Contingency Table - Measures of Association

Measures of Association

The degree of association between the two variables can be assessed by a number of coefficients: the simplest is the phi coefficient defined by

,

where χ2 is derived from Pearson's chi-squared test, and N is the grand total of observations. φ varies from 0 (corresponding to no association between the variables) to 1 or -1 (complete association or complete inverse association). This coefficient can only be calculated for frequency data represented in 2 x 2 tables. φ can reach a minimum value -1.00 and a maximum value of 1.00 only when every marginal proportion is equal to .50 (and two diagonal cells are empty). Otherwise, the phi coefficient cannot reach those minimal and maximal values.

Alternatives include the tetrachoric correlation coefficient (also only applicable to 2 × 2 tables), the contingency coefficient C, and Cramér's V.

C suffers from the disadvantage that it does not reach a maximum of 1 or the minimum of -1; the highest it can reach in a 2 x 2 table is .707; the maximum it can reach in a 4 × 4 table is 0.870. It can reach values closer to 1 in contingency tables with more categories. It should, therefore, not be used to compare associations among tables with different numbers of categories. Moreover, it does not apply to asymmetrical tables (those where the numbers of row and columns are not equal).

The formulae for the C and V coefficients are:

and

,

k being the number of rows or the number of columns, whichever is less.

C can be adjusted so it reaches a maximum of 1 when there is complete association in a table of any number of rows and columns by dividing C by (recall that C only applies to tables in which the number of rows is equal to the number of columns and therefore equal to k).

The tetrachoric correlation coefficient assumes that the variable underlying each dichotomous measure is normally distributed. The tetrachoric correlation coefficient provides "a convenient measure of correlation when graduated measurements have been reduced to two categories." The tetrachoric correlation should not be confused with the Pearson product-moment correlation coefficient computed by assigning, say, values 0 and 1 to represent the two levels of each variable (which is mathematically equivalent to the phi coefficient). An extension of the tetrachoric correlation to tables involving variables with more than two levels is the polychoric correlation coefficient.

The Lambda coefficient is a measure of the strength of association of the cross tabulations when the variables are measured at the nominal level. Values range from 0 (no association) to 1 (the theoretical maximum possible association). Asymmetric lambda measures the percentage improvement in predicting the dependent variable. Symmetric lambda measures the percentage improvement when prediction is done in both directions.

The uncertainty coefficient is another measure for variables at the nominal level.

The values range from -1 (100% negative association, or perfect inversion) to +1 (100% positive association, or perfect agreement). A value of zero indicates the absence of association.

• Gamma test: No adjustment for either table size or ties.
• Kendall tau: Adjustment for ties.
  • Tau b: For square tables.
  • Tau c: For rectangular tables.