## 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 **x**_{ij} be the i^{th} individualÂ’s (i=1, ..., n) reported position on the j^{th} issue (j = 1, ..., m) and let **X**_{0}be 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 **y**_{ik} be the i^{th} individualÂ’s position on the k^{th} (k = 1, ..., s) basic dimension. The model estimated is:

X_{0} = [Y W' + J_{n}__c__']_{0} + E_{0}

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, **J**_{n} is an n length vector of ones, and **E**_{0} 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 **E**_{0} 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 (.58 meg ZIP file)

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