Chapter Contents Previous Next
 The PRINQUAL Procedure

## Example 53.1: Multidimensional Preference Analysis of Cars Data

This example uses PROC PRINQUAL to perform a nonmetric multidimensional preference (MDPREF) analysis (Carroll 1972). MDPREF analysis is a principal component analysis of a data matrix with columns that correspond to people and rows that correspond to objects. The data are ratings or rankings of each person's preference for each object. The data are the transpose of the usual multivariate data matrix. (In other words, the columns are people instead of the more typical matrix where rows represent people.) The final result of an MDPREF analysis is a biplot (Gabriel 1981) of the resulting preference space. A biplot displays the judges and objects in a single plot by projecting them onto the plane in the transformed variable space that accounts for the most variance.

The data are ratings by 25 judges of their preference for each of 17 automobiles. The ratings are made on a 0 to 9 scale, with 0 meaning very weak preference and 9 meaning very strong preference for the automobile. These judgments were made in 1980 about that year's products. There are two additional variables that indicate the manufacturer and model of the automobile.

This example uses PROC PRINQUAL, PROC FACTOR, and the %PLOTIT macro. PROC FACTOR is used before PROC PRINQUAL to perform a principal component analysis of the raw judgments. PROC FACTOR is also used immediately after PROC PRINQUAL since PROC PRINQUAL is a scoring procedure that optimally scores the data but does not report the principal component analysis.

The %PLOTIT macro produces the biplot. For information on the %PLOTIT macro, see Appendix B, "Using the %PLOTIT Macro."

The scree plot, in the standard principal component analysis reported by PROC FACTOR, shows that two principal components should be retained for further use. (See the scree plot in Output 53.1.1 -there is a clear separation between the first two components and the remaining components.) There are nine eigenvalues that are precisely zero because there are nine fewer observations than variables in the data matrix. PROC PRINQUAL is then used to monotonically transform the raw judgments to maximize the proportion of variance accounted for by the first two principal components. The following statements create the data set and perform a principal component analysis of the original data. These statements produce Output 53.1.1.

   title 'Preference Ratings for Automobiles Manufactured in 1980';

data CarPref;
input Make $1-10 Model$ 12-22 @25 (Judge1-Judge25) (1.);
datalines;
Chevrolet  Chevette     0051200423451043003515698
Chevrolet  Citation     4053305814161643544747795
Chevrolet  Malibu       6027400723121345545668658
Ford       Fairmont     2024006715021443530648655
Ford       Mustang      5007197705021101850657555
Ford       Pinto        0021000303030201500514078
Honda      Accord       5956897609699952998975078
Honda      Civic        4836709507488852567765075
Lincoln    Continental  7008990592230409962091909
Plymouth   Gran Fury    7006000434101107333458708
Plymouth   Horizon      3005005635461302444675655
Plymouth   Volare       4005003614021602754476555
Pontiac    Firebird     0107895613201206958265907
Volkswagen Dasher       4858696508877795377895000
Volkswagen Rabbit       4858509709695795487885000
Volvo      DL           9989998909999987989919000
;

* Principal Component Analysis of the Original Data;
options ls=80 ps=65;
proc factor data=CarPref nfactors=2 scree;
ods select Eigenvalues ScreePlot;
var Judge1-Judge25;
title3 'Principal Components of Original Data';
run;


Output 53.1.1: Principal Component Analysis of Original Data

 Preference Ratings for Automobiles Manufactured in 1980 Principal Components of Original Data

