Correspondence analysis (CA) is a multidimensional factor analysis technique which falls within the category of multivariate methods (see Cornejo, 1988b, for an extensive description). Benzicri (e.g., 1972, 1976) was widely responsible for the application of CA to the social sciences, a trend enriched by the contributions of his disciples in what is known as the French school (e.g., Lebart, Morineau & Tabard, 1977; Lebart, Morineau & Warwick, 1984).[4] As Cornejo (1988b) explains:
"Its main advantages are the versatility and generality of its applications as an instrument for showing the simultaneous geometrical structure of the shared 'multispace' between individuals (rows) and variables (columns) as well as its capacity for dealing with all types qualitative and quantitative data" (p. 95).
Although CA functions much the same as PCA (reducing data to a few dimensions while accounting for the maximum variance possible), at least two aspects distinguish it from PCA: (1) the use of chi2 distances as a measure of similarity (as opposed to the contentious product-moment correlations), and (2) the simultaneous computation of elements and constructs in the same mathematical space. The latter is an important characteristic as the CA results in a single multispace that includes constructs as well as elements. The result is that a real joint representation can be produced. Recommended by Rivas (1981; Rivas & Marco, 1985), the CA has emerged as the only method employed in the analysis of the repertory grid that produces a joint and coherent mathematical treatment of constructs and elements. As a result, the graphs shown allow for an all-embracing and more precise interpretation enhanced by the plotting of left construct pole, right construct pole, and element labels.
The first data provided by CA are the eigenvalues (Figure 1) of the five axes calculated by the programme. These axes are equivalent to the factors or components found in PCA with the difference that they address the variance of both elements and constructs (as indicated earlier). The first line shows the analysis title and date. The second line shows the trace and the population ("Pob. total")[5]. This is then followed by the characteristics of each axis, such as eigenvalues and the percentage of variance that they account for, as well as the cumulative variance with the addition of each axis. A histogram is also provided to give a visual notion of the proportions between axes and the relative importance of each.
In Daniel's case, the first axis accounts for a total of 45.78% variance while the other two axes explain much less variance and bear similar size (accounting for 19.37% and 15.51% variance, respectively). This indicates that only three axes are responsible for 80% of the variance in the grid data. In this case, therefore, CA has successfully simplified the overall picture without losing too much information. Psychological meaning in terms of cognitive complexity and differentiation is then attributed to the percentage variance of the first axis, a consideration that will be discussed in the section on indices of cognitive structure.
As we can see in Table 3, the GRIDCOR programme also provides the coordinates and contributions for elements and for constructs (specified by their two poles). These data are the source for the construction of the graphics for the axes that the GRIDCOR programme provides (see below). In the case of Daniel, these tables show that the IDEAL, PREVIOUS THERAPIST, SELF and NON-GRATA elements are more determinant of the first axis (22, 22, 16, and 16, respectively). However, examination of the coordinates shows that the first two elements are placed on the opposite sides of the axes in relation to the last two. This indicates that, although the four elements are important in determining the first axis, they are so in opposing ways. We can also see elements that have no relation whatsoever to the first axis, although they may play a role in other axes. For example, the FATHER and FRIEND elements are decisive in the second axis (although in opposing ways). To interpret an element or construct such as, for example, the SELF element, it is necessary to examine the axis where it has the most weight (the first, the fourth and the fifth to a lesser degree), and the degree to which the axes account for it (relative contribution). It is probably inappropiate to interpret an element by referring to those axes where it has little weight (e.g., the second and third axes). We could examine this table in much more detail, as it contains important CA information, but for interpretative purposes it is better to focus on the graphic displays which give us the same information in a simpler and more visual form.
Dual diagrams simultaneously show both the elements and the constructs that are located on the same axis. Figure 2 shows the printout of the dual diagram of the first axis of Daniel's grid, which graphically shows the labels that he provided. We can see that the IDEAL and PREVIOUS THERAPIST elements are very close to each other and load in the same direction as the "happy," "relaxed" and "responsible" construct poles. Not far from this end of the axis is the element SPOUSE and the construct poles "committed" and "sexually healthy." On the opposite end of the axis, the elements SELF and NON-GRATA are strongly associated to the construct poles "intolerant," "irresponsible" and "perverse." As far as the remaining elements and constructs are concerned, it would be erroneous to infer their meaning from this axis as they neither contribute to it nor explain it significantly. The COR (coordinate), CA (weight) and CTR (relative contribution) values are shown to the left of every element label and to the right of each construct pole label.
Each axis must be understood as a dimension of meaning defined by the construct poles and by the elements that appear on the extremes of the poles. The structure of this first axis suggests that Daniel has a very negative self image. He places himself on the exact opposite of what he considers to be the IDEAL, defining himself as irresponsible, intolerant and sexually perverse. Similarly, the self is associated with the NON-GRATA element which corresponds to someone he does not like. The element SELF BEFORE THE CRISIS is closer to the IDEAL than the current self which indicates that the anxiety crisis has modified his self perception slightly, although it appears that it was already generally negative and unsatisfactory (at least as perceived by the subject and indicated by his having sought therapy before and participated in group therapy sessions).
The graphical representation of an axis also shows the relationships among constructs, between elements and constructs, and among elements. Although partially explained in the previous paragraph, it is worth mentioning, as an example, that the constructs "relaxed," "responsible," and "happy" are grouped together, as are "irresponsible," "intolerant," and "perverse." Some of these connections are of particular interest. We could infer, for example, that irresponsibility, as conceived by Daniel, is very closely associated to a "perverse" sexuality. This gives us a clue as to a very important area to explore with the client. On the other hand, the strong association between the IDEAL and PREVIOUS THERAPIST elements could suggest a degree of idealisation of the therapist, or transference issues, which would otherwise be difficult to assess from the psychoanalytic perspective.
Although the dimensions of meaning represented by the axes point to certain connections, the best place to look for general relationships among constructs or elements is the coorrelation or the distance matrices. GRIDCOR can display dual graphs for each axis, thereby making them easier to understand. Nevertheless, no other axis of the grid will be analysed in this text as the logic behind the exercise has already been shown. However, dual diagrams should be consulted for any axis that accounts for a substantial percentage of the variance (e.g., 15% or more).
The GRIDCOR programme also enables two axes to be put in graph form by placing them orthogonally. The joint graph of the two axes (Figure 3) creates a two-dimensional dotted space, including both construct and elements labels. This is basically a simplified version of the multispace created by CA. It is more valuable as a way of obtaining an overall picture of the axes rather than of giving a precise location for the dots it produces (which can already be seen in separate graph of each axis).
Although the conjoint representation of constructs and elements is the most common, the GRIDCOR programme gives us the option to plot just the elements. This element graph, represented in the two-dimensional space of the two axes accounting for the most of variance, is a kind of "spacegram" where the most significant people in the subject's world are located. An additional advantage of this graph is that it can be easily shared with the client for therapeutic discussion. In terms of this particular element graph (see Figure 4), a couple of points not so clearly visible in other graphs are worth discussing. The FATHER and SELF elements, for example, are in opposite quadrants. Also, the women of his generation (FEMALE FRIEND, SISTER and SPOUSE) are all located in the same quadrant.
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