Glossary

6.3 Estimation procedures in multistage sampling

3 min read•august 9, 2024

Multistage sampling is a complex method used in large-scale surveys. It involves selecting samples in stages, starting with larger units and progressively narrowing down to smaller ones. This approach balances cost-effectiveness with the need for representative data.

Estimation in multistage sampling requires special techniques to account for the complex design. These methods include using weighted estimators, calculating design effects, and applying variance estimation techniques like or resampling methods.

Multistage Sampling Units

Primary and Secondary Sampling Units

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  • (PSUs) form the first stage of selection in multistage sampling
    • Typically larger geographical areas (states, counties)
    • Selected using probability proportional to size (PPS) sampling
  • (SSUs) constitute the second stage of selection
    • Smaller units within PSUs (neighborhoods, schools)
    • Often selected using simple random sampling or systematic sampling
  • Tertiary and subsequent stages may follow depending on the study design
  • PSUs and SSUs create a hierarchical structure in the sample

Cluster Sampling and Its Applications

  • Cluster sampling involves selecting groups of elements rather than individual elements
  • Reduces costs associated with data collection and field operations
  • Natural clusters include households, schools, or hospital wards
  • Two-stage cluster sampling combines cluster and element sampling
    • First stage selects clusters (PSUs)
    • Second stage selects elements within chosen clusters (SSUs)
  • Cluster sampling can lead to increased sampling error due to homogeneity within clusters

Stratification in Multistage Sampling

  • Stratification divides the population into non-overlapping subgroups before sampling
  • Improves precision of estimates by reducing sampling variance
  • Can be applied at various stages of multistage sampling
    • PSUs can be stratified by region or urbanicity
    • SSUs can be stratified by demographic characteristics
  • Proportional allocation distributes sample size proportionally to strata sizes
  • Optimal allocation considers both strata sizes and variances for sample distribution

Variance Estimation Techniques

Design Effect and Intraclass Correlation

  • (DEFF) measures the efficiency of complex sampling designs
    • Ratio of the variance of an estimate under the complex design to the variance under simple random sampling
    • DEFF > 1 indicates loss of precision compared to simple random sampling
  • (ICC) quantifies the homogeneity within clusters
    • Ranges from 0 to 1, with higher values indicating greater similarity within clusters
    • Affects the precision of estimates in cluster sampling
  • Relationship between design effect and intraclass correlation: DEFF=1+(n−1)∗ICCDEFF = 1 + (n - 1) * ICC Where n is the average cluster size

Variance Estimation Methods

  • Taylor series linearization approximates complex estimators using linear functions
    • Widely used for its computational efficiency and broad applicability
    • Requires partial derivatives of the estimator with respect to survey variables
  • involves creating subsamples by removing one PSU at a time
    • Calculates the estimate for each subsample to assess variability
    • Useful for non-smooth estimators and small sample sizes
  • generates multiple resamples with replacement
    • Creates a distribution of estimates to assess variability
    • Versatile but computationally intensive for large datasets

Estimators and Adjustments

Horvitz-Thompson Estimator and Its Properties

  • provides unbiased estimation for complex survey designs
    • Incorporates sampling weights to account for unequal selection probabilities
    • General form: Y^=∑i=1nyiπi\hat{Y} = \sum_{i=1}^n \frac{y_i}{\pi_i} Where yiy_i is the observed value and πi\pi_i is the inclusion probability
  • Applicable to various sampling designs including multistage and unequal probability sampling
  • Variance of the Horvitz-Thompson estimator depends on joint inclusion probabilities

Weighting Adjustments and Their Impact

  • adjustments correct for non-response and
    • Non-response adjustments increase weights of respondents to represent non-respondents
    • Post-stratification aligns sample totals with known population totals
  • Calibration weighting adjusts weights to satisfy multiple constraints simultaneously
    • Improves precision and reduces in estimates
    • Uses auxiliary information from reliable external sources
  • Weight trimming reduces extreme weights to mitigate their influence on estimates
    • Balances between bias reduction and variance control

Finite Population Correction Factor

  • Finite population correction (FPC) adjusts variance estimates for sampling from finite populations
    • Becomes significant when the sampling fraction exceeds 5-10% of the population
    • FPC formula: fpc=N−nN−1fpc = \sqrt{\frac{N-n}{N-1}} Where N is the population size and n is the sample size
  • Applies to variance estimates at each stage of multistage sampling
  • Reduces the estimated variance, reflecting the increased precision from sampling a larger proportion of the population

Key Terms to Review (24)

