Other Indices of Fit - Browne-Cudeck Single Sample Cross-Validation Index

Browne and Cudeck (1989) proposed a single sample cross-validation index as a follow-up to their earlier (Cudeck & Browne, 1983) paper on cross-validation. Cudeck and Browne had proposed a cross-validation index which, for model Mk in a set of competing models is of the form FML(Sn,Sk(q)). In this case, F is the maximum likelihood discrepancy function, Sn is the covariance matrix calculated on a cross-validation sample, and Sk(q) the reproduced covariance matrix obtained by fitting model Mk to the original calibration sample. In general, better models will have smaller cross-validation indices.

The drawback of the original procedure is that it requires two samples, i.e., the calibration sample for fitting the models, and the cross-validation sample. The new measure estimates the original cross-validation index from a single sample.

The measure is


where nk is the number of free parameters in model k, p is the number of manifest variables, and N is the sample size.