Why Skewness Changes the Imputation Question
Continuous outcomes such as costs, waiting times, and laboratory measurements can be strongly right-skewed: many observations are relatively small, while a few are much larger. A normal regression imputation represents uncertainty with a bell-shaped distribution around a fitted value. For a skewed outcome, that shape may place too much probability in a thin lower tail and too little in the long upper tail. In some applications, it can also produce values that are impossible in context.
Predictive mean matching (PMM) takes a different route. It uses a regression-based predictive score to find observed cases that resemble a case with a missing value, then imputes an observed outcome from a selected donor. The earlier material on why PMM imputations stay within observed support explains the support property. Here the focus is narrower: what that property does and does not accomplish for skewed or outlier-prone continuous variables.
In a common version, the score is a fitted conditional mean from a linear regression. The score is used to rank donors; the donated value is the donor’s observed outcome, not a new value sampled from a normal distribution. This distinction can be valuable for skewed outcomes, but it does not make the outcome model, donor pool, or imputation automatically appropriate.
Assumptions Behind Using PMM for a Skewed Outcome
The central practical question is whether cases with similar predictive scores have relevant observed outcomes that can represent the recipient. This requires more than simply finding numerical neighbors. The following conditions support that interpretation:
- Relevant predictors are included. The predictors used in the imputation model should capture important differences in the outcome. The usual MICE interpretation also relies on a defensible missing-at-random assumption conditional on variables included in the analysis and imputation models.
- Recipients have comparable donors. There should be observed cases in the recipient’s score region. A close score is not a guarantee of similarity if important outcome predictors are omitted or the score does not distinguish relevant cases.
- The score is useful for ranking. A linear predictive score can be used even when the outcome distribution is skewed, but its form and included predictors must still rank suitable donors reasonably well. PMM does not remove misspecification in the relationship between predictors and outcome.
- Uncertainty is represented. For multiple imputation, the procedure should include appropriate model-parameter and donor-selection uncertainty rather than repeatedly returning one deterministic donor value. The number of imputations and the implementation should be adequate for the intended analysis.
Normality of the observed outcome is not required for the donated values themselves to be observed outcomes. However, the regression fit used to produce scores and parameter draws still matters. PMM is not a distribution-free cure for poor predictor choices, sparse donor regions, or implausible conditional relationships.
Worked Example: A Skewed Waiting-Time Outcome
Suppose a hypothetical clinic records waiting times in minutes. Among observed cases, values include 0.8, 1.0, 1.2, 1.5, 2.1, 2.8, 4.0, and 12.0. The long upper tail is evident. A recipient has a fitted predictive score of 3.0 minutes, and the three nearest eligible donors have scores and observed waiting times as follows:
| Donor | Predictive score (minutes) | Observed waiting time (minutes) | Distance from recipient score |
|---|---|---|---|
| A | 2.8 | 2.6 | 0.2 |
| B | 3.1 | 3.2 | 0.1 |
| C | 3.4 | 12.0 | 0.4 |
If the procedure uses the two closest donors, A and B are the pool because their distances, 0.2 and 0.1, are smaller than C’s distance of 0.4. With uniform random selection from these two donors, the imputation is 2.6 minutes with probability \(1/2\), or 3.2 minutes with probability \(1/2\). The expected imputed value over this donor draw is
This recipient’s imputation is an observed value, not an independently generated draw from a symmetric normal distribution. That can avoid a poorly shaped normal tail for this recipient. It does not mean that PMM reproduces the entire skewed distribution automatically: the donor pool, its size, and the scores determine which observed values are available to be selected.
For scale, a hypothetical normal regression draw for a positive waiting time might have distribution \(N(2.5,2^2)\), where the second parameter is the variance and the standard deviation is 2 minutes. Its probability of a negative value is
Thus this particular normal model assigns about 10.56% probability to a negative waiting time. The calculation illustrates a possible failure of that model for this outcome; it does not establish that PMM is preferable in every setting.
Why Donor Values Can Preserve Skew—and Why They Do Not Guarantee It
For a fixed recipient and a fixed pool of \(k\) eligible donors, suppose the procedure selects each donor with equal probability. Write their observed outcomes as \(y_1,\ldots,y_k\). The imputed value \(Y_{\mathrm{imp}}\) then has the empirical donor distribution:
when the donor outcomes are distinct. More generally, the probability of a particular outcome is its number of occurrences in the pool divided by \(k\). It follows that
This follows directly by summing each possible donor outcome multiplied by its selection probability. Because the distribution is made from observed outcomes, a large donor value can appear in the imputation distribution, and a skewed set of donor outcomes can retain some skew. But a small pool may omit most of the observed tail, while a pool containing an extreme donor may give that donor a substantial selection chance. PMM does not create an upper tail beyond the observed donor outcomes, nor does it guarantee that the conditional tail is correctly represented.
Worked Example: When an Extreme Donor Is Selected
Consider a hypothetical recipient with predictive score 4.25. Four observed cases have the following scores and outcomes:
| Donor | Score | Observed outcome | Absolute score distance |
|---|---|---|---|
| A | 4.18 | 4 | 0.07 |
| B | 4.22 | 5 | 0.03 |
| C | 4.27 | 40 | 0.02 |
| D | 4.31 | 6 | 0.06 |
With a three-donor pool, the closest scores belong to C, B, and D, with distances 0.02, 0.03, and 0.06. If one donor is selected uniformly, the outcome 40 is selected with probability \(1/3\). The expected donated value is
The value 40 may be a genuine observation, a data error, or a value from an unusual subgroup; the numbers alone do not decide which. Matching uses score distance, not distance between outcome values. Therefore, the unusual donor can enter the pool even though the other donated outcomes are much smaller. If only the closest donor were used, the imputation would be 40 every time in this example.
