Calibration with Bagging of the Principal Components on a Large Number of Auxiliary Variables

Calibration is a widely used method in survey sampling to adjust weights so that estimated totals of some chosen calibration variables match known population totals or totals obtained from other sources. When a large number of auxiliary variables are included as calibration variables, the variance of the total estimator can increase, and the calibration weights can become highly dispersed. To address these issues, we propose a solution inspired by bagging and principal component decomposition. With our approach, the principal components of the auxiliary variables are constructed. Several samples of calibration variables are selected without replacement and with unequal probabilities from among the principal components. For each sample, a system of weights is obtained. The final weights are the average weights of these different weighting systems. With our proposed method, it is possible to calibrate exactly for some of the main auxiliary variables. For the other auxiliary variables, the weights cannot be calibrated exactly. The proposed method allows us to obtain a total estimator whose variance does not explode when new auxiliary variables are added and to obtain very low scatter weights. Finally, our proposed method allows us to obtain a single weighting system that can be applied to several variables of interest of a survey.

💡 Research Summary

This paper addresses a critical challenge in survey sampling: the deterioration of calibration estimator performance when a large number of auxiliary variables are used. While calibration aims to improve estimate accuracy by adjusting sampling weights so that weighted sample totals match known population totals for auxiliary variables, an excessive number of such variables often leads to increased variance of the total estimator and highly dispersed, unstable weights.

The authors propose a novel methodology that hybridizes Principal Component Analysis (PCA) and a bagging-inspired technique. The core procedure unfolds in several stages. First, PCA is performed on the standardized matrix of auxiliary variables to obtain a set of orthogonal principal components (PCs) and their corresponding eigenvalues, which represent the proportion of variance each PC explains. This step serves as a dimensionality reduction and decorrelation pre-processing stage.

The innovative bagging-inspired process follows. Instead of bootstrapping observational units, the method repeatedly samples variables (the principal components). For a large number of iterations B, a subset of c principal components is selected without replacement, but with unequal probabilities proportional to their eigenvalues raised to a power α. This ensures components explaining more variance are selected more frequently. For each iteration, a standard calibration problem is solved using only this subset of PCs as the calibration variables, generating a unique set of calibrated weights. Finally, the B independent sets of weights are aggregated by simple averaging to produce the final calibration weights.

This approach offers several key advantages. By averaging over many weight systems derived from different, randomly selected combinations of PCs, the method reduces the variance of the final estimator—this is the classic bagging effect applied to the weighting process itself. The preliminary PCA step mitigates multicollinearity issues inherent in high-dimensional data. Consequently, the method successfully controls the “variance explosion” typically seen when adding many auxiliary variables and yields final weights with very low dispersion. Furthermore, because the weight generation process is independent of any specific variable of interest (y), a single set of final weights can be applied to estimate totals for multiple survey variables, enhancing practical utility.

The paper provides guidance on choosing the method’s parameters, suggesting the number of components selected per iteration (c) could be set to the square root of the sample size (√n) as a rule of thumb. An empirical evaluation using a simulation study based on real data from the Swiss Survey on Income and Living Conditions (SILC) demonstrates the method’s efficacy. The proposed Bagging-PCA (BP) estimator significantly reduces both the variability of the weights and the variance of the total estimator for several variables of interest, compared to standard calibration and calibration on a fixed set of principal components.

In conclusion, this research presents a robust and innovative solution to high-dimensional calibration problems in survey statistics. It effectively balances the bias-variance trade-off, enhances the stability of calibration weights, and provides a unified weighting system for multi-purpose surveys, marking a significant contribution to the field.