## 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)

**Common Space Scores Congresses 75 - 113 (6 January 2015)**

The Format of the Common Space Scores

Common Space Scores (Text File)

Common Space Scores (Excel File)

Common Space Scores (Stata 13 File)

Common Space Scores (Stata 12 File)

VOTEVIEW Blog

NOMINATE Data, Roll Call Data, and Software

Course Web Pages: University of Georgia (2010 - )

Course Web Pages: UC San Diego (2004 - 2010)

University of San Diego Law School (2005)

Course Web Pages: University of Houston (2000 - 2005)

Course Web Pages: Carnegie-Mellon University (1997 - 2000)

*Analyzing Spatial Models of Choice and Judgment with R*

*Spatial Models of Parliamentary Voting*

Recent Working Papers

Analyses of Recent Politics

About This Website

K7MOA Log Books: 1960 - 2016

Bio of Keith T. Poole

Related Links