## 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)

**Common Space Scores Congresses 75 - 111 (15 February 2010)**

The Format of the Common Space Scores

Common Space Scores (Text File)

Common Space Scores (Excel File)

Common Space Scores (Stata 8 File)

Common Space Scores (Stata 7 File)

## Spatial Maps of Common Space Scores 1937 - 2002

Each token represents a member of Congress. **D** is a Northern
(Non-Southern) Democrat, S is a Southern Democrat (11 States of
the Confederacy plus Kentucky and Oklahoma), **R** is a Republican, and
**P** is the President.

Below is a plot of the first (Liberal-Conservative) dimension of the Common Space Scores.
The histograms are for Democrats and Republicans in the two Chambers. Senator
Kerry (**D**-MA) is located at -.364 which is to the
left of the mean of the Senate Democrats, -.243 (standard deviation, .187). Senator
Edwards (**D**-NC) is located at -.238 which is almost
exactly on the mean. Senator Kennedy (**D**-MA) is
located at -.474. President Bush is located at .399 which is to the right of
the means of both the Senate Republicans, .271 (standard deviation, .187),
and the House Republicans, .300 (standard deviation, .158). Vice President
Cheney is located at .509 and former Senate Helms
(**R**-NC) is located at .648.

In the animation below the Houses and Senates that Senator
Kerry (**D**-MA) has served in are shown
in one picture. The flashing **K** indicates Senator Kerry's
location in the two dimensional map, the flashing **E** is
Senator Edwards' location, the flashing **B** is President
Bush's location, and the flashing **C** is Vice-President
Cheney's location.

House and Senate Images of Common Space
Scores 75^{th} to 107^{th} Congresses

VOTEVIEW Blog

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