If r and R denote correlation and multiple correlation coefficient for the data set for X ₁, X ₂and X ₃. Which option is correct?

Question

If r and R denote correlation and multiple correlation coefficient for the data set for X 1, X 2and X 3. Which option is correct?

Prepp Team · Accepted Answer

Understanding Correlation Coefficients

In statistics, correlation coefficients measure the strength and direction of a linear relationship between two or more variables. The question discusses two types: simple correlation and multiple correlation.

Simple Correlation Coefficient (r): This measures the linear relationship between two variables, say \(X_i\) and \(X_j\), denoted by \(r_{ij}\). Its value ranges from -1 to +1.
Multiple Correlation Coefficient (R): This measures the linear relationship between a dependent variable (say \(X_1\)) and a set of independent variables (say \(X_2\) and \(X_3\)). It is denoted by \(R_{1.23}\) and represents the correlation between \(X_1\) and the best linear combination of \(X_2\) and \(X_3\). Its value ranges from 0 to +1.

Key Properties of Multiple Correlation

A crucial property relating simple and multiple correlation is that the multiple correlation coefficient \(R_{1.23}\) is always greater than or equal to the absolute value of any simple correlation coefficient involving the dependent variable (\(X_1\) in this case). Mathematically, this means:

\(R_{1.23} \ge |r_{12}|\)

\(R_{1.23} \ge |r_{13}|\)

Also, the square of the multiple correlation coefficient (\(R^2\)) for \(X_1\) on \(X_2\) and \(X_3\) can be calculated using the formula:

\(R_{1.23}^2 = \frac{r_{12}^2 + r_{13}^2 - 2 r_{12} r_{13} r_{23}}{1 - r_{23}^2}\)

We can use these properties to evaluate the given options.

Analyzing the Given Options

Let's examine each option provided and check if the values are consistent with the properties of simple and multiple correlation coefficients.

Option 1: \(r_{12} = 0.69\), \(r_{13} = 0.22\), \(r_{23} = 0.23\), \(R_{1.23} = 0.69\)
- Check property: \(R_{1.23} \ge |r_{12}|\) and \(R_{1.23} \ge |r_{13}|\). \(0.69 \ge |0.69|\) (True) \(0.69 \ge |0.22|\) (True)
- Let's calculate \(R_{1.23}\) using the formula: \(r_{12}^2 = 0.69^2 = 0.4761\) \(r_{13}^2 = 0.22^2 = 0.0484\) \(r_{23}^2 = 0.23^2 = 0.0529\) \(r_{12} r_{13} r_{23} = 0.69 \times 0.22 \times 0.23 = 0.034814\) \(2 r_{12} r_{13} r_{23} = 2 \times 0.034814 = 0.069628\) \(1 - r_{23}^2 = 1 - 0.0529 = 0.9471\) \(R_{1.23}^2 = \frac{0.4761 + 0.0484 - 0.069628}{0.9471} = \frac{0.5245 - 0.069628}{0.9471} = \frac{0.454872}{0.9471} \approx 0.48028\) \(R_{1.23} = \sqrt{0.48028} \approx 0.693\) The calculated value (\(\approx 0.693\)) is very close to the given value (0.69). This option appears valid.
Option 2: \(r_{12} = 0.21\), \(r_{13} = 0.22\), \(r_{23} = 0.23\), \(R_{1.23} = 0.20\)
- Check property: \(R_{1.23} \ge |r_{12}|\). \(0.20 \ge |0.21|\) (False) This option is inconsistent with the properties, as the multiple correlation is less than one of the simple correlations involving the dependent variable.
Option 3: \(r_{12} = 0.24\), \(r_{13} = 0.22\), \(r_{23} = 0.23\), \(R_{1.23} = 0.21\)
- Check property: \(R_{1.23} \ge |r_{12}|\). \(0.21 \ge |0.24|\) (False) This option is also inconsistent with the properties.
Option 4: \(r_{12} = 0.69\), \(r_{13} = 0.22\), \(r_{23} = 0.23\), \(R_{1.23} = 0.21\)
- Check property: \(R_{1.23} \ge |r_{12}|\). \(0.21 \ge |0.69|\) (False) This option is inconsistent with the properties.

Based on the analysis, only Option 1 provides a set of values consistent with the properties and formula relating simple and multiple correlation coefficients.

Conclusion

Comparing the properties and calculations for each option, Option 1 is the only valid combination of simple and multiple correlation coefficients provided.

Option	\(r_{12}\)	\(r_{13}\)	\(r_{23}\)	Given \(R_{1.23}\)	Property \(R_{1.23} \ge \max(\|r_{12}\|, \|r_{13}\|)\) Check	Calculated \(R_{1.23}\)
1	0.69	0.22	0.23	0.69	\(0.69 \ge \max(0.69, 0.22) = 0.69\) (True)	\( \approx 0.693 \)
2	0.21	0.22	0.23	0.20	\(0.20 \ge \max(0.21, 0.22) = 0.22\) (False)	-
3	0.24	0.22	0.23	0.21	\(0.21 \ge \max(0.24, 0.22) = 0.24\) (False)	-
4	0.69	0.22	0.23	0.21	\(0.21 \ge \max(0.69, 0.22) = 0.69\) (False)	-

Revision Table: Correlation Concepts

Term	Description	Range	Key Use
Simple Correlation (r)	Measures linear association between two variables.	[-1, +1]	Bivariate analysis
Multiple Correlation (R)	Measures linear association between one variable and a set of others.	[0, +1]	Multiple regression goodness-of-fit
\(R^2\)	Proportion of variance in the dependent variable explained by independent variables.	[0, +1]	Goodness-of-fit measure for multiple regression

Additional Information: Interpreting Coefficients

Interpreting correlation coefficients is essential in statistical analysis:

A simple correlation \(r\) close to +1 indicates a strong positive linear relationship.
An \(r\) close to -1 indicates a strong negative linear relationship.
An \(r\) close to 0 indicates a weak or no linear relationship.
A multiple correlation \(R\) close to +1 indicates that the independent variables together explain a large proportion of the variability in the dependent variable.
An \(R\) close to 0 indicates that the independent variables together explain very little of the variability in the dependent variable.
\(R\) is always non-negative because it measures the strength of the overall relationship, not its direction (which is captured by the regression coefficients).
The square of the multiple correlation coefficient, \(R^2\), is often preferred as it has a direct interpretation as the proportion of the variance in the dependent variable that is predictable from the independent variables.