A new method for multivariate analysis of rank order data

Rank order data is common in social science applications, and a need exists for multivariate methods that can be used with such data. This research develops methodologies that generate principal component, factor analysis and canonical correlation solutions from Spearman and Kendall rank correlation matrices. A primary feature of these methodologies is that the sample rank correlations are transformed to make them more accurate estimates of the population correlations. These transformations are based on relationships between three population measures of correlation: product-moment correlation, grade correlation and population concordance. The rank-based solutions are obtained by substituting a rank correlation matrix (untransformed or transformed) for the Pearson correlation matrix in a standard multivariate algorithm. Both the population solutions and the Pearson solutions served as benchmarks for comparison. Monte Carlo simulation using specially structured population matrices, as well as analyses of real data sets, demonstrated that transformed Kendall matrices behave very much like Pearson correlation matrices when used in principal/factor analysis. Transformed Spearman matrices were also effective for certain population matrix structures. Results for canonical correlation analysis were less clear; Spearman matrices (transformed and untransformed) appeared to be most effective. One difficulty in canonical correlation analysis using transformed rank correlation matrices is that the transformations can produce a matrix that is not positive semi-definite. An algorithm was developed that was somewhat successful in correcting this problem. Overall, results are promising; multivariate analysis of rank order data is reasonable, and often produces results as good as or better than analyses of interval scale data.