HOW TO USE THE BLACK BOX
by
Keith T. Poole
Graduate School of Industrial
Administration
Carnegie-Mellon University
Pittsburgh, PA 15213
3 August 1998
Abstract
This paper is meant as a supplement to my August, 1998 AJPS article "Recovering a Basic Space From a Set of Issue Scales." I promised the editor and the reviewers of my article that I would provide and support the computer programs used in the article. Accordingly, this paper is part of a package of material that contains the FORTRAN code and the executables of the programs used in the article.
1. Introduction
This paper is meant as a supplement to my AJPS article "Recovering a Basic Space From a Set of Issue Scales. The body of the paper shows material deleted from the original paper to conserve journal space as well as additional empirical examples. Appendices A, B, and C show researchers how to use the various computer programs that implement the model shown in the article.
Section 2 shows how the method I develop in my AJPS article is related to Factor Analysis. Section 3 reports some Monte-Carlo work that was cut out of the final version of the paper. Section 4 shows the relationship between the method and Aldrich-McKelvey scaling (1977). In effect, the method can be used to perform an Aldrich-McKelvey scaling of an issue scale in more than one dimension. Finally, Section 5 shows some additional empirical applications.
2. Relationship With Standard Factor Analysis
A standard method of analyzing a rectangular data matrix is to compute a correlation matrix between variables (the columns of the data matrix) and then factor analyze (principal components or maximum likelihood) the correlation matrix. Factor analysis has its own special nomenclature and the method of its presentation varies from author to author. However, in its simplest form, principal components, it is simply eigenvalue/eigenvector decomposition and its connection to singular value decomposition is easily shown.
For example, suppose the n by m data matrix, X, has no missing data and is standardized such that each column sums to zero and the sum of the squared entries of the column sums to one. Note that this is the transformation
(1)
where
is the original
matrix entry, 
is the mean of the
jth column, and sj is the
standard deviation of the jth column.
Given the transformation given in equation (1), the Pearson correlation
matrix is simply R = X’X.
Alternatively, most authors (e.g., Harman, 1970; Van de Geer, 1971) assume that X is in standard deviation form. That is:
(2)
Written
in this form the Pearson correlation matrix is 
. This approach has the awkward result of
having the 1/n in various equations.
Accordingly, I use the simpler approach of equation (1). This has no material effect on the
discussion below except to simplify the expressions.
Let the singular value decomposition of X be ULV’, where U is an n by m matrix such that U’U = Im , L1/2 is a m by m diagonal matrix of singular values, and V is a m by m matrix such that V’V = Im.
To perform a principal components
analysis compute the correlation matrix,
R
= X’X = VL2V’
(2)
and then perform a standard eigenvalue/eigenvector decomposition of R. Note that the eigenvalues of R are the squared singular values of X. The factor matrix is the m by m matrix VL and the factor scores are the n by m matrix U, where U is from the singular value decomposition of X. Using the terminology of Harman (1970):
F = U
and A = VL
In terms of the model I use in the AJPS article,
X0
= [YW' + Jnc']0 + E0
let c' = 0, no missing data, and ignoring the error term for the moment, then
(3)
so that Y and W are simply related to the factor scores and the factor matrix, respectively.
Indeed, even when X is in the form of the AJPS
paper, [YW' + Jnc']0 + E0,
the estimated W matrix, 
, will be highly correlated with the factor matrix, A. For
example, Table 1 shows the SPSS output for a principal components analysis of
the 1980 issue scale example shown in the AJPS paper. The data set was read into SPSS and the
correlation matrix was computed using the pair-wise deletion option. The first part of the table shows the eigenvalue
table and the second part of the table shows the Factor Matrix, A, labeled “Component Matrix” in the
SPSS output.
The r-squares between the 3 columns
of the Factor Matrix shown in Table 1 and the columns of the 
shown in Table 4 of
the AJPS
article are .929, .802, and .223 respectively. In other words, the first two dimensions are
essentially the same. This makes sense
because I found that only two of the fourteen 
’s for the 3rd dimension to be statistically
significant whereas seven of the fourteen 
’s for the 2nd dimension and all for the 1st
dimension were statistically significant.
The individual placements appear to be at most two-dimensional.
Further evidence of the data being at most two-dimensional are the scree plots shown in Figure 1. The dotted line in the upper plot shows the eigenvalues of the correlation matrix from the SPSS output. The solid line in the upper plot shows the squared singular values for the data transformed as in equation (1). That is, let X* = X0 for the non-missing entries and let X* = YW’ (using three dimensions) for the missing entries. Column means were computed using the non-missing entries of X0, and each entry of X* was transformed as shown in equation (1). The singular values of this matrix were then squared to make them comparable to the eigenvalues extracted from the correlation matrix. These two series are virtually identical with the values of the eigenvalues/squared singular values falling off fairly smoothly from the elbow at the 3rd value through the 14th value. This is a clear indication that the data are most likely to be two-dimensional.
Table 1
SPSS Output for 1980 Issue Scale Example
Extraction Method: Principal Component
Analysis.
Total Variance Explained
|
|
Initial Eigenvalues |
|
|
Extraction Sums of Squared Loadings |
|
|
| Component |
Total |
% of Variance |
Cumulative % |
Total |
% of Variance |
Cumulative % |
| 1 |
4.432 |
31.654 |
31.654 |
4.432 |
31.654 |
31.654 |
| 2 |
2.055 |
14.677 |
46.331 |
2.055 |
14.677 |
46.331 |
| 3 |
1.155 |
8.252 |
54.583 |
1.155 |
8.252 |
54.583 |
| 4 |
.916 |
6.546 |
61.129 |
|
|
|
| 5 |
.879 |
6.281 |
67.410 |
|
|
|
| 6 |
.790 |
5.644 |
73.054 |
|
|
|
| 7 |
.780 |
5.570 |
78.625 |
|
|
|
| 8 |
.630 |
4.497 |
83.122 |
|
|
|
| 9 |
.599 |
4.278 |
87.399 |
|
|
|
| 10 |
.541 |
3.865 |
91.264 |
|
|
|
| 11 |
.409 |
2.919 |
94.183 |
|
|
|
| 12 |
.394 |
2.812 |
96.995 |
|
|
|
| 13 |
.223 |
1.595 |
98.590 |
|
|
|
| 14 |
.197 |
1.410 |
100.000 |
|
|
|
Figure 1
Eigenvalues vs. Singular Values
The lower plot shows the singular values of X* minus the actual column means, that is, X* – Jnc’, where c is the vector of actual column means. The column means are subtracted from X* because if the means are large numerically – as they are for issue scales – then the first singular value of X* will be very large compared to the others. This results from the fact that the sum of the squared singular values is equal to the sum of the squared values of the matrix. That is:
Here the singular values fall off a little less smoothly. The decline from the 2nd to the third value is not as dramatic as it is for the eigenvalues of the correlation matrix, but the plot clearly indicates that the data is at most three-dimensional (see the output for the 1980 issue scale example in Appendix A). In this context, a singular value decomposition of X yields essentially the same information as a principal components analysis of the correlation matrix.
3.
Additional Monte
Carlo Tests of the Model
This subsection was deleted from the final version of the paper in order to conserve journal space. The purpose of these tests is to show the ability of the procedure to estimate the Eckart-Young lower rank approximation matrix of an arbitrary matrix of real numbers with missing entries. The table and equation references are to those in the AJPS paper. The table deleted along with this subsection was the original Table 4. Here I will refer to it as 4* to avoid confusion with the AJPS article.
Estimating Eckart-Young Approximation Matrices
The Monte Carlo results reported in Tables 2 and 3 show that the procedure does an excellent job of estimating Y, W, and c when the observed data are in the form shown in equation (1). The purpose of this subsection is to show that the procedure also can be used as a general-purpose tool to obtain the Eckart-Young approximation matrix ULsV¢ of any rectangular matrix of real numbers with missing entries.
In this application 
must be used with
some caution. Recall that if a matrix X is of rank s, then subtracting off
the column means, X – Jnc¢ , in most
circumstances, does not change the rank.[1] However, the converse is not necessarily
true. By construction, 
has rank s and the
columns of 
sum to zero. Adding a vector of constants, 
, will usually increase the rank to s+1. However, the first s singular values of 
will be quite large
in comparison to the s+1st singular value. If the observed data are in form given by
equation (1), then the closer 
is to the true YW’ (which, by
construction, is X – Jnc¢), the smaller the
s+1st singular value. In
addition, if the column means are not of interest, then the Monte Carlo
results in Table 2 show that 
is an excellent
approximation of the true YW’
matrix even at substantial levels of error and missing data.
In order to test the ability of the
procedure to estimate the Eckart-Young approximation matrix of an arbitrary
matrix of real numbers of full rank, just the first s singular values of 
were utilized. That is, if the singular value
decomposition of 
is ULV’ where
L is an s+1 by s+1
matrix, then ULsV’
was used as the approximation matrix.
Similar to the method used for Table 2, to construct X of full rank m, U and V were obtained from a singular value decomposition of an n by m matrix of uniform [-1,1] random numbers. The first three singular values of X were set so that when the column means are subtracted from X, the singular values of the resulting matrix, YW' , were approximately 50, 35, 20, 5, 5, …, 5. This was done so as to approximate a situation where the first three dimensions largely account for the structure of the matrix but the matrix is of full rank.
Missing data was created in the same fashion as described for Table 2.
In these tests there is no error process so the only difference between trials is the pattern of missing data. Each entry in Table 4* is the average of 10 trials. The standard deviations are shown in parentheses.
