Glossary

13.2 Post-stratification and calibration

3 min read•august 9, 2024

and 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.

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

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  • 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 to each to match known population proportions
  • Improves precision of estimates by reducing
  • Helps correct for when response rates differ across strata
  • Calculation involves multiplying the sample mean of each stratum by its known
  • Formula for post-stratified estimator: Y^ps=∑h=1HWhyˉh\hat{Y}_{ps} = \sum_{h=1}^H W_h \bar{y}_h
    • Where WhW_h is the known population proportion for stratum h
    • yˉh\bar{y}_h is the sample mean for stratum h

Calibration and Regression Estimators

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

Calibration Constraints and Implementation

  • Calibration constraints ensure sample weights reproduce known population totals for auxiliary variables
  • Linear constraints most common: ∑i∈swixi=X\sum_{i \in s} w_i x_i = X
    • Where wiw_i are the calibrated weights, xix_i are auxiliary variable values, and XX 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
  • for calibrated estimators requires special techniques (e.g., linearization, replication methods)

Key Terms to Review (24)

Adjustment Factor: An adjustment factor is a numerical value used to modify survey estimates in order to correct for biases or to align sample data with known population characteristics. This concept is particularly important in ensuring that survey results accurately represent the population, especially when there are discrepancies in demographic distributions. By applying adjustment factors, researchers can enhance the reliability of their estimates and ensure better alignment with real-world data.
Adjustment techniques: Adjustment techniques are methods used to improve the accuracy of survey estimates by correcting for biases and discrepancies in the sample data. These techniques aim to align the survey results with known population characteristics, ensuring that the final estimates reflect the true distribution of the population. By utilizing processes such as post-stratification and calibration, adjustment techniques enhance the validity of survey findings and help mitigate potential sampling errors.
Auxiliary Variables: Auxiliary variables are additional data points that are not the main focus of a study but can provide valuable information to improve the estimation process. They help in adjusting survey estimates, leading to more accurate results, particularly when dealing with post-stratification and calibration methods. By incorporating auxiliary variables, researchers can reduce bias and enhance the efficiency of sample surveys.
Bias Reduction: Bias reduction refers to the techniques and strategies used to minimize systematic errors in data collection and analysis, ensuring that results more accurately reflect the true characteristics of the population. This concept is vital for improving the accuracy of statistical estimates and promoting more reliable conclusions. By addressing potential sources of bias, researchers can enhance the validity of their findings and ensure that their data better represents the diverse groups within the studied population.
Calibration: Calibration refers to the process of adjusting and verifying the accuracy of a measurement system or instrument to ensure it provides reliable and valid results. In the context of statistical sampling, calibration is particularly important for aligning survey estimates with known population characteristics, thus improving the quality and precision of the data collected.
Confidence Interval: A confidence interval is a range of values, derived from a data set, that is likely to contain the true population parameter with a specified level of confidence, often expressed as a percentage. It provides an estimate of uncertainty around a sample statistic, allowing researchers to make inferences about the larger population from which the sample was drawn.
Design-based inference: Design-based inference refers to a statistical approach that relies on the design of a survey to make valid conclusions about a population based on sample data. This method emphasizes the importance of how the sample is drawn, ensuring that it is representative of the entire population. By utilizing principles such as random sampling and stratification, design-based inference helps minimize bias and supports accurate estimation of population parameters.
General Regression Estimator: The general regression estimator is a statistical method used to obtain estimates of population parameters by combining data from various sources while adjusting for known characteristics. This technique helps in producing more accurate estimates by considering auxiliary information, which can improve the efficiency of estimators in survey sampling contexts.
Greg: In the context of post-stratification and calibration, 'greg' refers to a technique used to adjust survey weights in order to improve the accuracy and representativeness of survey estimates. This method is particularly useful when there are discrepancies between the sample and the target population, allowing researchers to correct for biases and enhance the reliability of the data collected.
Horvitz-Thompson Estimator: The Horvitz-Thompson estimator is a statistical method used to produce unbiased estimates of population parameters from survey data, particularly in complex sampling designs. This estimator is designed to account for unequal probabilities of selection, allowing for accurate estimation even when the sampling method varies, such as in cluster sampling or probability proportional to size. It plays a crucial role in multistage sampling and can be enhanced through techniques like post-stratification and calibration.
Iterative proportional fitting: Iterative proportional fitting is a statistical technique used to adjust the weights of survey data to align with known population margins. This method works by iteratively adjusting the values in a contingency table so that they match specified row and column totals, ensuring that the final weighted data is consistent with these margins. This approach is particularly useful in post-stratification and calibration processes, helping to improve the accuracy of survey estimates by aligning them with external population information.
