Why a Total Score Must Follow Its Items
A questionnaire total is often used as a predictor, outcome, or summary measure. If some items are missing, however, the total cannot be treated as though it were an independently observed quantity. Its value is determined by the item responses. In multiple imputation, imputing the missing items and then recomputing the total preserves that defining relationship in every completed data set.
The example throughout this tutorial uses three items, each scored from 0 to 4, with larger values indicating more of the measured construct. For a respondent with item responses \(Q_1,Q_2,Q_3\), the total score ranges from 0 to 12. The score is available as a complete-data value only when all three items have values, whether originally observed or imputed.
This is passive imputation of a derived variable: the item values are imputed through their conditional models, and the score is recalculated from those values. The score has no separate imputation model in this construction. Earlier material on passive imputation establishes the general principle; here the focus is to follow the arithmetic through a concrete chained-equations run.
Conditions for a Meaningful Score and Imputation
Before calculating a total, its scoring rule must be well-defined. In this example, each item is coded from 0 to 4, all items point in the same direction, and the declared score is their unweighted sum. If the questionnaire requires reverse coding, that recoding must be applied consistently before summing. The score’s range follows directly: the smallest possible sum is \(0+0+0=0\), and the largest is \(4+4+4=12\).
The imputation models must also be appropriate for the missing item values. For ordinal items with categories 0 through 4, an ordinal conditional model is one possible choice; a categorical model that respects the five allowed categories may also be appropriate. The conditional models should include useful observed information, such as other questionnaire items and relevant auxiliary variables, and should reflect important relationships in the intended analysis. The standard chained-equations assumptions include a defensible missing-at-random condition given the variables in the imputation procedure, adequate overlap in the observed data, and sufficiently well-specified conditional models. These assumptions are substantive and statistical; recalculating the score does not make them true.
Each imputed item must be a valid response in its allowed range. For example, a draw of 5 or a negative value is invalid for the 0-to-4 items and cannot be made acceptable by summing it. Using a model that can generate values outside the item’s response categories requires a justified procedure to enforce valid categories; simply treating the response as unrestricted continuous data and rounding is not automatically valid.
Worked Example: Defining the Score for Observed and Missing Items
Suppose a hypothetical respondent has \(Q_1=2\), \(Q_2=3\), and \(Q_3=1\). Under the stated scoring rule, the total is
The score is 6 points. Now suppose \(Q_2\) is missing while \(Q_1=2\) and \(Q_3=1\) are observed. The two observed responses sum to \(2+1=3\), but this is not the defined three-item total. It is a partial sum, and treating it as the full score would implicitly assign zero to the missing item. Once an imputed value is drawn, say \(Q_2=3\), the completed score is \(2+3+1=6\) points.
This distinction matters even when a partial sum is useful descriptively: it is not interchangeable with the specified total scale. The target being analyzed here is the sum of all three item responses.
Tracing a Score Through Chained-Equations Sweeps
Consider the following hypothetical data. A dash marks an item that is missing in the observed data. Respondent D has no missing items, so that person’s score remains fixed.
| Respondent | \(Q_1\) | \(Q_2\) | \(Q_3\) | Observed status |
|---|---|---|---|---|
| A | 2 | — | 1 | \(Q_2\) missing |
| B | — | 3 | 2 | \(Q_1\) missing |
| C | 4 | 1 | — | \(Q_3\) missing |
| D | 0 | 2 | 3 | All observed |
For illustration, suppose valid initial values are used to start the chain: \(Q_2=2\) for A, \(Q_1=2\) for B, and \(Q_3=2\) for C. The initial completed scores are therefore A: \(2+2+1=5\), B: \(2+3+2=7\), C: \(4+1+2=7\), and D: \(0+2+3=5\). These starting values are not treated as final imputations; the missing cells are updated by the chained conditional models.
A sweep updates the missing item values according to the specified chained-equations procedure. The following hypothetical draws are illustrative values, not results from fitted data. At the end of each sweep, the total is recalculated from the current three item values for each respondent.
| End of sweep | A: \((Q_1,Q_2,Q_3)\) | A: \(S\) | B: \((Q_1,Q_2,Q_3)\) | B: \(S\) | C: \((Q_1,Q_2,Q_3)\) | C: \(S\) | D: \(S\) |
|---|---|---|---|---|---|---|---|
| Initial values | (2, 2, 1) | 5 | (2, 3, 2) | 7 | (4, 1, 2) | 7 | 5 |
| 1 | (2, 3, 1) | 6 | (1, 3, 2) | 6 | (4, 1, 3) | 8 | 5 |
| 2 | (2, 1, 1) | 4 | (2, 3, 2) | 7 | (4, 1, 1) | 6 | 5 |
| 3 | (2, 2, 1) | 5 | (3, 3, 2) | 8 | (4, 1, 2) | 7 | 5 |
Worked Example: Recomputing Totals After Three Sweeps
For respondent A, \(Q_1=2\) and \(Q_3=1\) are observed. The missing \(Q_2\) draw is 3 at the end of sweep 1, so
At the end of sweep 2, the new draw for \(Q_2\) is 1, giving \(S_A^{(2)}=2+1+1=4\) points. At the end of sweep 3, its draw is 2, giving \(S_A^{(3)}=2+2+1=5\) points. The total changes because the imputed item changes; the observed items remain fixed.