__________________
Table 4* About Here
__________________
The first three columns of Table 4
show the number of rows, the number of columns, and the rank of the
approximation matrix. The fourth column, r-square with
Eckart-Young, shows the average squared Pearson correlation between the nm
elements in true Eckart-Young matrix and the reproduced Eckart-Young matrix. The fifth column shows the average squared
Pearson correlation between the true basic dimensions and the estimated basic
dimensions corresponding to the rank of the approximation; that is, the
average of the r-squares computed between each column of the true Y matrix, and its
corresponding column in 
.[2] Finally, the sixth column shows the
percentage of missing entries.
Table 4* shows that the procedure does a good job estimating lower rank approximations when a substantial portion of the matrix is missing. Not surprisingly, the lower the level of missing data and the larger the matrix, the better the approximation. With 25 percent missing data the r-squares all exceed .97. Even with 70 percent missing data the procedure will do a reasonable job if the size of the matrix is large enough.
4.
Empirical
Application Deleted From AJPS Article
This application was deleted from the final draft of the AJPS article to conserve journal space. In this application the scaling method is applied to a transposed matrix in which the number of columns is much larger than the number of rows. What I show below is that the general method developed in the AJPS article can be used to perform Aldrich-McKelvey scaling of an issue scale in more than one dimension.
Aldrich and McKelvey (1977) in effect solved the Likert scale problem. In my opinion, the Aldrich and McKelvey paper is the most under appreciated achievement in political methodology. In part this is due to the fact that, as I note below, the standard error of the estimate of their model is clearly biased downwards. However, as I have argued elsewhere (Palfrey and Poole, 1987), this is an advantage not a defect! Namely, the Aldrich-McKelvey scaling method can be used as a powerful filter. Respondents who see the political universe as backward (namely, Reagan to the left of Carter), clearly have a very low level of information about politics.
The big advantage of applications like the one shown below is that a researcher can check to see if the scale really is one-dimensional. That is, scales with labeled endpoints are designed to be one-dimensional. Performing the decomposition shown below can roughly test this. Namely, the r-square in one dimension should be very large and the increment to adding a second dimension should be quite small.
Note that because the number of political stimuli being placed on an issue scale is usually not too large, this application should be done with some caution. For the 1980 scale shown below, there are only 6 stimuli. Consequently, if respondents are included that placed only 4 of the 6 stimuli, then if 3 dimensions are estimated the r-square for these respondents will be 1.0! Because of this I only estimated 2 dimensions with two missing responses (see Appendix C for the output files).
The one-dimensional fit of the model was an r-square of .7541 and a standard error of the estimate of .9843. In two dimensions the r-square was .8645 and the standard error of the estimate was .8447. These fits plus the estimated configuration shown in Table 6* show, in my opinion, that the scale is indeed one-dimensional and that the second dimension is largely capturing respondent confusion about where to place Anderson on the scale.
Note
that in Table 6* I show the coordinates as 
. In the computer
code I simply write out the singular vectors to make it easier to compare with
the Aldrich-McKelvey coordinates (which is an eigenvector).
Analysis of 1980 Post-Election Liberal-Conservative Scale
The purpose of this application is to show the connection between the procedure developed here and the scaling method developed by Aldrich and McKelvey (1977) to analyze seven-point scales. The Aldrich-McKelvey scaling method is a one dimensional version of the model expressed in equation (1) – that is, the original model as expressed in equation (1B) applied to a transposed matrix where m > n. In this application I will analyze the responses to the 1980 Post-Election Liberal-Conservative seven-point scale. This is one of the fourteen scales analyzed in the previous subsection.
In the Aldrich-McKelvey framework, the matrix X is an n by m matrix where the rows are the respondents' perceived positions of the m stimuli on the scale.
The model they estimate is
(17)
where y
is an m length vector of underlying stimulus coordinates, W is a n by n diagonal matrix of weights, Jm is an m length vector of ones, c is an n length vector of constants, and E is an m by n matrix of error
terms. Aldrich and McKelvey assume
that the respondents correctly perceive the true underlying configuration
subject to some random perceptual error, E,
and report a linear transformation of that true configuration. Their scaling method estimates y using X, and W and c are
estimated using 
and X with ordinary least squares.
The Aldrich-McKelvey scaling method is, in effect, an s = 1 version of equation (1). Solving for X in (17) produces
(18)
where W*
= W-1Jn , c* = - W-1c , and E*
= EW
. Equation (18)
is identical to equation (1) the only difference being the reversal of the
roles of n and m.
Aldrich and McKelvey require that y¢y = 1 and missing entries are not allowed in X. The procedure outlined in Section 2 based on the model stated in equation (1) can be regarded as a generalization of the Aldrich-McKelvey scaling procedure to more than one dimension.
In
this application the rows of the data set are the political stimuli and the
columns are the respondents’ perceptions of where on the seven-point scale the
stimuli are. Consequently, there are n
political stimuli (note the reversal of role of n from the equations above)
and the basic space coordinates of the political stimuli are given in 
. Recall that, by
equation (1B), if X has no missing
entries and no error, then it has rank s.
However, because m > n, subtracting off the column means reduces the
rank of X by one provided that the
columns do not already sum to zero. In
this case the number of basic dimension is s-1 so that 
is an n by s-1 matrix, 
is m by s-1 and 
is an m length
vector where 
and 
are the linear
mappings for the m respondents.
Table
6* shows 
for two basic
dimensions along with the corresponding one dimensional vector estimated by
the Aldrich-McKelvey procedure. The
first basic dimension is the liberal-conservative dimension and the order of
the political stimuli – from Ted Kennedy at the far left to John Anderson near
the center of the spectrum to Ronald Reagan at the far right – is intuitively
appealing. The second basic dimension
essentially separates John Anderson from everyone else. The standard errors were computed using a
bootstrap procedure identical to that described earlier except now the columns
(respondents) are being sampled with replacement. The standard errors are based on 100 trials.
__________________
Table 6* About Here
__________________
These standard errors must be taken with a grain of salt, however, because, as Aldrich and McKelvey (1977) note, a respondent “…who sees things backwards … contributes to a better fit to the ‘true’ space” (p. 116). That is, respondents who perceive a mirror image of the true configuration improve the fit of the model so that the standard errors in Table 6* underestimate the true standard errors. However, Monte Carlo work done by Aldrich and McKelvey and Palfrey and Poole (1987), show that the recovery of the stimulus configuration is robust to violations of the error assumptions and is very accurate even when the error level is very high and a large number of respondents are reporting mirror or semi-mirror images.
The fourth column of Table 6* shows the first basic dimension normalized so that it can be directly compared to the Aldrich-McKelvey configuration shown in the fifth column. The two configurations are, not surprisingly, virtually identical. The differences are due to the slightly different samples analyzed by the two procedures. Of the 888 respondents used to estimate the two basic dimensions, 643 had no missing data and were used in the Aldrich-McKelvey procedure.
5. Additional Empirical Examples
a.
Recovering a Basic
Space From the 1992 Issue Scales
Table 2 shows an analysis of fifteen issue scales from the 1992 NES survey. The survey included a panel as well as a cross section with some respondents included in both groups. The table is laid out the same as Table 4 in the AJPS article.
__________________
Table 2 About Here
__________________
The results are similar to those
for the 1980 scales. The first
dimension is clearly liberal/conservative and the second dimension appears to
be picking up the abortion and women’s equal role questions. However, in contrast to the 1980 results,
the first dimension only accounts for about forty percent of the variance
(r-square of .394 versus .512 for 1980) and the second dimension is not as
strongly related to abortion and women’s rights as it was in 1980. If conventional statistical tests were
applied using the bootstrapped standard errors, then all the 
’s for the first dimension, fourteen of the fifteen for the
second, and eight of the fifteen for the third are statistically significant.
The data are clearly at most three-dimensional. Adding a fourth dimension improves the
overall r-square to .674 but only one
of the estimated 
’s is statistically significant.
Because the set of scales is not the same between 1980 and 1992 it is not possible to make claims about changes in the structure and the fit of the basic space over time. That requires a panel and would be an interesting topic for future research. However, it certainly appears that the basic space is low dimensional. It appears that only two basic dimensions – one capturing general liberalism/conservatism and one picking up issues related to the rights of women – give a good summary of mass attitudes.
References
Aldrich, John and Richard D. McKelvey. 1977. “A Method of Scaling with Applications
to the 1968 and 1972 Presidential Elections.” American
Political Science Review,
71:111-130
Eckart, Carl and Gale Young. 1936. “The Approximation of One Matrix By Another of
Lower Rank.” Psychometrika, 1: 211-218.
Harman, Harry H. 1970. Modern Factor Analysis. Chicago: University of Chicago Press.
Palfrey, Thomas R. and Keith T. Poole. 1987. “The Relationship Between Information,
Ideology, and Voting Behavior.” American Journal of Political Science,
31:511-530.
Schonemann, Peter H. 1966. “A Generalized Solution of the Orthogonal Procrustes
Problem.” Psychometrika, 31:1-10.
Schonemann, Peter H. and R.M. Carroll. 1970. “Fitting One Matrix to Another Under
Choice of a Central Dilation and Rigid Motion.” Psychometrika, 35:245-256
Van de Geer, John P. 1971. Introduction to Multivariate Analysis for the Social Sciences. San Francisco: W.H. Freeman and Company.
