Multistage sampling is a powerful technique for surveying large, diverse populations. It breaks down the sampling process into stages, starting with broad units and narrowing down to specific individuals or elements. This approach balances efficiency and representativeness.

The design involves selecting primary sampling units, then secondary units within those, and so on. Each stage can use different sampling methods like clustering, stratification, or probability proportional to size. This flexibility allows researchers to tailor the approach to their specific study needs and population characteristics.

Sampling Stages and Units

Multistage Sampling Structure

• Multistage sampling involves selecting samples in multiple stages, gradually narrowing down from larger units to smaller subunits
• Primary sampling units (PSUs) represent the first level of selection in a multistage design (states, counties, or large geographical areas)
• Secondary sampling units (SSUs) are selected within the chosen PSUs (cities, neighborhoods, or schools)
• Sampling stages can extend beyond two levels, depending on the complexity of the study design
• Subsampling occurs at each stage, selecting a portion of units from the previous stage

Hierarchical Selection Process

• Sampling frame consists of a list of all PSUs in the population
• First stage selects a sample of PSUs using probability sampling methods
• Subsequent stages involve selecting SSUs within the chosen PSUs
• Final stage typically selects individual respondents or elements of interest
• Each stage can employ different sampling techniques (simple random, stratified, or cluster sampling)

Sampling Techniques

Clustering in Multistage Sampling

• Clustering groups similar units together to improve sampling efficiency
• Reduces travel costs and time for data collection in geographically dispersed populations
• Natural clusters often exist in populations (schools, hospitals, or neighborhoods)
• Can lead to increased sampling error due to potential homogeneity within clusters
• Requires careful consideration of cluster size and number to balance efficiency and precision

Stratification and Probability Proportional to Size

• Stratification divides the population into homogeneous subgroups before sampling
• Improves representation of different population segments
• Enhances precision by reducing sampling variability
• Probability proportional to size (PPS) sampling allocates selection probabilities based on unit size
• PPS often used for selecting PSUs, giving larger units a higher chance of selection
• Combines well with stratification to ensure proper representation of diverse population segments

Sampling Frame Construction

• Sampling frame provides a comprehensive list of all sampling units at each stage
• Requires careful development to ensure coverage of the entire target population
• May use existing databases, registers, or specially constructed lists
• Updating frames regularly prevents issues with undercoverage or overcoverage
• Quality of the sampling frame directly impacts the validity of the multistage sample

Sampling Efficiency and Estimation

Design Effect and Intraclass Correlation

• Design effect measures the efficiency of a complex sampling design compared to simple random sampling
• Calculated as the ratio of the variance of an estimate under the complex design to the variance under simple random sampling
• Intraclass correlation quantifies the similarity of units within the same cluster
• Higher intraclass correlation leads to larger design effects and reduced precision
• Design effect influences sample size calculations and variance estimation in multistage designs

Sampling Weights and Variance Estimation

• Sampling weights account for unequal selection probabilities in multistage designs
• Calculated as the inverse of the overall selection probability for each sampled unit
• Weights adjust for non-response and post-stratification to improve representativeness
• Variance estimation in multistage sampling requires specialized techniques (Taylor series linearization, replication methods)
• Accounts for the complex design features such as stratification, clustering, and weighting
• Software packages (SUDAAN, STATA, R survey package) facilitate analysis of multistage sample data

Efficiency Considerations

• Sampling efficiency balances cost, precision, and representativeness in multistage designs
• Optimal allocation of sample sizes across stages maximizes efficiency
• Considers factors such as cost per unit at each stage and variability within and between clusters
• Trade-offs between number of PSUs and number of units sampled within each PSU
• Pilot studies or prior knowledge help inform efficient multistage sampling designs