Post-stratification and calibration are powerful tools for improving survey accuracy. They use known population info to adjust estimates, reducing bias from non-response and sampling errors. These techniques are crucial for getting reliable results from imperfect data.

Auxiliary variables play a key role in these methods. By leveraging data like demographics or geographic info, researchers can fine-tune their estimates. The choice between post-stratification and calibration depends on the type and quality of available auxiliary data.

Post-Stratification and Calibration Estimators

Post-Stratification Techniques

• Post-stratification adjusts survey estimates using population information known after data collection
• Divides the sample into groups (strata) based on characteristics like age, gender, or education level
• Assigns weights to each stratum to match known population proportions
• Improves precision of estimates by reducing sampling variability
• Helps correct for non-response bias when response rates differ across strata
• Calculation involves multiplying the sample mean of each stratum by its known population proportion
• Formula for post-stratified estimator: $\hat{Y}_{ps} = \sum_{h=1}^H W_h \bar{y}_h$
  • Where $W_h$ is the known population proportion for stratum h
  • $\bar{y}_h$ is the sample mean for stratum h

Calibration and Regression Estimators

• Calibration estimators adjust sample weights to match known population totals for auxiliary variables
• Raking ratio estimation iteratively adjusts weights to match marginal totals for multiple variables
• Uses iterative proportional fitting algorithm to converge on final weights
• Particularly useful when only marginal totals are available, not full cross-tabulations
• General regression estimator (GREG) extends calibration to use linear regression models
• GREG incorporates auxiliary information through a regression model of the survey variable
• Formula for GREG estimator: $\hat{Y}_{GREG} = \hat{Y}_{HT} + (\mathbf{X} - \hat{\mathbf{X}}_{HT})'\hat{\mathbf{B}}$
  • Where $\hat{Y}_{HT}$ is the Horvitz-Thompson estimator
  • $\mathbf{X}$ is the vector of known population totals for auxiliary variables
  • $\hat{\mathbf{X}}_{HT}$ is the Horvitz-Thompson estimator of auxiliary variable totals
  • $\hat{\mathbf{B}}$ is the estimated regression coefficient vector

Comparison and Applications

• Post-stratification works well with categorical auxiliary variables (age groups, regions)
• Calibration estimators handle both categorical and continuous auxiliary information
• GREG estimator often provides more precise estimates than post-stratification
• Raking useful for complex surveys with many auxiliary variables and marginal controls
• Applications include adjusting for non-response in political polls, improving official statistics
• Software packages (R, SAS, Stata) offer functions for implementing these estimators
• Choice of method depends on available auxiliary information and survey design complexity

Auxiliary Information

Types and Sources of Auxiliary Variables

• Auxiliary variables correlate with survey variables of interest or non-response patterns
• Demographic characteristics (age, gender, education level, income brackets)
• Geographic information (region, urban/rural classification, zip codes)
• Behavioral data (voting history, consumer spending patterns, internet usage)
• Administrative records (tax data, social security information, vehicle registrations)
• Previous survey results or census data provide reliable auxiliary information
• Big data sources (social media activity, mobile phone usage) offer new opportunities
• Quality of auxiliary information impacts effectiveness of calibration techniques

Population Totals and Their Importance

• Population totals refer to known aggregate values of auxiliary variables for the entire target population
• Examples include total population by age group, total households in each region, total registered voters
• Obtained from reliable sources like national statistical offices, government agencies, or trusted databases
• Accuracy of population totals crucial for effective calibration and bias reduction
• Totals should ideally come from the same time period as the survey to ensure relevance
• Misalignment between survey period and auxiliary data can introduce bias
• Population totals enable calculation of expansion weights in design-based inference

Calibration Constraints and Implementation

• Calibration constraints ensure sample weights reproduce known population totals for auxiliary variables
• Linear constraints most common: $\sum_{i \in s} w_i x_i = X$
  • Where $w_i$ are the calibrated weights, $x_i$ are auxiliary variable values, and $X$ is the population total
• Non-linear constraints possible but computationally more complex
• Constraints can be applied at different levels (national, regional, demographic subgroups)
• Over-constraining can lead to extreme weights or convergence issues
• Balance needed between utilizing available information and maintaining stable weights
• Diagnostic tools help assess impact of calibration on weight distribution and estimate precision
• Variance estimation for calibrated estimators requires special techniques (e.g., linearization, replication methods)