This is not a reason to delete a valid extreme observation simply because it is inconvenient. It is a reason to check data quality, the recipient’s context, and whether the predictors distinguish cases with different outcome behavior. An outlier can affect a donor-based imputation without being generated by a normal-tail draw.
Diagnostics and Comparisons
Diagnostics should ask whether the observed donor values selected for missing cases are plausible in context and whether their distribution is credible for those cases. Useful checks include:
- Compare observed and imputed distributions using plots and summaries that show both the central region and the upper tail, such as selected quantiles and counts above meaningful thresholds.
- Make comparisons within important predictor groups where sample sizes allow. A marginal comparison can hide differences in case mix between observed and missing outcomes.
- Inspect unusually influential donor values and how often they are selected. Check source records and substantive plausibility before treating an extreme value as either erroneous or representative.
- Review how many plausible donors are available around recipients’ scores. A numerical match can be close relative to the dataset yet still be poor in the relevant clinical or scientific context.
- Repeat checks across imputed data sets. Random donor selection means a single imputed data set may not reveal the range of plausible tail behavior.
Observed and imputed distributions are not expected to match mechanically. Missing cases can differ from observed cases, even under a missing-at-random assumption conditional on measured variables. The comparison is most informative when it accounts for those predictors and is used to identify implausible behavior, not to demand identical raw histograms.
A normal regression imputation is based on a parametric predictive distribution; when its assumptions and outcome scale are suitable, it can be a reasonable choice. For a strongly skewed outcome, an appropriate transformation or model may improve that representation, though analysis and back-transformation require care. PMM instead borrows actual observed outcomes, which can retain a skewed shape among available donors. Its limitation is that score matching may select inappropriate donors or miss tail behavior that is absent from the available donor pool. The choice should be based on the outcome, predictors, donor availability, and diagnostics—not on a rule that one method always handles skewness better.
Common Mistakes and Limitations
Other limitations deserve attention. First, observed support is a property of the donor pool, not proof that the imputed conditional distribution is correct. Second, an outlier may be influential because it is repeatedly selected, even though PMM does not generate new values beyond observed outcomes. Third, if missingness depends on the unobserved outcome even after conditioning on included variables, standard MAR-based imputation may not resolve that problem. Finally, an apparently good marginal distribution does not ensure that relationships among the outcome and other analysis variables are preserved.
A practical decision is therefore conditional: PMM can be useful when a normal predictive draw gives an implausible shape and observed cases with relevant scores provide credible donors. When the donor pool is sparse, score matches are questionable, or extreme donors dominate, the method’s advantages may be limited. Report and review the modeling choices rather than treating the method name as a guarantee.
Worked Example: Checking Upper-Tail Behavior Across Imputations
In a hypothetical diagnostic summary, eight observed waiting times in a relevant predictor group are 1.0, 1.2, 1.5, 1.8, 2.0, 2.2, 3.0, and 10.0 minutes. Four missing cases in that group receive PMM values in one completed data set: 1.8, 2.2, 3.0, and 10.0 minutes.
Using 5 minutes as an illustrative upper-tail threshold, one of the eight observed values exceeds it, so the observed fraction is \(1/8=0.125\), or 12.5%. One of the four imputations exceeds it, so the imputed fraction is \(1/4=0.25\), or 25%. The observed mean is
The mean of the four imputed values is
These summaries flag that the one completed data set has a relatively large upper-tail contribution. They do not, by themselves, show that PMM failed: four imputations are a small denominator, and these figures do not account for differences between observed and missing cases. The next checks would include other imputed data sets, relevant predictor patterns, and whether the 10-minute donor is valid and appropriate for these recipients.
Confirm that the skew and extreme observations are plausible and measured on the intended scale.
Check that predictors and the fitted score can identify donors comparable to recipients, with adequate observed cases nearby.
Compare observed and imputed values in context, across repeated imputations, and within relevant predictor groups.
Treat donor-based tail behavior as a consequence of available observed donors and matching, not as a guarantee of a correct conditional distribution.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A PMM recipient has two equally likely donors with outcomes 2 and 8. What is the probability of an imputation above 5, and what is the expected imputed value?
- Why can a valid extreme observed outcome still influence a PMM analysis?
- A normal regression draw for a positive outcome is \(N(10,4^2)\). Calculate the probability of a negative draw, using \(\Phi(-2.5)\approx0.00621\).
- Observed outcomes are skewed, but the imputed values show almost no upper tail. Give two possible explanations to investigate.
Solutions. (1) Only the outcome 8 exceeds 5, so the probability is \(1/2=0.5\). The expected value is \((2+8)/2=5\). (2) PMM selects donors using predictive-score distance, not whether their outcome is extreme; if an extreme case is in the donor pool, it can be selected and may be selected repeatedly. (3) Standardizing gives \(z=(0-10)/4=-2.5\), so \(P(Y<0)=\Phi(-2.5)\approx0.00621\), or 0.621%. (4) The donor pools may rarely include upper-tail cases because of score matching or limited observed tail data; alternatively, the matching score or predictors may not identify recipients who should receive upper-tail values. Check donor availability, predictor groups, and repeated imputations before concluding which explanation applies.