Sample Weighting - Calibration

Calibration

What is GREG calibration?

The Generalized Regression (GREG) approach is a model-assisted version of calibration. It assumes that the survey variable of interest, such as income or health status, is related to one or more auxiliary variables (like age, region, or education) through a regression-type relationship.

Rather than relying solely on the design weights, GREG calibration adjusts those weights using information from both:

1. The survey data — what we observe in the sample, and
   
2. The external data — known population totals for the auxiliary variables.

By combining these sources, the GREG estimator improves accuracy and reduces variance, especially when the auxiliary variables are good predictors of the study variable.

How it works in practice

In essence, GREG finds a new set of weights that:

• Stay as close as possible to the original design weights,
  
• And at the same time, make the weighted totals of the auxiliary variables match their known population values.

These new calibrated weights automatically correct for imbalances that might exist in the sample — for instance, if certain regions or demographic groups were slightly over- or under-represented.

This procedure can be thought of as fitting a weighted regression model using the auxiliary variables, and then using that model to fine-tune the survey estimates.
The resulting GREG estimates tend to be more stable and efficient than the simple design-based estimates, particularly when the auxiliary variables are strongly correlated with the study variable.

Why use GREG calibration?

GREG calibration offers several advantages: - Improved precision: by incorporating auxiliary information, estimates often have smaller standard errors. - Reduced bias: especially when the sample under- or over-represents certain groups. - Consistency: if the auxiliary variables perfectly explain the study variable, GREG reproduces the true population total. - Flexibility: it works with different distance functions and supports various forms of calibration, including bounded or non-linear adjustments.

In practice, you can apply GREG calibration in svy using the calibration or post-stratification utilities, providing your known totals and the variables to be aligned.

Practical guidance

• Choose auxiliaries carefully. The method works best when the auxiliary variables are strongly related to the outcome variable.
• Use reliable totals. Calibration will not improve accuracy if the reference totals are outdated or inconsistent.
• Monitor adjustment factors. Extremely large or small weight adjustments may indicate unstable calibration and should be reviewed.
• Combine with variance estimation. GREG calibration can be used with Taylor linearization or replicate-based methods that account for calibration.

TipImplementation in svy

Use Sample.calibration() or the CalibrationMatrix class to supply control totals and specify the calibration type (for example, “greg” or “raking”). The function automatically computes the adjusted weights and stores them in the sample object for downstream estimation.

Calibration adjusts survey weights so that weighted totals of selected auxiliary variables match known population totals. The Generalized Regression (GREG) class does this while keeping the adjusted weights as close as possible to the original design weights.

Let a sample \(s\) have design weights \(w_i\) and study variable \(y_i\). The direct estimator of the total is

\[ \hat{\mathbf{Y}} = \sum_{i \in s} w_i y_i. \]

Assume known population totals for auxiliary variables \(\mathbf{x}_i \in \mathbb{R}^p\),

\[ \mathbf{X} = \big(X_1, \ldots, X_p\big)^\top, \quad \hat{\mathbf{X}} = \sum_{i \in s} w_i \mathbf{x}_i. \]

Under the working model \(Y_i \mid \mathbf{x}_i = \mathbf{x}_i^\top \boldsymbol{\beta} + \varepsilon_i\), the GREG estimator of the total is

\[ \hat{\mathbf{Y}}_{\text{GR}} = \hat{\mathbf{Y}} + (\mathbf{X} - \hat{\mathbf{X}})^\top \hat{\mathbf{B}}, \]

where \(\hat{\mathbf{B}}\) is the weighted least squares estimate of \(\boldsymbol{\beta}\).

Calibration-weight formulation

Equivalently, GREG can be expressed through calibrated weights \(w_i^*\) that minimize a distance to \(w_i\) subject to calibration constraints:

\[ \begin{aligned} \text{minimize} \quad & \sum_{i \in s} c_i \, \phi\!\left(\frac{w_i^* - w_i}{w_i}\right) \\ \text{subject to} \quad & \sum_{i \in s} w_i^* \, \mathbf{x}_i = \mathbf{X}, \end{aligned} \]

where \(\phi(\cdot)\) measures deviation (e.g., quadratic/chi-square, entropy, logit) and \(c_i>0\) are scaling constants.

For the quadratic (chi-square) distance with \(q_i \equiv c_i = 1\),

\[ w_i^* \;=\; w_i \, \big(1 + \mathbf{x}_i^\top \boldsymbol{\lambda}\big), \]

with \(\boldsymbol{\lambda}\) solving

\[ \left(\sum_{i \in s} w_i \, \mathbf{x}_i \mathbf{x}_i^\top\right)\boldsymbol{\lambda} \;=\; \mathbf{X} - \hat{\mathbf{X}}. \]

More generally, with per-unit factors \(q_i>0\),

\[ w_i^* \;=\; w_i \, \big(1 + q_i\, \mathbf{x}_i^\top \boldsymbol{\lambda}\big), \qquad \left(\sum_{i \in s} w_i q_i\, \mathbf{x}_i \mathbf{x}_i^\top\right)\boldsymbol{\lambda} = \mathbf{X} - \hat{\mathbf{X}}. \]

Tip. The choice \(q_i\) can stabilize adjustments (e.g., \(q_i \propto 1/w_i\) or bounded by trims). Entropy/logit distances enforce positivity bounds on \(w_i^*\).