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Data integration of non-probability and probability samples with deterministic predictive mean matching

This paper proposes and validates deterministic predictive mean matching mass imputation estimators for integrating probability and non-probability samples, establishing their theoretical consistency and variance properties under both model specification and misspecification while demonstrating their effectiveness through simulations and an empirical application to job vacancy data.

Original authors: Aniela Czerniawska, Piotr Chlebicki, Łukasz Chrostowski, Maciej Beręsewicz

Published 2026-07-16

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CC BY 4.0

Original authors: Aniela Czerniawska, Piotr Chlebicki, Łukasz Chrostowski, Maciej Beręsewicz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out the average height of every student in a massive school. The most reliable way to do this is to pick a few students completely at random (a "probability sample") and measure them. Because the selection was random, you can mathematically guarantee that your average is close to the truth. But what if you don't have time or money to measure those random students? Instead, you have a giant list of volunteers who signed up online to join a "tall people club" (a "non-probability sample"). This list is huge, but it's biased: it's full of basketball players and models, so the average height here is way too high. You can't just use this list, or your school's average will be wrong.

This is a classic puzzle in the world of statistics, a field dedicated to making sense of data. The challenge is "data integration": how do you mix a small, perfectly fair list with a huge, messy, biased list to get a perfect answer? Usually, statisticians try to guess the "rules" that connect the two groups (like how height relates to age or gender) and use those rules to fix the biased list. But what if those rules are slightly wrong? Or what if the data is so complicated that the rules break down? This is where the paper by Czerniawska and her team steps in. They are tackling the problem of how to combine these two very different types of data lists to get a reliable answer, even when the math gets tricky.

The authors propose a clever new way to do this mixing called "Predictive Mean Matching" (PMM). Think of it like a high-tech game of "Find Your Twin." Imagine you have a student from your fair random list (let's call him Alex) whose height you don't know yet, but you do know his age and grade. You look at the huge, biased online list to find a student who looks just like Alex based on those details. Once you find that "twin," you borrow their height and give it to Alex. By doing this for every student in the random list, you create a complete, accurate picture of the whole school.

The paper explores two specific ways to play this "Find Your Twin" game. The first method, which they call PMM A, looks for twins based on what the computer predicts the height should be. If the computer thinks Alex should be 170cm, it finds someone in the biased list who the computer also predicts to be 170cm. The second method, PMM B, is a bit more direct: it looks for someone in the biased list whose actual, measured height matches what the computer predicts for Alex.

The researchers spent a lot of time proving that these methods actually work. They showed that if you use enough data, these methods will eventually give you the correct answer, even if your computer's prediction rules aren't perfect. Specifically, they proved that the PMM A method (the one using predicted values to find matches) is robust to model misspecification under certain conditions. This is a big deal because other popular methods (like simply finding the "nearest neighbor" based on raw data) can fail miserably if the data gets too complex or if the prediction rules are slightly off. The authors proved that their "Predictive Mean Matching" approach is much more robust; it doesn't break as easily when the math gets messy.

They also figured out how to measure how "sure" we can be about the answer. When you borrow data from a biased list, there's a little bit of extra uncertainty, like a hidden wobble in your scale. The authors developed a new formula to calculate this wobble and showed how to use computer simulations (called "bootstrapping") to measure it accurately. They tested their ideas with thousands of fake data scenarios in a computer lab. In these simulations, their new methods performed just as well as the best existing tools when the rules were perfect, but they were much better when the rules were wrong or the data was complicated.

Finally, the team tried their method on real-world data from Poland. They wanted to estimate how many job vacancies were specifically aimed at Ukrainian workers. They combined a small, official government survey (the fair list) with a massive database of job postings from public employment offices (the biased list). The results showed that their method could successfully merge these two very different sources to give a clear, reliable estimate, proving that their "Find Your Twin" strategy works in the real world, not just in computer simulations.

In short, this paper provides a sturdy, reliable toolkit for statisticians who need to mix fair but small data with huge but biased data. It offers a way to get accurate answers even when the math isn't perfect, ensuring that decisions based on this data are built on solid ground rather than shaky guesses.

Technical Summary: Data Integration of Non-Probability and Probability Samples with Deterministic Predictive Mean Matching

Problem Statement
The paper addresses the challenge of estimating population characteristics by integrating data from probability samples (which have known inclusion probabilities but often lack the target variable $Y$ ) and non-probability samples (which contain the target variable $Y$ but lack a known sampling mechanism). While existing literature proposes Inverse Probability Weighting (IPW), Prediction Estimators (PE), and Doubly Robust (DR) methods, this study focuses on Mass Imputation (MI) estimators. Specifically, it investigates Deterministic Predictive Mean Matching (PMM) as a technique to impute values for units in a probability sample using observed values from a non-probability sample. The authors aim to bridge the gap between parametric/non-parametric regression imputation and Nearest Neighbour (NN) imputation, providing rigorous theoretical properties for data integration contexts which have not been fully explored in previous literature.