 The FACTOR Procedure Initial Factor Method: Principal Components

 Eigenvalues of the Correlation Matrix: Total= 25 Average = 1 Eigenvalue Difference Proportion Cumulative 1 10.8857202 5.0349926 0.4354 0.4354 2 5.8507276 3.8077964 0.2340 0.6695 3 2.0429312 0.5207808 0.0817 0.7512 4 1.5221504 0.3078035 0.0609 0.8121 5 1.2143469 0.2564839 0.0486 0.8606 6 0.9578630 0.2197345 0.0383 0.8989 7 0.7381286 0.1497259 0.0295 0.9285 8 0.5884027 0.2117186 0.0235 0.9520 9 0.3766841 0.1091250 0.0151 0.9671 10 0.2675591 0.0773893 0.0107 0.9778 11 0.1901698 0.0463921 0.0076 0.9854 12 0.1437776 0.0349382 0.0058 0.9911 13 0.1088394 0.0607418 0.0044 0.9955 14 0.0480977 0.0056610 0.0019 0.9974 15 0.0424367 0.0202714 0.0017 0.9991 16 0.0221653 0.0221653 0.0009 1.0000 17 0.0000000 0.0000000 0.0000 1.0000 18 0.0000000 0.0000000 0.0000 1.0000 19 0.0000000 0.0000000 0.0000 1.0000 20 0.0000000 0.0000000 0.0000 1.0000 21 0.0000000 0.0000000 0.0000 1.0000 22 0.0000000 0.0000000 0.0000 1.0000 23 0.0000000 0.0000000 0.0000 1.0000 24 0.0000000 0.0000000 0.0000 1.0000 25 0.0000000 0.0000 1.0000

 Scree Plot of Eigenvalues | | | | | 12 + | | | | 1 | | 10 + | | | | | | 8 + E | i | g | e | n | v | a 6 + l | 2 u | e | s | | | 4 + | | | | | | 2 + 3 | | 4 | 5 | 6 7 | 8 | 9 0 1 2 0 + 3 4 5 6 7 8 9 0 1 2 3 4 5 | | | | -----+----+----+----+----+----+----+----+----+----+----+----+----+----+---- 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Number 

To fit the nonmetric MDPREF model, you can use the PRINQUAL procedure. The MONOTONE option is specified in the TRANSFORM statement to request a nonmetric MDPREF analysis; alternatively, you can instead specify the IDENTITY option for a metric analysis. Several options are used in the PROC PRINQUAL statement. The option DATA=CarPref specifies the input data set, OUT=Results creates an output data set, and N=2 and the default METHOD=MTV transform the data to better fit a two-component model. The REPLACE option replaces the original data with the monotonically transformed data in the OUT= data set. The MDPREF option standardizes the component scores to variance one so that the geometry of the biplot is correct, and it creates two variables in the OUT= data set named Prin1 and Prin2. These variables contain the standardized principal component scores and structure matrix, which are used to make the biplot. If the variables in data matrix X are standardized to mean zero and variance one, and n is the number of rows in X, then is the principal component model, where .The W and contain the eigenvectors and eigenvalues of the correlation matrix of X. The first two columns of V, the standardized component scores, and ,which is the structure matrix, are output. The advantage of creating a biplot based on principal components is that coordinates do not depend on the sample size. The following statements transform the data and produce Output 53.1.2.

   * Transform the Data to Better Fit a Two Component Model;
proc prinqual data=CarPref out=Results n=2 replace mdpref;
id model;
transform monotone(Judge1-Judge25);
title2 'Multidimensional Preference (MDPREF) Analysis';
title3 'Optimal Monotonic Transformation of Preference Data';
run;


Output 53.1.2: Transformation of Automobile Preference Data

 Preference Ratings for Automobiles Manufactured in 1980 Multidimensional Preference (MDPREF) Analysis Optimal Monotonic Transformation of Preference Data

 The PRINQUAL Procedure

 PRINQUAL MTV Algorithm Iteration History IterationNumber AverageChange MaximumChange Proportionof Variance CriterionChange Note 1 0.24994 1.28017 0.66946 2 0.07223 0.36958 0.80194 0.13249 3 0.04522 0.29026 0.81598 0.01404 4 0.03096 0.25213 0.82178 0.00580 5 0.02182 0.23045 0.82493 0.00315 6 0.01602 0.19017 0.82680 0.00187 7 0.01219 0.14748 0.82793 0.00113 8 0.00953 0.11031 0.82861 0.00068 9 0.00737 0.06461 0.82904 0.00043 10 0.00556 0.04469 0.82930 0.00026 11 0.00445 0.04087 0.82944 0.00014 12 0.00381 0.03706 0.82955 0.00011 13 0.00319 0.03348 0.82965 0.00009 14 0.00255 0.02999 0.82971 0.00006 15 0.00213 0.02824 0.82976 0.00005 16 0.00183 0.02646 0.82980 0.00004 17 0.00159 0.02472 0.82983 0.00003 18 0.00139 0.02305 0.82985 0.00003 19 0.00123 0.02145 0.82988 0.00002 20 0.00109 0.01993 0.82989 0.00002 21 0.00096 0.01850 0.82991 0.00001 22 0.00086 0.01715 0.82992 0.00001 23 0.00076 0.01588 0.82993 0.00001 24 0.00067 0.01440 0.82994 0.00001 25 0.00059 0.00871 0.82994 0.00001 26 0.00050 0.00720 0.82995 0.00000 27 0.00043 0.00642 0.82995 0.00000 28 0.00037 0.00573 0.82995 0.00000 29 0.00031 0.00510 0.82995 0.00000 30 0.00027 0.00454 0.82995 0.00000 Not Converged

