23.4 Unequal Probability Sampling

Unequal probability sampling assigns different selection probabilities to elements in the population. This approach is often used when certain units are more important, have higher variability, or require higher precision in estimation.

Common methods for unequal probability sampling include:

  • Probability Proportional to Size (PPS): Selection probability is proportional to a given auxiliary variable (e.g., revenue, population size).

  • Poisson Sampling: Independent selection of each unit with a given probability.

  • Systematic Sampling with Unequal Probabilities: Uses a systematic approach while ensuring different probabilities.

The following functions from the sampling package implement various unequal probability sampling methods:

library(sampling)

# Different methods for unequal probability sampling
UPbrewer()         # Brewer's method
UPmaxentropy()     # Maximum entropy method
UPmidzuno()        # Midzuno’s method
UPmidzunopi2()     # Midzuno’s method with second-order inclusion probabilities
UPmultinomial()    # Multinomial method
UPpivotal()        # Pivotal method
UPrandompivotal()  # Randomized pivotal method
UPpoisson()        # Poisson sampling
UPsampford()       # Sampford’s method
UPsystematic()     # Systematic sampling
UPrandomsystematic() # Randomized systematic sampling
UPsystematicpi2()  # Systematic sampling with second-order probabilities
UPtille()          # Tillé’s method
UPtillepi2()       # Tillé’s method with second-order inclusion probabilities

Each of these methods has specific use cases and theoretical justifications. For example:

  • Poisson sampling allows flexible control over sample size but may lead to variable sample sizes.

  • Systematic sampling is useful when population elements are arranged in a meaningful order.

  • Tillé’s method ensures better control over the sample’s second-order inclusion probabilities.