Assumption-Indexed Treatment Effects

Why Index an Estimate by Its Assumptions?

When follow-up outcomes are missing, the observed data alone do not generally determine the treatment effect for the full target population. An analysis therefore needs assumptions about the unobserved outcomes. The estimate is not fully described by a number such as “a difference of 1.8 points”: its interpretation also depends on which treatment contrast is targeted and what was assumed about the missing data.

Assumption-indexing makes that dependence explicit. It treats the estimate as a function: hold the observed data and target contrast fixed, state an assumption about unobserved outcomes, and calculate the estimate implied by that assumption. This is different from changing the estimand. The target can remain, for example, the mean difference between randomized treatment policies at a specified follow-up, while the assumptions used to identify that target vary.

The earlier material on estimands establishes why the population, treatment conditions, outcome, and handling of intercurrent events must be specified. Here the focus is narrower: once the target has been defined, how should its estimate be indexed and reported when some outcomes are missing?

Definition and Basic Formula

Consider a two-arm randomized trial with a continuous outcome \(Y\) measured at a fixed follow-up. Let \(A=a\) denote assignment to arm \(a\), for \(a=0,1\), and let \(R=1\) indicate that the follow-up outcome is observed and \(R=0\) that it is missing. In arm \(a\), let \(n_a\) be the number assigned, \(n_{a,\mathrm{obs}}\) the number with observed outcomes, and \(\bar{Y}_{a,\mathrm{obs}}\) their mean. Define the observed fraction as \(\hat{\pi}_a=n_{a,\mathrm{obs}}/n_a\).

An assumption index \(\eta\) specifies a mean \(m_{a,\mathrm{mis}}(\eta)\) for the missing outcomes in arm \(a\). In this simple mean-based pattern-mixture representation, the estimated full-arm mean is the observed fraction’s contribution plus the missing fraction’s assumed contribution:

For a treatment-minus-control mean difference, the assumption-indexed estimate is

This formula is a plug-in estimate, not a claim that each missing person’s outcome is known. It uses the assumed missing-outcome mean to represent the unobserved part of each arm. Its simple form also has a scope: it assumes the chosen arm-level means are adequate for the target. If assumptions are made conditional on baseline covariates, or the analysis uses a different estimator, the corresponding calculation must represent those assumptions and the target population rather than silently substituting this formula.

Conditions for Interpreting the Estimate

The formula is useful only when its ingredients match a clearly stated question. For a causal interpretation as a randomized-treatment contrast, the treatment conditions and outcome must be well-defined, the randomized assignment must support comparison of the arms, and observed outcomes must correspond to outcomes under assigned treatment in the target estimand. The endpoint, follow-up time, population, and handling of intercurrent events must also match that estimand.

The missing-outcome component requires an additional assumption. For each arm, the value \(m_{a,\mathrm{mis}}(\eta)\) must be a defensible representation of the mean among participants whose outcomes are missing under the stated assumption. It is not estimated directly from those missing outcomes. If an assumption is given in terms of a shift from an observed mean, that shift and its reference mean must be stated. If it is given as a direct assumed mean, the mean and its units must be stated. A bare label such as “adjusted” does not identify either basis.

The target contrast also needs a label. “Treatment effect” alone is incomplete: state whether it is a treatment-minus-control mean difference, a ratio, or another contrast, and give its outcome units where applicable. State the direction of subtraction. For a randomized trial, the causal language should refer to the specified treatment conditions and population; an estimated association from a nonrandomized comparison should not be presented as causal merely because the same arithmetic was used.

Derivation: Why the Assumed Mean Enters With Its Fraction

The weighting in the formula follows from partitioning the arm’s full outcome mean by whether the outcome is observed. In population notation, let \(\pi_a=P(R=1\mid A=a)\), and let \(m_{a,\mathrm{obs}}=E(Y\mid A=a,R=1)\). Under an assumption \(\eta\) that supplies \(m_{a,\mathrm{mis}}(\eta)=E(Y\mid A=a,R=0)\), the law of total expectation gives

Replacing the population observed fraction and observed mean with their sample counterparts yields the plug-in expression for \(\hat{\mu}_a(\eta)\). Subtracting the two arm means then yields the stated treatment-minus-control contrast. The key point is that the observed outcomes supply only the first component; the second component depends on the assumption. The calculation does not turn the assumption into observed evidence.

This derivation also clarifies a frequent ambiguity around “assuming missing outcomes equal observed outcomes.” At the arm-mean level, that assumption sets \(m_{a,\mathrm{mis}}(\eta)=m_{a,\mathrm{obs}}\), giving \(\hat{\mu}_a(\eta)=\bar{Y}_{a,\mathrm{obs}}\). It is a particular mean restriction. It should not be called a general missing-at-random assumption without qualification: conditional missing-at-random assumptions can require covariates and appropriate adjustment, and do not in general imply equality of the two unconditional arm means.

Reporting the Target and Assumed-Data Basis

A report should let a reader identify both the question being estimated and the missing-data assumption used to obtain the number. At minimum, include the target population or trial population, treatment conditions, outcome and follow-up, contrast and direction, and the assumption used for missing outcomes. For a mean-based calculation, report the arm-specific observed fractions and the assumed missing-outcome means, or refer to a clearly defined model that supplies them.

For example: “For the randomized population, the estimated treatment-minus-control difference in mean [outcome] at [follow-up] was 1.85 points under the assumption that missing outcomes had means of 11.0 points in treatment and 9.5 points in control; the observed fractions were 80% and 90%, respectively.” This phrasing exposes the result’s basis without implying that the missing means were observed or that the assumption is empirically verified.

An assumption index can be a descriptive label, such as \(\eta_{\mathrm{equal}}\), or a parameter that defines a model. Either way, define it in words and specify its role. A label with no definition is not reproducible. Conversely, a full list of mathematical details without naming the target contrast can leave the reader unsure what the estimate means.

Diagnostics, Alternatives, and Limitations

Before interpreting an assumption-indexed estimate, check that the arithmetic and the label agree. The observed fractions must be between zero and one and must correspond to the stated arm counts. The assumed missing means must use the outcome’s units and refer to the correct arm and follow-up. The contrast must use the stated direction. If all outcomes are observed in an arm, its missing fraction is zero, so the assumed missing mean has no effect on that arm’s estimate. If no outcomes are observed in an arm, the estimate for that arm depends entirely on the assumption; the data provide no observed mean for that arm at that follow-up.

This mean-based calculation is one approach to missing outcomes, not a universal replacement for other analyses. A complete-case analysis uses observed outcomes and targets or estimates a quantity based on that observed subset unless additional assumptions justify extending its interpretation. Likelihood-based methods, inverse-probability weighting, and multiple imputation are alternative approaches whose validity depends on their own models and conditions. They may use covariate information or the outcome distribution in ways this simple arm-mean formula does not. Whatever approach is used, the treatment contrast and the assumptions needed to interpret it must remain clear.

A common mistake is to call an estimate “the treatment effect” and treat a missing-data assumption as a harmless technical detail. The correction is to report the estimate as \(\hat{\tau}(\eta)\), define \(\eta\) in substantive terms, and name the target contrast. Another mistake is to infer that an assumption is supported simply because the resulting estimate looks plausible. Plausibility is not verification; missing outcomes are unobserved, so the assumed part cannot be checked directly from those outcomes.