Implementing Delta Adjustments Within Chained Equations

Why Implementation Within the Chain Matters

A delta adjustment represents a specified departure from the missing-at-random (MAR) reference distribution for missing values. In chained equations, the implementation question is not only what shift to use, but when to apply it. An adjusted value can become a predictor in later conditional models, so applying the delta during the chain may affect subsequent imputations as well as the variable’s own completed values.

The central rule is to fit the conditional model using the available observed responses, generate a new reference draw for the missing responses, and then apply the chosen delta to that draw in the appropriate way. The observed responses are not shifted. The delta is a fixed sensitivity parameter, not a coefficient learned from the observed data.

Definition: Let \(Y_j\) be an incomplete variable and let \(X_{-j}\) denote its predictors in a chained-equation update. An implementation of a delta adjustment replaces the MAR reference conditional distribution for the missing \(Y_j\) values with a specified shifted distribution. At each update, the adjustment is applied to the newly generated missing-value draw; observed \(Y_j\) values are retained exactly as recorded.

Where the Adjustment Enters an Update

For imputation \(m\) and chain iteration \(t\), write \(Y_{j,\mathrm{obs}}\) and \(Y_{j,\mathrm{mis}}\) for the observed and missing cells of variable \(Y_j\). The conditional model is fitted from the observed responses, using the current values of its predictors. Some predictors may themselves be incomplete; in that case their current imputed values from the chain are used. A proper imputation procedure also represents uncertainty in the fitted model parameters and in the predictive draw.

After fitting, generate a reference draw for the missing responses from the MAR conditional distribution. Apply the delta to that draw before moving to the next variable in the chain. The adjusted values then become the current values of \(Y_j\) used wherever \(Y_j\) appears as a predictor in later updates and in the next iteration.

Fit the conditional model.
Use observed responses for the target variable and current chain values for its predictors. The delta does not alter the observed response data used for this fit.

Generate a fresh reference draw.
For the missing cells, draw from the fitted MAR reference predictive distribution, including the uncertainty appropriate to the imputation method.

Adjust the missing draw.
Apply the specified delta on the selected scale, using the outcome-appropriate operation. Do not change any observed cell.

Continue the chain.
Store the adjusted missing values as the current state of this variable, then use that state in later conditional updates.

For a continuous outcome with an additive shift, if \(Y_{j,\mathrm{mis}}^{0,(m,t)}\) is the fresh MAR reference draw, the adjusted draw is

Y_{j,\mathrm{mis}}^{(m,t)} = Y_{j,\mathrm{mis}}^{0,(m,t)}+\delta.

This is a draw-by-draw operation: each new reference draw receives the specified shift. It is not an instruction to add \(\delta\) repeatedly to the old adjusted value.

Worked Examples of the Update

Worked Example: An Additive Shift for a Continuous Variable

Suppose a hypothetical study measures a continuous biomarker in units per liter. Its conditional model is fitted to observed biomarker values using the current values of the predictors. The sensitivity analysis specifies an additive delta of \(-1.5\) units per liter. At iteration 8, the model generates a reference draw of \(12.4\) units per liter for one missing case.

The adjustment is applied to the newly generated value:

12.4+(-1.5)=10.9\text{ units per liter}.

The value stored for this missing case at this update is \(10.9\) units per liter. If the case’s observed predictors and the subsequent chain state produce a fresh reference draw of \(11.8\) at iteration 9, the correct new adjusted value is \(11.8-1.5=10.3\), not \(10.9-1.5=9.4\). Reusing and shifting the previous adjusted value would accumulate the delta rather than apply it to a new draw. The observed biomarker values remain unchanged.

This implementation assumes that an additive shift is a defensible sensitivity model on the biomarker’s scale and that the shifted values remain in a meaningful range. Here the example gives no indication of a boundary violation; in practice, the range and distribution should be checked.

Worked Example: A Log-Odds Shift for a Binary Variable

Consider a hypothetical binary outcome indicating whether a patient experiences an event. For a missing case, suppose the fitted MAR conditional model gives an event probability of \(0.25\). Let the specified delta be \(\log(2)\), a positive log-odds shift.

The MAR odds are \(0.25/(1-0.25)=0.3333\). A log-odds shift of \(\log(2)\) multiplies these odds by \(2\), giving adjusted odds \(0.3333(2)=0.6667\). Converting back to a probability gives

p_\delta=\frac{0.6667}{1+0.6667}\approx 0.4000.

Equivalently, \(p_\delta=\operatorname{expit}\{\operatorname{logit}(0.25)+\log(2)\}=0.4000\). The chain then draws a binary value from a Bernoulli distribution with probability \(0.4000\). For example, if the random draw used for this case is \(u=0.37\), the imputed value is 1 because \(0.37<0.4000\); if it is \(u=0.62\), the value is 0.

The delta must change the event probability before the Bernoulli draw. Adding a number to an already drawn 0 or 1 would not produce a valid binary value. This example assumes that the binary conditional model is appropriate and that the delta is specified on the log-odds scale.

Worked Example: Adjusting a Predictive Mean Matching Draw

Suppose a hypothetical continuous variable is imputed by predictive mean matching (PMM). For one missing case, PMM selects a donor whose observed value is \(10\) units. The chosen sensitivity adjustment is an additive \(\delta=2\) units.

