Coherent Delta Shifts for Repeated Outcomes

Why the Joint Trajectory Matters

Repeated outcomes are not merely a collection of separate measurements. Values at nearby visits are often related: a participant with a relatively high outcome at one visit may tend to remain relatively high at the next. When outcomes are missing, a sensitivity adjustment should therefore specify not only how each missing visit changes, but also how the adjusted values fit together as a trajectory.

A delta can differ by visit without treating visits as independent. The key distinction is between shifting coordinates of a joint distribution and adjusting each visit’s marginal distribution and then generating the visits independently. The first retains the dependence represented by the reference model; the second generally replaces it with a different dependence structure, potentially producing trajectories inconsistent with the reference model or clinical expectations.

Definition: Let $Y=(Y_1,\ldots,Y_T)$ be a participant’s repeated outcomes, let $O$ denote the observed visits, and let $M$ denote the missing visits for a particular missingness pattern. A reference model specifies the joint conditional distribution $f_0(y_M\mid y_O,x)$, where $x$ denotes covariates. An additive delta vector $\delta_M=(\delta_t:t\in M)$ defines the adjusted distribution by $$ f_\delta(y_M\mid y_O,x) = f_0(y_M-\delta_M\mid y_O,x). $$ Equivalently, draw the missing outcomes jointly from the reference conditional distribution and add the corresponding visit-specific delta to each draw. The observed values remain fixed.

This definition assumes the reference conditional distribution is a suitable model for the missing outcomes before sensitivity adjustment, and that the chosen delta vector represents the intended departures from that reference. The delta values are sensitivity parameters: the observed data do not, by themselves, identify the unobserved outcomes well enough to determine them. For the additive expression to define outcomes on the intended scale, translated values must also be meaningful on that scale. When this is not plausible, a different scale or transformation may be needed.

The shift is constant within the pattern and covariate stratum being described. It may vary across visits, patterns, arms, or covariate strata if those differences are part of the stated sensitivity model. For example, $\delta_2$ need not equal $\delta_3$. What matters for coherence is that the vector is applied to a joint draw, not that all its entries are identical.

Why a Joint Shift Preserves Dependence

For a fixed observed history and covariate value, write $Y_M^0$ for a random draw from the reference conditional distribution. The adjusted draw is $Y_M^\delta=Y_M^0+\delta_M$. Adding constants changes conditional means but does not change conditional covariances. For any two missing visits $j$ and $k$,

\operatorname{Cov}(Y_j^\delta,Y_k^\delta\mid y_O,x) = \operatorname{Cov}(Y_j^0+\delta_j,Y_k^0+\delta_k\mid y_O,x) = \operatorname{Cov}(Y_j^0,Y_k^0\mid y_O,x).

The equality follows because covariance is unchanged by adding constants to either variable. More generally, translating a joint distribution preserves its dependence structure: the same joint draw, ranks, and residual co-movement are retained, while the location of each coordinate changes. The claim is conditional on the observed history and covariates, and on a fixed delta vector. If delta varies across patterns or covariate strata, the mixture across those strata can have a different overall covariance.

This result does not say that every adjusted trajectory is clinically plausible. It says that a coordinate-wise translation of a joint draw preserves the reference model’s within-stratum dependence. Plausibility also depends on the delta values, the outcome scale, the reference model, and the clinical setting.

Worked Example: Shifting a Joint Distribution

Suppose a hypothetical study has two missing visits. Conditional on the observed history, the reference model assigns probability $0.5$ to each of two trajectories: $(10,12)$ and $(20,22)$, with outcome units at both visits. Consider the visit-specific delta vector $\delta_M=(0,-5)$ units.

Apply the shift to each joint draw. The first trajectory becomes $(10+0,12-5)=(10,7)$. The second becomes $(20+0,22-5)=(20,17)$. Thus the adjusted joint distribution puts probability $0.5$ on each of $(10,7)$ and $(20,17)$. Every adjusted draw is generated from one reference trajectory, with the same visit-specific adjustments applied.

