  Measuring the (causal) direction and strength of dependence between two
variables (events), Xi and Xj , is fundamental for all science. Our survey of
decades-long literature on statistical dependence reveals that most assume
symmetry in the sense that the strength of dependence of Xi on Xj exactly
equals the strength of dependence of Xj on Xi. However, we show that such
symmetry is often untrue in many real-world examples, being neither necessary
nor sufficient. Vinod's (2014) asymmetric matrix R* in [-1, 1] of generalized
correlation coefficients provides intuitively appealing, readily interpretable,
and superior measures of dependence. This paper proposes statistical inference
for R* using Taraldsen's (2021) exact sampling distribution of correlation
coefficients and the bootstrap. When the direction is known, proposed
asymmetric (one-tail) tests have greater power.
