This article proposes a model that facilitates the analysis of triadic relationships among three objects. Recently, interest in studies of three-way data models has increased, and many significant contributions have been made in this area. However, one-mode, three-way models have yet to be considered. This study focuses on a one-mode, three-way model in which three-way distances are explained as the subtraction of the smallest squared distance among the three squared distances from the sum of these squared distances. This formulation is used to illustrate the idea that relationships with many differences carry more information than relationships with few differences. Moreover, the distance between two objects is weighted more heavily when two objects differ greatly, and this weight indicates the salience of their dyadic distance. Finally, the model and algorithm are applied to purchase data for convenience stores. The proposed model of multidimensional scaling clearly identifies differences among groups of objects.
We study the properties of the power-transformation model to improve the non-additivity in regression, proposed by Goto (1992). This power additive transformation (PAT) model is an extension of the Box and Cox power transformation (BCPT) model and then includes the Bleasdale's simplified model and the one-compartment model as special cases. We describe the procedure to obtain the maximum likelihood estimates of the PAT model and discuss some issues in the maximum likelihood estimation, especially the consistency of the estimates and the effect of the error variance on the parameter estimation. We also provide two examples to illustrate the aspect of the PAT model, compared with the BCPT model. The results suggest that the PAT model provides reasonable transformations for improving the non-additivity in the data and is useful for identifying the nonlinear function form even when there is no strong knowledge on the data-generating process.
Two test theoretical approaches to item analysis are compared, an approach based on homogeneity analysis and one based on item response theory. The literature on the relationship between the two approaches is briefly reviewed. The paper contains a contribution to the relationship between the two approaches for the case that the scores are dichotomous and a single latent variable is assumed to underlie the data. A loss function is proposed for modelling item response functions with two parameters, one for discrimination and one for difficulty. It turns out that the loss of the proposed loss function is related to loss of homogeneity. Demonstrations with simulated data are used to evaluate the proposed method.
Two methods are typically used for the analysis of multivariate longitudinal data: multivariate random-effects growth curve models and latent curve models. Despite their popularity, in the former method an entire set of basis functions should a priori be specified and in the latter the estimates of random-effects or latent variable scores are not uniquely determined. An extended multivariate random-effects growth curve model is proposed to overcome the limitations of the two methods. The proposed method extends the existing multivariate random-effects growth curve model in such a way that it does not need to specify all basis functions in advance. It also offers the unique estimates of random-effects. Furthermore, the method can deal with unbalanced response variables, so that they do not have to be measured at the same time points, nor the same number of time points. An example is given to illustrate the method.
The growth curve model is useful for the analysis of longitudinal data. It helps investigate an overall pattern of change in repeated measurements over time and the effects of time-invariant explanatory variables on the temporal pattern. The traditional growth curve model assumes that the matrix of covariances between repeated measurements is unconstrained. This unconstrained covariance matrix often appears unattractive. In this paper, the generalized estimating equation method is adopted to estimate parameters of the growth curve model. As a result, the proposed method allows a more variety of constrained covariance structures than the traditional growth curve model. An empirical application is provided so as to illustrate the proposed method.