First, the PMM procedure selects the donor using its usual fitted-model and donor-selection steps. The delta is then applied to the selected donor value:

Y_{\mathrm{mis}}^{0}=10,\qquad Y_{\mathrm{mis}}=10+2=12\text{ units}.

The adjusted imputation is \(12\) units. This order keeps the donor-selection mechanism distinct from the sensitivity adjustment. The adjusted value need not equal an observed donor value, so it no longer necessarily retains PMM’s observed-value support property. Check that \(12\) is plausible for the variable; do not silently replace it with a different donor value unless that is part of a separately specified method. The donor’s observed value remains \(10\) in the data.

Why the Order Is Important

For an additive shift, the distributional meaning is direct. If the MAR reference draw has conditional density \(f_0(y\mid x)\), shifting it by \(\delta\) gives the adjusted density \(f_\delta(y\mid x)=f_0(y-\delta\mid x)\). Its conditional mean is shifted by \(\delta\), while its conditional variance is unchanged when the shift is constant. This follows by writing \(Y_\delta=Y_0+\delta\): \(E(Y_\delta\mid x)=E(Y_0\mid x)+\delta\), and \(\operatorname{Var}(Y_\delta\mid x)=\operatorname{Var}(Y_0\mid x)\).

For a binary outcome, shifting the log odds gives a different but similarly explicit distribution. If \(p_0\) is the MAR event probability, then

\operatorname{logit}(p_\delta)=\operatorname{logit}(p_0)+\delta, \qquad p_\delta= \frac{\exp\{\operatorname{logit}(p_0)+\delta\}} {1+\exp\{\operatorname{logit}(p_0)+\delta\}}.

These operations alter the predictive distribution for missing responses; they do not redefine the likelihood contribution of observed responses. The conditional models are still estimated from observed target-variable values. If the target variable is used as a predictor elsewhere, however, its adjusted imputations can affect later updates. That feedback is a key reason to apply the adjustment inside the chain rather than only to the final completed data.

Assumptions, Diagnostics, and Failure Modes

A delta-adjusted chained-equation analysis is a sensitivity analysis, not an estimate of the delta from the observed data. Its interpretation depends on the chosen MAR reference models and on the specified departure from those models. The conditional models should be suitable for the variable types and relationships, and the imputation procedure should account for parameter and predictive uncertainty. The chosen delta must also have a meaningful scale and produce values compatible with the outcome’s support.

Chained equations specify a sequence of conditional updates. In general, those conditionals need not correspond to a single compatible joint distribution. The usual practical requirement is that the implemented chain be sufficiently stable for its intended summaries; convergence is not guaranteed merely because a fixed number of iterations was run. Inspect traces and distributions of imputed values across iterations and chains, and compare results across independent starts. Also check that all imputations were generated under the intended delta and that observed cells are identical to the original data.

Repeated accumulation: Adding \(\delta\) to the previous adjusted value at every sweep produces a drift. Generate a fresh reference draw and apply the delta once to that draw.
Adjusting observed values: Shifting the entire column changes recorded data and the model fit. Restrict the adjustment to cells that are missing.
Adjusting after the chain when feedback matters: A final-only shift does not let adjusted values influence later conditional updates. It may therefore yield different results from an in-chain adjustment when the target variable predicts other incomplete variables.
Shifting a binary realization: A delta on the log-odds scale changes the probability before the Bernoulli draw; it is not added to a realized zero or one.
Ignoring support or implausibility: A shifted draw can fall outside a valid range or become scientifically implausible. Diagnose this explicitly rather than treating a numerical output as automatically valid.

Post-imputation adjustment can be simpler and may coincide with in-chain adjustment in special cases, such as when the adjusted variable has no role in other conditional updates and no subsequent model step depends on it. When feedback is present, in-chain adjustment propagates the specified change through the sequence. Refitting the conditional model to include a chosen delta as if it were an ordinary predictor is not equivalent: the delta is fixed sensitivity input, not a data-estimated regression effect.

A further limitation is that the data alone generally cannot select a uniquely correct delta for departures from MAR. Results should therefore be reported across substantively defensible delta values, with the reference assumptions and implementation made explicit. A stable chain does not establish that the sensitivity model is true; it only indicates that the computation appears to have settled under the specified model and adjustment.

Key takeaway: At each chained-equation update, fit the reference conditional model, generate a new draw for missing cells, apply the delta on the chosen scale, leave observed cells untouched, and carry adjusted imputations forward. Never repeatedly shift an old adjusted value.

Check Your Understanding

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

A continuous missing value has a fresh reference draw of \(6.2\) units and an additive delta of \(0.8\). What value is stored at this update?
For a binary outcome, the MAR probability is \(0.20\) and the log-odds delta is \(\log(3)\). Find the adjusted probability, to three significant figures.
Why should a delta not be added to the previously adjusted value at every iteration?
In what situation can adjusting only the final completed data differ from applying the delta within the chain?

Solutions. (1) \(6.2+0.8=7.0\) units. (2) The MAR odds are \(0.20/0.80=0.25\); adjusted odds are \(0.25(3)=0.75\), so \(p_\delta=0.75/(1+0.75)=0.429\). (3) Each iteration generates a fresh reference draw. Shifting the old adjusted value again accumulates the delta and creates unintended drift. (4) They can differ when the adjusted variable is used in later conditional updates, because the in-chain adjustment feeds into those updates whereas a final-only adjustment does not.