Reference Groups in Stratified Randomized Trials

Why the Reference Group Must Match the Randomization Stratum

In a stratified randomized trial, treatment assignment is carried out separately within levels of one or more baseline factors. Those strata may be prognostic: even under the reference treatment, participants in different strata can have different expected outcomes. If a missing outcome is to be predicted from a reference group, the relevant reference distribution is therefore generally the one for the participant’s own stratum, not an undifferentiated average across the reference arm.

This distinction matters in reference-based imputation. A reference-based assumption specifies how the unobserved outcomes of participants assigned to the active treatment are related to outcomes under the reference treatment after missingness. Once that assumption identifies a reference distribution, the randomization stratum identifies which reference distribution is used. Stratification does not by itself justify a particular post-dropout assumption, and randomization alone does not prove that a reference-based assumption is true.

The observed history can include earlier outcomes and baseline covariates used in the predictive model. The exact history and the way it enters the reference distribution depend on the reference-based rule and model already specified for the analysis. The focus here is the stratum index: the model and its predictions must preserve the distinction between \(s\) and other strata when the stratum-specific distributions differ.

Conditional Predictions Within a Stratum

Suppose a reference-arm model estimates the expected missing outcome at a future visit conditional on observed history. Write this conditional mean as \(\mu_{0,s,m}(y_{\mathrm{obs}})=E(Y_m\mid A=0,S=s,Y_{\mathrm{obs}}=y_{\mathrm{obs}})\), where \(m\) identifies the missing visit. Under a reference-based assumption that uses this conditional reference distribution for the participant, the corresponding conditional mean prediction is \(\mu_{0,s,m}(y_{\mathrm{obs}})\). The prediction is specific to both the participant’s history and stratum.

For example, if a normal reference-arm model is used, a conditional mean may be written as an intercept and a history adjustment:

where \(c\) is a chosen centering value and the coefficients are estimated from reference-arm data in stratum \(s\). A model can instead use several observed outcomes or covariates. The essential feature is that the prediction uses the reference-arm relationship appropriate to that stratum. A conditional mean alone does not specify the full predictive distribution: residual variation and uncertainty in estimated model parameters also matter when generating imputations.

The stratum-specific construction rests on several conditions. The stratum must be defined from baseline information used in the randomization scheme; the reference arm must have relevant observations in that stratum; and the conditional reference model must be adequate for the histories at which predictions are required. Applying that model to missing active-arm outcomes additionally relies on the chosen reference-based assumption. In particular, randomization makes treatment groups comparable in expectation within strata at baseline, but it does not establish that observed reference outcomes after conditioning on post-randomization histories are interchangeable with missing active-arm outcomes.

Why Pooling Can Change the Imputation

A pooled reference model combines information across strata. If the reference outcome distributions differ by stratum, that model can produce a prediction that is appropriate for neither the individual stratum nor its conditional history. With a known stratum, averaging across strata is not a neutral simplification: it changes the reference distribution being used.

The distinction can be seen from the law of total probability. A pooled conditional distribution is a mixture of stratum-specific distributions, with weights determined by the stratum composition among the reference observations at the relevant history. Its mean is the corresponding weighted mean of the stratum-specific conditional means. For a participant known to be in stratum \(s\), however, the target is the distribution conditional on \(S=s\), not this mixture.

This expression describes a pooled distribution; its weights need not equal the overall proportions randomized to each stratum, because conditioning on observed history can change the stratum composition. If all strata have the same conditional reference distribution, pooling may leave predictions unchanged. If they do not, a pooled prediction generally answers a different question.

A short justification follows directly from conditioning. For a participant whose stratum is known to be \(s\), the relevant conditional expectation under a stratum-specific reference assignment is \(E(Y_m\mid Y_{\mathrm{obs}},S=s,A=0)\). The pooled expectation instead averages over possible strata:

Unless the stratum-specific conditional expectations are equal, or the participant’s stratum is genuinely unknown and mixture prediction is intended, the pooled expectation does not equal the target conditional expectation for that participant. The same conditioning logic applies to the full predictive distributions, not only their means.

Worked Checks and Sparse Strata

When a stratum has little or no reference-arm support, the problem is substantive as well as computational. A fitted regression may be unstable, and predictions at histories poorly represented in that stratum may be extrapolations. If a stratum has no reference participants, its reference distribution cannot be estimated from that stratum’s observed data alone. Any fallback, such as a pooled or partially pooled model, adds assumptions and should be reported as such rather than described as ordinary stratum-specific borrowing.

Diagnostics, Alternatives, and Limitations

Useful checks focus on whether the within-stratum reference model has enough data and whether it predicts plausible outcomes over the histories requiring imputation. Examine sample sizes and observed history ranges by treatment arm and stratum. Compare fitted conditional means with observed reference outcomes within each stratum, and check for conspicuous differences in residual spread or systematic patterns. A model that fits well overall can still fit poorly in a smaller stratum if the pooled fit is driven by the larger one.

Also check whether the active-arm histories at dropout are represented in the reference arm within the same stratum. Predictions outside that support depend on extrapolation, even when a regression formula returns a finite value. A stratum-specific model may be unstable if there are few reference participants, few observed outcomes at later visits, or too many predictors for the available information. These problems can affect both the estimated mean and the predictive distribution.

The main alternatives answer different questions. A pooled reference model assumes, or models in a way that effectively permits, sharing across strata; it can improve stability but can obscure genuine stratum differences. A stratum-specific reference model preserves those differences but can be imprecise when strata are small. A treatment-arm-only regression predicts from active-arm outcomes and does not, by itself, impose a reference-based post-dropout distribution. These alternatives should not be treated as interchangeable merely because they produce predictions for the same missing values.

The introductory material on reference-based imputation distinguishes the substantive post-dropout assumptions from the predictive model used to implement them. Stratification refines the source distribution; it does not decide whether jump-to-reference, copy-reference, or another rule is scientifically appropriate. Nor does a within-stratum fit resolve uncertainty about the missingness mechanism. Results can remain sensitive to the chosen rule, model form, and available reference data.

In reporting, state which baseline strata were used, how the reference distribution was estimated within them, what happened when a stratum had sparse or absent reference data, and whether predictions required extrapolation. This makes clear which part of the result comes from the reference-based assumption and which part comes from the stratum-specific model.