Bias: Bias refers to a systematic error that leads to an inaccurate representation of a population in sampling or survey results. It can occur in various forms, affecting the validity and reliability of research findings. Understanding bias is crucial as it influences sampling designs, estimation processes, and ultimately the interpretation of data.
Bootstrap method: The bootstrap method is a resampling technique used to estimate the distribution of a statistic by repeatedly sampling with replacement from the observed data. This technique is particularly useful in situations where the underlying distribution is unknown or when traditional parametric assumptions are not valid. By creating multiple simulated samples, the bootstrap method allows for more robust inference and helps in constructing confidence intervals for estimates derived from complex survey designs.
Central Limit Theorem: The Central Limit Theorem states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's original distribution. This theorem is essential because it allows researchers to make inferences about population parameters from sample statistics, making it a cornerstone of statistical theory and practice.
Cluster Estimation: Cluster estimation is a statistical method used in survey sampling where the population is divided into groups, or clusters, and a sample of these clusters is selected for analysis. This approach is particularly useful when dealing with large populations, allowing researchers to simplify data collection by focusing on a smaller number of clusters rather than attempting to sample the entire population. It enhances efficiency while maintaining accuracy in estimating population parameters.
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 Effect: The design effect is a measure used in survey sampling that quantifies how much the variance of an estimator increases due to the sampling design, particularly in cluster sampling. It helps in understanding how different sampling strategies, such as cluster sampling or multistage sampling, impact the efficiency of the survey and the precision of estimates.
Finite population correction factor: The finite population correction factor is a statistical adjustment used in survey sampling to account for the fact that a sample is drawn from a finite population, rather than an infinite one. This factor helps to reduce the standard error of estimates when the sample size is a significant proportion of the total population size. By adjusting for this correction, researchers can obtain more accurate estimates and confidence intervals, making it crucial in both simple random sampling and multistage sampling methods.
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.
Intraclass Correlation: Intraclass correlation (ICC) is a statistical measure used to assess the reliability or consistency of measurements made by different observers measuring the same quantity. It is particularly important in cluster and multistage sampling designs because it helps evaluate the degree of similarity or agreement among units within the same group or cluster, influencing sample size calculations and estimation procedures.
Jackknife method: The jackknife method is a resampling technique used to estimate the bias and variance of a statistical estimator by systematically leaving out one observation at a time from the dataset. This method helps assess the stability and reliability of the estimates derived from data, making it particularly useful in situations like multistage sampling where complex sampling designs can complicate the estimation process. By creating multiple estimates based on subsets of data, the jackknife method allows for a clearer understanding of how much influence individual data points have on overall results.
Mean Squared Error: Mean Squared Error (MSE) is a statistical measure that quantifies the average of the squares of the errors—that is, the average squared difference between estimated values and the actual value. It is an essential concept for evaluating the accuracy of estimations, particularly in the context of sampling methodologies, where precision and bias are critical in the estimation procedures employed.
Nested designs: Nested designs refer to a sampling framework where units are organized in a hierarchical structure, allowing for different levels of analysis. This approach is commonly used in multistage sampling to understand the relationships between groups at various levels, such as individuals within clusters or schools within districts. By using nested designs, researchers can effectively account for variability at different hierarchical levels, leading to more accurate estimations and insights.
Point Estimate: A point estimate is a single value or statistic that serves as a best guess or approximation of an unknown population parameter. It provides a simple way to summarize data and make inferences about a larger group based on a sample. Point estimates are commonly used in various sampling methods, providing a foundation for further statistical analysis and decision-making.
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.
Primary Sampling Units: Primary sampling units (PSUs) are the first level of sampling units selected in a multistage sampling design, serving as the foundational blocks for further sampling stages. They are crucial in determining the initial strata from which subsequent samples will be drawn, impacting the representativeness and efficiency of the overall sampling process. By carefully choosing PSUs, researchers can effectively manage costs and improve the accuracy of their estimates.
Sampling distribution of the sample mean: The sampling distribution of the sample mean is the probability distribution that describes the means of all possible samples of a specific size taken from a population. This concept highlights how sample means can vary from one sample to another, providing insight into the accuracy and reliability of estimates derived from sample data. Understanding this distribution is crucial for making inferences about a population based on sample statistics, especially in complex sampling designs like multistage sampling.
Sampling frame: A sampling frame is a list or database from which a sample is drawn for a study, serving as the foundation for selecting participants. It connects to the overall effectiveness of different sampling methods and is crucial for ensuring that every individual in the population has a known chance of being selected, thus minimizing bias and increasing representativeness.
Sampling Unit: A sampling unit is the basic element or individual item selected from a population for the purpose of a survey or study. It can be a single individual, a household, a group, or even an entire cluster depending on the study design, and its selection plays a crucial role in determining the quality and accuracy of the data collected. The concept of sampling units is integral to understanding how to gather representative data and minimize errors in various research designs.
Secondary sampling units: Secondary sampling units are the groups or elements that are selected from the primary sampling units in a multistage sampling process. These units help refine the sample by providing additional layers of selection, enabling researchers to manage the complexity of data collection while maintaining efficiency and representativeness in their sampling design.
Stratified Estimation: Stratified estimation is a statistical technique used in sampling surveys where the population is divided into distinct subgroups, known as strata, to improve the accuracy and efficiency of estimates. By ensuring that each stratum is adequately represented in the sample, stratified estimation helps to reduce sampling error and provides more precise estimates of population parameters. This method is particularly useful in multistage sampling where different strata may have different characteristics that are essential for accurate overall estimates.
Taylor Series Linearization: Taylor Series Linearization is a mathematical technique used to approximate complex functions by linear functions around a specific point, using the function's derivatives. This approach allows statisticians to simplify the estimation procedures, making it easier to analyze data in multistage sampling, especially when the true distribution of the estimator is unknown or complex.
Three-stage sampling: Three-stage sampling is a complex multistage sampling method that involves selecting samples in three distinct stages. The process typically begins with a random selection of larger clusters, followed by the selection of smaller groups within those clusters, and finally, the selection of individual units from these groups. This method helps researchers efficiently gather data from large populations while controlling costs and improving feasibility.
Two-stage sampling: Two-stage sampling is a statistical sampling technique where the population is divided into clusters, and then a random sample of these clusters is selected. Within each selected cluster, a second stage of sampling occurs where elements or units are randomly chosen for the study, allowing researchers to effectively manage large populations and improve efficiency in data collection.
Weighting: Weighting is a statistical technique used to adjust the results of a survey or study to better reflect the overall population. This process involves assigning different levels of importance, or weights, to various responses based on certain characteristics such as demographics, ensuring that the sample accurately represents the target population.
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