The ghosts of departed quantities: approaches to dealing with observations below the limit of quantitation.

A common but not necessarily logical requirement in drug development is that a 'limit of quantitation' be set for chemical assays and that observations that fall below the limit should not be treated as real data but should be labelled as below the limit and set aside for special treatment. We examine five of seven approaches to analysing such data considered by Beal in 2001, concentrating in particular on two: one that treats the data as a truncated sample and another that treats them as a censored sample. In fact, using a pattern-mixture framework, one can show that the former consists of using the conditional distribution of the 'acceptable values' and the latter adds the information from the marginal mixing distribution. We illustrate these approaches with a real example, concentrating in particular on the two likelihood-based methods, provide various formulae that may be used to compare these and other approaches, check these formulae using simulations and make some recommendations as to which approach one should use.