Missing Inputs to Deterministic Transformations

When Can a Derived Value Be Calculated?

A deterministic transformation has a fixed rule: once its inputs are specified, its output is determined. The practical complication in multiple imputation is that one or more inputs may be missing for a record. Until those inputs have values, the transformation may not be evaluable. Once the inputs have been imputed, the output can be calculated separately in each completed dataset.

For instance, a total score defined as the sum of three items cannot ordinarily be calculated for a person if one item is missing. If that item receives different imputed values in different completed datasets, the total will also differ. The derived value is not an independently imputed quantity; it is the result of applying the stated rule to each dataset’s source values.

Definition: Let \(X_{ij}\) denote source variable \(j\) for record \(i\), and let \(S\) be the set of source variables used by a deterministic transformation \(g\). The derived value for record \(i\) is \(D_i=g((X_{ij})_{j\in S})\) whenever the required inputs have values in the domain of \(g\). In completed imputed dataset \(m\), calculate \(D_i^{(m)}=g((X_{ij}^{(m)})_{j\in S})\). A missing source value must first be supplied by the imputation procedure unless the output is already determined from the available information.

Availability is a record-level property. A derived value may be calculable for one record but not another, depending on which of that record’s inputs are observed or imputed. In a completed dataset, all required source values have been assigned, so the transformation can be evaluated for every record for which its inputs are valid.

Availability During Imputation

Let \(R_{ij}=1\) indicate that source value \(X_{ij}\) was missing before imputation, and \(R_{ij}=0\) indicate that it was observed. Observed inputs remain the recorded inputs; imputation supplies values for missing inputs. During an iterative algorithm, a missing input becomes available once its conditional imputation step has assigned it a value. The transformation can then be evaluated using the current source values.

For a transformation requiring all variables in \(S\), define \(A_i^{(t,m)}\) to be 1 if every required input for record \(i\) has a current value at iteration \(t\) of imputation \(m\), and 0 otherwise. In the usual case where the output depends on all of its inputs, it is evaluable when \(A_i^{(t,m)}=1\). At completion, write the derived value as:

D_i^{(m)}=g\left((X_{ij}^{(m)})_{j\in S}\right) \quad\text{when all required } X_{ij}^{(m)} \text{ are available.}

This rule is about calculation, not a claim that any imputation model is adequate. The source-variable imputations still need to be appropriate for the analysis and the missing-data problem. Availability itself is an algebraic matter: it does not require a missing-at-random assumption. Inferential conclusions from the completed datasets do depend on the imputation and missingness assumptions underlying the analysis.

If a function does not actually depend on one of its listed inputs for a particular record, its value may sometimes be known earlier. For example, if a product includes a source value known to be zero, the product is zero even when another input is not yet assigned. Such special cases follow from the function itself. A reliable general workflow is to wait until all declared source variables have current values, then calculate the transformation, rather than building special-case availability rules into the imputation process.

Worked Example: A Total With One Missing Item

Suppose a hypothetical questionnaire total is defined by \(T_i=A_i+B_i+C_i\), where each item is a whole-number score from 0 to 5. For one participant, the recorded values are \(A_i=4\), \(B_i\) missing, and \(C_i=2\). The total is not yet calculable from the recorded inputs because \(B_i\) is required.

In completed dataset 1, suppose the imputed value is \(B_i^{(1)}=3\). Then:

T_i^{(1)}=A_i+B_i^{(1)}+C_i=4+3+2=9.

In completed dataset 2, suppose \(B_i^{(2)}=5\). Then:

T_i^{(2)}=A_i+B_i^{(2)}+C_i=4+5+2=11.

The two totals differ because the imputed source value differs. The observed items stay at 4 and 2 in both datasets. Both derived totals are calculated from their own completed source values, not copied from one dataset to the other.

Transformations With Several Incomplete Inputs

When multiple inputs can be missing, one missing value is enough to prevent ordinary evaluation of a function that needs all of them. Imputing just one input does not make the result available if another required input remains missing. Once every source value is assigned, apply the same function to the values in that particular completed dataset.

The imputation process may draw different values for different missing sources and different datasets. Therefore, even when the transformation itself is fixed, its output can vary across imputations. What must remain fixed is the rule \(g\), not the numerical derived value.

Worked Example: A Product With Two Incomplete Sources

Consider a hypothetical analysis term \(P_i=X_iZ_i\). For one record, \(X_i=2\) is observed and \(Z_i\) is missing. Suppose \(Z_i\) is imputed as 1.5 in the first completed dataset and 3.0 in the second. The product is available in each dataset after the missing input has been assigned:

P_i^{(1)}=2(1.5)=3.0, \qquad P_i^{(2)}=2(3.0)=6.0.

For a second record, suppose \(X_i\) is missing and \(Z_i=4\) is observed. If the imputed \(X_i\) values are 0.8 and 1.1, the calculations are:

P_i^{(1)}=0.8(4)=3.2, \qquad P_i^{(2)}=1.1(4)=4.4.

In both records, there is a single missing input, but it is a different input. The product becomes available only after that record’s missing source has a value. If both \(X_i\) and \(Z_i\) were missing, both would need values before the ordinary product calculation could be made.

The same logic applies to a function of three or more variables. For example, a weighted score \(W_i=2U_i+V_i+3Q_i\) requires values for \(U_i\), \(V_i\), and \(Q_i\). If two inputs are missing, imputing just one leaves the usual calculation incomplete. Once all three are present, use the formula as written; do not replace the missing inputs by zero or silently omit their terms unless that alternative rule is part of the defined quantity.

