6.2 Probability proportional to size (PPS) sampling

Probability proportional to size (PPS) sampling is a game-changer in survey design. It gives bigger, more important units a better shot at being picked, which can lead to more accurate results. This method is super useful when you're dealing with units of different sizes or importance.

PPS sampling is a key player in multistage sampling. It helps researchers pick primary sampling units (PSUs) in a way that balances representation and efficiency. This approach can save time and money while still giving reliable data.

PPS Sampling Basics

Understanding PPS and Measure of Size

• Probability proportional to size (PPS) sampling selects units with probabilities proportional to their size or importance
• Measure of size (MOS) quantifies the relative importance of each unit in the population
• MOS correlates strongly with the variable of interest improves precision of estimates
• Common MOS includes population counts, land area, or sales volume
• Sampling frame contains list of all units in the population with their corresponding MOS values

Unequal Probability and Size Bias

• Unequal probability sampling assigns different selection probabilities to units based on their MOS
• Larger units have higher chances of selection enhances representation of important elements
• Size bias occurs when larger units are overrepresented in the sample
• PPS sampling corrects for size bias by adjusting selection probabilities
• Balances representation of small and large units in the final sample

PPS Sampling Methods

Cumulative Total and Systematic PPS

• Cumulative total method creates a running sum of MOS values across all units
• Random number generated between 0 and the total cumulative MOS
• Unit selected when cumulative total exceeds the random number
• Systematic PPS sampling divides the cumulative total into equal intervals
• Random start point chosen within the first interval determines subsequent selections

With and Without Replacement PPS

• With replacement PPS (PPSWR) allows units to be selected multiple times
• PPSWR simplifies calculations and analysis
• Without replacement PPS (PPSWOR) ensures each unit appears only once in the sample
• PPSWOR increases efficiency by avoiding duplication
• PPSWOR requires more complex sampling algorithms (Brewer's method or Hanurav-Vijayan algorithm)

PPS Sampling Properties

Inclusion Probabilities and Self-Weighting Samples

• Inclusion probability represents the chance of a unit being selected in the sample
• Calculated as the ratio of unit's MOS to the total MOS of the population
• Self-weighting samples have equal weights for all selected units
• PPS can create self-weighting samples when MOS is proportional to the variable of interest
• Self-weighting simplifies analysis and reduces the need for complex weighting procedures

Efficiency Gains and Precision

• PPS sampling often yields more precise estimates than simple random sampling
• Efficiency gains result from incorporating auxiliary information through MOS
• Reduces sampling variance by allocating more resources to important units
• Particularly effective when MOS strongly correlates with the variable of interest
• Can lead to smaller sample sizes for the same level of precision

PPS Estimators

Horvitz-Thompson Estimator and Applications

• Horvitz-Thompson estimator provides unbiased estimates for population totals in PPS sampling
• Calculated by summing the ratio of observed values to inclusion probabilities
• Formula: $\hat{Y} = \sum_{i \in s} \frac{y_i}{\pi_i}$ where $\hat{Y}$ is the estimated total, $y_i$ is the observed value, and $\pi_i$ is the inclusion probability
• Accounts for unequal selection probabilities in the estimation process
• Widely used in complex surveys and multi-stage sampling designs

Variance Estimation and Confidence Intervals

• Variance of Horvitz-Thompson estimator depends on joint inclusion probabilities
• Exact variance calculation complex for PPSWOR designs
• Approximation methods (linearization, replication) often used for variance estimation
• Confidence intervals constructed using estimated variance and normal distribution assumptions
• Bootstrap methods provide alternative approach for variance and confidence interval estimation in PPS sampling