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On PCQ for WindowsTHE EASE of computation afforded by computer programs today is a mixed blessing for students and researchers who use Q technique. Manipulating many numbers, of course, is the forte of electronic computers. But the facility provided is superficial in the sense that students and researchers can mount a Q study without much knowledge or understanding of either Q technique or Q methodology. In other words, we risk forgetting, or at least becoming complacent, about how and why our tools work. Students of factor analysis had to know their mathematics in earlier days. They could rightly take pride in becoming masters of doing the arithmetic and of finding and perfecting efficient algorithms for hand calculation. Today, on the other hand, we do not have to worry very much about anything that happens after the data have been entered correctly. Today we can avoid knowing about what happens to the data between the time we collect the sorts and the final report generated by the software. Since a primary supposition of Q Methodology is that communication and communicability are indeterminant and generative, it follows that analytical tools and procedures should be also. The design philosophy of PCQ for Windows extends the premise of indeterminancy. The student and researcher will encounter it throughout the program. For example, centroid factors are extracted rather than principal components because a centroid factor solution is inherently indeterminant and generative. This is to say centroid produces an infinite number of solutions, any of which are mathematically correct. By contrast principal components produces only one mathematical solution and can therefore be said to be deterministic and reductionistic. Similarly, Judgmental Rotation is included in PCQ for Windows because there are an infinite number of ways to rotate factors. Varimax Rotation yields only one mathematical solution. In PCQ, theory is always the researcher's guide.
On Factor AnalysisTHE EPISTEMOLOGY of Q Methodology requires that Q technique adhere to the principles of integration and indeterminancy at every stage of a Q study. This essay elaborates in nonmathematical language how the activities that comprise the analysis stages of Q Technique  sorting, correlating, factor analyzing, and rotating  are best thought of in terms of being children of Q Methodology. The general line of argument is that
SortingEVERY STAGE of a Q study can be expressed geometrically. This is so because linearity is assumed to apply to data in all the varieties of factor analysis. But, please note that until a sorter completes sorting, the Q sample is not linear. How does this work?


Before sorting, the Q sample is indeterminant, which means that the statements have no values associated with them and that they have equal potential in terms of eventual meaning. Each statement can become either positive or negative, and each one has potential to be expressed as very important or not. Until the sorter places them into ranks  the piles  we can say they have no value, no valence, no variance. As individual members of a group they are all alike. Their state is zero. The example to the left shows a Q sample with symmetrical distribution, five shells with pile values ranging from 4 to +4. The red numbers identify the five shells, and only details of the positive hemisphere are shown. Statements are represented by the white numbers, and all of them are shown to be in the innermost shell, i.e., with zero values.


At the sorting stage, a person changes the state of the collection of statements. By positioning each statement in a pile, what was before indeterminant now becomes determinant, and what was a zero state becomes the state representing the sorter's interpretation of them. The image to the left represents the positioning of the same numbers included in the image above, and again only details of the positive hemisphere are shown. Now, however, each has been assigned to one of the five shells. For example, most weight has been given to statements 31 and 18 by placing them in the +4 shell. Seven of the statements are in the 0 shell. They are 5, 33, 26, 25, 30, 20 and 15.


There is an important distinction displayed here. Please note that before sorting, these same seven statements had no value. But now  since they have been evaluated by the sorter  we can say they have values equal to zero, just as we can say that statements 31 and 18 have values of +4. A completed Q sort is the physical record of the measurements of a Q sample as produced by the sorter. What was undetermined before sorting is now determined via the sorter's measurements of the statements in relation to each other. Taken as a collection of measurements, a completed sort is a quantified record of one possible state of the Q sample. Thus, we can say the act of sorting transforms the Q sample from an indeterminant state to a determinant one. Every Q sort shares this quality. Once all the Q sorts are in hand, the Q sample has been tested, and, because these events have transpired, we now have a set of experimental data  the collection of Q sorts  that can be analyzed.


CorrelatingTHE COLLECTION of Q sorts is the collection of tested states of the Q sample. It is multidimensional, with the number of dimensions equal to the number of sorts. But, before we can factor, we need to know how the sorts compare to each other in terms of how each of the items was scored in them. We can do so in a precise way by calculating the correlation coefficients of each sort with all the others. Another way to think about this would be to think of holding aside for the moment the details of each Q sort and concentrating on each sort's degree of similarity or dissimilarity with all the others. What is a correlation coefficient, in nontechnical terminology? It is single, precise numerical expression of a sort’s rankings in relationship to the rankings in another sort. But to arrive at this numerical expression, the ranking scores of each Q sort are changed into a linear format. Once this has been accomplished, all the mathematical rules that apply to lines can be brought to bear. So, while a correlation coefficient is a single numerical value, all the correlations can be compared directly because the values are expressions of linear relationships between all the sorts. From this point in factor analysis until very near to the end, numerical relationships are only expressed with linear equations. This is a powerful idea, as Steve Brown gracefully lays out in Political Subjectivity.


