This paper presents a new stochastic multidimensional scaling procedure for the analysis of three-mode, three-way pick any/J data. The method provides either a vector or ideal-point model to represent the structure in...
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This paper presents a new stochastic multidimensional scaling procedure for the analysis of three-mode, three-way pick any/J data. The method provides either a vector or ideal-point model to represent the structure in such data, as well as "floating" model specifications (e.g., different vectors or ideal points for different choice settings), and various reparameterization options that allow the coordinates of ideal points, vectors, or stimuli to be functions of specified background variables. A maximum likelihood procedure is utilized to estimate a joint space of row and column objects, as well as a set of weights depicting the third mode of the data. An algorithm using a conjugate gradient method with automatic restarts is developed to estimate the parameters of the models. A series of Monte Carlo analyses are carried out to investigate the performance of the algorithm under diverse data and model specification conditions, examine the statistical properties of the associated test statistic, and test the robustness of the procedure to departures from the independence assumptions. Finally, a consumer psychology application assessing the impact of situational influences on consumers' choice behavior is discussed.
Given a matrix of dissimilarities, it has been debated whether researchers should perform multidimensional scaling on this original matrix or on a new one derived by comparing rows in the original matrix. Careful comp...
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Given a matrix of dissimilarities, it has been debated whether researchers should perform multidimensional scaling on this original matrix or on a new one derived by comparing rows in the original matrix. Careful comparison studies (Drasgow & Jones, 1979;Van der Kloot gr Van Herk, 1991) in the context of sorting data indicated that most of the initial enthusiasm for the derivative approach was unfounded. The current work, a Monte Carlo study of structured binarydata derived from known two-dimensional configurations using ALSCAL, complements and extends the previous studies. We discuss a weakness in the squared difference (Delta) row-comparison rule used previously and propose an alternative row-comparison measure based on the Jaccard coefficient. Scaling the binarydata directly gave better performance, as gauged by Procrustes statistics, than did scaling Delta data across a range of noise levels. The quality of solutions obtained by scaling Jaccard data was always better or essentially equal to that from scaling Delta data, and in certain parameter regions improved upon that of direct scaling. Another alternative approach, applying the Delta rule after first row-centering the binarydata, was found to be generally ineffective. These findings are pertinent to the analysis not just of stimulus sorting data but of coarse dissimilarities generally, for example from direct pairwise judgment tasks and in fields outside statistical psychology.
Unidimensionality is one of the important assumptions the data should satisfy in order to apply unidimensional item response models. There are several methodologies available to date to assess the dimensionality of th...
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Unidimensionality is one of the important assumptions the data should satisfy in order to apply unidimensional item response models. There are several methodologies available to date to assess the dimensionality of the latent space underlying binary item responses. Rosenbaum (1984) and Holland and Rosenbaum (1986) have proved theorems concerning conditional associations that can be applied to assess dimensionality. Holland and Rosenbaum's method has been applied to assess dimensionality by Zwick (1987), Ben-Simon and Cohen (1990), and Nandakumar (1991) to various test situations. This article outlines the Holland and Rosenbaum's (1986) methodology to assess unidimensionality, illustrates the procedure through a simulated data set, and describes how to interpret the results.
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