At visit 1, the mean is $(10+20)/2=15$ units. At visit 2, the adjusted mean is $(7+17)/2=12$ units. In the reference distribution, the second visit equals the first visit plus 2 units for every draw. After adjustment, it equals the first visit minus 3 units for every draw. The shift changes the visit-to-visit level difference by 5 units while retaining the perfect positive association in this simple example.

The reference covariance is $0.5(10-15)(12-17)+0.5(20-15)(22-17)=25$ squared units. The adjusted covariance is $0.5(10-15)(7-12)+0.5(20-15)(17-12)=25$ squared units. This confirms numerically that the joint translation leaves covariance unchanged.

How Independent Marginal Shifts Can Break Trajectories

A different procedure is to form a shifted marginal distribution at each visit and sample from those marginals independently. Even if each visit’s distribution is correct by itself, the resulting joint distribution need not be correct. It discards the dependence that links the repeated outcomes.

Worked Example: Independent Visits Create Unrepresented Combinations

Use the two-trajectory reference distribution from the previous example, but apply a delta of $-5$ units at both visits. Joint shifting gives $(5,7)$ and $(15,17)$, each with probability $0.5$. Both trajectories preserve the reference pattern that visit 2 is 2 units above visit 1.

If instead the two shifted marginal distributions are sampled independently, visit 1 is 5 or 15 units with equal probability, and visit 2 is 7 or 17 units with equal probability. Independence assigns probability $0.25$ to each of $(5,7)$, $(5,17)$, $(15,7)$, and $(15,17)$.

The independent procedure has introduced $(5,17)$ and $(15,7)$, neither of which appears in the reference model after a joint translation. In particular, the second trajectory decreases by 8 units between visits, despite every reference trajectory increasing by 2 units. The marginals at both visits are unchanged by this mistake, so checks of visit-specific means alone would not reveal it.

For another diagnostic, the reference covariance is 25 squared units, and a constant shift at both visits leaves it at 25. Independent sampling gives covariance 0 because the generated visits are independent. This is a different joint model, not a computationally equivalent way of implementing the same delta adjustment.

Independently chosen visit-specific delta values are not automatically incoherent. A vector such as $(0,-5)$ is a valid coordinate-wise adjustment if it is applied to joint draws. The incoherence arises when visit outcomes are generated independently, or when separate marginal adjustments are used without specifying and preserving a suitable joint dependence structure.

Conditioning on Observed Visits

The relevant joint distribution is usually conditional on the outcomes already observed for the participant. A missing outcome at a later visit may be strongly related to an earlier observed value; the reference model’s conditional distribution represents that relationship. A coherent delta adjustment changes the missing coordinates of draws from that conditional distribution while keeping the observed coordinates fixed.

Worked Example: Adjusting Two Missing Visits Given an Observed Outcome

Suppose a hypothetical participant has an observed outcome of 8 units at visit 1, with visits 2 and 3 missing. Conditional on this observed value, assume the reference model assigns equal probability to the missing-outcome pairs $(9,10)$ and $(13,16)$. Consider the delta vector $(1,-2)$ units for visits 2 and 3.

Apply the shifts to the pairs jointly: $(9+1,10-2)=(10,8)$ and $(13+1,16-2)=(14,14)$, each with probability $0.5$. The observed visit-1 value remains 8 units. The adjusted conditional means are 12 units at visit 2 and 11 units at visit 3, obtained as $(10+14)/2=12$ and $(8+14)/2=11$.

The reference covariance between visits 2 and 3 is $0.5(9-11)(10-13)+0.5(13-11)(16-13)=6$ squared units. The adjusted covariance is $0.5(10-12)(8-11)+0.5(14-12)(14-11)=6$ squared units. Thus the adjustment retains the conditional dependence while changing each visit’s location by its specified amount.

This calculation depends on the assumed reference conditional distribution given the observed outcome. If that distribution does not adequately represent how repeated outcomes relate to observed history, preserving its dependence will preserve the wrong structure. A delta shift cannot repair a poorly specified reference model.

