If the regression coefficient of x on y and y on x are \(- \frac{1}{2}\) and \(- \frac{1}{8}\) respectively, then what is the correlation coefficient between x and y?
Understanding Regression and Correlation Coefficients
In statistics, regression coefficients measure the change in the dependent variable for a one-unit change in the independent variable. We are given two regression coefficients:
- Regression coefficient of x on y, denoted as \(b_{xy}\)
- Regression coefficient of y on x, denoted as \(b_{yx}\)
The correlation coefficient, denoted as \(r\), measures the strength and direction of the linear relationship between two variables, x and y. It ranges from -1 to +1.
Relationship Between Regression and Correlation Coefficients
There is a direct relationship between the correlation coefficient and the two regression coefficients. The correlation coefficient is the geometric mean of the two regression coefficients. The formula connecting them is:
\[r = \pm \sqrt{b_{xy} \times b_{yx}}\]The sign of the correlation coefficient (\(r\)) is always the same as the sign of the two regression coefficients (\(b_{xy}\) and \(b_{yx}\)). This means if both regression coefficients are positive, the correlation coefficient is positive. If both are negative, the correlation coefficient is negative. It is important to note that \(b_{xy}\) and \(b_{yx}\) must always have the same sign.
Calculating the Correlation Coefficient
We are given the following values:
- Regression coefficient of x on y, \(b_{xy} = -\frac{1}{2}\)
- Regression coefficient of y on x, \(b_{yx} = -\frac{1}{8}\)
Both regression coefficients are negative, which means the correlation coefficient \(r\) must also be negative.
Now, we can substitute these values into the formula:
\[r = - \sqrt{\left(-\frac{1}{2}\right) \times \left(-\frac{1}{8}\right)}\]First, multiply the two regression coefficients:
\[\left(-\frac{1}{2}\right) \times \left(-\frac{1}{8}\right) = \frac{1 \times 1}{2 \times 8} = \frac{1}{16}\]Now, take the square root of the result and apply the negative sign:
\[r = - \sqrt{\frac{1}{16}}\] \[r = - \frac{\sqrt{1}}{\sqrt{16}}\] \[r = - \frac{1}{4}\]Thus, the correlation coefficient between x and y is \(-\frac{1}{4}\).
Conclusion on Correlation Coefficient Value
The calculated correlation coefficient is \(-\frac{1}{4}\). This value falls between -1 and +1, which is consistent with the range of a correlation coefficient. Since the value is negative, it indicates a negative linear relationship between x and y; as one variable increases, the other tends to decrease.
Let's compare this result with the given options:
| Option | Value |
|---|---|
| 1 | \(-\frac{1}{4}\) |
| 2 | \(-\frac{1}{{16}}\) |
| 3 | \(\frac{1}{{16}}\) |
| 4 | \(\frac{1}{4}\) |
Our calculated value \(-\frac{1}{4}\) matches Option 1.
Revision Table: Key Concepts in Regression and Correlation
| Concept | Description | Notation | Relationship to Others |
|---|---|---|---|
| Regression Coeff. of x on y | Measures change in x for unit change in y | \(b_{xy}\) | \(r = \pm \sqrt{b_{xy} b_{yx}}\) \(b_{xy}\) and \(b_{yx}\) have same sign as \(r\) |
| Regression Coeff. of y on x | Measures change in y for unit change in x | \(b_{yx}\) | |
| Correlation Coeff. | Measures strength and direction of linear relationship | \(r\) | \(r^2 = b_{xy} \times b_{yx}\) -1 \(\le\) r \(\le\) 1 |
Additional Information: Properties of Correlation and Regression
- The correlation coefficient \(r\) is unitless and symmetric, meaning the correlation between x and y is the same as the correlation between y and x.
- The regression coefficients \(b_{xy}\) and \(b_{yx}\) are generally not equal and depend on the units of measurement of x and y.
- The product of the two regression coefficients is equal to the square of the correlation coefficient: \(b_{xy} \times b_{yx} = r^2\).
- If the correlation coefficient is 0, it indicates no linear relationship between the variables.
- If the absolute value of the correlation coefficient is 1 (\(r=1\) or \(r=-1\)), it indicates a perfect linear relationship, and the regression lines are identical.
- Regression analysis helps predict the value of one variable based on the value of another, while correlation analysis quantifies the association between them.
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