Applying Delta Within Sequential Imputation Models

Why Update Order Matters

When several outcomes are incomplete, an imputation for one outcome may be a predictor in the model for another. A delta shift applied to the first outcome can therefore affect subsequent imputations. The implementation must specify not only the delta assigned to each outcome, but also the order in which outcomes are updated and which current values enter each conditional model.

The earlier material on multiple imputation and delta adjustment establishes the basic idea: begin with a reference model for the unobserved outcome, then shift its distribution according to a sensitivity parameter. Here the focus is on carrying out those shifts within a sequence of conditional models. The examples use continuous outcomes and additive shifts on their original measurement scales.

The delta \(\delta_j\) is in the units of \(Y_j\). Its sign indicates the direction of the imposed departure from the reference conditional distribution. A single \(\delta_j\) may be used for all missing cases in an outcome, or the shift may be specified by a missingness pattern or other prespecified subgroup. The latter choice creates more conditional distributions to check. This tutorial uses fixed outcome-specific shifts to make the update mechanics clear; it does not determine which values are clinically plausible.

Assumptions and a Sequential Update

A sequential implementation relies on a set of conditional reference models that can generate draws for the outcomes being imputed. It also requires a defined update order, a rule for handling observed values, and a fixed delta specification for the sensitivity analysis. If model parameters are treated as estimated rather than known, parameter uncertainty should be represented in the imputation procedure; the arithmetic examples below condition on fitted parameters to isolate the delta updates.

For an outcome \(Y_j\) updated at a particular step, predictors should be the current values available at that step: observed values and, for missing outcomes already updated in the current sweep, their newly generated delta-adjusted values. Missing outcomes not yet updated must have starting values from initialization or the previous sweep. The procedure then continues through the chosen order, and sweeps are repeated as required by the algorithm.

This is an implementation rule, not a claim that any collection of conditionals defines a coherent joint distribution. A sequence can be executed even when its conditional models are mutually incompatible. In that case, the result may depend on update order or on the behavior of the iterative algorithm, so compatibility and convergence deserve explicit attention.

Compatibility: Do the Conditionals Describe One Joint Model?

A set of conditional models is compatible if there exists a joint distribution whose conditional distributions are exactly the specified models. Compatibility is not required merely to compute a single pass through the algorithm. It matters when interpreting the sequence as generating draws from a coherent joint distribution, and incompatibility can contribute to order sensitivity or unstable behavior.

A useful check is available for two Gaussian conditional models. Suppose their variances are constant, with

If these are conditionals of a nondegenerate bivariate normal joint distribution, their regression slopes and conditional variances must obey a specific equality. Write the joint normal precision matrix (the inverse covariance matrix) as \(Q\). In a bivariate normal distribution, the conditional variance of \(Y_1\) given \(Y_2\) is \(1/Q_{11}\), and the conditional slope is \(-Q_{12}/Q_{11}\). Therefore \(b_{12}/v_1=-Q_{12}\). Similarly, \(b_{21}/v_2=-Q_{12}\). Equating these expressions gives the required condition:

For these two linear Gaussian conditionals, this equality is not by itself enough: the implied precision matrix must also be positive definite. Equivalently, the conditional slopes must satisfy \(b_{12}b_{21}<1\). When the equality holds and this positivity condition holds, the specified slopes and conditional variances can arise from a nondegenerate bivariate normal joint model; intercepts then determine its means.

A delta shift changes more than a reported mean when the shifted outcome is used as a predictor later in the sequence: it can change later conditional predictions. Moreover, independently altering several conditionals can make them incompatible even if the reference conditionals were compatible. In the Gaussian example, shifts of conditional means change intercepts; intercepts alone do not violate the ratio condition. But shifts applied to outcomes in a pattern-specific way, or to conditional models with different slopes or variances, may create a set of distributions that does not correspond to one common joint model. The appropriate check is on the conditional distributions actually used, not just on the unadjusted reference models.

Diagnostics, Alternatives, and Limitations

Diagnostics should address both the mechanics of the updates and the distributions they produce. Check that observed outcomes are unchanged; each missing outcome receives its intended shift exactly once per update; and later models use current, not stale, imputed predictors. Compare summaries across sweeps and imputation chains, including traces of imputed means or selected quantiles. Persistent drift, poor mixing, or strong sensitivity to the initial values can signal instability. Also inspect distributions by outcome and missingness pattern for values outside the outcome's support or implausible combinations across outcomes.

A useful order check is to run a prespecified alternative update order as a diagnostic and compare the resulting imputations and analysis summaries. Material differences are a warning to investigate, not a reason to select whichever order gives a preferred conclusion. Check conditional-model fit using observed data where feasible, and assess compatibility when the model family makes a formal check available. Compatibility checks do not establish that the reference models or delta assumptions are scientifically appropriate.

Sequential conditional imputation is often convenient when a multivariate joint model is difficult to specify. A joint-model approach instead starts with one joint distribution and can make coherence explicit, but may impose restrictive distributional assumptions. Separate marginal delta adjustments are simpler to describe, but can fail to represent how one incomplete outcome predicts another. Sequential imputation makes those predictor relationships operational, at the cost of requiring careful order specification and attention to compatibility and convergence. None of these approaches identifies the unobserved outcomes from observed data alone; delta values encode sensitivity assumptions.

A common mistake is to add each delta to the final completed data set after all conditional models have run. That procedure leaves later conditional predictions based on unshifted values, so it is not equivalent to applying the shifts within the sequence. The correction is to shift each missing outcome at its update and pass that shifted value to subsequent models. Another mistake is to treat different one-sweep results as proof that the algorithm has converged to different distributions. A one-sweep comparison is a diagnostic of order effects, not a convergence result.