Assumptions, Diagnostics, and Alternatives

A coherent additive adjustment relies on several explicit choices. First, the reference model must specify a proper joint conditional distribution for the missing visits, given the observed visits and relevant covariates. Second, the delta vector must be defined on the chosen outcome scale and for the relevant missingness pattern or stratum. Third, the analyst must intend the sensitivity departure to act as a translation of the reference joint distribution; this is an assumption about the form of departure, not a consequence of the observed data.

Useful diagnostics focus on trajectories, not only separate visit summaries. Compare reference and adjusted simulated trajectories conditional on common observed histories. Check whether adjacent changes, ordering across visits, and ranges of plausible trajectories accord with the application. Examine whether a visit-specific shift creates abrupt jumps or reversals that are hard to defend. Also verify that the implementation uses the same joint draw for all missing visits in a participant. Agreement of marginal means is not evidence that the joint distribution is coherent.

A common implementation failure is to fit or use a separate marginal distribution at each visit, shift each one, and draw each missing visit independently. The resulting marginal means may match the intended deltas, but the dependence can be lost. Another failure is to apply a single shift to a participant’s entire trajectory when the stated model calls for different visit-specific shifts. In either case, the simulated trajectories do not implement the specified joint adjustment.

Approach	What it specifies	Implication for repeated outcomes
Joint additive delta shift	A delta vector applied to draws from the reference conditional joint distribution	Retains the reference conditional dependence while shifting locations
Independent shifted marginals	A shifted distribution for each visit, sampled independently	Usually changes dependence and can generate combinations absent from the reference model
Joint model with a changed dependence structure	A new joint distribution, including a stated change to association	Can represent altered dependence, but requires assumptions beyond a location shift

The joint additive shift is useful when the sensitivity question concerns changes in outcome levels while keeping the reference dependence structure. It does not represent every plausible departure. For example, a sensitivity assumption that missing outcomes become more variable, or that dependence between visits weakens, needs a model that changes more than location. Nor does the translation guarantee valid values for bounded or otherwise constrained outcomes. The model and the outcome scale must support the proposed transformation.

Common mistake: “I used the correct delta at each visit, so the repeated outcomes are correctly adjusted.” Correct visit-specific shifts do not determine a joint distribution on their own. The correction is to specify how the visits co-vary and, if the aim is to retain the reference dependence, apply the delta vector to joint conditional draws. Separate marginal shifts plus independent draws generally answer a different sensitivity question.

Key takeaway: For repeated outcomes, specify the delta adjustment jointly. A visit-specific delta vector can shift each missing outcome by a different amount while preserving the reference conditional dependence, provided it is applied to joint draws and the translation is meaningful on the chosen scale.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A reference conditional distribution assigns equal probability to $(4,6)$ and $(8,10)$. What trajectories result from applying the delta vector $(2,-1)$ jointly?
In the setting of question 1, what covariance is retained after the joint shift? Show the calculation.
Why can independent sampling from the two shifted marginals create trajectories not present under a joint shift?
Does a visit-specific delta vector necessarily require the delta to be identical at every visit? Explain.
Name two limitations of a joint additive delta shift.

Solutions. (1) Add 2 to the first coordinate and subtract 1 from the second: $(6,5)$ and $(10,9)$, each with probability $0.5$. (2) The reference means are 6 and 8. Its covariance is $0.5(4-6)(6-8)+0.5(8-6)(10-8)=4$ squared units. Adding constants leaves covariance unchanged, so the adjusted covariance is also 4 squared units; directly, the adjusted means are 8 and 7 and the covariance is $0.5(6-8)(5-7)+0.5(10-8)(9-7)=4$. (3) Independent sampling combines each visit-1 value with either visit-2 value, including cross-pairs that did not occur in the joint reference distribution. (4) No. The delta may vary by visit; coherence requires applying the vector to joint draws, not using a common delta. (5) It preserves the reference dependence even if that dependence is inappropriate for the sensitivity scenario, and it may produce values outside the meaningful outcome range. It also changes location, not dependence or variability, unless the model is extended.