Designing an FCS Update Schedule

What an FCS Update Schedule Specifies

A fully conditional specification (FCS) run repeatedly updates the missing values of one variable at a time. An update schedule records which incomplete variable is updated at each step, which other variables predict it, and which current values of those predictors are used. It also states when the chain’s completed data are saved. These details make an algorithm reproducible: saying only “we used chained equations” does not specify the sequence of computations.

The schedule does not, by itself, determine whether each conditional model is appropriate or whether the resulting conditionals correspond to one joint distribution. Those are separate questions. Here the goal is to specify the computation correctly and make its state transitions auditable.

For a schedule to be well-defined, it must identify the target variables, their order, and each target’s predictor set. A fully observed variable may be a predictor but is not an incomplete target. A variable omitted from a target’s predictor set is not used in that update, even if its values are available. In ordinary single-target FCS, each incomplete target occurs once per sweep.

Writing the Within-Sweep Conditioning Sets

Suppose the incomplete targets are \(Y_1,\ldots,Y_p\), and choose a fixed order represented by the permutation \(\pi=(\pi_1,\ldots,\pi_p)\). Let \(Y_j^{(t-1)}\) be the completed values at the end of sweep \(t-1\). During sweep \(t\), write \(Y_j^{(t,k)}\) for the values after step \(k\) has been completed. Thus, at the start of sweep \(t\), \(Y_j^{(t,0)}=Y_j^{(t-1)}\) for every target. At step \(k\), target \(Y_{\pi_k}\) is updated; earlier targets in the order already have their sweep-\(t\) values, while later targets still have their sweep-\((t-1)\) values.

Let \(\mathcal{C}_{\pi_k}\) be the chosen predictor set for target \(Y_{\pi_k}\), drawn from other incomplete variables and fully observed predictors. The correct conditioning set uses the version of each predictor according to its position in the schedule:

Here \(f_{\pi_k}\) denotes the specified conditional imputation model; \(Y_{\pi_k,\mathrm{obs}}\) denotes the target’s observed values used in fitting or otherwise informing that model, and \(X_{\mathcal{C}_{\pi_k}}\) denotes the fully observed predictors included in its predictor set. The sets in the display are restricted to predictors actually included in \(\mathcal{C}_{\pi_k}\). The target’s observed values are not overwritten. After all \(p\) updates, define \(Y_j^{(t)}=Y_j^{(t,p)}\).

This indexing is particularly useful when the order is not the numerical order of the variable labels. For example, if \(\pi=(2,3,1)\), then at the first step both later targets are still at their previous-sweep values: \(Y_3^{(t-1)}\) and \(Y_1^{(t-1)}\). The schedule position, not the target’s variable number, determines whether its value is current or previous.

Worked Example: A Three-Target Sweep

Justification: Why the Schedule Uses Mixed Iteration Indices

The indexing follows directly from the sequential algorithm. At the start of sweep \(t\), no target has yet been updated, so every incomplete predictor has its value from \(t-1\). After step 1, the first target has a new value; after step 2, the first two targets have new values, and so on. Therefore, immediately before step \(k\), a predictor \(Y_{\pi_r}\) with \(r<k\) has been updated and must use its current value. A predictor with \(r>k\) has not been updated and must use the previous completed state. This accounts for every scheduled predictor without assuming that the numerical index of a variable matches its position in the order.

The target itself is not included as its own predictor. After its missing entries are drawn, its observed entries are retained, so the update changes only the missing part of that target. Applying this rule at every step yields a complete sweep: every target is updated once, and every predictor value is traceable to either an earlier update in the same sweep or the completed state at its start.

Worked Example: A Non-Numerical Update Order

Choosing and Recording the Schedule

A practical schedule should be explicit enough that another analyst could reproduce its update sequence. For each target, record its position in the sweep, predictor set, and the rule for handling observed target values and fixed covariates. The order is generally a design choice rather than a claim that one permutation is universally best. Earlier material on sensitivity to variable ordering and randomized ordering motivates treating the chosen order as something to document and, when appropriate, examine for robustness.

Here is a compact record for the first worked example:

Worked Example: Defining Which Sweeps to Save

Diagnostics, Alternatives, and Limitations

Schedule auditing begins with bookkeeping. For a small test run, write down the order and check each update against the state used for every predictor. Confirm that every intended incomplete target appears exactly once per sweep, observed entries never change, fully observed predictors remain fixed, and saved records are end-of-sweep states. A trace indexed by update number rather than sweep number can help reveal accidental omissions or duplicated target updates.

• Stale-value error: using the previous-sweep value for a target that has already been updated ignores the specified sequential schedule. Conversely, using a current-sweep value for a target not yet updated uses a value that does not yet exist in that sweep.
• Wrong target order: confusing a variable’s label with its position in the permutation can assign incorrect conditioning values, especially for a non-numerical order.
• Partial saving: retaining a state after only some targets have been updated produces an inconsistent record of a full sweep.
• Unintended predictor set: an available variable is not automatically a predictor. Compare the implemented model matrix or model specification with the recorded set for each target.

A fixed systematic order is straightforward to implement and audit. A randomized order changes the permutation across sweeps and can be used as an ordering robustness check, but its randomization rule and realized order should be recorded. Updating all targets from only the previous sweep’s values is a different, simultaneous-style scheme; it is not the sequential sweep described here. Neither a careful schedule nor a particular ordering guarantees good mixing, valid conditional models, or a compatible joint distribution. These are limitations of what schedule design alone can establish; the chain still requires appropriate model choices and diagnostics.