Stage vs phase

When two-phase study designs started being used in epidemiology and biostatistics there was a period of conflict. Survey statisticians insisted on the term “two-phase” and biostatisticians (following survey textbooks in some cases) wanted to call these “two-stage” designs. Like the correct pronounciation of ‘Scheveningen’¹, the terminology identified communities.

In a \(K\)-stage survey design we have sampling units (clusters) at stage 1, smaller ones at stage 2, and so on. You can compute a probability \(\pi_i=\pi_{i,1}\times \pi_{i,2|1}\times \cdots\times\pi_{i,K|K-1}\), where \(\pi_{i,1}\) is the probability that unit \(i\) is sampled at stage 1, \(\pi_{i,2|1}\) is the probability that unit \(i\) is sampled at stage 2 given that it is sampled at stage 1, and so on. The probabilities are all known constants and \(\pi_i\) is the marginal probability that unit \(i\) is sampled.

In a \(K\)-phase survey design we have sampling units (clusters) at stage 1, other units at stage 2, and so on. You can compute a number \(\pi^*_i=\pi_{i,1}\times \pi^*_{i,2|1}\times \cdots\times\pi^*_{i,K|K-1}\), where \(\pi_{i,1}\) is the probability that unit \(i\) is sampled at phase 1, \(\pi^*_{i,2|1}\) is the probability that unit \(i\) is sampled at stage 2 given the phase-1 data and so on. The probabilities may depend on the entire data for the previous phases and so are random variables, so \(\pi^*_i\) is (in general) not the marginal probability that unit \(i\) is sampled.

It’s easy to see that multistage sampling is a special case of multiphase sampling; it’s what you get if you use only the information unit \(i\) was in phase \(k-1\) in defining \(\pi^*_{i,k|k-1}\). The simplest application of two-phase sampling that isn’t two-stage is when you want to stratify on variables that aren’t available for the whole population. You can measure those variable at phase 1 and then stratify the sampling of phase 2 on them. That’s how two-phase sampling is typically used in health research.

In some ways the distinction doesn’t matter. Suppose we write \(R_i\) for the indicator that unit \(i\) is sampled. The key property of multi-phase sampling is that \(E[R_i/\pi^*_i]=1\), just as \(E[R_i/\pi_i]=1\) for multistage sampling. The computational formulas for multiphase sampling are conceptually quite different from those for multistage sampling, but practically very similar: you get them by simply putting *s on all the probabilities.

This does raise one modestly interesting question: if \(\pi_{i}\) and \(\pi_i^*\) are different, can we say anything about which one is better? This is a theoretical question: in practice you usually can’t compute \(\pi_i\) because it involves averaging over all possible samples at intermediate phases. It’s still an interesting question. You could argue that using \(\pi^*\) was better because handwaving about conditioning, or you could argue that using \(\pi\) was better because handwaving about random variation. The answer doesn’t seem to be known.