Modified FF tests for assessing the multiple correlation between one spatial process and several others

The modified t   test presented by Dutilleul [1993. Modifying the tt test for assessing the correlation between two spatial processes. Biometrics 49, 305–314] to assess the simple correlation between two spatial processes is built on the sample correlation coefficient calculated from partial realizations, and requires the estimation of an effective sample size appropriately defined. The autocovariances of processes are taken into account via the effective sample size, both in the evaluation of the test statistic and in the calculation of the probability of significance.In this article, we present modified F   tests to assess the multiple correlation between one spatial process and several (i.e., qq) others, from partial realizations collected at the same sampling locations. An effective sample size is used in all these modified F tests. Its theoretical expression is obtained from the expected value of the coefficient of determination in the absence of multiple correlation, under the bivariate Gaussian and first-order stationarity assumptions for the ‘dependent process’ and the ‘predicted process’ (i.e., the ordinary least-squares (OLS) regression predictor defined from the q other processes). The modified F tests presented differ in the procedure (i.e., model-based versus model-free) followed to estimate the autocovariance matrices appearing in the expression of the effective sample size. The validity and power of the new testing procedures are studied through simulations, for various sampling schemes and combinations of the autocovariance structure and number of sampling locations. An example with real data is given. In closing, we discuss the theoretical and practical aspects of our modified F tests in multiple correlation analysis, in comparison with those used in the repeated measures ANOVA.