Transformations which preserve exchangeability and application to permutation tests

Abstract

Exchangeability of observations is a key condition for applying permutation tests. We characterize the linear transformations which preserve exchangeability, distinguishing second-moment exchangeability and global exchangeability; we also examine non-linear transformations. When exchangeability does not hold one may try to find a transformation which achieves approximate exchangeability; then an approximate permutation test can be done. More specifically, consider a statistic T = φ( Y ); it may be possible to find V such that Ỹ = V ( Y ) is exchangeable and to write T = φ¯(Ỹ). In other cases we may be content that Ỹ has an exchangeable variance matrix, which we denote second-moment exchangeability. When seeking transformations towards exchangeability we show the privileged role of residuals. We show that exact permutation tests can be constructed for the normal linear model. Finally we suggest approximate permutation tests based on second-moment exchangeability. In the case of an intraclass correlation model, the transformation is simple to implement. We also give permutational moments of linear and quadratic forms and show how this can be used through Cornish-Fisher expansions.