Worked Example: A Weighted Score With Two Missing Inputs

Suppose a hypothetical index is defined as \(W_i=2U_i+V_i+3Q_i\). For one record, \(U_i=3\) is observed, while \(V_i\) and \(Q_i\) are missing. In one completed dataset, let \(V_i^{(1)}=2\) and \(Q_i^{(1)}=1\). Substitution gives:

W_i^{(1)}=2(3)+2+3(1)=6+2+3=11.

In a second completed dataset, let \(V_i^{(2)}=4\) and \(Q_i^{(2)}=2\). Then:

W_i^{(2)}=2(3)+4+3(2)=6+4+6=16.

Before both missing inputs have assigned values, this score is not ordinarily evaluable. Afterward, each completed dataset gets its own score, calculated from its own imputed values and the same observed value \(U_i=3\).

Why Calculation Must Follow the Source Values

The justification for recalculating a deterministic quantity is direct. A deterministic rule assigns one output to a given set of inputs. If two completed datasets have different inputs, applying the rule separately gives the outputs corresponding to those different inputs. Using an output calculated from some other set of inputs would violate the defining relationship.

(X_{ij}^{(m)})_{j\in S}\ne (X_{ij}^{(r)})_{j\in S} \ \text{may imply}\ g((X_{ij}^{(m)})_{j\in S})\ne g((X_{ij}^{(r)})_{j\in S}),

where \(m\) and \(r\) index completed datasets. The word “may” matters: different inputs can still yield the same output, but that cannot be assumed in general. Thus, the required consistency condition is that every derived value agree with the function applied to that dataset’s source values.

This also explains why a derived variable should not ordinarily receive an unrelated imputation draw. A separate draw could produce a value that does not equal the defined function of its sources. The deterministic relationship is maintained by calculation, while the uncertainty in the result arises through the imputed source values.

Checks, Alternatives, and Limitations

After calculating derived values, check that the source values used were the intended ones, that observed inputs were preserved, and that the transformation was evaluated within each completed dataset. For a small set of records, recompute a few values by hand. For a larger data set, verify the identity \(D_i^{(m)}=g((X_{ij}^{(m)})_{j\in S})\) across records and imputations. This catches both coding errors and values left over from an earlier state of the source data.

Unassigned inputs: Confirm that no required source remains missing when the transformation is calculated.
Stale outputs: Recalculate after any source value changes; an old derived value may no longer match its inputs.
Wrong source set: Check that the function uses all and only the variables in its definition.
Unexpected values: Investigate outputs that do not match hand calculations or that cannot arise from the stated function and plausible source values.

A tempting alternative is to impute the derived variable directly, independently of its sources. That does not, by itself, preserve the deterministic definition. Another alternative is to omit records whose sources are incomplete; this avoids imputing those inputs but discards incomplete records and changes the information used. Calculating the derived quantity after imputing its sources preserves the defined relationship, but it does not guarantee that the source-variable imputation models are adequate.

A known derived value may sometimes provide a constraint on its incomplete sources. However, a single output need not identify the individual inputs: for example, many pairs of numbers can have the same sum. Do not treat a derived value as a uniquely recoverable substitute for its missing sources without a model or additional information that justifies that recovery. The function must also be defined for the imputed source values; handling transformations with restricted domains requires separate care.

Common Mistake

A common error is to compute a total using only observed items and treat that partial sum as the completed total. For example, if \(A_i=4\), \(B_i\) is missing, and \(C_i=2\), reporting \(4+2=6\) as the total silently changes the definition from \(A_i+B_i+C_i\) to a different quantity. The correction is to impute \(B_i\) under an appropriate model and then calculate the defined total in each completed dataset. If the intended score instead allows a specified prorating or partial-item rule, that rule must be explicitly defined and used consistently; it is not interchangeable with the original sum.

Key takeaway: A deterministic derived value becomes available when the required source values for that record have been assigned. Calculate it from the source values in each completed dataset, and verify that the resulting value still obeys the defining function.

Check Your Understanding

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

A score is defined as \(S_i=K_i+L_i\). For one record, \(K_i=7\) is observed and \(L_i\) is missing. Is \(S_i\) available from the recorded data alone? What should be done in each completed dataset?
For \(D_i=A_iB_i\), suppose \(A_i=0\) is observed and \(B_i\) is missing. What is the product under ordinary arithmetic, and why is waiting until all inputs are assigned still a sound general workflow?
A quantity is defined as \(Q_i=2X_i+Y_i\). In one completed dataset \(X_i=3\), \(Y_i=5\); in another \(X_i=4\), \(Y_i=1\). Calculate both values and state what must remain the same across datasets.
Why is independently imputing a derived score generally not equivalent to calculating it after imputing its sources?

Solutions. (1) No. Calculate \(S_i^{(m)}=7+L_i^{(m)}\) separately for each completed dataset after imputing \(L_i\). (2) The product is \(0\times B_i=0\) for any ordinary numerical value of \(B_i\). Nevertheless, waiting for all declared inputs to be assigned gives a consistent general procedure and avoids special-case rules that may not apply to other functions or records. (3) The first value is \(2(3)+5=11\); the second is \(2(4)+1=9\). The function \(Q_i=2X_i+Y_i\) remains the same, even though its output differs. (4) An independently imputed score need not equal the function of the completed source values, so it can violate the deterministic definition. Calculating the score from those sources preserves that definition.