Cramér’s V

Tests of Nominal-Variable Associations

In order to understand the utility of Cramér’s V, it is important to understand the ways in which statistical tests of significance differ from measures of association for categorical variables. The chi-squared (χ2) test for independence provides a statistical test of association between two categorical (nominal) variables from a single population. The test is used to determine whether the association between two variables is significant, with the null hypothesis being that the two variables are not dependent on one another.

In chi-square analysis, expected frequencies are generated following the null hypothesis and compared against the observed frequencies. The chi-square statistic tests whether one variable is independent of the other variable. If the fit is good—that is, if the difference between the expected frequencies and the observed frequencies is small—the chi-square statistic will be small and one would conclude that the two variables are independent. Conversely, a poor fit yields a large chi-square statistic and rejection of the null hypothesis, and suggests the two variables are related.

Although a significant chi-squared statistic suggests a relationship exists between two variables, it does not describe the strength of association. Variation in the size of the chi-square statistic influences the level of confidence in rejecting or retaining null hypotheses (p values), but higher and lower chi-square statistics do not necessarily correspond with varying strengths of associations. Because the chi-square statistic is sensitive to sample size, very weak relationships can produce very large chi-square values in large sample sizes. For example, flipping a coin only 10 times and correctly predicting two of four heads-up results and five of six tails-up results yields a chi-square statistic of 1.27 and p-value of .260, whereas flipping a coin 1,000 times with results of the same proportions—i.e., correctly predicting 200 of the 400 heads and 500 of the 600 tails—yields a chi-square statistic of approximately 127 and p-value of less than .001. As this example illustrates, despite the proportions of frequencies remaining the same, the chi-square statistic and its corresponding probability vary as a function of sample size.

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