Sequential Delta Adjustments for Multiple Incomplete Variables

Why Adjustment Order Matters

When several variables have missing values, a delta adjustment to one variable can affect the imputation of another. The effect arises when the later conditional model uses the earlier variable as a predictor. Thus, it is not enough to specify a delta for each variable in isolation: the imputation sequence determines which adjusted values are available when each conditional model is used.

This tutorial focuses on that sequence. It does not imply that changing the order must change the final result. In a well-behaved chained-equation procedure with compatible conditional models, different update orders can target the same joint distribution. But adjusted conditionals need not correspond to a compatible joint distribution, and finite chains can also show order-dependent behavior. The practical question is therefore whether and how the specified sequence changes the imputations and conclusions.

Each delta must be interpreted on the scale of its conditional model. For a continuous outcome modeled on its original scale, an additive \(\delta_Y\) shifts its conditional mean by that amount. For a binary variable, an adjustment may instead be defined on the log-odds or probability scale. The deltas for different variables may therefore have different units and meanings; they should not be added directly unless their scales make that operation meaningful.

A Linear Derivation of the Propagated Shift

Consider a simple conditional model for a continuous \(Y\), measured in mmHg, given a continuous \(X\):

Suppose an MAR reference draw for a missing \(X\) is \(x^{\mathrm{MAR}}\), and its additive adjustment is \(\delta_X\). The adjusted draw is \(x^{\mathrm{adj}}=x^{\mathrm{MAR}}+\delta_X\). If the conditional model for \(Y\) is updated afterward, and its own additive adjustment is \(\delta_Y\), then the conditional mean used for missing \(Y\) is

The difference from the MAR conditional mean before either adjustment is \(\beta\delta_X+\delta_Y\). If we compare two imputations that both use the same \(\delta_Y\), the difference attributable to changing \(X\) is only \(\beta\delta_X\): the \(Y\) adjustment is common to both. This distinction prevents a common error in describing sequential effects.

This calculation concerns the conditional mean. In a stochastic imputation, parameter uncertainty and residual variation remain; the formula does not mean every individual draw changes by exactly the same amount in every fitted model. The calculation also assumes the stated linear conditional mean is appropriate over the relevant values of \(X\), and that the adjustment to \(Y\) is additive on the model’s outcome scale.

What the Update Sequence Does

In a chained-equation sweep, each incomplete variable is updated from its conditional model using the current values of the other variables. With an order \(X\) then \(Y\), the imputation of missing \(Y\) can use the newly updated, delta-adjusted \(X\). With an order \(Y\) then \(X\), the \(Y\) update uses the current \(X\) values before the new \(X\) update. The two sequences therefore use different predictor values during that sweep whenever the relevant values change.

Missingness Patterns and Which Values Are Adjusted

A delta applies only to imputations for the variable to which it belongs. If \(X\) is observed, its value is not shifted by \(\delta_X\), even when \(Y\) is missing. If \(Y\) is observed, it remains observed even when an earlier adjustment changes the value of an \(X\) predictor. The conditional model for a missing variable can still use observed or imputed predictor values as appropriate.

Assumptions, Diagnostics, and Limitations

Sequential delta adjustment is a sensitivity analysis, not a way to identify the unobserved values from the observed data alone. Its interpretation depends on the stated reference conditional models and on the chosen deltas. The reference models should include appropriate predictors and represent the intended MAR reference analysis; the sensitivity parameters should have meaningful scales and signs. Conditional models must also be suitable for their variable types, and adjustments must preserve valid values where relevant.

The order comparison is informative only when the update is implemented as described. A delta adjustment must be applied at the intended stage, and downstream conditional models must use the adjusted current values. If software applies the shifts only after completing all imputations, those shifts cannot affect earlier or later conditional draws within that sweep.

• Check chain behavior. Inspect trace plots or other appropriate convergence summaries for imputed quantities and analysis estimates. Poor mixing, drift, or persistent dependence on starting values makes a sequential comparison unreliable.
• Check conditional predictions. Look for implausible values, sparse predictor regions, extrapolation, or instability in the conditional model after earlier variables are shifted.
• Compare sequences deliberately. Repeat the sensitivity analysis with the specified alternative update order while holding the reference models, deltas, observed data, and other settings fixed. Distinguish differences within a sweep from differences in stabilized imputations and estimates.
• Check scale and constraints. A shift appropriate on one scale may be inappropriate on another. For example, an additive shift can produce values outside a variable’s allowable range unless the model and adjustment account for that constraint.

There is no general rule that an order effect is evidence of a coding error. It can arise from the way conditional models interact, from finite-chain behavior, or because separately adjusted conditionals are not compatible with a single joint distribution. Conversely, finding little order sensitivity does not establish that the deltas are correct or that the missingness assumptions are true. The results describe consequences of the specified assumptions, not proof of them.

A practical comparison is between a joint sequential implementation and adjustments made independently to completed variable draws. Independent post-processing may be useful for a stated sensitivity question, but it cannot propagate an adjusted predictor into a later conditional imputation. Sequential adjustment does propagate that value; it also makes the update sequence and chain diagnostics especially important.