Methodology
The authors propose and analyze two semi-parametric deterministic PMM algorithms for matching units from a probability sample ( $S_P$ ) to a non-probability sample ( $S_{NP}$ ):

PMM A ( $\hat{y}-\hat{y}$ matching): Units in $S_P$ are matched to units in $S_{NP}$ based on the similarity of their predicted values ( $\hat{y}$ ) derived from a regression model $E[Y|X] = m(X, \beta)$ . The matching distance is $d(\hat{y}_i, \hat{y}_j)$ .
PMM B ( $\hat{y}-y$ matching): Units in $S_P$ are matched to units in $S_{NP}$ based on the similarity of the predicted value of the $S_P$ unit and the observed value of the $S_{NP}$ unit. The matching distance is $d(\hat{y}_i, y_j)$ .

In both approaches, for each unit $i \in S_P$ , the $k$ nearest neighbours in $S_{NP}$ are identified, and the imputed value is the average of the observed $Y$ values of these neighbours. The final estimator for the population mean is a weighted average of these imputed values using design weights from the probability sample.

Theoretical analysis relies on a superpopulation model and a sequence of finite populations. Key assumptions include:

Existence of a consistent estimator $\hat{\beta}$ for the regression parameters.
Asymptotic vanishing of regression residuals (A2).
Local Lipschitz continuity of the regression function (A3).
A mild regularity condition (A4) regarding the representativeness of the non-probability sample, replacing the strict "positivity" assumption often required in NN methods.
Specific conditions for PMM A regarding the stability of donor weights (B3) and the influence function of $\hat{\beta}$ (B2).

Key Contributions

Theoretical Consistency: The authors prove the consistency of both PMM A and PMM B estimators under correct model specification. Crucially, they establish the consistency of the PMM A ( $\hat{y}-\hat{y}$ ) estimator even under model misspecification, provided the working model satisfies certain convexity and compactness conditions on the support.
Extension to Non-Parametric Models: The paper demonstrates that the consistency of the PMM B ( $\hat{y}-y$ ) estimator extends to non-parametric regression techniques (e.g., Nadaraya–Watson kernel regression, neural networks), provided the regression function estimator converges uniformly.
Variance Decomposition: A novel variance decomposition is derived, splitting the total variance into:
- $V_1$ : The design-based variance component (analogous to the Horvitz-Thompson variance of the imputed values).
- $V_2$ : An additional variance component arising from the randomness of the imputation process (donor selection and regression estimation).
Variance Estimation: The authors propose:
- A partially analytic estimator for $V_1$ .
- A "mini-bootstrap" procedure (Algorithm 2) to estimate the covariance terms required for $V_2$ .
- A full bootstrap procedure (Algorithm 3) combining both components.
- They show that the proposed analytical variance expression for $V_1$ can also be applied to the standard Nearest Neighbour (NN) estimator, extending previous results.
Distinction from NN: The paper rigorously distinguishes PMM from NN, noting that they are identical only in the specific case of a linear model with a single covariate. PMM is shown to be less susceptible to the "curse of dimensionality" and does not require the strict positivity assumption.

Results

Simulation Studies: Extensive simulations with $N=10^5$ $N = 1 0^{5}$ compared PMM estimators against Nearest Neighbour (NN) and Regression Prediction (GLM) estimators under Simple Random Sampling.
- Correctly Specified Models: PMM estimators performed similarly to GLM. PMM B ( $\hat{y}-y$ ) generally exhibited lower standard errors than PMM A ( $\hat{y}-\hat{y}$ ).
- Model Misspecification: PMM A demonstrated superior robustness, maintaining lower bias compared to NN and GLM when the linear model was misspecified for a non-linear data generating process.
- Variance Components: The contribution of the $V_2$ component (imputation uncertainty) was found to be non-negligible in finite samples, increasing the total variance of matching estimators by approximately 40–60% in the empirical study.
- Coverage Rates: Analytical and bootstrap variance estimators achieved nominal 95% coverage rates.
Empirical Study: The methods were applied to integrate a Job Vacancy Survey (probability sample) with a Central Job Offers Database (non-probability sample) in Poland to estimate the share of vacancies for Ukrainian workers.
- Point estimates across methods (GLM, NN, PMM, IPW, DR) were consistent, hovering around 22%.
- The inclusion of the $V_2$ component via mini-bootstrap significantly increased standard errors, highlighting the importance of accounting for matching uncertainty.
- PMM A and PMM B yielded indistinguishable results in this specific application.

Significance and Claims
The paper claims that deterministic PMM offers a robust alternative for data integration, particularly the PMM A variant which retains consistency under model misspecification—a property not guaranteed for standard regression imputation or NN in high dimensions. The authors emphasize that their rigorous approach to variance decomposition and estimation provides a practical framework for uncertainty quantification in mass imputation, addressing a gap in existing literature where the variance of NN and PMM estimators in non-probability contexts was either unknown or lacked a decomposition separating design and imputation errors. The availability of open-source software (nonprobsvy package) facilitates the application of these methods. The authors remain modest regarding the "mini-bootstrap" for $V_2$ , characterizing it as a heuristic finite-sample correction rather than a generally consistent estimator, and note that further research is needed for multiply robust PMM procedures and optimal selection of the $k$ hyperparameter.

Technical Summary: Data Integration of Non-Probability and Probability Samples with Deterministic Predictive Mean Matching

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