Estimating a Basic Space From A Set of Issue Scales

American Journal of Political Science, 42 (July 1998), pp. 954-993.

Abstract

This paper develops a scaling procedure for estimating the latent/unobservable dimensions underlying a set of manifest/observable variables. The scaling procedure performs, in effect, a singular value decomposition of a rectangular matrix of real elements with missing entries. In contrast to existing techniques such as factor analysis that work with a correlation or covariance matrix computed from the data matrix, the scaling procedure shown here analyzes the data matrix directly.
The scaling procedure is a general-purpose tool that can be used not only to estimate latent/unobservable dimensions but also to estimate an Eckart-Young lower-rank approximation matrix of a matrix with missing entries. Monte Carlo tests show that the procedure reliably estimates the latent dimensions and reproduces the missing elements of a matrix even at high levels of error and missing data.

The Model

Let xij be the ith individualís (i=1, ..., n) reported position on the jth issue (j = 1, ..., m) and let X0be the n by m matrix of observed data where the "0" subscript indicates that elements are missing from the matrix -- not all individuals report their positions on all issues. Let yik be the ith individualís position on the kth (k = 1, ..., s) basic dimension. The model estimated is:

X0 = [Y W' + Jnc']0 + E0

where Y is the n by s matrix of coordinates of the individuals on the basic dimensions, W is an m by s matrix of weights, c is a vector of constants of length m, Jn is an n length vector of ones, and E0 is a n by m matrix of error terms. W and c map the individuals from the basic space onto the issue dimensions. The elements of E0 are assumed to be random draws from a symmetric distribution with zero mean.

The decomposition is accomplished by a simple alternating least least squares procedure coupled with some long established techniques for extracting eigenvectors. The estimation procedure is covered in great detail in the AJPS article.

The paper How to Use the Black Box (Updated, 4 August 1998) is in Adobe Acrobat (*.pdf) format and explains how to use the software used in the AJPS article. (If you do not have an Adobe Acrobat reader, you may obtain one for free at http://www.adobe.com.)

The files below contain the FORTRAN programs, input files, and executables that perform the analyses shown in the AJPS article. These files are documented in the "How To Use the Black Box" paper above.

Programs and Input Files From AJPS Article (.6 meg LHA file)
Programs and Input Files From AJPS Article (.58 meg ZIP file)