When both variables are dichotomous, which correlation coefficient is appropriate?

When both variables are dichotomous, which correlation coefficient is appropriate?

• A. Spearman Correlation
• B. Pearson Correlation
• C. Phi-Coefficient
• D. Point-Biserial Correlation

Answer: (C)

When both variables are dichotomous, the Phi coefficient is the natural measure of association because it is specifically designed for two binary variables. It can be viewed as the Pearson correlation computed on data coded as 0 and 1, so it shares the familiar interpretation of a correlation that ranges from -1 to 1 and reflects the strength and direction of the association. The calculation uses the four cells of the 2x2 table (the counts where both are 1, where the first is 1 second is 0, where the first is 0 second is 1, and where both are 0) and is given by phi = (ad - bc) / sqrt((a+b)(c+d)(a+c)(b+d)). This formulation captures how far the observed co-occurrence deviates from what would be expected if the variables were independent. This makes Phi the right choice here, whereas Spearman relies on ranks (more appropriate for ordinal data), Pearson assumes continuous, normally distributed variables, and the point-biserial correlation is used when one variable is binary and the other is continuous.

When both variables are dichotomous, the Phi coefficient is the natural measure of association because it is specifically designed for two binary variables. It can be viewed as the Pearson correlation computed on data coded as 0 and 1, so it shares the familiar interpretation of a correlation that ranges from -1 to 1 and reflects the strength and direction of the association.

The calculation uses the four cells of the 2x2 table (the counts where both are 1, where the first is 1 second is 0, where the first is 0 second is 1, and where both are 0) and is given by phi = (ad - bc) / sqrt((a+b)(c+d)(a+c)(b+d)). This formulation captures how far the observed co-occurrence deviates from what would be expected if the variables were independent.

This makes Phi the right choice here, whereas Spearman relies on ranks (more appropriate for ordinal data), Pearson assumes continuous, normally distributed variables, and the point-biserial correlation is used when one variable is binary and the other is continuous.