Table 4*
Monte Carlo Tests of Eckart-Young Lower
Rank Approximation
(Singular Values of YW’: 50, 35, 20, 5, 5, 5, …., 5)
|
N |
M |
Rank of Approximation |
R2 With E-Young |
R2 With True Y |
Percent Missing |
|
100 |
25 |
1 |
.932 (.015) |
.949 (.016) |
50 |
|
100 |
25 |
2 |
.954 (.007) |
.959 (.012) |
50 |
|
100 |
25 |
3 |
.961 (.004) |
.947 (.017) |
50 |
|
500 |
25 |
1 |
.962 (.005) |
.966 (.004) |
50 |
|
500 |
25 |
2 |
.966 (.003) |
.965 (.003) |
50 |
|
500 |
25 |
3 |
.970 (.001) |
.953 (.004) |
50 |
|
1000 |
25 |
1 |
.822 (.032) |
.828 (.031) |
70 |
|
1000 |
25 |
2 |
.845 (.019) |
.843 (.020) |
70 |
|
1000 |
25 |
3 |
.915 (.006) |
.873 (.022) |
70 |
|
100 |
50 |
3 |
.989 (.001) |
.986 (.004) |
25 |
|
250 |
50 |
2 |
.992 (.000) |
.993 (.001) |
25 |
|
500 |
15 |
2 |
.987 (.001) |
.986 (.003) |
25 |
Table 6*
1980
Liberal-Conservative Scale
|
Political Stimulus |
First Basic Dimension |
Second Basic Dimension |
Normalized First Basic Dimension |
Aldrich- McKelvey |
|
Jimmy Carter |
-2.234 (0.088) |
-3.203 (0.301) |
-0.229 |
-0.232 |
|
Ronald Reagan |
5.685 (0.059) |
-0.773 (0.298) |
0.582 |
0.582 |
|
Ted Kennedy |
-4.703 (0.091) |
0.007 (0.539) |
-0.482 |
-0.485 |
|
John Anderson |
-0.757 (0.099) |
6.756 (0.833) |
-0.078 |
-0.066 |
|
Republican Party |
5.089 (0.077) |
-0.759 (0.137) |
0.521 |
0.517 |
|
Democratic Party |
-3.080 (0.095) |
-2.029 (0.277) |
-0.315 |
-0.317 |
Table
2A
Overall Fit Statistics for Fifteen 1992 Issue Scales
|
S |
N |
% Missing |
R2 |
Standard
Error of
Estimate |
Singular
Values of
|
|
1 |
1264 |
17.6 |
.394 |
1.392 |
113.526 |
|
2 |
1264 |
17.6 |
.519 |
1.300 |
86.803 |
|
3 |
1264 |
17.6 |
.604 |
1.242 |
76.235 |
Table
2B
Fit
Statistics by Issue
(Bootstrapped
Standard Errors in Parentheses)
|
Issue |
|
|
|
|
|
r-square 1 2 3 |
|
Liberal/Conservative |
924 |
4.19 (0.04) |
-2.56 (0.16) |
0.82 (0.42) |
-0.79 (0.89) |
.301 |
.327 |
.372 |
|
Women’s Equal Role |
606 |
2.51 (0.07) |
-3.25 (0.24) |
4.32 (0.39) |
-3.29 (1.09) |
.188 |
.553 |
.749 |
|
Defense Spending |
1158 |
3.59 (0.05) |
-2.48 (0.16) |
1.54 (0.32) |
3.09 (0.75) |
.202 |
.385 |
.529 |
|
Government Jobs |
1143 |
4.17 (0.06) |
-3.48 (0.17) |
-4.09 (0.36) |
-1.23 (0.86) |
.419 |
.650 |
.661 |
|
Govt Help Minorities |
1191 |
4.51 (0.06) |
-3.44 (0.19) |
-2.38 (0.45) |
0.78 (1.15) |
.420 |
.457 |
.489 |
|
Govt Provide Services |
1122 |
4.36 (0.05) |
1.98 (0.16) |
2.18 (0.43) |
3.22 (1.08) |
.179 |
.319 |
.478 |
|
Abortion |
1239 |
2.87 (0.03) |
0.97 (0.11) |
-1.94 (0.21) |
0.66 (0.39) |
.027 |
.215 |
.306 |
|
Liberal/Conservative |
627 |
4.44 (0.07) |
-2.83 (0.23) |
1.45 (0.60) |
-0.48 (1.27) |
.328 |
.382 |
.412 |
|
Defense Spending |
846 |
4.01 (0.06) |
-2.87 (0.16) |
1.37 (0.35) |
3.37 (0.69) |
.269 |
.442 |
.584 |
|
Govt Provide Services |
1063 |
4.17 (0.05) |
2.35 (0.19) |
2.06 (0.42) |
2.46 (1.02) |
.243 |
.374 |
.447 |
|
Defense Spending |
1135 |
3.53 (0.04) |
-2.29 (0.13) |
1.28 (0.21) |
2.90 (0.53) |
.204 |
.371 |
.529 |
|
Government Jobs |
1141 |
4.29 (0.05) |
-3.12 (0.17) |
-3.02 (0.31) |
-0.13 (0.82) |
.380 |
.506 |
.512 |
|
Abortion |
1233 |
2.95 (0.03) |
1.22 (0.11) |
-1.91 (0.20) |
0.76 (0.36) |
.061 |
.247 |
.348 |
|
Urban Unrest |
972 |
3.35 (0.07) |
-3.88 (0.24) |
-0.91 (0.81) |
3.22 (2.30) |
.383 |
.398 |
.526 |
|
Women’s Equal Role |
1225 |
2.32 (0.05) |
-2.89 (0.17) |
3.29 (0.42) |
-2.30 (0.88) |
.160 |
.399 |
.631 |
Appendix
A: BLACKBOX.FOR
1.
Introduction
BLACKBOX.FOR is a FORTRAN program that implements the model discussed in “Recovering a Basic Space From a Set of Issue Scales.” The program reads a "control card" file (BTSTR.DAT) and the data file (usually an NES dataset) and writes three output files: BLACK23.DAT, BLACK24.DAT, and BLACK28.DAT. BLACK23.DAT is a file that contains various information about the estimation, BLACK24.DAT is the output file for the respondent parameters (the p by s Y matrix for k=1,2,…,s), and BLACK28.DAT is the output file for the issue scale parameters (the n by s W matrix and the vector of constants, c).
The program has been compiled for both the Pentium P5 processor as well as the Pentium P6 (Pentium II) processor. These executables are BLACK5.EXE and BLACK6.EXE respectively. They will run under both Windows 95 and Windows NT.
The example below is from the AJPS article and it uses the 14 issue scales from 1980 NES cross-sectional survey data set -- NES1980.DAT.
2.
Input File: BTSTR.DAT
The first line of the input file (see next page) gives the name of the data set being analyzed. In this case, the 1980 NES data is in the subdirectory \NES. That is, the program, BLACK6.EXE is in the root directory. Note that if NES1980.DAT could be placed in the same directory as the program the first line would simply be NES1980.DAT.
The second line of the input file is the title of the scaling
The third line contains, in order, the number of basic dimensions to be estimated, the number of issue scales, the maximum number of missing data values for the issue scales, and the minimum number of responses for a respondent to be included in the analysis. Note that this is fixed format, namely, in FORTRAN syntax, 4I5. Hence, if you change any of the numbers be sure to not change the spacing. For example, if you want 2 basic dimensions and there are 10 stimuli, 11 missing data values, and a minimum number of 5 responses, this line would be:
2 10 11 5
The fourth line is the format statement for the 1968 NES data file. The I4 is the respondent ID number and the I1's (there are 14 of them) are the respondents' self-reported positions on the 14 issue scales. The "X"s indicate spaces in FORTRAN format syntax. This format statement can be figured out by using the ICPSR codebook for the election study. If you have problems figuring out how to do this, just send me E-Mail at KPoole@uh.edu.
The remaining 28 lines contain the title of each issue scale and the missing value codes for the scale. For the 7-Point scales these are 0, 8, and 9 but for the two abortion scales the missing values are 0, 7, 8, 9. Note that these numbers are also fixed format.
BLACKB\NES\NES1980.DAT
DECOMPOSITION OF 14 1980 7-POINT SCALES
3
14 4 8
(8X,I4,527X,I1,13X,I1,11X,I1,11X,I1,11X,I1,13X,I1,1217X,I1,36X,I1,17X,I1,17X,I1,17X,I1,18X,I1,5X,I1,2X,I1)
LIBERAL/CONSERVATIVE
0
8 9
DEFENSE
0
8 9
GOVT SERVICES
0
8 9
INFLATION
0
8 9
ABORTION
0
7 8 9
TAX CUTS
0
8 9
LIBERAL/CONSERVATIVE
0
8 9
GOVT HELP
MINORITIES
0
8 9
RUSSIA
0
8 9
WOMENS EQUAL
ROLE
0
8 9
GOVT JOBS
0
8 9
EQUAL RIGHTS
AMEND
0
8 9
BUSING
0
8 9
ABORTION
0
7 8 9
3.
Output files for 1980 Issue Scale Example Shown in AJPS Article
a.
BLACK23.DAT File
This file is the primary output file for the program. For ease of exposition I will annotate this file for the convenience of the reader. My comments will be preceded by #### signs.
The first part of the output file just echoes the lines in BTSTR.DAT. This is convenient because you can glance at this to make sure the starting file is configured and being read correctly.
\BLACKB\NES\NES1980.DAT
DECOMPOSITION OF 14 1980 7-POINT SCALEs
3 14 4 8
(8X,I4,527X,I1,13X,I1,11X,I1,11X,I1,11X,I1,13X,I1,1217X,I1,36X,I1,17X,I1,17X,I1,17X,I1,18X,I1,5X,I1,2X,I1)
LIBERAL/CO
0 8 9 0
DEFENSE
0 8 9 0
GOVT SERVI
0 8 9 0
INFLATION
0 8 9 0
ABORTION
0 7 8 9
TAX CUTS
0 8 9 0
LIBERAL/CO
0 8 9 0
GOVT HELP
0 8 9 0
RUSSIA
0 8 9 0
WOMENS EQU
0 8 9 0
GOVT JOBS
0 8 9 0
EQUAL RIGH
0 8 9 0
BUSING
0 8 9 0
ABORTION
0 7 8 9
####
####
Here 1270 respondents have been included in the analysis. The
####
program requires that a respondent place himself/herself on at
####
least 8 (see input file) scales.
