Correlating variables with different scale types: A new framework based on matrix comparisons

Ecological variables may be expressed on four basic measurement scales (nominal, ordinal, interval or ratio), whereas circular variables and those combining a nominal state with other scale types are also common. However, existing methods are not suited to calculate correlations between all pairwise combinations of such variables, preventing the application of standard multivariate techniques. The essence of the new approach is to derive a so-called difference semimatrix for all pairs of observations for each variable, and then to calculate the matrix correlation based on two such semimatrices. The advantage of this function, termed d-correlation, is that comparisons are made on the same logical basis regardless of the measurement scale, allowing for the use of principal components analysis to visualize interrelationships among many variables simultaneously. Further advantages are that missing values in the data are tolerated and that the Gower index of dissimilarity between objects may also be computed. The use of the method is demonstrated on a small toy matrix, an artificial plant trait matrix and a large dataset summarizing ecological features of all vascular plant species of Sardinia, Italy. The source code in R and FORTRAN, and applications for three different operation systems, are provided for computations with results serving as input for other statistical software. The new computational framework will allow the comparison of any types of ecological traits in a mathematically meaningful manner. This option was not available earlier in the field of multivariate statistics, and the method is expected to receive applications in other subject areas as well in which many objects are described in terms of variables expressed on different measurement scales.