### Coefficients for Measuring Association

The following are a few of the many measures of association used with chi-square and other contingency table analyses. When using the chi-square statistic, these coefficients can be helpful in interpreting the relationship between two variables once statistical significance has been established. The logic for using measures of association is as follows:

Even though a chi-square test may show statistical significance between two variables, the relationship between those variables may not be substantively important. These and many other measures of association are available to help evaluate the relative strength of a statistically significant relationship.  In most cases, they are not used in interpreting the data unless the chi-square statistic first shows there is statistical significance (i.e., it doesn't make sense to say there is a strong relationship between two variables when your statistical test shows this relationship is not statistically significant).

Nominal and Ordinal Variables

Phi

Only used on 2x2 contingency tables. Interpreted as a measure of the relative (strength) of an association between two variables ranging from 0 to 1. Pearson's Contingency Coefficient (C)

It is interpreted as a measure of the relative (strength) of an association between two variables. The coefficient will always be less than 1 and varies according to the number of rows and columns. Cramer's V Coefficient (V)

Useful for comparing multiple X2 test statistics and is generalizable across contingency tables of varying sizes. It is not affected by sample size and therefore is very useful in situations where you suspect a statistically significant chi-square was the result of large sample size instead of any substantive relationship between the variables. It is interpreted as a measure of the relative (strength) of an association between two variables. The coefficient ranges from 0 to 1 (perfect association). In practice, you may find that a Cramer's V of .10 provides a good minimum threshold for suggesting there is a substantive relationship between two variables. where    q = smaller # of rows or columns

Describing Strength of Association

Characterizations

>.5                   high association

.3 to .5             moderate association

.1 to .3             low association

0 to .1              little if any association

Proportional Reduction of Error (PRE)

Lambda

This is a proportional reduction in error (PRE) measure that ranges from 0 to 1.  Lambda indicates the extent to which the independent variable reduces the error associated with predicting the value of a dependent variable. Multiplied by 100, it represents the percent reduction in error.

Ordinal Variables Only

Gamma

Another PRE measure ranging from -1 to 1 that estimates the extent errors are reduced in predicting the order of paired cases. Gamma ignores ties.

Kendall’s Tau b

Similar to Gamma but includes ties.  Ranges from -1 to 1 but since standardization is different from Gamma, it provides no clear explanation of PRE.

Inter-rater Agreement

Cohen’s Kappa

Measures agreement beyond chance. Although a negative value is possible, it commonly ranges from 0 to 1 (perfect agreement). This measure requires a balanced table where the number of rows is the same as the number of columns.  The diagonal cells represent agreement. Web www.acastat.com