 WARNING: Failed to converge, however criterion change is less than 0.0001.

The iteration history displayed by PROC PRINQUAL indicates that the proportion of variance is increased from an initial 0.66946 to 0.82995. The proportion of variance accounted for by PROC PRINQUAL on the first iteration equals the cumulative proportion of variance shown by PROC FACTOR for the first two principal components. In this example, PROC PRINQUAL's initial iteration performs a standard principal component analysis of the raw data. The columns labeled Average Change, Maximum Change, and Variance Change contain values that always decrease, indicating that PROC PRINQUAL is improving the transformations at a monotonically decreasing rate over the iterations. This does not always happen, and when it does not, it suggests that the analysis may be converging to a degenerate solution. See Example 53.2 for a discussion of a degenerate solution. The algorithm does not converge in 30 iterations. However, the criterion change is small, indicating that more iterations are unlikely to have much effect on the results.

The second PROC FACTOR analysis is performed on the transformed data. The WHERE statement is used to retain only the monotonically transformed judgments. The scree plot shows that the first two eigenvalues are now much larger than the remaining smaller eigenvalues. The second eigenvalue has increased markedly at the expense of the next several eigenvalues. Two principal components seem to be necessary and sufficient to adequately describe these judges' preferences for these automobiles. The cumulative proportion of variance displayed by PROC FACTOR for the first two principal components is 0.83. The following statements perform the analysis and produce Output 53.1.3:

   * Final Principal Component Analysis;
proc factor data=Results nfactors=2 scree;
ods select Eigenvalues ScreePlot;
var Judge1-Judge25;
where _TYPE_='SCORE';
title3 'Principal Components of Monotonically Transformed Data';
run;


Output 53.1.3: Principal Components of Transformed Data

 Preference Ratings for Automobiles Manufactured in 1980 Multidimensional Preference (MDPREF) Analysis Principal Components of Monotonically Transformed Data

 The FACTOR Procedure Initial Factor Method: Principal Components

 Eigenvalues of the Correlation Matrix: Total= 25 Average = 1 Eigenvalue Difference Proportion Cumulative 1 11.5959045 2.4429455 0.4638 0.4638 2 9.1529589 7.9952554 0.3661 0.8300 3 1.1577036 0.3072013 0.0463 0.8763 4 0.8505023 0.1284323 0.0340 0.9103 5 0.7220700 0.2613540 0.0289 0.9392 6 0.4607160 0.0958339 0.0184 0.9576 7 0.3648821 0.0877851 0.0146 0.9722 8 0.2770970 0.1250945 0.0111 0.9833 9 0.1520025 0.0506622 0.0061 0.9894 10 0.1013403 0.0292763 0.0041 0.9934 11 0.0720640 0.0200979 0.0029 0.9963 12 0.0519661 0.0336675 0.0021 0.9984 13 0.0182987 0.0027059 0.0007 0.9991 14 0.0155927 0.0093669 0.0006 0.9997 15 0.0062258 0.0055503 0.0002 1.0000 16 0.0006755 0.0006755 0.0000 1.0000 17 0.0000000 0.0000000 0.0000 1.0000 18 0.0000000 0.0000000 0.0000 1.0000 19 0.0000000 0.0000000 0.0000 1.0000 20 0.0000000 0.0000000 0.0000 1.0000 21 0.0000000 0.0000000 0.0000 1.0000 22 0.0000000 0.0000000 0.0000 1.0000 23 0.0000000 0.0000000 0.0000 1.0000 24 0.0000000 0.0000000 0.0000 1.0000 25 0.0000000 0.0000 1.0000

