From Candidate Donors to an Imputed Value
Predictive mean matching (PMM) has two distinct decisions. First, it identifies observed cases whose matching scores are close to the score for a case with a missing outcome. Second, it selects a donor from that candidate set and copies the donor’s observed outcome into the missing cell. Finding the closest cases does not, by itself, determine which outcome is imputed.
This distinction matters because the matching step is based on scores, while the imputation step uses an observed response. If several cases are similarly close, choosing among them at random lets the imputed value vary across completed data sets. A fixed choice of the closest case would remove this particular source of variation.
The score and the distance rule are treated as specified inputs here. In the PMM setting, scores can depend on a fitted model and its parameter draw; consequently, the candidate set can also change when that draw changes. The random donor choice is a further step, conditional on the set currently obtained.
Uniform sampling means every member of the candidate set has the same selection probability, not that every observed case in the data has that probability. Cases outside \(D_i\) have probability zero of supplying this imputation under the current set. The donor’s observed response is copied as recorded; the random choice does not generate a new response from a regression distribution.
Assumptions and Interpretation
The arithmetic of random selection is straightforward, but its statistical usefulness depends on the candidate set being suitable. The observed donors should have outcomes that are informative for the target case, given the information used to construct the matching scores. In practical terms, the matching model should rank cases so that nearby observed cases are plausible sources of the missing outcome, and there should be observed cases in the relevant part of the score distribution.
These conditions do not follow merely from selecting the smallest distances. A close score is a tool for locating possible donors, not proof that their outcomes are exchangeable with the target’s unobserved outcome. Valid inference also depends on the assumptions for the missing-data analysis and the imputation models; donor sampling alone cannot establish those assumptions or repair a poorly specified model.
For the calculations below, the donor set is treated as fixed and the selection is uniform. Thus the probabilities and variances describe donor-sampling variation conditional on that set. In a full MICE run, parameter draws can alter scores and donor sets as well. That model-related variation is distinct from the random choice made after a particular set has been formed.
Worked Example: Ranking Cases and Sampling a Donor
Suppose a hypothetical imputation model gives the following scores for four observed cases and one target case. The outcome is a laboratory concentration in micrograms per litre (µg/L). For this illustration, closeness is measured by absolute score difference, and the two nearest observed cases form the candidate set.
| Case | Score | Observed outcome (µg/L) | Absolute distance from target score 2.4 |
|---|---|---|---|
| A | 1.8 | 8 | 0.6 |
| B | 2.1 | 11 | 0.3 |
| C | 2.6 | 16 | 0.2 |
| D | 3.4 | 20 | 1.0 |
The two smallest distances are 0.2 for case C and 0.3 for case B. Therefore \(D_i=\{B,C\}\). This step identifies the candidates; it does not yet impute 16 µg/L simply because C is the closest.
With uniform random sampling, each candidate has probability \(1/2\). The imputed value is 11 µg/L if B is selected and 16 µg/L if C is selected. Its conditional expected value is
The conditional variance, in squared concentration units, is
The actual imputation in one run is either 11 or 16 µg/L, not 13.5 µg/L. The expectation summarizes repeated random selections from the same set; it is not itself the value copied into the missing cell.
Why Random Selection Creates Variation
For a fixed set \(D_i\) of \(k\) donors and their observed outcomes \(y_j\), uniform random sampling gives each possible imputed value a probability of \(1/k\). The conditional mean is the average of donor outcomes. The conditional variance is the average squared deviation of those outcomes from that mean, using divisor \(k\), because these \(k\) values constitute the full discrete distribution for the selection.
If every donor has the same outcome, this variance is zero even though the selected donor identity is random. If donor outcomes differ, the imputed value can vary. The size of that variation depends on the outcomes in the candidate set, not only on how many donors it contains.
The calculation follows directly from the definition of a discrete random variable. Each value \(y_j\) occurs with probability \(1/k\), so its expectation is \(\sum_{j\in D_i}(1/k)y_j\). Applying the definition of variance gives \(\sum_{j\in D_i}(1/k)(y_j-E(Y_i^{\mathrm{imp}}\mid D_i))^2\), which yields the displayed expression. This is a probability variance, not the usual sample variance of the donor outcomes: the latter uses divisor \(k-1\) because it estimates a larger population variance.
Worked Example: Variation Across Repeated Imputations
Consider a fixed hypothetical candidate set with observed outcomes 4, 7, and 13 units. In each imputation, suppose one donor is sampled uniformly and independently from this set. The probabilities of imputing 4, 7, and 13 are each \(1/3\), and the expected imputed value is
The conditional variance is
Suppose six repeated imputations happen to select donors with outcomes 7, 4, 13, 7, 7, and 4 units. Their empirical frequencies are \(3/6\) for 7, \(2/6\) for 4, and \(1/6\) for 13. The mean of these six imputed values is
This realised mean differs from the conditional expectation of 8 units; a finite number of random draws need not reproduce the theoretical probabilities exactly. With six independent draws, the probability that the donor with outcome 13 is never selected is \((2/3)^6=64/729\approx0.08779\). Random sampling can therefore produce noticeably different completed values across imputations, especially when the candidate outcomes differ.
