APPLICATION OF THE PARAMETRIC BOOTSTRAP TO MODELS THAT INCORPORATE A SINGULAR-VALUE DECOMPOSITION

Simulation is a standard technique for investigating the sampling dist ribution of parameter estimators. The bootstrap is a distribution-free method of assessing sampling variability based on resampling from the empirical distribution; the parametric bootstrap resamples from a fit ted parametric model. However, if the parameters of the model are cons trained, and the application of these constraints is a function of the realized sample, then the resampling distribution obtained from the p arametric bootstrap may become badly biased and overdispersed. Here we discuss such problems in the context of estimating parameters from a bilinear model that incorporates the singular value decomposition (SVD ) and in which the parameters are identified by the standard orthogona lity relationships of the SVD. Possible effects of the SVD parameter i dentification are arbitrary changes in the sign of singular vectors, i nversion of the order of singular values and rotation of the plotted c o-ordinates. This paper proposes inverse transformation or 'filtering' techniques to avoid these problems. The ideas are illustrated by asse ssing the variability of the location of points in a principal ed-ordi nates diagram and in the marginal sampling distribution of singular va lues. An application to the analysis of a biological data set is descr ibed. In the discussion it is pointed out that several exploratory mul tivariate methods may benefit by using resampling with filtering.