PMM for Bounded and Integer-Valued Outcomes

Why the Outcome’s Support Matters

A bounded measurement cannot fall below its physical or defined minimum, and a count takes integer values. An imputation method that produces a value of \(-2.4\) for a nonnegative count, or 103 for a score bounded above by 100, has produced an impossible value. Predictive mean matching (PMM) avoids that particular problem by imputing an observed donor’s outcome rather than a newly generated outcome from a regression distribution.

For these outcomes, the key distinction is between the prediction used to find donors and the outcome copied from a donor. A prediction can be fractional or even outside the outcome’s allowed range: it is a matching score, not the imputation. The donor outcome, by contrast, is an actual observed value.

Definition: Let \(Y\) be an outcome with observed values \(y_i\) for cases \(i\) in the observed set \(O\), and let \(X\) denote predictors observed for both observed and incomplete cases. PMM assigns each case a predictive score, such as a fitted conditional mean \(\hat m_i\). For a case \(j\) with \(Y\) missing, it identifies a set of observed donors with scores close to \(\hat m_j\), then draws a donor \(d(j)\) from that set and imputes \(Y_j=y_{d(j)}\). The matching score guides the donor choice; the donor’s observed outcome is the imputed value.

This tutorial focuses on what changes when \(Y\) is bounded or integer-valued. The general PMM algorithm, parameter draws, and donor selection are treated in the earlier material on Predictive Mean Matching. In practice, the score model should reflect the outcome and predictors well enough to rank cases with similar expected outcomes. The donor pool must also contain observed cases relevant to the incomplete case. As with other imputation methods, credible inference requires an appropriate missingness and imputation-model setup; PMM’s support-preserving property alone does not establish that setup.

What PMM Guarantees—and What It Does Not

Suppose a bounded outcome is known to lie in a set \(S\), and every observed donor outcome belongs to \(S\). Because PMM copies one of those donor outcomes, the imputed outcome belongs to \(S\) as well. The same argument applies when \(S\) is the set of integer values observed in the donor pool.

Y_j^{\mathrm{imp}}=y_{d(j)}, \qquad y_{d(j)}\in S \quad\Longrightarrow\quad Y_j^{\mathrm{imp}}\in S .

This is a direct consequence of donor copying, not a consequence of the fitted regression being correctly bounded. The fitted value can be non-integer even when \(Y\) is a count. Nor does PMM necessarily reproduce the full theoretically possible support: it can only copy values available among observed donors. If the only observed donors have counts from 0 to 4, PMM cannot impute a 5, even if 5 is possible in the population.

For a bounded measurement, distinguish the known allowable interval from the range actually present in the donor data. If the scale is defined from 0 to 100, a donor value will be within that interval when the observed data are valid. But a donor-based imputation will not create unobserved values in the interval. It can therefore preserve allowable bounds while still underrepresenting a region of the possible scale.

Worked Examples

Worked Example: A Measurement Bounded from 0 to 100

In a hypothetical assessment, a measurement is recorded on a scale from 0 to 100. Four cases have observed outcomes and fitted PMM scores. A fifth case has a missing outcome and a score of 46.5. Use two nearest donors and suppose the random donor draw selects the closest one.

Case	Observed outcome	Fitted score	Distance from 46.5
A	20	12.0	34.5
B	43	51.0	4.5
C	48	68.0	21.5
D	70	96.0	49.5

The distances are absolute differences between the missing case’s score and each donor score. For example, case B has distance \(|46.5-51.0|=4.5\). The two nearest donors are B and C, with distances 4.5 and 21.5. Under the stated draw, B is selected and the missing measurement is imputed as 43, not as the fitted score 46.5.

The imputation is within the allowed range \([0,100]\) because the donor outcome 43 is within that range. If the random draw instead selected C, the imputation would be 48. The score determines donor proximity, while the observed donor value supplies the imputed measurement.

Worked Example: A Count Bounded by a Fixed Denominator

Consider a hypothetical count of correct answers on an 8-item test. The outcome is an integer from 0 through 8. A binomial score model is used, with score \(\hat m_i=8\hat p_i\), where \(\hat p_i\) is the fitted probability of a correct answer for a case with that case’s predictors. This score has the interpretation of an expected count under the model. A binomial model is reasonable only if its assumptions, including a fixed number of trials and an adequate account of trial dependence, are tenable; a different suitable count model may be needed otherwise.

For a case with a missing count, suppose the fitted expected count is 3.2. The available donor values and scores are shown below.

Donor	Observed count	Fitted expected count	Distance from 3.2
A	0	1.2	2.0
B	2	2.1	1.1
C	3	3.0	0.2
D	5	4.2	1.0
E	7	6.1	2.9

The distances use \(|3.2-\hat m_i|\). With three donors, the nearest are C, D, and B, at distances 0.2, 1.0, and 1.1. Suppose the random draw selects D. The imputation is 5 correct answers. The score 3.2 is not rounded to 3 and inserted; it is used to locate plausible donors. The imputed value is an observed integer in the allowed set \(\{0,1,\ldots,8\}\).

This procedure does not require the fitted score itself to be an integer. It does require that the score ranking be informative: cases close in score should have donor outcomes that are reasonable substitutes for the missing outcome, given the predictors used.

Worked Example: Why Arbitrary Category Codes Are Not Count Outcomes

Suppose a hypothetical survey records a person’s preferred color as one of three nominal categories: red, blue, or green. The labels are coded 1, 2, and 3 for data storage. A tempting but inappropriate approach is to fit a numerical outcome model and use its predicted code as a PMM score.

