Mathematical Analysis

Once the grid has been administered, the resulting data matrix is potentially as complex as the rating system allows, and its size depends on the final number of elicited or provided constructs and elements used. These data represent the answers to many questions (as many as the boxes in the grid) and contain a great deal of information as to how the subject construes the elements used. We are therefore faced with a great deal of information which must be synthesized so that the basic structure can be retained without too much loss of information.

From the non-parametric factor analysis proposed by Kelly himself (1955) to the more modern multivariate analyses, many efforts have been made to mathematically synthesize the basic structure of rep-grid data so that it is useful to the investigator. This chapter briefly covers the most significant methods proposed and the resulting computer programmes. The contribution of correspondence analysis to the analysis of the repertory grid will be covered given the clear advantages that it has over other methods. Its inclusion in the GRIDCOR programme also allows for other analyses of great potential for use in the clinical field. Finally, Daniel's case will be studied, allowing for an appreciation of how psychological meaning can be derived from this mathematical data analysis.

Brief History of Mathematical Repertory Grid Analysis

Kelly (1955) proposed the first system for summarising grid data using dichotomous scores. As mentioned previously, this involves a form of non-parametric factor analysis that can be done by hand (his son, J.V. Kelly, created a computer programme in 1964 which is not currently operable on modern computers). Although we consider this to be a creative and original mathematical contribution on Kelly's behalf, it is limited to the dichotomically scored grid used then.

Besides suggesting the use of ordinal data, Bannister (1965a) proposed a very simple type of cluster analysis called the "anchor method" based on Spearman's rho correlations between constructs which can also be computer calculated. In a latter version of this method, included in the GAB programme (Higginbotham & Bannister, 1983), Pearson's product-moment "r" correlation is used to analyse grids with ordinal, interval or dichotomous data. The procedure consists of finding the construct that correlates the most with the remaining constructs (first component or factor) and using it as the horizontal axis. The vertical axis is the construct accounting for the next highest amount of variance but which does not correlate with the construct chosen as the horizontal axis (at a 5% significance level). The remaining constructs can then be plotted on the graph according to the coordinates taken from their correlations with the axis or constructs. The same procedure can be followed by calculating the correlations between columns or elements (see Fransella & Bannister, 1977). This method, though simple and ingenious, lacks the mathematical and descriptive power of the methods which we are about to discuss. Furthermore, it analyses constructs and elements separately, which makes it very difficult to relate them to each other.

Perhaps the most significant contribution to the history of repertory grid mathematical analysis was made by Slater and his INGRID programme (Slater, 1972). This programme involves the application of the Principal Components Analysis (PCA) to rep-grid data with interval scores. Basically, this mathematical procedure translates a number of variables (elements or constructs) into a lesser number of hypothetical variables (components or factors) which explain the maximum possible variance. The resulting components can be used as the axes where the constructs are plotted according to their factor loadings. It can analyse constructs and elements separately but does not perform joint mathematical analyses. Although the INGRID programme projects the elements onto the construct graph, this is merely a statistical artifact. This is one of the main disadvantages of PCA. There are three main problems related to its being based on product-moment correlations:

  1. Lineal correlations detect the similarities between two variables based on their general pattern but not their proximity.
  2. The same product-moment correlation can hide very distinct data distributions as a scatter analysis of the data would reveal. For example, the results can be very heavily influenced by outliers (scattered values).
  3. Some authors question the appropiateness of using product-moment correlations on interval data (they accept it only for ratio data).

Despite this, correlational techniques and particularly factor analysis/PCA are so extensively used in psychological studies (and their results generally considered valid within the scientific and psychological communities) that the PCA has been the most widely used type of analysis for the repertory grid. Apart from INGRID, other programmes such as CIRCUMGRIDS (Chambers & Grice, 1986), G-PACK (Bell 1987), FLEXIGRID (Tschudi, 1993) and REPGRID (Shaw, 1989) have been created and include PCA within their options as do more general statistical packages such as the SPSS and the BMDP (now also available for PCs).

Within multivariate methods, cluster analysis and multidimensional scaling have also been suggested (besides PCA) as appropiate for the analysis of grid data. Although the latter has not been widely adapted to computer programmes and not widely used, the former has been the focus of considerable attention. As its name suggests, cluster analysis shows the ways in which variables (in this case constructs or elements) are grouped as cluster trees. As in PCA, constructs and elements are analysed separately. However, cluster analysis differs from PCA in that it uses distance coefficients (Euclidian, "city block," etc.) as a measure of the association between variables. Despite the fact that the use of distance measures is more mathematically appropiate for measuring the proximity between variables (i.e., constructs or elements), it creates a serious dilemma when constructs are concerned: it handles what is really a bipolar dimension (the construct) as if it were a single unit. As they are bipolar dimensions, two highly negatively correlated constructs (which indicates a strong similarity between the right pole of one construct and the left pole of the other construct, and viceversa) would appear to have a high distance coefficient, thus indicating independence between the constructs. Therefore, high distance coefficients make it difficult to determine whether they indicate inverted association or independence.

Unlike G-PACK (and other general statistical packages such as SPSS, BMDP and CLUSTAN) which simply analyse the clusters, FOCUS (Thomas and Shaw, 1976) and REPGRID (Shaw, 1989) programmes invert the two poles of the constructs by replacing scores (e.g., a "1" becomes a "7") in order to avoid this problem by establishing a single direction of construing.[3] The GRIDCOR programme is structured in the same way, checking the constructs and redirecting them in order to facilitate their analysis.

Actually, inverting the construct poles is part of what is known as grid focusing. This procedure, which is included in the FOCUS programme, (1) inverts the same construct poles to prevent changes in direction, and (2) rearranges the rows (constructs) and the columns (elements) so that the most similar (as determined by a previous cluster analysis) are grouped together. The results of these alterations can be seen on Table 2, which shows a grid with dichotomous scores where the C and E constructs have been inverted and the constructs and elements are rearranged to make it easier to spot similarities without having to pour over mathematical data.

This "clean" presentation is based on a two-way cluster analysis of rows and columns. The main advantage of this procedure is the clarity with which any similarities appear without the aid of any graphic or mathematical support. What remains is the protocol created by the subject during the interview which has been rearranged to make its structure clearer. This is the ideal type of analysis to use when the results are to be shared with the interviewee, as it does not require any more comprehension than that which is necessary for the completion of the protocol. It is, therefore, potentially useful in clinical as well as educational or business contexts. Data focusing has been included in the GRIDCOR programme due to the growing interest in this procedure.