Transformations which preserve exchangeability and application to permutation tests. (English) Zbl 1054.62055

Commenges, Daniel

J. Nonparametric Stat. 15, No. 2, 171-185 (2003).

Summary: 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 \(T=\phi(Y)\); it may be possible to find \(V\) such that \(\tilde Y=V(Y)\) is exchangeable and to write \(T=\bar{\phi}(\bar Y)\). In other cases we may be content that \(\tilde Y\) 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.

MSC:

62G10	Nonparametric hypothesis testing
60G09	Exchangeability for stochastic processes

Keywords:

Exchangeability; Permutation test; Homogeneity test; Residuals; Score test; Transformation

References:

[1]	DOI: 10.1002/(SICI)1098-2272(1996)13:6<559::AID-GEPI3>3.0.CO;2-W
[2]	DOI: 10.1111/1467-9868.00061
[3]	Cox D. R., Theoretical Statistics (1974) · Zbl 0334.62003
[4]	Cox D. R., J. Roy. Stat. Soc. B 30 pp 248– (1968)
[5]	de Finetti B., Theory of Probability (1990)
[6]	DOI: 10.1007/BF00354037 · Zbl 0596.60022
[7]	DOI: 10.1016/0024-3795(90)90339-E · Zbl 0712.15002
[8]	DOI: 10.1111/j.1467-9574.1990.tb01269.x · Zbl 0715.62081
[9]	Eaton M. L., Multivariate Statistics: A Vector Space Approach (1983) · Zbl 0587.62097
[10]	DOI: 10.1002/(SICI)1097-0258(19970615)16:11<1283::AID-SIM532>3.0.CO;2-G
[11]	Good P., Permutations Tests (1992)
[12]	Mantel N., Cancer Research, 27, Part 1 pp 209– (1967)
[13]	McCullagh P., Generalized Linear Models, 2. ed. (1989) · Zbl 0744.62098
[14]	McCune E. D., Encyclopedia of Statistical Sciences pp 188– (1981)
[15]	DOI: 10.2307/2288275 · Zbl 0547.62029
[16]	Robinson J., J. Roy. Statist. Soc. B pp 91– (1982)
[17]	DOI: 10.2307/2290003 · Zbl 0706.62047
[18]	DOI: 10.2307/2291013 · Zbl 0810.62047
[19]	DOI: 10.2307/2283406
[20]	DOI: 10.2307/2283844
[21]	DOI: 10.2307/2290084
[22]	DOI: 10.2307/2290009 · Zbl 0704.62032