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 ${\pi }_i^{\ast }={\pi }_{i,1}\times {\pi }_{i,2|1}^{\ast }\times \cdots \times {\pi }_{i,K|K-1}^{\ast }$, where ${\pi }_{i,1}$ is the probability that unit $i$ is sampled at phase 1, ${\pi }_{i,2|1}^{\ast }$ 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 ${\pi }_i^{\ast }$ 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 ${\pi }_{i,k|k-1}^{\ast }$. 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 $E\left[R_i/{\pi }_i^{\ast }\right]=1$, just as $E\left[R_i/{\pi }_i\right]=1$ 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 ${\pi }_i^{\ast }$ 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 ${\pi }^{\ast }$ 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.