Why Predictor Use Requires Synchronization
A derived variable can carry useful information about a nonlinear or combined relationship. For example, a squared predictor can help an imputation model represent curvature, and a product can represent an interaction. But a passive variable is not an additional independently observed quantity: it is fixed by its source variables. If a source changes during a chained-equations update, the derived term must change with it before a later conditional model uses that term.
The key decision is conditional: is this term a suitable predictor for the variable being updated, and are its source values current? A term that is useful when imputing one variable may be inappropriate when imputing one of its own sources. Passive updating maintains the deterministic relationship; it does not by itself guarantee that every conditional imputation model is compatible with the substantive analysis.
Suppose \(X\) is incomplete and \(Q=X^2\). When imputing a different variable \(Y\), \(Q\) can be a useful predictor if a quadratic relationship is plausible and the conditional model is specified accordingly. When imputing \(X\), however, \(Q\) is a function of the very value being drawn. Treating an old \(Q\) as an ordinary fixed predictor of \(X\) creates a target-dependent, out-of-date input. The usual passive approach is to omit such a term from the conditional model for its own source, draw the source, and then calculate the term again.
When to Include a Passive Predictor
A passive term is a candidate predictor when it is calculated from the current completed values of its sources, is available for the cases being modeled, and represents information relevant to the conditional distribution of the target. Its inclusion should also be consistent with the analysis plan: if the substantive analysis uses an interaction or nonlinear term, omitting that structure from relevant imputation models can weaken preservation of the relationship of interest.
These are modeling judgments, not automatic consequences of making a variable passive. The conditional model still needs an appropriate form for the target, and its assumptions must be considered. For example, a linear regression imputation model assumes a suitable conditional mean relationship and an adequately modeled residual distribution; a logistic model requires a binary target and a suitable logit specification. Including a transformed predictor does not repair misspecification elsewhere, ensure valid missing-data assumptions, or prove compatibility with the substantive model.
Before using a passive term as a predictor, check the following:
- Source availability: Have the source variables been observed or imputed for the records that need the prediction?
- Update order: Has the term been recalculated since any of its sources last changed?
- Target dependence: Does the term depend on the variable currently being imputed? If so, do not use a stale value as an ordinary predictor in that update.
- Domain and scale: Is the function defined for the current source values, and does its scale make sense in the model?
- Model purpose: Does the term represent a relationship relevant to the conditional model, rather than merely adding complexity?
Worked Example: A Squared Predictor for Imputing an Outcome
In a hypothetical data set, \(X\) is a continuous exposure and \(Y\) is a continuous outcome. Some values of both are missing. The analysis model uses \(X\) and \(X^2\) to allow curvature. Consider a record whose current imputed exposure is \(X=2\), and suppose the imputation model for \(Y\) is \(Y=10+3X-0.5X^2+\varepsilon\), where \(\varepsilon\) has mean zero and standard deviation \(2\). This example assumes the squared term is scientifically plausible, the conditional mean is adequately represented by the stated model, and the residual assumptions are acceptable for this illustration.
The passive predictor is \(Q=X^2\). At the current value \(X=2\), \(Q=2^2=4\), so the fitted conditional mean is \(10+3(2)-0.5(4)=10+6-2=14\). A draw of \(Y\) for this record is then generated around a mean of 14 under the stated residual model.
Suppose a later conditional update changes the imputed exposure to \(X=3\). The correct passive value is now \(Q=3^2=9\). The outcome model’s fitted mean, if evaluated at this updated exposure, is \(10+3(3)-0.5(9)=10+9-4.5=14.5\). Keeping \(Q=4\) would instead give \(10+9-0.5(4)=17\), which combines the new \(X\) with the old \(Q\) and does not obey \(Q=X^2\).
Thus \(Q\) is a reasonable predictor when imputing \(Y\), subject to the modeling assumptions, but it must be recalculated after \(X\) changes and before a subsequent model uses it. When imputing \(X\), this same \(Q\) is target-dependent and should not be carried into the update as though it were an independent, fixed predictor.
Keeping the Derived Term Current
Let \(S=(S_1,\ldots,S_k)\) denote the sources and \(T=f(S)\). At any point in the chain, algebraic consistency requires the equality \(T=f(S)\) to hold for each record where the inputs are defined. If one source changes from \(S_j^{\text{old}}\) to \(S_j^{\text{new}}\), the term required by that condition changes from \(f(S_1,\ldots,S_j^{\text{old}},\ldots,S_k)\) to \(f(S_1,\ldots,S_j^{\text{new}},\ldots,S_k)\). Leaving the previous value in place breaks the defining equality.
This is why update order matters. A safe sequence is to draw a source variable, recalculate every passive term that depends on it, and only then fit or evaluate a later conditional model that uses those terms. When several passive variables depend on one another, calculate them in dependency order: update a term’s sources first, then the term, then any downstream term. This is a bookkeeping requirement for the data being passed between conditional models, not a separate stochastic draw.
Record which source variables define each passive term and which conditional models use that term.
For the variable being imputed, draw a value from its specified conditional model using suitable predictors.
