Linear indices in nonlinear structural equation models: best fitting proper indices and other composites

The recent advent of nonlinear structural equation models with indices poses a new challenge to the measurement of scientific constructs. We discuss, exemplify and add to a family of statistical methods aimed at creating linear indices, and compare their suitability in a complex path model with linear and moderating relationships. The composites used include principal components, generalized canonical variables, partial least squares, factor extraction ('LISREL'), and a newly developed method: best fitting proper indices. The latter involves the construction of linear combinations of indicators that maximize the fit of (non-)linear structural equations in terms of these indices; the weights as well as the loadings of the indicators are sign restricted so that each indicator contributes to as well as reflects its own index in a predefined way. We use cross-validation to evaluate the methods employed, and analyze the most general situation with a complete interaction specification using the bootstrap. The methods are exemplified using an empirical data set. An additional novel feature is the use of simulations to delineate the range of the possible parameter estimates.