For B at the end of sweep 2, the imputed \(Q_1\) is 2, while the other responses are 3 and 2. Thus \(S_B^{(2)}=2+3+2=7\) points. For C at the end of sweep 1, the imputed \(Q_3\) is 3, so \(S_C^{(1)}=4+1+3=8\) points. D’s score is \(0+2+3=5\) points at every sweep because all three of D’s item values are observed.
In general, for imputation \(m\) and sweep \(t\), the score is calculated from the item values then present:
The superscripts identify the imputed data set and the point in the chain. They do not imply that a separate random draw was made for \(S_i\). After a final completed data set has been retained, its score is still exactly the sum of its three items.
Why Recalculation Preserves the Intended Relationship
The justification is direct. Let the current item vector for respondent \(i\) be \(\mathbf{Q}_i=(Q_{i1},Q_{i2},Q_{i3})\), and define the score function \(f(\mathbf{Q}_i)=Q_{i1}+Q_{i2}+Q_{i3}\). If a chained update replaces one missing component, it changes the vector from \(\mathbf{Q}_i\) to a new vector \(\mathbf{Q}_i'\). By the score’s definition, the corresponding new value must be \(f(\mathbf{Q}_i')\), not the former value \(f(\mathbf{Q}_i)\).
For example, if only \(Q_{i2}\) changes by \(\Delta\), then the updated score must change by the same amount:
Thus recalculation is not a cosmetic step. It is what ensures that every record in each completed data set satisfies the deterministic score definition. If a score is retained from an earlier sweep after an item has been updated, the record violates the rule used to define the score.
The item draws also determine the score’s uncertainty. For random completed item values, the variance of the sum is
The covariance terms show why separately plausible item imputations are not enough if they fail to represent relationships among items. Passive recomputation correctly carries each completed item vector into a score, but it cannot repair conditional models that generate an unrealistic joint pattern of responses.
Worked Example: Score Variation Across Completed Data Sets
Suppose the same hypothetical respondent has fixed observed items \(Q_1=2\) and \(Q_3=1\), while three retained imputations for the missing \(Q_2\) are 1, 2, and 3. The corresponding completed scores are
The mean of these three illustrative score draws is \((4+5+6)/3=5\) points. Their sample variance is
This arithmetic describes variation among these three example draws; three imputations are not being recommended as an adequate number for an actual analysis. The key point is that each score is computed within its own completed item vector. Replacing all three scores with their average of 5 before analysis would erase imputation-to-imputation variation and create a fractional summary rather than preserve the separate completed records.
Diagnostics, Alternatives, and Limitations
Useful checks distinguish errors in the deterministic calculation from problems in the imputation models. For every completed data set, verify the identity \(S_i=Q_{i1}+Q_{i2}+Q_{i3}\) for every respondent. Check that every item belongs to \(\{0,1,2,3,4\}\), every score is an integer from 0 through 12, and originally observed items have not been overwritten. These are direct consistency checks; they should hold exactly, not approximately.
Then examine whether the imputed item values and combinations are plausible for the questionnaire and analysis. Inspect the distributions of observed and imputed responses, the frequency of each response category, and relationships among items and relevant predictors. Across iterations, trace summaries of the imputed items and derived score. Persistent drift, implausible category frequencies, or markedly different patterns across chains can indicate poor mixing, inadequate model specification, or too little information in some response patterns. A stable-looking score trace alone does not establish that the item models are appropriate.
A common mistake is to update a missing item but leave the total from the previous sweep. For A, if \(Q_2\) changes from 2 to 3, retaining the old total \(2+2+1=5\) is inconsistent; the current total is \(2+3+1=6\). Recompute from the current item values after updates. Another mistake is to impute a score independently and also impute its items. That can produce a score that disagrees with their sum. The correction is to decide which quantities are primary and, for this questionnaire total, derive the score deterministically from the completed items.
Directly imputing the total can be an alternative when the total itself is the target and item-level completed values are not needed, but it does not by itself produce item responses consistent with that total. Complete-case analysis avoids imputing items but discards respondents with any missing item and may be inefficient or biased under plausible missingness mechanisms. Neither alternative automatically resolves problems in the missingness assumptions or model specification. For a score defined from individual items and analyses that need those items, imputing the components and passively recomputing the score preserves the score’s definition while allowing uncertainty to flow from the item draws.
The limitation is equally important: deterministic recomputation guarantees algebraic consistency, not validity of the scale, adequacy of the item imputation models, or compatibility with every planned analysis. The questionnaire’s scoring rule must be justified, and the imputation procedure must represent the relevant item relationships and missingness process. The score’s range and arithmetic can be checked exactly; the quality of its imputed distribution requires statistical and substantive diagnostics.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For items scored 0 to 4, a respondent has completed values 3 and 2, with the third item missing. What is the full-score range, and what is the observed partial sum?
- A three-item score is defined as the sum of its components. One component changes from 1 to 4 after an update. By how much must the score change if the other components are fixed?
- Give two exact checks for a completed data set containing this 0-to-12 total.
- Why can a plausible marginal distribution for each item still be insufficient to guarantee plausible total scores?
Solutions. (1) The full score ranges from \(0+0+0=0\) to \(4+4+4=12\); the observed partial sum is \(3+2=5\), not a completed three-item total. (2) The component increases by \(4-1=3\), so the sum must increase by 3 points. (3) Verify that each item is one of 0, 1, 2, 3, or 4 and that every stored score equals the sum of its three items; also verify that each score lies from 0 to 12. (4) Marginal checks do not establish that the joint item patterns or item-to-item associations are realistic. Since the sum depends on combinations of responses, unrealistic dependence can yield implausible totals even when each item’s marginal distribution looks reasonable.