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Judgmental Rotation Tutorial



PERHAPS THE EASIEST way to approach learning Judgmental Rotation is by way of analogy. Since the purpose of this tutorial is to introduce you to Judgmental Rotation in practical terms, let us consider a musician, say, a violinist. What is the objective of tuning a violin? What does the reference tone for tuning it sound like?

We say the violin is in tune when the four strings are in precise relationship with each other in terms of a given tone. The choice of that tone -- it is almost universally the 440-cycles-per-second "A" in the middle of a piano keyboard -- is partly convention and partly arbitrary:

  • It is a convention in the sense that 440-A is known to and accepted by musicians everywhere because it is in the middle register of almost all instruments, including the female voice;
  • It is arbitrary in the sense that on a given day, in a given place, an orchestra may really tune to, say, a 439-A or a 441-A, because a combination of heat and humidity physically limits the capability of the most difficult instrument to tune, that being the oboe. This is why musicians tune their instruments to the middle-register A of the oboe.

So, what does one do to tune a violin? First, pick up the bow and adjust the tension. Pick up the instrument and pull it up and under the chin. Pluck the "A" string, comparing its tone with the reference tone, 440-A. Tighten or loosen the string until the sound it produces matches the reference tone. The violinist listens to the combined tones and adjusts the string until it produces a pure sound in combination with the reference tone. The process is repeated with the other three strings: Pluck, listen, tighten or loosen until the strings produce a pure sound in combination. This works because well-tuned violin strings are precisely relative to each other and precisely relative to the reference tone. Now draw the bow across the strings to produce continuous tones because it is easier to make fine adjustments when hearing continuous tones.

An expert can tune an instrument quickly because the ear and fingers have become trained through experience. But, for untrained ears and fingers, many minutes may be needed. Indeed, beginning string players struggle as they learn to listen and to hear a pure combination of tones.

A person unfamiliar with music might think this analogy is obscure and wonder what is going on here. As a matter of fact, though, engineers tune ships, buildings, bridges and computer structures in ways parallel to tuning a violin or an entire orchestra. Or, consider a photographer focusing a camera lens until the image is sharp. Or, consider adding spices to a recipe -- the ubiquitous "salt to taste". In all of these there are reference points and adjustment processes that are used to bring complicated components into balance.

And so it is, too, in Judgmental Rotation. Much as with tuning a violin, Judgmental rotation can be described as both abstract and concrete at the same time. The logic of it, though, is completely theoretical. The numbers are theoretical; the graphical representation is nothing more than a geometric projection of the factor loadings, which are also theoretical. In a way, judgmental rotation is an example of pure communication play, to apply Stephenson's concept, in the sense that the researcher is in search of balancing the factors as guided by theory, tuning them in terms of the preferred reference points.

  • What is the objective of rotating the factors?
  • What are the reference points?
  • How does one bring the sorts into sharp focus, balance them, tune them?

Visualizing the factors on a graph helps the researcher see the relationships between the sorts as they are individually and collectively associated with the factors. But it is a pointless exercise if there are no reference points, in the physical sense of graphing, and no guides, in the abstract sense of theoretical considerations. PCQ provides the physical reference points in the graph plotted on the screen. The researcher provides the guiding theory.

The program shows the sorts in relation to one another in the form of a graph having two dimensions, called axes. By convention, the two dimensions are labeled Factor X, read vertically, and Factor Y, read horizontally. Please note that the dimensions are always at right angles, that is, at 90 angles, because mathematically this maintains a condition begun in the factoring stage wherein the two factors are uncorrelated.

In PCQ all the factors can be visualized, but only two at a time. The process is parallel to the violinist listening to the sounds produced by two strings at once. With PCQ, though, rotating the X axis and Y axis is the visual parallel to the tightening and loosening adjustments the violinist makes. But adjustments in rotating factors involves many more components. Remember that when the factors were extracted, each sort was assigned a numerical value establishing its relationship to each factor. The values are called "factor loadings". For example, if a study has 9 factors and 33 sorts, each sort has 9 loadings associated with it. This means there are (9 * 33 = 297) relationships involved in the complete factor structure.

A common objective in Judgmental Rotation -- there are others -- is to account for as many of the sorts as possible in as few factors as possible. Or, put another way, one seeks to have fewer dimensions which are identified by all the sorts. The result of rotating the Lipset Q study data, for example, produced four factors accounting for all the sorts.

This section contains two major parts: (1) a tutorial introducing the practical qualities, commands and processes of Judgmental Rotation, and (2) a step-by-step example showing how Judgmental Rotation can become a  tool for the researcher.


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Practical Qualities

Judgmental Rotation in PCQ has these components: (1) Cartesian graph, (2) commands, and (3) tables. They are inter-related and are designed to provide you with a real-time working environment. The thumbnail shows the three screen areas.


fullrotationscrn.gif (2821 bytes)  

The graph displays the x, y positions of each sort. The vertical axis is Factor X, and the horizontal axis is Factor Y.

Factor X and Factor Y loadings are listed in the table to the left of the graph.



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Rotation controls and the results they produce

Here is the command area, as shown on screen.


rotatecmds.gif (4743 bytes)



Once arrived at the graphic rotation screen, use these commands.


Select factors

selectfacs.gif (1532 bytes) 


Click on the left window and select Factor X, then click on the other window and select Factor Y. To display the factors on the graph, click on Start.

Set rotation angle

anglecmd.gif (1242 bytes)


After clicking on Start, you have three ways to select an angle of rotation.

  1. You may type in a positive or negative number;
  2. Or, left click with the mouse and hold on the Up button to increase angle of rotation positively; left click with the mouse and hold on the Down button to move the angle of rotation toward the negative.
  3. In the graph window, position the mouse and RIGHT click with the mouse at a desired position. Then choose either the positive or negative angle.

The program will respond automatically and move the X by Y axes to the position you have selected.

Please note that only the axis lines move; the points representing each sort remain stationary.


Operation commands

startaccept.gif (1573 bytes)


Start: Click on this button to display loadings of the selected factors.

Undo: Click here to Undo the most recently Accepted rotation. Click it again to undo the previous rotation.

Cancel: Click here if you want to select other factors.

Preview: Click here to display the loadings that would result if you Accept the rotation of the currently selected factors.

Accept: When you are satisfied, click to record the rotation.


Show or Hide sort numbers

showloadings.gif (1317 bytes)


This feature is useful for visual inspection, particularly with a large number of sorts. You may choose to show the sort numbers for all sorts, only the sorts with significant loadings, or only the sorts with less than significant loadings.


Change significance level

siglevelcmd.gif (1164 bytes)


Default is the theoretical value


General Commands

historyfinish.gif (1343 bytes)


History: A pop-up listing of rotations performed in the current session.

Labels: A pop-up listing of sort labels that were entered with the Q sort data.

Matrix: Click on this command to examine the status of the rotation. The Matrix displays three tables:

  1. sort loadings on all factors;
  2. sorts with significant loadings on more than one factor -- referred to as "Confounded";
  3. sorts having no significant loading.

While the Matrix is on the screen, you may choose to show the sort labels.

Please note that the Matrix shows values as they were after the last rotation was accepted. Use the Preview command to display the effect of rotating the two factors displayed in the graph.

Finish: Click here when all done. You will be returned to Rotation Dialog. For example, suppose you want to start over or abandon rotation, click Finish.



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Copyright 2000, 2001, 2002, 2003, 2004 Michael Stricklin & Ricardo Almeida (All Rights Reserved)

Last update on 27 April 2004.