## 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 (.58 meg ZIP file)

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