 Scree Plot of Eigenvalues | | | | | 12 + | 1 | | | | | 10 + | | | 2 | | | 8 + E | i | g | e | n | v | a 6 + l | u | e | s | | | 4 + | | | | | | 2 + | | | 3 | 4 5 | 6 | 7 8 9 0 + 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 | | | | -----+----+----+----+----+----+----+----+----+----+----+----+----+----+---- 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Number 

The remainder of the example constructs the MDPREF biplot. A biplot is a plot that displays the relation between the row points and the columns of a data matrix. The rows of V, the standardized component scores, and , which is the structure matrix, contain enough information to reproduce X. The (i,j) element of X is the product of row i of V and row j of .If all but the first two columns of V and are discarded, the (i,j) element of X is approximated by the product of row i of V and row j of .

Since the MDPREF analysis is based on a principal component model, the dimensions of the MDPREF biplot are the first two principal components. The first principal component is the longest dimension through the MDPREF biplot. The first principal component is overall preference, which is the most salient dimension in the preference judgments. One end points in the direction that is on the average preferred most by the judges, and the other end points in the least preferred direction. The second principal component is orthogonal to the first principal component, and it is the orthogonal direction that is the second most salient. The interpretation of the second dimension varies from example to example.

With an MDPREF biplot, it is geometrically appropriate to represent each automobile (object) by a point and each judge by a vector. The automobile points have coordinates that are the scores of the automobile on the first two principal components. The judge vectors emanate from the origin of the space and go through a point with coordinates that are the coefficients of the judge (variable) on the first two principal components.

The absolute length of a vector is arbitrary. However, the relative lengths of the vectors indicate fit, with the squared lengths being proportional to the communalities in the PROC FACTOR output. The direction of the vector indicates the direction that is most preferred by the individual judge, with preference increasing as the vector moves from the origin. Let v' be row i of V, u' be row j of ,|v| be the length of v, |u| be the length of u, and be the angle between v and u. The predicted degree of preference that an individual judge has for an automobile is .Each car point can be orthogonally projected onto the vector. The projection of car i on vector j is u((u'v)/(u'u)) and the length of this projection is .The automobile that projects farthest along a vector in the direction it points is that judge's most preferred automobile, since the length of this projection, , differs from the predicted preference, , only by |u|, which is constant within each judge.

To interpret the biplot, look for directions through the plot that show a continuous change in some attribute of the automobiles, or look for regions in the plot that contain clusters of automobile points and determine what attributes the automobiles have in common. Those points that are tightly clustered in a region of the plot represent automobiles that have the same preference patterns across the judges. Those vectors that point in roughly the same direction represent judges who tend to have similar preference patterns.

The following statement constructs the biplot and produces Output 53.1.4:

   title3 'Biplot of Automobiles and Judges';
%plotit(data=results, datatype=mdpref 2);


The DATATYPE=MDPREF 2 option indicates that the coordinates come from an MDPREF analysis, so the macro represents the scores as points and the structure as vectors, with the vectors stretched by a factor of two to make a better graphical display.

Output 53.1.4: Preference Ratings for Automobiles Manufactured in 1980

In the biplot, American automobiles are located on the left of the space, while European and Japanese automobiles are located on the right. At the top of the space are expensive American automobiles (Cadillac Eldorado, Lincoln Continental) while at the bottom are inexpensive ones (Pinto, Chevette). The first principal component differentiates American from imported automobiles, and the second arranges automobiles by price and other associated characteristics.

The two expensive American automobiles form a cluster, the sporty automobile (Firebird) is by itself, the Volvo DL is by itself, and the remaining imported autos form a cluster, as do the remaining American autos. It seems there are 5 prototypical automobiles in this set of 17, in terms of preference patterns among the 25 judges.

Most of the judges prefer the imported automobiles, especially the Volvo. There is also a fairly large minority that prefer the expensive cars, whether or not they are American (those with vectors that point towards one o'clock), or simply prefer expensive American automobiles (vectors that point towards eleven o'clock). There are two people who prefer anything except expensive American cars (five o'clock vectors), and one who prefers inexpensive American cars (seven o'clock vector).

Several vectors point toward the upper-right corner of the plot, toward a region with no cars. This is the region between the European and Japanese cars on the right and the luxury cars on the top. This suggests that there is a market for luxury Japanese and European cars.

 Chapter Contents Previous Next Top