Achieving equal probability of selection under various random sampling strategies

The underlying objective of epidemiological investigations is to extrapolate results from a sample to the relevant population. The simplest way of achieving this is to adopt a sampling strategy in which each individual in the population has the same chance of being selected--that is, to employ an 'equal probability of selection method' (epsem). The easiest ways of achieving this are to use simple random sampling or stratified random sampling with a constant sampling fraction. These strategies are often impracticable, however, particularly in large investigations covering a wide geographical area where resource implications dictate a more complex approach such as multi-stage or cluster sampling. Following detailed definitions and appropriate illustrations of these terms, the main purpose of this paper is to provide a working guide of how to achieve epsem using these various random sampling techniques. In brief, for multi-stage sampling with the rare feature of equal-sized first stage units, epsem is achieved by applying the above simple or stratified approaches to the first stage units. Even in the more realistic scenario of unequal first stage units, the same options apply provided that a fixed proportion of second stage units are to be selected (cluster sampling is in fact just one example of this, with 100% sampling of second stage units). If on the other hand a fixed number of second stage units are to be selected then for epsem the first stage units should be selected with each one having a probability proportionate to its size.