####
NUMBER OF CASES 1270
******************************************************************************
******************************************************************************
******************************************************************************
####
The program first estimates a one dimensional model, then two
####
dimensions, etc.
NUMBER OF DIMENSIONS= 1
***********************************************************************
####
This information is only given once.
####
The matix is 1270 by 14 and contains 17,780 entries. There
####
are 2509 missing entries or [2509/(14*1270)]*100 = 14.11% missing
####
entries. The Sum of Squares is
computed around the grand mean of
####
the matrix. Hence, it is the sum of
the squared differences between
####
the 15,271 non-missing entries and the matrix mean.
NUMBER OF ROWS
= 1270
NUMBER OF COLUMNS
= 14
TOTAL NUMBER OF DATA ENTRIES = 15271
NUMBER MISSING ENTRIES =
2509
PERCENT MISSING DATA
= 14.11136
SUM OF SQUARES GRAND MEAN
= 52705.13281
******************************************************************************
####
This is the iteration record for the first basic dimension.
####
REG1 estimates W and c and REG2 estimates y.
DIMENSION= 1 TOTAL SSE
REG1= 26094.3320
DIMENSION= 1 TOTAL SSE
REG2= 25818.1855
DIMENSION= 1 TOTAL SSE
REG1= 25742.5508
DIMENSION= 1 TOTAL SSE
REG2= 25718.7637
DIMENSION= 1 TOTAL SSE
REG1= 25710.2461
DIMENSION= 1 TOTAL SSE
REG2= 25706.7617
DIMENSION= 1 TOTAL SSE
REG1= 25705.1016
DIMENSION= 1 TOTAL SSE
REG2= 25704.3594
DIMENSION= 1 TOTAL SSE
REG1= 25704.0059
DIMENSION= 1 TOTAL SSE
REG2= 25703.9063
DIMENSION= 1 TOTAL SSE
REG1= 25703.7559
DIMENSION= 1 TOTAL SSE
REG2= 25703.6621
DIMENSION= 1 TOTAL SSE
REG1= 25703.6855
DIMENSION= 1 TOTAL SSE
REG2= 25703.5859
DIMENSION= 1 TOTAL SSE
REG1= 25703.6719
DIMENSION= 1 TOTAL SSE
REG2= 25703.5996
DIMENSION= 1 TOTAL SSE
REG1= 25703.6699
DIMENSION= 1 TOTAL SSE
REG2= 25703.5859
####
The singular values for the one dimensional estimation are reported
####
below. Only the first s+3 singular values are shown.
SINGULAR VALUES OF ESTIMATED MATRICES
FIRST COLUMN: ORIGINAL
MATRIX WITH FILLED IN MISSING ENTRIES
SECOND COLUMN: REPRODUCED MATRIX -- PSI*W + Jc
THIRD COLUMN: ORIGINAL
MATRIX WITH FILLED IN MISSING ENTRIES MINUS THE ORIGINAL COLUMN MEANS
FOURTH COLUMN: PSI*W
1 545.737 544.149
111.761 111.757
2 97.628 93.424
68.908 0.000
3 68.464 0.000
57.681 0.000
4 57.486 0.000
54.781 0.000
******************************************************************************
####
Below are the constraint checks discussed in the AJPS article --
####
namely, the sum of the columns of y
must equal zero, and:
####
y'y = W'W = L where
L
is the s by s diagonal matrix of the
####
singular values of yW' which
is the least squares estimate of
####
[X0.- Jnc'].
CONSTRAINT CHECKS ON PSI AND W
SUM OF COLUMNS OF PSI
0.0000
PSI-TRANSPOSE*PSI
1 111.7571
W-TRANSPOSE*W
1 111.7571
####
Here yW' + Jnc'
is constructed and the r-square between the elements
####
of yW'
+ Jnc' and the original data
matrix, X0 ,is computed
as a
####
check on the estimation.
####
R-SQUARE 15271 0.512
####
Similar to the above, as a check, a singular value decomposition
####
of the estimated matrix, yW', is performed. The rank of the
####
matrix is reported (here it is one) along with the singular values.
R-SQUARE 15271 0.512
RANK CHECK OF PSI*W 1
1 111.7571
2 0.0000
3 0.0000
4 0.0000
5 0.0000
6 0.0000
####
####
The program now estimates all the above for two dimensions --
####
s = 2.
####
******************************************************************************
NUMBER OF DIMENSIONS= 2
******************************************************************************
DIMENSION= 1 TOTAL SSE
REG1= 26094.3320
DIMENSION= 1 TOTAL SSE
REG2= 25818.1855
DIMENSION= 1 TOTAL SSE
REG1= 25742.5508
DIMENSION= 1 TOTAL SSE
REG2= 25718.7637
DIMENSION= 1 TOTAL SSE
REG1= 25710.2461
DIMENSION= 1 TOTAL SSE
REG2= 25706.7617
DIMENSION= 1 TOTAL SSE
REG1= 25705.1016
DIMENSION= 1 TOTAL SSE
REG2= 25704.3594
DIMENSION= 2 TOTAL SSE
REG1= 21380.9219
DIMENSION= 2 TOTAL SSE
REG2= 21119.8066
DIMENSION= 2 TOTAL SSE
REG1= 20949.8340
DIMENSION= 2 TOTAL SSE
REG2= 20838.7109
DIMENSION= 2 TOTAL SSE
REG1= 20766.0664
DIMENSION= 2 TOTAL SSE
REG2= 20717.7871
DIMENSION= 2 TOTAL SSE
REG1= 20685.3633
DIMENSION= 2 TOTAL SSE
REG2= 20663.1270
DIMENSION= 2 TOTAL SSE
REG1= 20646.6641
DIMENSION= 2 TOTAL SSE
REG2= 20547.0156
DIMENSION= 2 TOTAL SSE
REG1= 20538.1191
DIMENSION= 2 TOTAL SSE
REG2= 20532.5488
DIMENSION= 2 TOTAL SSE
REG1= 20528.8223
DIMENSION= 2 TOTAL SSE
REG2= 20526.2266
DIMENSION= 2 TOTAL SSE
REG1= 20524.4395
DIMENSION= 2 TOTAL SSE
REG2= 20523.1094
DIMENSION= 2 TOTAL SSE
REG1= 20522.2148
DIMENSION= 2 TOTAL SSE
REG2= 20521.5879
####
The singular values for the two dimensional estimation are reported
####
below. Only the first s+3 singular values are shown. Hence, there
####
are now 5 rows being printed out.
SINGULAR VALUES OF ESTIMATED MATRICES
FIRST COLUMN: ORIGINAL
MATRIX WITH FILLED IN MISSING ENTRIES
SECOND COLUMN: REPRODUCED MATRIX -- PSI*W + Jc
THIRD COLUMN: ORIGINAL
MATRIX WITH FILLED IN MISSING ENTRIES MINUS THE ORIGINAL COLUMN MEANS
FOURTH COLUMN: PSI*W
1 546.795 545.439
110.997 110.981
2 97.860 95.370
77.571 77.559
3 74.628 71.390
57.259 0.000
4 57.217 0.000 54.304 0.000
5 47.271 0.000
47.049 0.000
******************************************************************************
####
####
Below are the constraint checks discussed in the AJPS article --
####
namely, the sum of the columns of y
must equal zero, and:
####
y'y = W'W = L where
L
is the s by s diagonal matrix of the
####
singular values of yW' which
is the least squares estimate of
####
[X0.- Jnc']. Now two by two matrices are being
printed out.
####
CONSTRAINT CHECKS ON PSI AND W
SUM OF COLUMNS OF PSI
0.0000 0.0000
PSI-TRANSPOSE*PSI
1 110.9815 0.0000
2 0.0000 77.5591
W-TRANSPOSE*W
1 110.9814 0.0000
2 0.0000 77.5591
R-SQUARE 15271 0.611
RANK CHECK OF PSI*W 2
1 110.9815
2 77.5591
3 0.0000
4 0.0000
5 0.0000
6 0.0000
####
####
Three dimensions -- the maximum set in the input file -- are
####
now estimated.
