# Nominal vs. Nominal

## Part 3c: Effect size

We saw earlier that there is a significant association between the gender and marital status. However it was not the case that all men for example were married, and all women were divorced. To indicate the strength of the association Cramér's V (Cramér, 1946) is often used.

As for the interpretation for Cramér's V various rules of thumb exist but one of them is from Cohen (1988) who let's the interpretation depend on the degrees of freedom, shown in Table 1.

df* | negligible | small | medium | large |
---|---|---|---|---|

1 |
0 < .10 |
.10 < .30 |
.30 < .50 |
.50 or more |

2 |
0 < .07 |
.07 < .21 |
.21 < .35 |
.35 or more |

3 |
0 < .06 |
.06 < .17 |
.17 < .29 |
.29 or more |

4 |
0 < .05 |
.05 < .15 |
.15 < .25 |
.25 or more |

5 |
0 < .05 |
.05 < .13 |
.13 < .22 |
.22 or more |

The degrees of freedom used here is not the one from the chi-square test, but it is the minimum of the number of rows, and number of columns, then minus one. In the example we have 5 rows and 2 columns, so the minum of those is the 2. Then minus 1 gives the degrees of freedom (df*) as 1. With Cramér's V of .0094, this would make it negligible. We could add this to our report:

Gender and marital status showed to have a significant but negligible association, *χ*^{2}(4, *N* = 1941) = 16.99, *p* < .001, *V* = .09. A pairwise z-test post hoc analysis with Bonferroni correction revealed that only for widowed there was a significant difference between the male and female percentage, *p *< .05.

**Click here to see how to obtain Cramer's V, with SPSS, R (Studio), Excel, Python, an Online Calculator, or Manually**.

**with SPSS**

**with R (Studio)**

**with Excel**

**with Python**

**Online calculator**

Enter the requested information below:

**Manually (formulas and example)**

**Formulas**

The formula for Cramer's V is:

In this formula χ^{2} is the chi-square value, *n* the total sample size, *r* the number of rows (or categories in the 1st variable), and *c* the number of columns (or categories in the 2nd variable). MIN(r,c) simply indicates to take the minimum from r and c, so the lowest of the two.

**Example**

*Note* this is a different example than the one used in the rest of this section, but the same as the one used in the example of the manual calculation of the test.

We are given the following table with observed frequencies.

Brand | Red | Blue |
---|---|---|

Nike |
10 |
8 |

Adidas |
6 |
4 |

Puma |
14 |
8 |

There are three rows, so *r* = 3, and two columns, so *c* = 2. The total sample size is:

The chi-square value has also been calculated (see example in manual calculation of the test in the previous section):

The minimum of the rows and columns is 2 (the columns):

We can now fill out the formula for Cramer's V:

Now we can complete the report on our analysis on the next page.

**Two nominal variables**

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