Factor analyzingWHILE WE usually think of the collection of sorts as the raw data in Q technique, it is the table of correlation coefficients which is factored. In other words, we factor the numerical relationships between the sorts and not the data within the sorts. The results are reported in a table of numerical relationships called the factor loadings. Since all the sorts are related, each sort has a loading for each factor. How are factors identified? A centroid can be thought of as a kind of grand average of the relationships between all the sorts as they are represented by their correlation coefficients. Centroid factor analysis, then, is a way of defining centers of gravity embedded in a correlation matrix and expressing these centers in precise terms. In physics, a center of gravity turns out to be where the weight tends to fall on average. For us this concept can be represented as a vector that spans the longest dimension of the data space. The factor loadings, then, are values expressing each sort’s relationship with the centroid. Each loading represents a sort’s contribution to the length of the centroid, and thus can be expressed as the correlation of that sort with the centroid. The higher the correlations the more the sorts have in common with each other, the longer the centroid when expressed as a vector. If enough variance remains after subtracting out the influence of the first centroid, other centroids may be extracted. Each centroid, then, represents a different linear dimension hitherto unobserved in the correlations. Centroid factoring satisfies an important requisite of Q methodology in that it provides a means to integration in an indeterministic framework by finding which sorts have the most in common. If more than one centroid can be extracted, the factor structure is multidimensional, and each dimensional is uncorrelated with the others.


RotatingSINCE THERE are an infinite number of ways to rotate factors, the researcher's goals have special importance. Another way to phrase this would be to ask, "What reference frame for the factor structure is appropriate, given the goals of the Q study?" The most generalized way of choosing a reference frame is based upon the principle of parsimony, or, in other words, to account for as many of the sorts in as few factors as the data will permit. The images below illustrate the unrotated and rotated Lipset data, as reported by Steven Brown.


Notice that before rotation the centroid factors  labeled F1, F2 and F3  are not centered on the sort loadings. This is very often the case with unrotated factor loadings. There is no visually apparent reference frame. 

After Judgmental Rotation, Brown settled on the reference frame shown in the image at left. Now the centroid factors pass through the sorts associated with them. 

The Q methodology principles of indeterminancy and integration are also qualities of centroid factor analysis and of Judgmental Rotation, thus Brown was able to account for all but one of the nine sorts with three factors.
Factor scoresTHE FACTORS extracted and rotated are not the final stage in Q technique. When we have the rotated factors, we are still two steps away from what we seek, namely, one or more ideal Q sorts, the arrays of factor scores. It is significant that a sort's factor loading can be defined as the correlation of that sort with the centroid. This works out because the factor loadings are proportional to the factor, and we use these values to transform the results of the factor analysis back into the multidimensional space from which we started. How are the factor score arrays calculated? A factor score array has the form of the original Q sample. The sorts with statistically significant loadings on a factor are used to calculate an array. The other sorts  those associated with other factors, those with no association with a factor, and those associated with more than one factor  are not used.


===================================== sort
label factors 1 2
3

In the table at left, the array for Factor 3 will be calculated using the Q sorts of persons 1 and 2. The array for Factor 2 will be calculated using the Q sorts of persons 3, 4 and 5. The array for Factor 1 will be calculated using the Q sorts of persons 6 and 7. The Q sort for person 8 is not used because it has significant loadings on both Factor 1 and Factor 3. The Q sort for person 9 does not have a significant loading on any factor.

The arrays of factor scores result from simple numerical integrations of the original Q sorts. Thus, although we often refer to an array of factor scores as an ideal sort, it is better and more precise to say that an array is theoretical. The contribution of a Q sort to a factor array is proportional to its factor loading. In the example shown above, Q sort 6 is given more weight than Q sort 7 in calculating the array for Factor 3, Q sort 2 more than Q sort 1 in calculating the array for Factor 1, while Q sorts 3 and 4 are essentially equal, but Q sort 5 will be given more in calculating Factor 2. Constructing theoretical Q sorts in this way has more than mathematical elegance, and surely it is elegant to begin with many dimensions and end with few. But Q methodology shuns reductionism of the variety usually associated with factor analysis studies. Q technique, though, is not reductionistic because it does not end with the mathematics. The theoretical Q sorts are the opening of the final stage of a Q study. That stage becomes available only after they are revealed, after the preceding steps have been completed. They are the results of the experiment. They offer us moments of discovery. Stephenson called them the summum bonum  the good end  of Q technique because interpreting the factor arrays is at heart an integrative and generative experience.


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Copyright © 2000, 2001, 2002, 2003, 2004 Michael Stricklin & Ricardo Almeida (All Rights Reserved) 
Last update on 27 April 2004. 
