Describe Point-Biserial Correlation and Phi Coefficient

Introduction

Correlation is a measure of the relationship between two variables. In psychological research, when dealing with specific types of variables—particularly dichotomous ones (variables that take only two values)—special correlation methods are used. Two such techniques are the point-biserial correlation and the phi coefficient. These are valuable tools when analyzing the relationship between categorical and continuous data (point-biserial) or between two categorical dichotomous variables (phi coefficient).

Point-Biserial Correlation

The point-biserial correlation (r_pb) is used when one variable is continuous (e.g., test score) and the other is dichotomous (e.g., gender coded as 0 and 1). It is a special case of the Pearson product-moment correlation. The formula used is:

r_pb = (M₁ – M₀) / s_p * √(pq)

• M₁ and M₀ are the means for the two groups (coded 1 and 0)
• p is the proportion of the group coded 1
• q is the proportion of the group coded 0
• s_p is the pooled standard deviation

This correlation is appropriate when the dichotomous variable is naturally occurring (e.g., gender, passed/failed). Values range from -1 to +1, with 0 indicating no correlation.

Application

For example, in a study analyzing whether gender (male = 0, female = 1) is associated with academic performance, the point-biserial correlation would determine the strength and direction of this relationship.

Phi Coefficient (φ)

The phi coefficient is used when both variables are dichotomous. It is equivalent to the Pearson correlation coefficient but specifically adapted for binary variables. The formula is:

φ = (AD – BC) / √((A+B)(C+D)(A+C)(B+D))

Where A, B, C, and D are the frequencies from a 2×2 contingency table.

Application

For example, in a study to determine the relationship between smoking (yes/no) and disease status (present/absent), the phi coefficient quantifies the association.

Conclusion

Both the point-biserial correlation and phi coefficient are essential for analyzing relationships involving dichotomous variables. While point-biserial is ideal for continuous-dichotomous relationships, the phi coefficient suits dichotomous-dichotomous associations. Their correct use ensures valid conclusions in psychological and behavioral research, particularly in categorical data analysis.