Separating Donor Sampling From Parameter Draws
A PMM imputation can vary for more than one reason. Even before the donor is sampled, a different parameter draw can change the target and observed scores and hence the candidate set. Conditional on a particular parameter draw, the donor is then selected from that draw’s set. A short variance decomposition makes the distinction explicit.
This identity is the law of total variance. To justify it, write \(Y-E(Y)=[Y-E(Y\mid\Theta)]+[E(Y\mid\Theta)-E(Y)]\), square both sides, and take expectations. The cross-product has expectation zero because \(E[Y-E(Y\mid\Theta)\mid\Theta]=0\). The first squared term has expectation \(E\{\operatorname{Var}(Y\mid\Theta)\}\), and the second has expectation \(\operatorname{Var}\{E(Y\mid\Theta)\}\), giving the result.
Worked Example: Two Sources of Variation
For a hypothetical concentration outcome, suppose two possible parameter draws are equally likely. Under the first draw, uniform sampling from the candidate set gives outcomes 2 and 6 units. Under the second, it gives outcomes 8 and 12 units. Within either set, the conditional mean is the midpoint and the conditional variance is 4 units squared:
The average within-set variance is 4 units squared. The two conditional means, 4 and 10, occur with equal probability, so their variance is
The total variance is therefore \(4+9=13\) units squared. Directly, the four equally likely outcomes are 2, 6, 8, and 12; their mean is 7, and their variance is \((25+1+1+25)/4=13\) units squared. The matching parameter draw accounts for variation between conditional means, while random donor choice accounts for variation within each set.
Diagnostics, Alternatives, and Limitations
Diagnostics should examine both stages rather than treating every imputed value as an undifferentiated model output. For a target case, check that the selected donor belongs to the intended candidate set and that the imputed value exactly equals that donor’s observed outcome. Across imputations, inspect how often each donor is selected and whether the resulting values vary. Repeated selection of one donor can be a consequence of the candidate sets or random variation; by itself, it does not establish an error.
Warning signs include a candidate set whose members have very different outcomes despite close scores, a target for which suitable observed cases are scarce, or imputed values that barely change when donor sampling is intended to be random. The first two may indicate that the score does not locate useful donors or that observed support is limited. The last may reflect identical candidate outcomes, a fixed selection rule, or a random-number implementation problem. These checks diagnose the selection mechanism; they do not prove that the missing-data assumptions hold.
| Approach | What determines the imputed value? | Key distinction |
|---|---|---|
| Uniform random selection within a nearest-donor set | An observed response from a randomly selected candidate | Allows donor-sampling variation while keeping values observed |
| Deterministic nearest-donor selection | The response of a fixed closest case | Removes random choice among candidates and can make imputations less variable |
| Random selection from all observed cases | An observed response chosen without restricting to the nearest set | Does not use the local candidate restriction described here |
| Parametric regression imputation | A model-based generated or predicted response | Does not necessarily copy an observed response |
Random donor sampling does not guarantee that the selected response is a good substitute for the missing one. It cannot create information in regions with few relevant observed cases, and it cannot correct an inadequate matching model. Conversely, always choosing the single closest donor is not automatically preferable: that choice suppresses selection variation and makes the result depend on a fixed selection rule. The appropriate distinction is procedural: candidate construction defines which observed values are eligible, and the donor-sampling rule defines how one eligible value is chosen.
Common Pitfall
Wrong reasoning: “The closest case is the donor, so its outcome must be imputed.” This confuses identifying a nearest-donor set with selecting a donor. Under uniform sampling from a set of several nearest cases, the closest case is only one candidate and need not be selected.
Correction: State both steps separately. First list the eligible nearest donors under the specified score and distance rule. Then apply the stated sampling rule to that set and copy the selected donor’s observed response. For uniform sampling from \(k\) candidates, each has probability \(1/k\); the closest one does not receive probability one merely because it ranks first.
Check Your Understanding
Answer each question using the distinction between forming a candidate set and sampling from it.
- A fixed candidate set has outcomes 3, 9, and 12 units. Under uniform sampling, what is the probability of imputing 9 units, and what is the conditional expected imputed value?
- For the same candidate set, calculate the conditional variance of the imputed value in squared units.
- Two parameter draws produce different candidate sets, but each set has identical donor outcomes. Can donor sampling still vary the imputed value within either set? Explain.
- Why is a realised average over a small number of imputations not required to equal the conditional expected value?
Solutions: (1) The probability is \(1/3\), and the expected value is \((3+9+12)/3=8\) units. (2) The variance is \(\{(3-8)^2+(9-8)^2+(12-8)^2\}/3=(25+1+16)/3=14\) units squared. (3) No. If every donor outcome within a set is identical, every donor choice copies the same value, so the conditional variance is zero; different parameter draws can still change the value between sets. (4) A finite sequence of random donor choices has realised frequencies that may differ from their probabilities, so its average can differ from the theoretical conditional expectation.