####
******************************************************************************
NUMBER OF DIMENSIONS= 3
******************************************************************************
DIMENSION= 1 TOTAL SSE
REG1= 26094.3320
DIMENSION= 1 TOTAL SSE
REG2= 25818.1855
DIMENSION= 1 TOTAL SSE
REG1= 25742.5508
DIMENSION= 1 TOTAL SSE
REG2= 25718.7637
DIMENSION= 1 TOTAL SSE
REG1= 25710.2461
DIMENSION= 1 TOTAL SSE
REG2= 25706.7617
DIMENSION= 1 TOTAL SSE
REG1= 25705.1016
DIMENSION= 1 TOTAL SSE
REG2= 25704.3594
DIMENSION= 2 TOTAL SSE
REG1= 21380.9219
DIMENSION= 2 TOTAL SSE
REG2= 21119.8066
DIMENSION= 2 TOTAL SSE
REG1= 20949.8340
DIMENSION= 2 TOTAL SSE
REG2= 20838.7109
DIMENSION= 2 TOTAL SSE
REG1= 20766.0664
DIMENSION= 2 TOTAL SSE
REG2= 20717.7871
DIMENSION= 2 TOTAL SSE
REG1= 20685.3633
DIMENSION= 2 TOTAL SSE
REG2= 20663.1270
DIMENSION= 3 TOTAL SSE
REG1= 17706.8184
DIMENSION= 3 TOTAL SSE
REG2= 17619.4512
DIMENSION= 3 TOTAL SSE
REG1= 17548.3496
DIMENSION= 3 TOTAL SSE
REG2= 17489.4707
DIMENSION= 3 TOTAL SSE
REG1= 17440.7461
DIMENSION= 3 TOTAL SSE
REG2= 17400.1348
DIMENSION= 3 TOTAL SSE
REG1= 17365.9629
DIMENSION= 3 TOTAL SSE
REG2= 17336.9277
DIMENSION= 3 TOTAL SSE
REG1= 17296.1191
DIMENSION= 3 TOTAL SSE
REG2= 17058.3926
DIMENSION= 3 TOTAL SSE
REG1= 17023.2207
DIMENSION= 3 TOTAL SSE
REG2= 16995.1074
DIMENSION= 3 TOTAL SSE
REG1= 16971.8438
DIMENSION= 3 TOTAL SSE
REG2= 16952.2773
DIMENSION= 3 TOTAL SSE
REG1= 16935.7246
DIMENSION= 3 TOTAL SSE
REG2= 16921.6406
DIMENSION= 3 TOTAL SSE
REG1= 16909.6445
DIMENSION= 3 TOTAL SSE
REG2= 16899.3594
SINGULAR VALUES OF ESTIMATED MATRICES
FIRST COLUMN: ORIGINAL
MATRIX WITH FILLED IN MISSING ENTRIES
SECOND COLUMN: REPRODUCED MATRIX -- PSI*W + Jc
THIRD COLUMN: ORIGINAL
MATRIX WITH FILLED IN MISSING ENTRIES MINUS THE ORIGINAL COLUMN MEANS
FOURTH COLUMN: PSI*W
1 546.947 546.331
111.614 111.561
2 99.310 98.669
78.557 78.521
3 75.528 75.265
70.796 70.679
4 63.612 52.554
53.402 0.000
5 46.900 0.000
46.409 0.000
6 46.021 0.000
45.545 0.000
******************************************************************************
####
####
Below are the constraint checks discussed in the AJPS article --
####
namely, the sum of the columns of y
must equal zero, and:
####
y'y = W'W = L where
L
is the s by s diagonal matrix of the
####
singular values of yW' which
is the least squares estimate of
####
[X0.- Jnc']. Note that the matrices are now 3 by 3.
####
CONSTRAINT CHECKS ON PSI AND W
SUM OF COLUMNS OF PSI
0.0000
0.0000 0.0000
PSI-TRANSPOSE*PSI
1 111.5610 0.0000
0.0000
2 0.0000 78.5212
0.0000
3 0.0000 0.0000
70.6788
W-TRANSPOSE*W
1 111.5609 0.0000
0.0000
2 0.0000 78.5211
0.0000
3 0.0000 0.0000
70.6788
R-SQUARE 15271 0.679
RANK CHECK OF PSI*W 3
1 111.5610
2 78.5211
3 70.6788
4 0.0000
5 0.0000
6 0.0000
****************************************************************************
####
####
Below is the estimation summary. Note
that the error and the
####
explained always add to the total sum of squares. The standard
####
error of the estimate is also computed for each dimension.
####
The singular values above and the entries below are reported
####
in Table 4A of the AJPS
article. The slight differences
####
between these numbers and those in the article are due to some
####
marginal improvements made in the program code.
####
ITERATION RECORD
DIM ERROR
EXPLAINED PERCENT CUM PERCENT R-SQUARE STD ERR EST
1 25703.6367 27001.4961 51.2312
51.2312 0.5123 1.3549
2 20521.4863 32183.6465 9.8323 61.0636 0.6106
1.2703
3 16899.3535 35805.7813 6.8724 67.9360 0.6794
1.2158
****************************************************************************
****************************************************************************
b. BLACK24.DAT
Below is the output of the respondent parameters -- Y -- for one, two, and three dimensions. The first integer is a simple counter and runs, in this instance, from 1 to 1270 -- the number of respondents in the analysis. The second number is the actual case number in the NES1980.DAT file.
1
1 -0.281
2
2 0.595
3
3 0.517
4
5 -0.111
5
7 0.398
6
8 0.368
7
9 -0.364
8
10 -0.095
9
11 0.090
10
12 0.148
etc.
etc.
1261
1756 -0.264
1262
1757 0.024
1263
1758 -0.158
1264
1759 -0.419
1265
1760 -0.324
1266 1761 -0.011
1267
1762 -0.332
1268
1763 0.296
1269
1764 -0.123
1270
1765 -0.033
1
1 -0.282 -0.390
2
2 0.607 -0.255
3
3 0.675 -0.375
4
5 -0.099 0.170
5
7 0.390 0.023
6 8 0.383 -0.120
7
9 -0.378 0.267
8
10 -0.108 0.253
9
11 0.072 -0.703
10
12 0.145 -0.004
etc.
etc.
1261
1756 -0.267 0.050
1262
1757 0.023 -0.013
1263
1758 -0.162 0.327
1264
1759 -0.456 0.385
1265
1760 -0.338 0.419
1266
1761 -0.018 0.111
1267
1762 -0.297 -0.142
1268
1763 0.296 -0.264
1269
1764 -0.129 0.320
1270
1765 -0.054 0.189
1
1 -0.281 0.392 0.024
2
2 0.600 0.213
0.197
3
3 0.602 -0.053 0.787
4
5 -0.126 -0.200 0.208
5
7 0.416 -0.002 -0.275
6
8 0.368 0.128 -0.097
7
9 -0.380 -0.259 -0.060
8
10 -0.116 -0.215 -0.204
9
11 0.075 0.701
0.350
10
12 0.170 -0.031 0.113
etc.
etc.
1261
1756 -0.275 -0.093 0.206
1262
1757 -0.005 -0.028 0.170
1263
1758 -0.166 -0.345 0.055
1264
1759 -0.449 -0.334 -0.234
1265
1760 -0.337 -0.363 -0.291
1266
1761 -0.017 -0.083 -0.151
1267
1762 -0.160 0.353 -0.302
1268
1763 0.285 0.215
0.260
1269
1764 -0.135 -0.328 -0.005
1270
1765 -0.046 -0.161 -0.292
c.
BLACK28.DAT
Below is the output for the issue scale parameters -- the W matrix and the vector c -- for one, two, and three dimensions. After the name of the issue, the first number is the number of respondents placing themselves on the issue, the next number is the constant, c, and the last number is r-square. The numbers between c and the r-square are the elements of W. Note that this is the source for the parameter estimates shown in Table 4B in the AJPS article. The slight differences between these numbers and those in the article are due to some marginal improvements made in the program code. The slight differences in the 2nd and 3rd dimensions are due to a slight rotation. The r-squares are unchanged except for a few rounding differences.
LIBERAL/CO 875 4.280 -3.028 0.414
DEFENSE
1163 5.210 -1.753 0.123
GOVT SERVI
1119 4.323 4.306
0.451
INFLATION 816 4.106 2.018
0.159
ABORTION
1238 2.856 0.624
0.030
TAX CUTS 836 2.839
-1.074 0.055
LIBERAL/CO 949 4.369 -2.755 0.414
GOVT HELP
1160 4.542 -3.400 0.412
RUSSIA 1152 3.891
-3.034 0.231
WOMENS EQU
1223 2.845 -2.859 0.204
GOVT JOBS
1131 4.377 -4.491 0.518
EQUAL RIGH
1144 2.663 -3.295 0.381
BUSING
1219 6.051 -2.699 0.256
ABORTION
1246 2.675 0.722
0.047
LIBERAL/CO 875 4.300 -2.966
-0.954 0.424
DEFENSE
1163 5.214 -1.779 -0.899 0.147
GOVT SERVI
1119 4.368 4.331 -3.042 0.617
INFLATION 816 4.152 2.088 -2.940 0.393
ABORTION
1238 2.856 0.512
2.211 0.290
TAX CUTS 836 2.818
-1.103 0.667 0.071
LIBERAL/CO 949 4.377 -2.758
-0.459 0.423
GOVT HELP
1160 4.535 -3.456 0.119
0.424
RUSSIA
1152 3.887 -3.140 -0.241 0.247
WOMENS EQU
1223 2.872 -2.466 -6.007 0.771
GOVT JOBS
1131 4.350 -4.595 2.417
0.635
EQUAL RIGH
1144 2.673 -3.148 -2.438 0.491
BUSING
1219 6.049 -2.741 -0.059 0.263
ABORTION
1246 2.676 0.629
2.112 0.318
LIBERAL/CO 875 4.294 -2.976 0.708
1.180 0.448
DEFENSE
1163 5.200 -1.806 1.586 -2.562 0.315
GOVT SERVI
1119 4.410 4.295
3.707 -2.929 0.778
INFLATION 816 4.169 1.998
3.286 -1.111 0.451
ABORTION
1238 2.856 0.497 -2.004 -1.174 0.312
TAX CUTS 836 2.813 -1.049
-0.902 0.891 0.091
LIBERAL/CO 949 4.367 -2.785 0.265
0.557 0.437
GOVT HELP
1160 4.534 -3.457 0.140
-0.961 0.440
RUSSIA
1152 3.831 -3.255 1.558 -5.590 0.695
WOMENS EQU
1223 2.891 -2.372 5.602
2.868 0.805
GOVT JOBS
1131 4.341 -4.632 -2.176
-1.392 0.648
EQUAL RIGH
1144 2.680 -3.159 1.860
2.372 0.563
BUSING
1219 6.042 -2.819 0.329 -1.282 0.306
ABORTION
1246 2.675 0.587 -1.980 -0.906 0.329
Appendix B: BMC2.FOR (Bootstrap Program for BLACKBOX.FOR)
1.
