We study the class of multivariate distributions in which all bivariate regressions can be linearized by separate transformation of each of the variables. This class seems more realistic than the multivariate normal or the elliptical distributions, and at the same time its study allows us to combine the results from multivariate analysis with optimal scaling and classical multivariate analysis. In particular a two-stage procedure which first scales the variables optimally, and then fits a simultaneous equations model, is studied in detail and is shown to have some desirable properties.

Current computer programs for analyzing linear structural models will apparently handle only two types of constraints: fixed parameters, and equality of parameters. An important constraint not handled is inequality; this is particularly crucial for preventing negative variance estimates. In this paper, a method is described for imposing several kinds of inequality constraints in models, without the necessity for having computer programs which explicitly allow such constraints. The examples discussed include the prevention of Heywood cases, extension to inequalities of parameters to be greater than a specified value, and imposing ordered inequalities.

An interdependent multivariate linear relations model based on manifest, measured variables as well as unmeasured and unmeasurable latent variables is developed. The latent variables include primary or residual common factors of any order as well as unique factors. The model has a simpler parametric structure than previous models, but it is designed to accommodate a wider range of applications via its structural equations, mean structure, covariance structure, and constraints on parameters. The parameters of the model may be estimated by gradient and quasi-Newton methods, or a Gauss-Newton algorithm that obtains least-squares, generalized least-squares, or maximum likelihood estimates. Large sample standard errors and goodness of fit tests are provided. The approach is illustrated by a test theory model and a longitudinal study of intelligence.

Noncrystalline aluminosilicates termed allophane and imogolite are common constituents of spodosols, soils derived from volcanic ash, and many inceptisols. The surface charge characteristics of their synthetic analogues may be used to better understand their ion retention properties. In this study, we determined the point of zero salt effect (PZSE) by potentiometric titration of allophanes with Al/Si ratios of 1.12, 1.52, and 2.04 and of imogolite with an Al/Si ratio of 2.02. We also used microelectrophoresis to determine the point of zero charge (PZC) at the particle shear plane for the same materials in CI solutions of Li, Na, Cs, and tetramethyl ammonium. The PZSE decreased with decreasing Al/Si ratio for the allophanes, but the imogolite PZSE was much lower than that of the allophane with 2.04 Al/Si. The PZC was always higher than the PZSE of the same material, especially for imogolite. The results are best explained if cations reside within the hollow tubes of imogolite. This conclusion is supported by a fluorescence study that showed that only quenchers smaller than the inner diameter of the imogolite tube could fully quench Ce-imogolite.

X-ray diffraction shows that ordered interlayer structures form when high-charged Llano vermiculite is reacted with HDTMA-Br or HDTMA-acetate, but the structures differ from those given by low-charged vermiculites. The differences arise in two ways; firstly when both HDTMA+ cations and HDTMA-Br (or -acetate) ion pairs are present as interlayer guest species, a superstructure forms which is less commensurate with the host structure of high-charge vermiculite than it is for lower-charged material. Secondly, although a commensurate 3a × b interlayer superstructure forms when HDTMA+ cations are the only interlayer guests, an increasing number of charge-balancing cations must be accommodated as the layer charge increases. This results in some structural disorder which is only observed in the diffraction pattern of HDTMA+-intercalated, high-charge vermiculite. Organic ion pairs tend to order along widely spaced rows corresponding to the intersections of two sets of scattering planes. For the acetate form, one set of planes has spacings of 3.52 Å, but for the bromide form the value is 3.67 Å. Scattering planes in the second set have spacings of 4.02 Å, independently of counter-anion size. These two sets of planes diverge from one another by ∼10°. Ordering is more apparent in the presence of acetate anions than it is for the smaller Br− anions. In the high-charged intercalates, from which organic ion pairs have been removed, HDTMA+ cations occupy positions on a centered two-dimensional superlattice and also randomly, some interstitial positions as necessitated by layer charge.