Effect size for contingency tables

While the P value from a chi-square or Fisher's test tells you whether an association between variables is statistically significant, it doesn't tell you how strong that association is. P values are heavily influenced by sample size: a weak association can be statistically significant with a large enough sample, while a strong association might not reach significance with a small sample.

Effect size measures like Cramér's V and the Phi coefficient quantify the strength of association between categorical variables, independent of sample size. These values help you determine whether a statistically significant result might also be scientifically meaningful.

The Phi coefficient for 2×2 tables

For 2×2 contingency tables, Prism provides the option to report the Phi coefficient. The Phi coefficient ranges from 0 to 1, where:

•0 indicates no association between the variables

•1 indicates a perfect association between the variables

the Phi coefficient can be calculated directly from the cell frequencies. For a 2×2 table with cells labeled as:

Phi is calculated as:

The Phi coefficient is mathematically related to the chi-square statistic by the following relationship:

where N is the total sample size.

However, because Phi can be calculated directly from the cell frequencies in your data table, Prism will report the same value of Phi for the same data whether you're performing a chi-square test or Fisher's exact test.

Cramér's V for larger tables

For contingency tables larger than 2×2, Prism reports Cramér's V. Like Phi, Cramér's V ranges from 0 to 1 and measures the strength of association between the row and column variables.

Cramér's V is calculated directly from the data in your table, adjusted for the table dimensions. While Cramér's V is related to chi-square through the formula:

where N is the total sample size, r is the number of rows, and c is the number of columns. This calculation ultimately only depends on the cell frequencies in your table. Like Phi, Cramér's V can be reported with P values from either the chi-square or Fisher's exact tests.

Note that for 2×2 tables, Cramér's V is equal to the Phi coefficient, so Prism only reports Phi. While for larger tables, Prism only reports Cramér's V as this adjusts for the dimensions of the table making it a more appropriate measure.

Interpreting the values

Generally, values of Phi and Cramér's V greater than 0.5 are considered to indicate a meaningful association between the column and row variables of your table. However, as with all effect sizes, these results should be considered only in the context of the other results of the analysis (including the P values, confidence intervals, and other values such as odds ratios, or relative risk values).

Note that - unlike P values - Phi and Cramér's V are not affected by sample size. If you doubled the number of subjects in each cell of your table while maintaining the same proportions, your chi-square statistic would double and your P value would become smaller, but Phi and Cramér's V would remain unchanged. This demonstrates why effect sizes are particularly valuable for comparing the strength of association between variables across different studies with potentially different sample sizes.