Generalizing Univariate Predictive Mean Matching to Impute Multiple Variables Simultaneously

Abstract

Predictive mean matching (PMM) is an easy-to-use and versatile univariate imputation approach. It is robust against transformations of the incomplete variable and violation of the normal model. However, univariate imputation methods cannot directly preserve multivariate relations in the imputed data. We wish to extend PMM to a multivariate method to produce imputations that are consistent with the knowledge of derived data (e.g., data transformations, interactions, sum restrictions, range restrictions, and polynomials). This paper proposes multivariate predictive mean matching (MPMM), which can impute incomplete variables simultaneously. Instead of the normal linear model, we apply canonical regression analysis to calculate the predicted value used for donor selection. To evaluate the performance of MPMM, we compared it with other imputation approaches under four scenarios: 1) multivariate normal distributed data, 2) linear regression with quadratic terms; 3) linear regression with interaction terms; 4) incomplete data with inequality restrictions. The simulation study shows that with moderate missingness patterns, MPMM provides plausible imputations at the univariate level and preserves relations in the data.

Appendix

The MPMM algorithm with multiple missing patterns:

• 1. Sort the rows of $\boldsymbol{Y}$ into S missing data patterns $\boldsymbol{Y_{[s]}}, s=1,\cdots ,S$.

• 2. Initialize $\boldsymbol{Y_{mis}}$ by a reasonable starting value.

• 3. Repeat for $T=1,\cdots ,t$.

• 4. Repeat for $S=1,\cdots ,s$.

• 5. Impute missing values by steps 1–8 of PMM-CRA algorithm proposed in Sect. 2.3.

• 6. Repeat steps 1–5 m times and save m completed datasets.