Introduction
BMC2.FOR is a FORTRAN program that is identical to BLACKBOX.FOR only it performs a bootstrap analysis to estimate the standard errors of the elements of W and c. The program samples respondents with replacement and re-runs the estimation for each constructed matrix. For example, below 100 bootstrap process is repeated 100 times and the standard errors for W and c are obtained by computing the sum of squared differences between the actual W and c and the 100 W/c's from the bootstrap analysis. Because W is only defined up to an arbitrary rotation, this was removed before the standard error computation. That is, each W from the bootstrap was rotated to best fit the actual W before the squared difference was computed.
2.
Input File for
BMC2.FOR: BMCSTR.DAT
This file is identical to the start file for BLACKBOX.FOR -- BTSTR.DAT -- shown above in Appendix A. The only difference is in the third line where the 5th number is the number of bootstrap iterations.
\BLACKB\NES\NES1980.DAT
DECOMPOSITION OF 14 1980 7-POINT SCALES
3
14 4 8
100
(8X,I4,527X,I1,13X,I1,11X,I1,11X,I1,11X,I1,13X,I1,1217X,I1,36X,I1,17X,I1,17X,I1,17X,I1,18X,I1,5X,I1,2X,I1)
LIBERAL/CONSERVATIVE
0
8 9
DEFENSE
0
8 9
GOVT
SERVICES
0
8 9
INFLATION
0
8 9
ABORTION
0
7 8 9
TAX CUTS
0
8 9
LIBERAL/CONSERVATIVE
0
8 9
GOVT HELP
MINORITIES
0
8 9
RUSSIA
0
8 9
WOMENS EQUAL
ROLE
0
8 9
GOVT JOBS
0
8 9
EQUAL RIGHTS
AMEND
0
8 9
BUSING
0
8 9
ABORTION
0
7 8 9
3. Output files for Bootstrap Program
a. BMC223.DAT File
This output file is the same as BLACK23.DAT except that it will contain the output for 300 runs of the program -- 100 one dimensional outputs, 100 two dimensional outputs, and 100 three dimensional outputs. Consequently this file will be rather large and is only useful for debugging purposes if there is some oddity in the data set.
Iseed= 93600
\BLACKB\NES\NES1980.DAT
DECOMPOSITION OF 14 1980 7-POINT SCALEs
3
14 4 8
101
(8X,I4,527X,I1,13X,I1,11X,I1,11X,I1,11X,I1,13X,I1,1217X,I1,36X,I1,17X,I1,17X,I1,17X,I1,18X,I1,5X,I1,2X,I1)
LIBERAL/CO
0
8 9 0
DEFENSE
0
8 9 0
GOVT
SERVI
0
8 9 0
INFLATION
0
8 9 0
ABORTION
0
7 8 9
TAX
CUTS
0
8 9 0
LIBERAL/CO
0
8 9 0
GOVT
HELP
0
8 9 0
RUSSIA
0
8 9 0
WOMENS
EQU
0
8 9 0
GOVT
JOBS
0
8 9 0
EQUAL
RIGH
0
8 9 0
BUSING
0
8 9 0
ABORTION
0
7 8 9
NUMBER OF CASES 1270
******************************************************************************
******************************************************************************
******************************************************************************
NUMBER OF DIMENSIONS= 1
******************************************************************************
NUMBER OF ROWS = 1270
NUMBER OF COLUMNS = 14
TOTAL NUMBER OF DATA ENTRIES = 15271
NUMBER MISSING ENTRIES =
2509
PERCENT MISSING DATA = 14.11136
SUM OF SQUARES GRAND MEAN =
52705.13281
******************************************************************************
Etc., Etc., Etc.
DIMENSION=
1 TOTAL SSE REG1= 25390.1230
DIMENSION=
1 TOTAL SSE REG2= 25098.9512
DIMENSION=
1 TOTAL SSE REG1= 25014.0723
DIMENSION=
1 TOTAL SSE REG2= 24985.0508
DIMENSION=
1 TOTAL SSE REG1= 24973.4492
DIMENSION=
1 TOTAL SSE REG2= 24968.3242
DIMENSION=
1 TOTAL SSE REG1= 24965.9434
DIMENSION=
1 TOTAL SSE REG2= 24964.7285
DIMENSION=
2 TOTAL SSE REG1= 20486.9941
DIMENSION=
2 TOTAL SSE REG2= 20256.3594
DIMENSION=
2 TOTAL SSE REG1= 20113.1563
DIMENSION=
2 TOTAL SSE REG2= 20025.7383
DIMENSION=
2 TOTAL SSE REG1= 19972.8457
DIMENSION=
2 TOTAL SSE REG2= 19940.5918
DIMENSION=
2 TOTAL SSE REG1= 19920.7480
DIMENSION=
2 TOTAL SSE REG2= 19908.3516
DIMENSION=
3 TOTAL SSE REG1= 17202.4023
DIMENSION=
3 TOTAL SSE REG2= 17016.2832
DIMENSION=
3 TOTAL SSE REG1= 16902.0215
DIMENSION=
3 TOTAL SSE REG2= 16823.1484
DIMENSION=
3 TOTAL SSE REG1= 16767.6309
DIMENSION=
3 TOTAL SSE REG2= 16726.9844
DIMENSION=
3 TOTAL SSE REG1= 16697.0645
DIMENSION=
3 TOTAL SSE REG2= 16674.2578
DIMENSION=
3 TOTAL SSE REG1= 16647.0781
DIMENSION=
3 TOTAL SSE REG2= 16432.5684
DIMENSION=
3 TOTAL SSE REG1= 16404.9863
DIMENSION=
3 TOTAL SSE REG2= 16382.3418
DIMENSION=
3 TOTAL SSE REG1= 16363.2178
DIMENSION=
3 TOTAL SSE REG2= 16345.4531
DIMENSION=
3 TOTAL SSE REG1= 16330.0176
DIMENSION=
3 TOTAL SSE REG2= 16315.5293
DIMENSION=
3 TOTAL SSE REG1= 16302.9590
DIMENSION=
3 TOTAL SSE REG2= 16291.3379
SINGULAR VALUES OF ESTIMATED MATRICES
FIRST COLUMN: ORIGINAL MATRIX WITH FILLED IN MISSING ENTRIES
SECOND COLUMN: REPRODUCED MATRIX -- PSI*W +
Jc
THIRD COLUMN: ORIGINAL MATRIX WITH FILLED IN MISSING ENTRIES MINUS THE
ORIGINAL COLUMN MEANS
FOURTH COLUMN: PSI*W
1
546.964 545.703 115.836
115.807
2
97.009 94.903 93.010
92.977
3
91.542 90.323 76.499
76.470
4
74.872 73.101 52.415 0.000
5
47.111 0.000 47.085 0.000
6
44.479 0.000 43.660 0.000
******************************************************************************
CONSTRAINT CHECKS ON PSI AND W
SUM OF COLUMNS OF PSI
0.0000 0.0000 0.0000
PSI-TRANSPOSE*PSI
1
115.8070 0.0000 0.0000
2
0.0000 92.9768 0.0000
3
0.0000 0.0000 76.4702
W-TRANSPOSE*W
1
115.8070 0.0000 0.0000
2
0.0000 92.9768 0.0000
3
0.0000 0.0000 76.4701
R-SQUARE
15234 0.684
RANK CHECK OF PSI*W 3
1
115.8070
2
92.9769
3
76.4702
4
0.0000
5
0.0000
6
0.0000
RANK OF SCHOENMANN ROTATION MATRIX 3
COVARIANCE MATRIX
1
112.4401 0.9420 -0.3295
-0.0632
2
76.7452 -0.0914 -0.4333
0.8966
3
64.1138 0.3228 0.8389
0.4383
******************************************************************************
ITERATION RECORD
DIM
ERROR EXPLAINED PERCENT CUM PERCENT
R-SQUARE STD ERR EST
1
25646.3789 25988.1484 50.3309 50.3309
0.5141 1.3526
2
20455.4668 31179.0605 10.0532 60.3841
0.6048 1.2736
3
16291.4453 35343.0820 8.0644 68.4485
0.6845 1.1956
******************************************************************************
******************************************************************************
b. BMC228.DAT File
This output file is the same as BLACK28.DAT except that it will contain the output for 300 runs of the program -- 100 one dimensional outputs, 100 two dimensional outputs, and 100 three dimensional outputs.
LIBERAL/CO 875 4.280 -3.028 0.414
DEFENSE
1163 5.210 -1.753 0.123
GOVT SERVI
1119 4.323 4.306
0.451
INFLATION 816 4.106 2.018
0.159
ABORTION
1238 2.856 0.624
0.030
TAX CUTS 836 2.839
-1.074 0.055
LIBERAL/CO 949 4.369 -2.755 0.414
GOVT HELP
1160 4.542 -3.400 0.412
RUSSIA
1152 3.891 -3.034 0.231
WOMENS EQU
1223 2.845 -2.859 0.204
GOVT JOBS
1131 4.377 -4.491 0.518
EQUAL RIGH
1144 2.663 -3.295 0.381
BUSING
1219 6.051 -2.699 0.256
ABORTION
1246 2.675 0.722
0.047
Etc.
Etc.
LIBERAL/CO 875 4.300 -2.966
-0.954 0.424
DEFENSE
1163 5.214 -1.779 -0.899 0.147
GOVT SERVI
1119 4.368 4.331 -3.042 0.617
INFLATION 816 4.152 2.088 -2.940 0.393
ABORTION
1238 2.856 0.512
2.211 0.290
TAX CUTS 836 2.818
-1.103 0.667 0.071
LIBERAL/CO 949 4.377 -2.758
-0.459 0.423
GOVT HELP
1160 4.535 -3.456 0.119
0.424
RUSSIA
1152 3.887 -3.140 -0.241 0.247
WOMENS EQU
1223 2.872 -2.466 -6.007 0.771
GOVT JOBS
1131 4.350 -4.595 2.417
0.635
EQUAL RIGH
1144 2.673 -3.148 -2.438 0.491
BUSING
1219 6.049 -2.741 -0.059 0.263
ABORTION
1246 2.676 0.629
2.112 0.318
Etc.
