## 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} individual’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} individual’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 (.6 meg LHA file)

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