Introduction

In linear regression analysis, the coefficient of determination, denoted as $R^2$, is a crucial metric that measures the proportion of the variability in the dependent variable that can be explained by the independent variable(s). $R^2$ provides valuable insights into the strength and quality of the relationship between variables. Beyond its interpretation as a measure of explained variance, $R^2$ can also be used to estimate the correlation coefficient between the dependent and independent variables, particularly in the case of linear regressions with a single explanatory variable. This chapter delves into the method of estimating the correlation coefficient using the $R^2$ measure and explores its significance in understanding the linear relationship between variables.

Estimating the Correlation Coefficient from $R^2$

When dealing with simple linear regression, where there’s only one explanatory variable $(X)$ and one dependent variable $(Y)$, the relationship between $R^2$ and the correlation coefficient $(r)$ is straightforward. The formula for calculating the correlation coefficient from the $R^2$ measure is as follows: $$r=\sqrt{R^2}$$ Where:

• $r$ represents the correlation coefficient between the dependent variable $(Y)$ and the independent variable (X)
• $R^2$ is the coefficient of determination obtained from the regression analysis

Example: Suppose we have a dataset that examines the relationship between the number of hours studied $(X)$ and the score achieved in an exam ($Y$). After performing a linear regression analysis, we obtain an $R^2$ value of 0.64. To estimate the correlation coefficient between the hours studied and the exam score, we apply the formula:

$$r=\sqrt{0.64}=0.8$$

Here, the calculated correlation coefficient (r) is 0.8, indicating a strong positive linear relationship between hours studied and exam score.

Significance and Interpretation:

Estimating the correlation coefficient from $R^2$ provides additional insight into the direction and strength of the linear relationship between variables. A correlation coefficient (r) close to 1 signifies a strong positive linear relationship, while a value close to -1 indicates a strong negative linear relationship. Values closer to 0 suggest a weak linear relationship.