause of any observed variation, or rule out differences too imprecisely estimated to detect. (3) There are \(3-1=2\) interaction coefficients, jointly tested with 2 degrees of freedom. (4) Without both treatment and control in that period, the treatment contrast is confounded with the period-specific mean; the model has no within-period comparison to identify it without additional assumptions.