Tests of randomness: unidimensional and multidimensional

A variety of tests for randomness are reviewed based on simple product-moment statistics defined between two matrices, {<i>a_ij</i>} and {<i>b_ij</i>}. Typically, the first matrix, {<i>a_ij</i>}, contains proximity data on the spatial placement of <i>n</i> observations, {<i>x</i>₁, ..., <i>x_n</i>}; the second matrix, {<i>b_ij</i>}, is obtained from the relationships among the <i>n</i> observations themselves. Depending on the definitions of {<i>a_ij</i>} and {<i>b_ij</i>}, a variety of well-known tests of randomness and/or trend (those attributed to Mann, Daniels, Moore and Wallis, Cox and Stuart, and Goodman and Grunfeld) as well as various approaches to serial correlation and spatial autocorrelation can be encompassed. Finally, these notions of randomness can be extended to an assessment of spatial association between two variables. A numerical example is given within this latter context.