Association coefficients consist of a number used to indicate the degree to which two variables
or attributes are related. Two basic types are covariation measures and dis/similarity measures. Covariation measures (such as the Pearsonian product-moment correlation, r
) are based upon the product (multiplication) of the data values
, and indicate the extent to which the variables are associated (0 = none and 1 = perfect), and the direction in which the variables vary with each other (positive, where one variable increases with another, or negative where one variable decreases as the other increases). Association measures should be chosen by reference to the level of measurement
of the variables involved, and most measures of association retain the same value if the data values are legitimately transformed in accord with that level; thus, Kendall's rank-order correlation, s
, retains the same value if the data are given new values but in the same order as the original ones. Dis/similarity measures cover both similarity (or proximity) measures, where a high value means considerable likeness between the variables, and dissimilarity measures where a high value denotes considerable difference. Most association measures have the form basic value/norming factor
, as in u2 = v2/N
. The basic value represents the property of interest, and the norming factor is defined as the maximum value which the basic value can assume, thus ensuring the overall measure achieves a maximum of 1. The zero value normally represents either statistical independence (as in q, v2, u
, and s
) or absence of any common property (as in dis/similarity measures).
Statistical associations and measures of correlation
are not in themselves evidence of causal relationships, which must be identified by theoretical reasoning and models. In practice, statistical associations are often treated as equivalent to establishing causal links, with textbooks thus warning against spurious correlations
. Arguably, statistical measures become less important in the research process as our knowledge of causal mechanisms becomes more complete, although they remain useful for assigning quantitative values to a model of a causal process.