We have measured hysteresis losses, W (ho, hb' H bO ) ' in ribbons ofVTi and Nb subjected to oscillating magnetic fields of various amplitudes ho directed transverse to a static bias magnetic field H bO ' hence the name, noncollinear regime. A second bias field hb along ho may also be introduced. HbO and ho are directed along the length and width of the ribbon respectively or vice versa (90· rotation). The families of data curves are compared with the predictions of double criticalstate models where (i) dB I dx = ± F p (B )1 B governs the critical gradients of the magnetic induction and (ii) dO Idx = ± Fp(B )F(O )/B 2, the spatial variation of the orientation of the sheets offlux lines, with 0 = OalongH bo . The effect on W(ho,hb ,H bO ) ofchoosingF(O ) = l/tan 0, k and k I( 1 + 10 1)2 is illustrated and the latter is seen to yield the best agreement. The locus of the magnetization along the length and width of the samples, (M II ) and (M 1 ), have been monitored separately and simultaneously during the cycles of ho(t) for various ho andH bo with hb = ° or h o . These observations, in particular the curves of the magnetization along H bO ' constitute severe tests of any model. The corresponding curves generated by the model using F (0) = k I( 1 + 10 1)2 are presented. The physical origin of F(O) is explored.