Apply each deterministic definition to the updated source values, checking that the function is defined.
Fit or evaluate the next conditional model using passive terms only when their sources are current and the term is suitable for that target.
Worked Example: A Product Term and Two Different Targets
Suppose hypothetical variables \(X\) and \(Z\) are predictors of an incomplete outcome \(W\), and the analysis allows an interaction \(P=XZ\). At one record, the current values are \(X=2\) and \(Z=5\), so the passive product is \(P=2\times5=10\). The conditional model for \(W\) may use \(X\), \(Z\), and \(P\), if the interaction is relevant and the model assumptions are reasonable.
Now consider the next updates. If \(X\) is drawn as 3 while \(Z\) remains 5, recompute \(P=3\times5=15\). If \(Z\) is then drawn as 4, recompute again: \(P=3\times4=12\). Using \(P=10\) after either source update would make the recorded interaction inconsistent with the current pair of source values.
The predictor decision changes by target. For the \(W\) update, \(P\) is formed from the current \(X\) and \(Z\), which are predictors rather than the target. For the \(X\) update, \(P=XZ\) depends on the target \(X\); for the \(Z\) update it depends on target \(Z\). In ordinary passive updating, do not insert the old product as a fixed predictor in either source’s own conditional model. Recompute it after the draw, so it is ready for later updates such as the one for \(W\).
Domain Checks and Failure Modes
Some derived terms are not defined for every possible source value. For example, mathematically \(\log(A/B)\) is defined when \(B\ne0\) and \(A/B>0\). If the measurements are restricted to positive values, impose the additional requirement \(A>0\) and \(B>0\). A chain that generates values outside the required domain cannot evaluate the passive term as specified; the remedy is a valid domain-handling strategy, not silently retaining a stale term or substituting an arbitrary number.
Worked Example: Checking a Log-Ratio Predictor
Consider a hypothetical rate comparison using \(L=\log(A/B)\). A current draw gives \(A=12\) units and \(B=3\) units. Here \(B\ne0\), \(A/B=12/3=4>0\), and \(L=\log(4)\approx1.386\). The passive term is defined.
If a later draw gives \(A=-10\) units and \(B=-2\) units, then \(A/B=5>0\), so the mathematical expression is still defined and \(L=\log(5)\approx1.609\). It would be incorrect to reject it solely because both values are negative. However, if \(A\) and \(B\) are measurements that are substantively restricted to positive values, this draw violates the additional measurement-domain restriction and requires a model or imputation strategy that respects that restriction. If instead \(A=10\) and \(B=-2\), the ratio is \(-5\), so the real logarithm is undefined. Domain checks must follow the actual function and the scientific constraints separately.
Common warning signs include passive terms that remain unchanged after source draws, missing values in predictors that should have been recomputed, and conditional models that use a source-dependent term to predict that same source. Another warning sign is an imputation model that includes every available transformation without checking whether it is useful or identifiable. A passive term can be mathematically synchronized and still be a poor predictor because the conditional model is misspecified, the term is weakly supported by the data, or the chosen model cannot estimate the required relationship reliably.
Alternatives and Limits
There are three common choices. First, include a passive term in conditional models for suitable targets and update it deterministically after source changes. Second, omit it when the relationship is not needed or the term is not appropriate for a given target; this is simpler, but may fail to represent a relationship important to the analysis. Third, impute the derived quantity independently. That third choice is generally not passive imputation: it can produce completed records where \(T\ne f(S)\), contradicting the variable’s definition.
A deterministic update solves the consistency problem, but it does not guarantee that the resulting chained conditional models preserve every feature of a substantive analysis. Nor does adding a passive predictor alone establish a correct missingness assumption or make an unsuitable conditional distribution valid. Treat passive variables as carefully constructed predictors, not as automatic corrections for model incompatibility. Diagnostics should include checking deterministic identities in each completed data set, checking domains and missing predictor values, reviewing conditional model fit, and comparing whether important analysis relationships are represented.
Check Your Understanding
Answer each question, then compare your reasoning with the short solutions below.
- For \(Q=X^2\), \(X\) changes from \(-2\) to \(3\). What is the correct updated value of \(Q\), and why is retaining the old value wrong?
- A passive product \(P=XZ\) is used to impute \(Y\). Is it automatically suitable for imputing \(X\)? Explain.
- For \(L=\log(A/B)\), are \(A=-6\) and \(B=-3\) enough to make the mathematical expression undefined? State the calculation.
- Name two checks to perform if a passive term is used as a predictor during a chain.
Solutions. (1) The updated value is \(Q=3^2=9\). Retaining \(4\), the former value, violates the definition \(Q=X^2\) for the current \(X\). (2) No. \(P\) depends on target \(X\), so its old value is not an ordinary fixed predictor for that update; draw \(X\), then recompute \(P\). (3) The ratio is \((-6)/(-3)=2>0\), so the real logarithm is defined and \(L=\log(2)\approx0.6931\). A separate scientific restriction to positive measurements could rule out those values. (4) Check that the term was recomputed after relevant source updates and that its function is defined for the current source values. Also assess whether it is appropriate for the target model.