Regression and correlation are two interrelated statistical tools used to analyze the strength and direction of a linear relationship between two variables. While regression focuses on predicting the value of one variable from another, correlation measures the degree of association between them. This article explains how the correlation coefficient (r) is derived mathematically from the regression coefficients of X on Y and Y on X.
Key Concepts Recap
- Regression Coefficient (β): Measures the rate of change of the dependent variable with respect to the independent variable.
- Correlation Coefficient (r): Measures the strength and direction of a linear relationship between two variables. Ranges from -1 to +1.
Mathematical Derivation of r from Regression Coefficients
The correlation coefficient can be obtained using the formula:
r = √(βyx × βxy)
Where:
- βyx is the regression coefficient of Y on X
- βxy is the regression coefficient of X on Y
This formula is valid when both regression coefficients have the same sign. If they have opposite signs, the result is not valid in real-world linear regression analysis, as it indicates a contradiction in the direction of relationship.
Assumptions
For the derivation and use of this formula to be valid, the following assumptions are made:
- There exists a linear relationship between the variables X and Y.
- The variables are measured on an interval or ratio scale.
- Each regression line is based on minimizing the sum of squares of vertical deviations (least squares method).
- Regression coefficients are calculated from the same dataset.
- The direction of causality is not implied in correlation.
Step-by-Step Derivation
We know from the definition:
βyx = r × (σy / σx)
βxy = r × (σx / σy)
Multiplying both:
βyx × βxy = r² × (σy / σx) × (σx / σy)
βyx × βxy = r²
Therefore,
r = √(βyx × βxy)
Example
Given:
- Regression coefficient of Y on X, βyx = 0.6
- Regression coefficient of X on Y, βxy = 0.75
To Find: Correlation coefficient (r)
r = √(0.6 × 0.75)
r = √(0.45)
r ≈ 0.6708
Interpretation of Result
The correlation coefficient r ≈ 0.6708 indicates a moderately strong positive linear relationship between the variables X and Y.
Merits and Demerits
Merits:
- Provides a simple and efficient way to estimate correlation when regression coefficients are known.
- Reinforces the theoretical connection between regression and correlation.
- Reduces computational effort when working from raw or summary data.
Demerits:
- Only valid when both regression coefficients are calculated from the same dataset.
- Assumes linearity, which may not always be true in practical scenarios.
- Fails if one of the regression coefficients is zero or undefined.
- Cannot interpret causality, only association.
Practical Application Cases
- Economics: To relate income (X) and consumption (Y), and to understand how consumption is associated with changes in income.
- Marketing: Correlation between advertising expenditure (X) and sales (Y).
- Education: To evaluate the association between hours of study (X) and academic performance (Y).
- Healthcare: Correlation between patient age (X) and recovery time (Y).
Conclusion
The derivation of the correlation coefficient from regression coefficients is a powerful illustration of how deeply connected the two concepts are. By understanding the relationship between βyx, βxy, and r, students can not only solve numerical problems efficiently but also gain a conceptual understanding of how statistical variables interact in the real world. This topic is crucial for UGC NET Commerce and provides a solid foundation for further topics in Business Statistics.