With this coding, a prediction of 2.1 is closer to code 2 (blue) than to code 1 (red). But the numbers do not measure an amount: there is no meaningful claim that blue is one unit above red, or that green lies between red and a fourth category. If the codes were changed to red = 1, blue = 100, and green = 2, the numerical distances and donor rankings could change even though the recorded preferences did not.

That dependence on arbitrary labels shows why these codes are not a numerical outcome for PMM. A donor draw would return a valid category label, but that fact does not make the numerical score or its matching distances meaningful. Use a categorical imputation model appropriate to nominal outcomes, rather than treating the codes as quantitative counts.

Choosing a Score and Checking Its Use

A bounded or integer-valued outcome does not by itself dictate one score model. A score can be based on a suitable conditional mean—for example, a model for a bounded count with a known denominator, or a count model when the count structure calls for one. The count-model material earlier in the course covers choices such as binomial, negative-binomial, and zero-inflated or hurdle models. For PMM, the practical question is whether the resulting score ranks observed donors usefully for the incomplete cases.

A linear regression score is also sometimes used for numerical outcomes. Its fitted values need not respect the outcome bounds or be integers; that is not automatically a problem because scores are not imputed outcomes. However, a poor score model may rank donors badly. For instance, a model that misses a nonlinear relationship, a strong zero-generating mechanism, or important predictors can place cases with quite different outcome distributions near each other.

Useful checks are tied to the outcome and the actual donor choices:

Confirm that every imputed bounded value is within the defined limits and every imputed count is an allowed integer.
Compare observed and imputed frequencies at important values, especially zeros, upper-bound values, and other scientifically meaningful endpoints.
Inspect score distances for selected donors. Large distances indicate that a nominally “nearest” donor may still be a poor match.
Check whether a small number of observed cases supply many imputations. Repeated use can make the imputed distribution depend heavily on a few donors.
Compare distributions across relevant predictor patterns. Plausible overall bounds do not ensure plausible conditional relationships.

These checks do not prove that the imputation model is adequate. They can, however, expose impossible values, poor score ranking, or a donor pool that does not represent the cases with missing outcomes. The later tutorial on cases where predictive scores fail to rank suitable donors treats score-ranking failure in greater depth.

Comparison and Limitations

Parametric regression imputation draws outcomes from an explicitly specified conditional distribution. If that distribution is suitable for the outcome and correctly parameterized, it can represent values not observed in a finite donor pool. But a model that ignores a lower bound, upper bound, or integer support may generate impossible values. A suitable bounded or count distribution can address that issue, although its assumptions must be assessed.

Rounding a continuous regression imputation is not generally equivalent to imputing a count. Rounding may create boundary problems, distort probabilities of individual counts, and fail to account for the count-generating process. PMM instead copies an observed value, avoiding the need to round a fractional score. For a nominal category, however, choosing a categorical imputation model is more appropriate than assigning arbitrary numeric codes and applying numerical PMM.

PMM’s strength is also a limitation: it cannot generate a value absent from its donor pool. If high counts or boundary values are rare among observed cases, the imputed values may not represent their true frequency among incomplete cases. Nor does matching a single score ensure similarity on every predictor or guarantee that donor outcomes are exchangeable with the missing case’s outcome. PMM preserves observed support; it does not by itself guarantee unbiased estimates, correct conditional distributions, or valid uncertainty.

Common Mistakes

Using the fitted score as the imputation. A score such as 3.2 for a count is used to find donors, not inserted as the missing count. Impute the selected donor’s observed integer value.
Assuming observed bounds guarantee realistic imputations. A value can be within the scale limits and still be implausible for the case’s predictors. Assess donor similarity and conditional distributions as well as bounds.
Treating every integer code as a numerical measurement. Counts have meaningful arithmetic; nominal labels encoded as 1, 2, and 3 do not. Arbitrary recoding can change numerical distances without changing the categories.
Assuming PMM can recover unobserved values. A donor method cannot select an outcome value that is absent from the available donor outcomes. Consider whether this limitation matters for the estimand and missingness pattern.

Key takeaway: For bounded measurements and integer-valued counts, PMM uses a score to identify donors and copies an observed donor outcome. This preserves the donor’s observed value and support, but useful imputation still depends on informative scores, suitable donors, and an appropriate imputation setup.

Check Your Understanding

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

A bounded measurement has an allowable range from 0 to 50. A PMM score for a case with a missing value is 26.4, and the selected donor’s observed value is 31. What is imputed, and why is 26.4 not inserted?
A missing count has a fitted score of 2.8. Two selected donors have observed counts 2 and 4. If the random donor draw selects the second donor, what value is imputed?
Why can PMM fail to impute a theoretically possible count of 9 even if the count’s allowed range includes 9?
A nominal outcome is coded as 1, 2, and 3. Explain why numerical PMM based on those codes may be inappropriate even if every donor draw produces one of the valid labels.
Name one check that can reveal a problem even when every imputed value is within its allowed bounds.

Solutions. (1) Impute 31, the selected donor’s observed outcome. The score 26.4 locates donors; it is not the imputed measurement. (2) Impute 4, the selected donor’s observed count. (3) PMM can copy only values available in its donor pool; if no observed donor has count 9, it cannot produce 9. (4) The codes may be arbitrary labels, so their order and distances have no substantive meaning and can change under recoding. (5) For example, inspect score distances or compare observed and imputed frequencies at important counts or boundaries; valid bounds alone do not establish plausible donor matches.