Margin of Error: The margin of error is a statistical measure that expresses the amount of random sampling error in a survey's results. It indicates the range within which the true value for the entire population is likely to fall, providing an essential understanding of how reliable the results are based on the sample size and variability.
Non-response bias: Non-response bias occurs when certain individuals selected for a survey do not respond, leading to a sample that may not accurately represent the overall population. This bias can distort survey results, as the characteristics of non-respondents may differ significantly from those who participate, affecting the validity of conclusions drawn from the data.
Nonresponse adjustment: Nonresponse adjustment is a statistical technique used to correct for bias in survey results caused by individuals who do not respond to the survey. This adjustment helps to ensure that the sample reflects the characteristics of the target population more accurately by weighing the responses of those who did participate, often based on known demographics or behaviors. This method is crucial for maintaining the integrity of survey data, particularly when post-stratification and calibration techniques are applied to improve accuracy.
Population proportion: Population proportion refers to the fraction or percentage of a specific characteristic within a population. It is a key concept in understanding the makeup of a population, especially when analyzing survey results or demographic data. Population proportions are often used to adjust and calibrate estimates, ensuring that they accurately reflect the diversity and characteristics of the entire population.
Population Totals: Population totals refer to the complete count or estimate of individuals within a defined group or demographic at a given point in time. Understanding population totals is crucial for analyzing data and making informed decisions, particularly when applying weighting adjustments and post-stratification methods to ensure that survey results are representative of the entire population.
Post-stratification: Post-stratification is a statistical technique used to adjust survey estimates by dividing the sample into subgroups after data collection, allowing for more accurate representations of a population. This method improves the precision of estimates, especially when certain demographic groups are underrepresented in the sample, and it helps reduce bias in survey results.
Raking: Raking is a statistical technique used to adjust survey weights so that the sample aligns more closely with known population characteristics. This method ensures that the survey results accurately represent the broader population by correcting any disparities in demographic distribution, which may occur during data collection. Raking helps to improve the quality and reliability of survey estimates by making them more reflective of actual population parameters.
Raking Ratio Estimation: Raking ratio estimation is a statistical technique used to adjust survey weights to align sample estimates with known population totals across various dimensions, such as demographics. This method is particularly useful when the sample does not reflect the characteristics of the population accurately, ensuring that the estimates are more reliable and valid. Raking helps mitigate bias by iteratively adjusting weights so that the sample proportions match those of the population in terms of specified variables.
Sampling error: Sampling error is the difference between the results obtained from a sample and the actual values in the entire population. This error arises because the sample may not perfectly represent the population, leading to inaccuracies in estimates such as means, proportions, or totals.
Sampling variability: Sampling variability refers to the natural differences that occur when different samples are taken from the same population. This concept highlights how the estimates derived from these samples can vary due to random chance, which ultimately impacts the accuracy and reliability of statistical inferences. Understanding sampling variability is crucial for evaluating the effectiveness of sampling methods and addressing potential biases that can arise in various sampling designs.
Stratum: A stratum is a subset of a population that shares a specific characteristic, which is used in stratified sampling to ensure representation across different segments. Each stratum is formed based on key attributes like age, income, or education level, helping to provide a more accurate reflection of the population. This division allows for tailored sampling methods that enhance the precision of estimates and analyses.
Variance estimation: Variance estimation is a statistical method used to measure the variability or dispersion of a set of data points, allowing researchers to understand how much the data points differ from the mean. This concept is crucial in survey sampling as it helps assess the precision of estimates derived from various sampling techniques, ultimately influencing the reliability of conclusions drawn from the data. Accurate variance estimation is especially important when dealing with complex sampling designs like cluster and multistage sampling, where understanding the sources of variability can lead to more informed decision-making.
Weights: Weights are numerical factors applied to data points in surveys to adjust for unequal probabilities of selection or to ensure representation of different subgroups in the population. They help to correct biases and provide more accurate estimates for survey results by reflecting the true proportions of various characteristics in the population. This adjustment is crucial in methodologies like post-stratification and calibration, as it enhances the validity of conclusions drawn from the survey data.
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