Cross-correlational analysis represents an experimental and exploratory form of data analysis based upon the manipulation, comparison and inter-correlation of different correlational matrices that are derived the correlation coefficients which measure the degree to which equal sized sets of variables move together, regardless of their relative magnitude. In short, cross-correlational analysis takes two or more correlation matrices of equal dimensionality, and rearranges the data in order to construct a correlational matrix based upon the aligned first order matrices--alternatively, it takes different, equal sized sections of the same correlational matrix, and compares these by a second order correlation matrix. Sometimes third and even fourth order matrices can be constructed out of previous intercorrelation matrices, especially if such analysis is strongly motivated by theory and the occurrence of a significant patterning of the correlation. The first and second order correlation matrix can be reconstructed in different ways, or analyzed in alternative ways, enabling different kinds of intercorrelational matrices to be derived. For instance, rows and columns of a first order matrix, if equal in number, may be compared in a subsequent table based upon their intercorrelation--they may be rank-ordered and compared in this manner in an alternative form of correlational matrix. Techniques of cross-correlation lead to different ways of representing data and the relationships between sets of data that enable us to draw inferences systematically and statistically evaluate certain kinds of inferences about such data, as if the data were nominal, ordinal or rank order. It enables us to see patterns of interrelationship between sets of data which would not otherwise be apparent, and to infer from such patterns the likelihood and degree to which a functional relationship may cohere between the data points.

In a sense, the information contained within an intercorrelational matrix is fundamentally different from that of the usual correlational matrix, as it demonstrates the degree to which each part is related to and representative of the whole set of variables--information about the total set of relationships between all the different points is contained within such a matrix, and this information itself can be rank-ordered or arranged in meaningful ways that allow us to make inferences about the data.

Any set of points may be correlated with any other set of points, whether an actual relationship exists between such points or not. The only stipulation of different sets of data in their correlation is that they be of the same magnitude in terms of equal number of degrees of freedom. Though anything may be correlated with anything else, it is also the case that when and if a functional relationship exists between two or more sets of data, then definite patterns of correlation will be apparent and will tend to be statistically significant, and will take on certain characteristic patterns that will not be evident among unrelated and basically random sets of data. Thus, techniques of intercorrelation can be systematically exploited for the discovery of hidden functional relationships between different kinds of data, and for providing insight into the causal direction and theoretical significance of such relationships. Correlation and intercorrelation matrices can also be converted into discrimination tables or histograms, or used in different forms of data analysis and representation, such as multidimensional scaling, cluster analysis or factor analysis.

Cross-correlational analysis can be time-consuming and therefore costly. The number of possible intercorrelational matrices with can be constructed and derived grows exponentially with the size and number of sets of data. It can yield a great deal of information at a level of complexity which is sometimes difficult to decipher or interpret theoretically. To some extent these draw backs can be offset by the introduction of systematic means of cross-correlational search and analysis, by the close coordination of such analysis with theoretical interpretation and inference derivation, and, finally, by the construction of an integrated data-base or program which can take over and exhaustively conduct such analysis on raw data.

But the reward of conducting such analysis and search is not to be measured in terms of its cost or complexity, but in term of its productivity of theoretical insights, inferences and in providing a systematic means by which to construct a sophisticated rule-based inference engine derived from the patterning of the data. These advantages outweigh the disadvantages.

Cross-correlational analysis provides a context for the integration of quantitative and qualitative forms of information, and for the inductive construction, operationalization and validation of theory and topical domains which are otherwise nonquantitative and only qualitatively represented. To the extent to which such a form of analysis is ultimately tethered to a specific set or range of data sets which empirically rooted and measurable, such analysis can be claimed to be empirically consistent and representative of actual, if hidden, relationships of patterning in the data.

The possibility of cross-correlational analysis rests upon the following premises:

1. a strong correlation may represent a functional relationship, but a weak correlation represents a probable lack of a direct functional relationship.

2. a stronger correlation is more likely to represent such a relationship than a weaker one.

3. correlational sets of a matrix may be grouped and described statistically in a meaningful way--they form a curve the characteristics of which (mode, median, mean, etc.) may be evaluated.

4. a correlational matrix can be reorganized in different ways, resulting in different descriptions of the resulting tables--these values can be compared (i.e. rows versus columns).

5. The rows and columns of a correlational matrix represent hypothetical dimensions of analysis which can be topically characterized.

6. In a complex and large correlational matrix, even low correlational values may be significant indicators of a functional relationship.

A correlational matrix forms an uneven landscape of values which can be topographically mapped. It is the pattern of this landscape which yields insight into the functional organization of the data. This patterning is carried over into second and subsequent order cross-correlational matrices.

Basics of correlation

Table manipulation and Intercorrelation

Theoretical Interpretation

Fishers Conversion

Statistical procedures.

Graphing and representation

Functional inference and discrimination networks.

A cross-correlational database, search and inference engine.

Blanket Copyright, Hugh M. Lewis, © 2005. Use of this text governed by fair use policy--permission to make copies of this text is granted for purposes of research and non-profit instruction only.

Last Updated: 04/19/05