Preface
After writing up all my notes on Thomas Lumley’s excellent book on his complex survey software package, I found myself still interested in reading and writing more about the subject, particularly those areas that weren’t explicitly covered in Lumley’s text. I found myself reading Lumley’s main theoretical reference for his book, (Särndal, Swensson, and Wretman 2003) and enjoying it immensely It also didn’t hurt that these topics overlapped with my own professional work considerably.
So I thought I’d continue writing on the subject. In particular, this post looks to make a more thorough review of the types of sampling one might encounter in a complex survey sample - prior to the analysis of the data. Additionally, while certain sampling types — sampling without replacement for example — are defined in statistical terms by their sampling distribution, the implementation of any algorithm that produces the desired sampling distribution is another matter that can have implications on how easily one is able to draw a sample on, say, a very large data set. Details here can only get more complicated depending on what’s being measured, or how quickly samples need to be drawn. I’ll address some of these issues and refer to my own experience in working in this area to highlight what I think are the best strategies.
Sampling Designs and Survey Estimators for Sampling Population Elements
(Särndal, Swensson, and Wretman 2003) provides a formal definition for the technical requirements of a sampling design. In this post, it’ll suffice to say that a sampling design provides some procedure by which elements are drawn probabilistically from a population. The use of the word “probabilistically” here is key, as it denotes that the realized sample was but one of possible samples and grounds our inference as a method that can provided unbiased estimators with quantifiable uncertainty. A further distinction worth noting here is that we’re sampling the elements of the population themselves upon which we’ll construct our measurement. This distinction is worth making because more complex survey designs will sample clusters of elements before the elements themselves to reduce the overall cost of the design and/or improve efficiency of estimate’s precision.
(Särndal, Swensson, and Wretman 2003) lists the following eight element sampling designs in chapter 3:
- Bernoulli Sampling
- Simple random sampling with replacement
- Simple random sampling without replacement.
- Systematic Sampling
- Poisson Sampling
- Probability proportional to size sampling with replacement
sampling R package.
- Probability proportional to size sampling without replacement
- Stratified Sampling.
Each of these have various advantages and disadvantages that we’ll now walk through, with special attention to their properties for estimating the total of some measure of the population via the Horvitz Thompson or \(\pi\) estimator, as (Särndal, Swensson, and Wretman 2003) refers to it in this section.
Bernoulli Sampling
Simple Random Sampling
Systematic Sampling
Poisson Sampling
PPS Sampling
Stratified Sampling
The benefits of stratified sampling have already been covered in my notes on Lumley’s text here, but to recap briefly, stratified samples gain their power from identifying some discrete measure that is associated with the outcome of interest. By ensuring that a sufficient number of elements are sampled within this group, a better estimate of the overall population value can be obtained. Additionally, if the strata represent some domain of interest - race, income group or other area of interest, stratified sampling can ensure that the measurement within that domain attains the desired estimate precision.