Etc.
LIBERAL/CO 875 4.294 -2.976 0.708
1.180 0.448
DEFENSE
1163 5.200 -1.806 1.586 -2.562 0.315
GOVT SERVI
1119 4.410 4.295
3.707 -2.929 0.778
INFLATION 816 4.169 1.998
3.286 -1.111 0.451
ABORTION
1238 2.856 0.497 -2.004 -1.174 0.312
TAX CUTS 836 2.813 -1.049
-0.902 0.891 0.091
LIBERAL/CO 949 4.367 -2.785 0.265
0.557 0.437
GOVT HELP
1160 4.534 -3.457 0.140 -0.961 0.440
RUSSIA
1152 3.831 -3.255 1.558 -5.590 0.695
WOMENS EQU
1223 2.891 -2.372 5.602
2.868 0.805
GOVT JOBS
1131 4.341 -4.632 -2.176
-1.392 0.648
EQUAL RIGH
1144 2.680 -3.159 1.860
2.372 0.563
BUSING
1219 6.042 -2.819 0.329 -1.282 0.306
ABORTION
1246 2.675 0.587 -1.980 -0.906 0.329
b. BMC229.DAT File
This is the important file. This file reports the original W's and c's for one, two, and three dimensions (note that these will be identical to BLACK28.DAT above), the bootstrap standard errors, and the corresponding t-values for the null hypotheses that the parameters are all equal to zero. The r-squares are not shown. As I noted above, the slight differences between these values and those shown in the AJPS article for the 2nd and 3rd dimensions are due to a slight rotational difference due to some efficiencies introduced into the program code.
For one dimension, the first number is c, the second is w, the third and fourth are the standard errors, and the fifth and sixth are the t-values. The output for two and three dimensions is in the same order.
1
4.280 -3.028 0.050 0.143 85.378 -21.137
2
5.210 -1.753 0.044
0.207 119.542 -8.458
3
4.323 4.306 0.057
0.191 76.194 22.539
4
4.106 2.018 0.055
0.223 74.261 9.061
5
2.856 0.624 0.028
0.134 101.583 4.667
6
2.839 -1.074 0.047
0.189 60.889 -5.674
7
4.369 -2.755 0.034
0.138 129.931 -19.985
8
4.542 -3.400 0.045
0.151 100.279 -22.482
9
3.891 -3.034 0.054
0.218 72.384 -13.888
10
2.845 -2.859 0.050
0.271 57.178 -10.558
11
4.377 -4.491 0.055
0.130 79.867 -34.511
12
2.663 -3.295 0.043
0.124 61.819 -26.644
13
6.051 -2.699 0.043
0.222 139.673 -12.131
14
2.675 0.722 0.024
0.124 110.989 5.807
1
4.300 -2.966 -0.954
0.050 0.161 0.264
85.190 -18.441
-3.609
2
5.214 -1.779 -0.899
0.045 0.213 0.477 116.915 -8.355
-1.886
3
4.368 4.331 -3.042
0.052 0.168 0.458
84.058 25.740
-6.640
4
4.152 2.088 -2.940
0.052 0.201
0.312 79.912 10.364
-9.428
5
2.856 0.512 2.211
0.032 0.124 0.152
90.355 4.127
14.512
6
2.818 -1.103 0.667
0.054 0.198 0.393
52.389 -5.579
1.698
7
4.377 -2.758 -0.459
0.042 0.144 0.226 104.907 -19.149
-2.032
8
4.535 -3.456 0.119
0.049 0.147 0.307
93.011 -23.427
0.388
9
3.887 -3.140 -0.241
0.053 0.320 0.915
72.717 -9.802
-0.263
10
2.872 -2.466 -6.007
0.049 0.143 0.243 58.883 -17.215
-24.716
11
4.350 -4.595 2.417
0.054 0.125 0.303
80.519 -36.869
7.979
12
2.673 -3.148 -2.438
0.045 0.141 0.352
58.948 -22.276
-6.935
13
6.049 -2.741 -0.059
0.045 0.230 0.380 135.150 -11.931
-0.156
14
2.676 0.629 2.112
0.029 0.106 0.130
91.571 5.951
16.201
1
4.294 -2.976 0.708
1.180 0.046 0.178
0.509 0.844
92.487 -16.709 1.391
1.399
2
5.200 -1.806 1.586
-2.562 0.049 0.441
1.520 2.899
106.219 -4.099 1.043 -0.884
3
4.410 4.295 3.707
-2.929 0.064 0.220
1.092 2.305
68.705 19.511 3.396 -1.270
4
4.169 1.998 3.286
-1.111 0.052 0.253
0.879 1.495
80.561 7.900 3.737
-0.743
5
2.856 0.497 -2.004
-1.174 0.028 0.136
0.388 0.633
101.767 3.642 -5.162 -1.854
6
2.813 -1.049 -0.902
0.891 0.049 0.316
0.776 1.057
56.947 -3.321 -1.162 0.842
7 4.367 -2.785
0.265 0.557 0.043
0.162 0.286 0.611
102.567 -17.144 0.925
0.912
8
4.534 -3.457 0.140
-0.961 0.053 0.219
0.867 1.506
85.959 -15.770 0.161
-0.638
9
3.831 -3.255 1.558
-5.590 0.098 0.398
0.995 4.111
38.911 -8.175 1.566 -1.360
10
2.891 -2.372 5.602
2.868 0.054 0.170
0.541 1.335
53.952 -13.988 10.348
2.149
11
4.341 -4.632 -2.176
-1.392 0.053 0.230
0.824 1.201
81.543 -20.106 -2.641
-1.159
12
2.680 -3.159 1.860
2.372 0.051 0.145
0.455 1.026
52.695 -21.720 4.085
2.312
13
6.042 -2.819 0.329
-1.282 0.046 0.355
1.333 2.688
132.768 -7.949 0.247 -0.477
14
2.675 0.587
-1.980 -0.906 0.028
0.128 0.298 0.474
94.502 4.582 -6.638 -1.913
Appendix
C: BLACKT.FOR
1.
Introduction
BLACKT.FOR is a FORTRAN program that can be used to do an Aldrich-McKelvey scaling in more than one dimension. The program reads a "control card" file (BTSTRT.DAT) and the data file (usually an NES data set) and writes three output files: BLACKT23.DAT, BLACKT24.DAT, and BLACKT28.DAT. The program has been compiled for both the Pentium P5 processor as well as the Pentium P6 (Pentium II) processor. These executables are BLACKT5.EXE and BLACKT6.EXE respectively. They will run under both Windows 95 and Windows NT.
The example below uses the liberal-conservative 7-point scale from the 1980 NES cross-sectional survey data set -- NES1980.DAT.
2.
Input File for the Transpose Program:
BTSTRT.DAT
The first line of the input file (see next page) gives the name of the data set being analyzed. In this case, the 1980 NES data is in the subdirectory \NES. That is, the program, BLACKT6.EXE is in the root directory. Note that if NES1980.DAT could be placed in the same directory as the program the first line would simply be NES1980.DAT.
The second line of the input file is the title of the scaling
The third line contains, in order, the number of basic dimensions, the number of stimuli being placed on the 7-point scale, and the number of missing data values. Note that this is fixed format, namely, in FORTRAN syntax, 3I5. Hence, when you analyze a different scale be sure to not change the spacing. For example, if you want 3 basic dimensions and there are 10 stimuli and 3 missing data values, this line would be:
3 10 3
The fourth line contains the values of the missing data. In this case 0, 8, and 9. Note that these numbers are also fixed format.
The fifth line is the format statement for the data file. The 4I1 and the 2I1 are the six stimuli and the I4 is the respondent ID number. This format statement can be figured out by using the ICPSR codebook for the election study. If you have problems figuring out how to do this, just send me E-Mail at KPoole@uh.edu.
Finally, the last group of lines are the names of the stimuli. In this case, the four names of the important presidential candidates in 1980 along with the two political parties.
Note that this starting file differs from those for BLACKBOX.FOR and BMC2.FOR in that this program only analyzes a single issue scale. Hence, there is no need to repeat the missing data entries.
\NES\NES1980.DAT
DECOMPOSITION OF 1980 LIBERAL-CONSERVATIVE
7-POINT SCALE
2
6 3
0
8 9
(8X,I4,1810X,4I1,11X,2I1)
CARTER
REAGAN
KENNEDY
ANDERSON
REPUB
DEMO
3.
Output files for Transpose X0 Example
a. BLACKT23.DAT File
This file is the primary output file for the transpose program. For ease of exposition I will annotate this file for the convenience of the reader. My comments will be preceded by #### signs.
The first part of the output file just echoes the lines in BTSTRT.DAT. This is convenient because you can glance at this to make sure the starting file is configured correctly.
\BLACKB\NES\NES1980.DAT
DECOMPOSITION OF 1980 LIBERAL-CONSERVATIVE
7-POINT SCALE
2
6 3
0
8 9
(8X,I4,1810X,4I1,11X,2I1)
CARTER
REAGAN
KENNEDY
ANDERSON
REPUB
DEMO
####
Here 888 respondents have been included in the analysis. The
####
program requires that a respondent place at least s+2 (where s is
####
the number of basic dimensions being estimated) stimuli on the
####
scale.
