Strength of Association

The strength of association shows how much two variables covary and the extent to which the INDEPENDENT VARIABLE affects the DEPENDENT VARIABLE.

In summarizing the relationship between two VARIABLES in a single summary STATISTIC, the strength of the association is shown by a value between 0 and 1. The larger the absolute value of the association measure, the greater the association between the variables. A value of 1 signifies a perfect association between the variables according to the association model that is being used. A value of 0 shows that there is a NULL relationship between the variables. Negative values for ordinal and numeric data occur when larger values of one variable correspond to smaller values of the other variable.

The strength of association between two variables corresponds to the TREATMENT difference in EXPERIMENTS. As such, it can be measured using ASYMMETRIC MEASURES, although it is more commonly measured with symmetric COEFFICIENTS whose value is the same regardless of which variable is the independent variable.

Many statistical measures of RELATIONSHIP have been proposed, and they differ in their assessment of strength of association. The “gold standard” is PEARSON'SR for numeric data, whose squared value shows the proportion of variance in one variable that can be accounted for by linear prediction from the other variable. Several measures for non-numeric variables can similarly be interpreted as PROPORTIONAL-REDUCTION-IN-ERROR measures (Costner, 1965). Whereas some of these measures attain their maximum value of 1 when there is a strong MONOTONIC relationship between the variables, others (such as YULE'S Q) require only a weak monotonic relationship, as when larger values on one variable correspond to equal or larger values on the other variable.

Pearson's r interprets statistical INDEPENDENCE as the condition for no relationship, as do such other coefficients as Goodman and Kruskal's TAU and Yule's Q. Weisberg (1974) distinguishes other null conditions for some coefficients. The LAMBDA COEFFICIENT for nominal variables, for example, has a value of 0 when the modal category on the dependent variable is the same for every category of the independent variable.

There are no strict criteria for evaluating whether an association is large or small, although relationships above .70 are usually considered large, and those below .30 are usually considered small.

The strength-of-association concept can be generalized to more than two variables. For example, a CANONICAL CORRELATION shows the strength of association between two sets of variables. The R-SQUARED statistic shows the strength of association represented by a REGRESSION EQUATION.