Left orthogonally invariant (l.o.i.) vector and matrix laws are considered in modelling the errors in ANOVA and MANOVA. Those which do not admit the F-test in ANOVA or Hotelling’s \(T^ 2\)-test in MANOVA as UMP invariant are shown to violate a finite extendibility property. In practical problems, however, finite extendibility of the data is generally taken for granted.
The logical conclusion is that the practically important l.o.i. error laws must admit the usually normal theory tests as UMP invariant. Within a multiple decision framework, the usual forward variable selection procedure based on \(T^ 2\)-values is shown to be the uniformly best invariant Bayes decision procedure when a practically important l.o.i. error law is assumed.