The Mantel test versus Pearson's correlation analysis : Assessment of the differences for biological and environmental studies

The space-time clustering procedure of Mantel was originally designed to relate a matrix of spatial distance measures and a matrix of temporal distance measures in a generalized regression approach. The procedure, known as the Mantel test in the biological and environmental sciences, includes any analysis relating two distance matrices or, more generally, two proximity matrices. In this paper, we discuss the extent to which a Mantel type of analysis between two proximity matrices agrees with Pearson's correlation analysis when both methods are applicable (i.e., the raw data used to calculate proximities are available). First, we demonstrate that the Mantel test and Pearson's correlation analysis should lead to a similar decision regarding their respective null hypothesis when squared Euclidean distances are used in the Mantel test and the raw bivariate data are normally distributed. Then we use fish and zooplankton biomass data from Lake Erie (North American Great Lakes) to show that Pearson's correlation statistic may be nonsignificant while the Mantel statistic calculated on nonsquared Euclidean distances is significant. After small-size artificial examples, seven bivariate distributional models are tried to simulate data reproducing the difference between analyses, among which three do reproduce it. These results and some extensions are discussed. In conclusion, particular attention must be paid whenever relations established between proximities are backtransposed to raw data, especially when these may display patterns described in the body of this paper.