NUMBER OF CASES 888
***********************************************************************
***********************************************************************
***********************************************************************
####
The program first estimates a one dimensional model, then two
####
dimensions, etc.
NUMBER OF DIMENSIONS= 1
***********************************************************************
####
This information is only given once.
####
The matix is 6 by 888 and contains 4973 entries. Hence, there
####
are 355 missing entries or [355/(6*888)]*100 = 6.66291% missing
####
entries. The Sum of Squares is
computed around the grand mean of
####
the matrix. Hence, it is the sum of
the squared differences between
####
the 4973 non-missing entries and the matrix mean.
NUMBER OF ROWS = 6
NUMBER OF COLUMNS = 888
TOTAL NUMBER OF DATA ENTRIES = 4973
NUMBER MISSING ENTRIES =
355
PERCENT MISSING DATA = 6.66291
SUM OF SQUARES =
13967.70215
***********************************************************************
####
This is the iteration record for the first basic dimension.
####
REG1 estimates W and c and REG2 estimates y.
DIMENSION=
1 TOTAL SSE REG1= 3438.4109
DIMENSION=
1 TOTAL SSE REG2= 3435.7090
DIMENSION=
1 TOTAL SSE REG1= 3435.3950
DIMENSION=
1 TOTAL SSE REG2= 3435.3579
DIMENSION=
1 TOTAL SSE REG1= 3435.3455
DIMENSION=
1 TOTAL SSE REG2= 3435.3394
DIMENSION=
1 TOTAL SSE REG1= 3435.3452
DIMENSION=
1 TOTAL SSE REG2= 3435.3481
DIMENSION=
1 TOTAL SSE REG1= 3435.3479
DIMENSION=
1 TOTAL SSE REG2= 3435.3486
####
The singular values for the one dimensional estimation are reported
####
below. Only the first s+3 singular values are shown.
SINGULAR VALUES OF ESTIMATED MATRICES
FIRST COLUMN: ORIGINAL MATRIX WITH FILLED IN MISSING ENTRIES
SECOND COLUMN: REPRODUCED MATRIX -- PSI*W +
Jc
THIRD COLUMN: PSI*W
1
301.190 301.097 95.444
2
65.736 64.998 0.000
3
36.408 0.000 0.000
4
31.730 0.000 0.000
***********************************************************************
####
Below are the constraint checks discussed in the AJPS article --
####
namely, the sum of the columns of y
must equal zero, and:
####
y'y = W'W = L where
L
is the s by s diagonal matrix of the
####
singular values of yW' which
is the least squares estimate of
####
[X0.- Jpc'].
CONSTRAINT CHECKS ON PSI AND W
SUM OF COLUMNS OF PSI
0.0000
PSI-TRANSPOSE*PSI
1
95.4437
W-TRANSPOSE*W
1
95.4438
####
Here yW' + Jpc'
is constructed and the r-square between the elements
####
of yW'
+ Jpc' and the original data
matrix, of X0 is
computed as a
####
check on the estimation.
####
R-SQUARE CHECK 4973 0.754
####
Similar to the above, as a check, a singular value decomposition
####
of the estimated matrix, yW', is performed. The rank of the
####
matrix is reported (here it is one) along with the singular values.
RANK CHECK OF PSI*W 1
1
95.4437
2
0.0000
3
0.0000
4 0.0000
5
0.0000
6
0.0000
***********************************************************************
NUMBER OF DIMENSIONS= 2
***********************************************************************
####
This is the iteration record for two basic dimensions. The first
####
basic dimension is extracted first followed by the second basic
####
dimension.
DIMENSION=
1 TOTAL SSE REG1= 3438.4109
DIMENSION=
1 TOTAL SSE REG2= 3435.7090
DIMENSION=
1 TOTAL SSE REG1= 3435.3950
DIMENSION=
1 TOTAL SSE REG2= 3435.3579
DIMENSION=
1 TOTAL SSE REG1= 3435.3455
DIMENSION=
1 TOTAL SSE REG2= 3435.3394
DIMENSION=
1 TOTAL SSE REG1= 3435.3452
DIMENSION=
1 TOTAL SSE REG2= 3435.3481
DIMENSION=
2 TOTAL SSE REG1= 1915.0280
DIMENSION=
2 TOTAL SSE REG2= 1912.3396
DIMENSION=
2 TOTAL SSE REG1= 1910.8296
DIMENSION=
2 TOTAL SSE REG2= 1909.8190
DIMENSION=
2 TOTAL SSE REG1= 1909.0571
DIMENSION=
2 TOTAL SSE REG2= 1908.4723
DIMENSION=
2 TOTAL SSE REG1= 1908.0084
DIMENSION=
2 TOTAL SSE REG2= 1907.6342
DIMENSION=
2 TOTAL SSE REG1= 1894.3529
DIMENSION=
2 TOTAL SSE REG2= 1893.7955
DIMENSION=
2 TOTAL SSE REG1= 1893.4442
DIMENSION=
2 TOTAL SSE REG2= 1893.1843
DIMENSION=
2 TOTAL SSE REG1= 1892.9827
DIMENSION=
2 TOTAL SSE REG2= 1892.8132
DIMENSION=
2 TOTAL SSE REG1= 1892.6793
DIMENSION=
2 TOTAL SSE REG2= 1892.5618
DIMENSION=
2 TOTAL SSE REG1= 1892.4615
DIMENSION=
2 TOTAL SSE REG2= 1892.3750
####
The singular values of the estimated matrices are printed out
####
below. Note that if there were no
missing data, the first s+1
####
singular values in the first two columns would be identical.
SINGULAR VALUES OF ESTIMATED MATRICES
FIRST COLUMN: ORIGINAL MATRIX WITH FILLED IN MISSING ENTRIES
SECOND COLUMN: REPRODUCED MATRIX -- PSI*W +
Jc
THIRD COLUMN: PSI*W
1
301.159 301.131 95.383
2
65.403 65.256 61.191
3
60.451 60.393 0.000
4
30.425 0.000 0.000
5
24.299 0.000 0.000
************************************************************************
CONSTRAINT CHECKS ON PSI AND W
SUM OF COLUMNS OF PSI
0.0000 0.0000
PSI-TRANSPOSE*PSI
1 95.3833 0.0000
2
0.0000 61.1914
W-TRANSPOSE*W
1
95.3833 0.0000
2
0.0000 61.1914
R-SQUARE CHECK 4973 0.865
RANK CHECK OF PSI*W 2
1
95.3833
2
61.1914
3
0.0000
4
0.0000
5
0.0000
6
0.0000
***********************************************************************
ITERATION RECORD
DIM
ERROR EXPLAINED PERCENT CUM PERCENT R-SQUARE
1
3435.3513 10532.3506 75.4050 75.4050 0.7541
2
1892.3855 12075.3164 11.0467 86.4517 0.8645
***********************************************************************
b. BLACKT24.DAT
Rather than writing out Y, for purposes of comparison with the Aldrich-McKelvely procedure, the singular vectors are written out instead. Namely, Y = UL1/2, where U is p by s matrix such that U'U = Is , and L is the s by s diagonal matrix of singular values of YW'. Here, just U is written out for one and two dimensions.
CARTER 0.242
REAGAN
-0.580
KENNEDY
0.478
ANDERSON
0.059
REPUB
-0.519
DEMO
0.321
CARTER
0.229 0.409
REAGAN
-0.582 0.099
KENNEDY
0.482 -0.001
ANDERSON
0.077 -0.864
REPUB
-0.521 0.097
DEMO
0.315 0.259
c. BLACKT28.DAT
This file contains the respondent linear transformation parameters -- W and c. The first number is the respondent's identification number from the NES1980.DAT file. The second number is the number of responses, the third number is c, and the last number is the r-square. Between c and the r-square are the W values.
1
6 4.333 -0.317 0.351
8
6 5.167 -0.207 0.463
9
6 3.833 -0.371 0.884
10
6 4.167 -0.059 0.048
11
6 5.167 0.121
0.084
13 6 5.000 -0.321 0.613
14
6 4.167 -0.375 0.903
16
6 4.333 -0.407 0.679
17
4 3.651 0.204
0.364
19
6 4.333 -0.310 0.686
20
6 3.333 -0.407 0.913
etc
etc
1756
6 3.667 -0.573 0.888
1757
6 3.500 -0.115 0.134
1758
6 3.500 -0.442 0.866
1759
6 3.500 -0.533 0.722
1760
5 3.363 -0.322 0.879
1761
6 4.333 -0.104 0.140
1762
5 2.586 -0.229 0.283
1764
6 3.833 -0.387 0.962
1765
5 4.020 0.169
0.340
1
6 4.333 -0.311 -0.364 0.636
8
6 5.167 -0.202 -0.281 0.989
9
6 3.833 -0.371 0.041
0.893
10
6 4.167 -0.064 0.229
0.521
11
6 5.167 0.119
0.093 0.111
13
6 5.000 -0.323 0.059
0.635
14
6 4.167 -0.372 -0.145 0.979
16
6 4.333 -0.410 0.209
0.805
17
4 3.845 0.202
0.283 0.985
19
6 4.333 -0.306 -0.180 0.817
20
6 3.333 -0.408 0.011
0.914
etc
etc
1756
6 3.667 -0.569 -0.252 0.984
1757
6 3.500 -0.110 -0.282 0.630
1758
6 3.500 -0.445 0.186
0.978
1759
6 3.500 -0.525 -0.427 0.999
1760
5 2.827 -0.349 0.386
0.969
1761
6 4.333 -0.108 0.222
0.559
1762
5 2.597 -0.219 -0.380 0.948
1764
6 3.833 -0.388 0.075
0.992
1765
5 3.173 0.127
0.628 0.636
Footnotes