Calculating the Square of the Multiple Correlation Coefficient

The multiple correlation coefficient measures the linear relationship between a dependent variable and a set of independent variables. The square of the multiple correlation coefficient, denoted as \(R^2\), represents the proportion of the variance in the dependent variable that is predictable from the independent variables.

For three variables, say X, Y, and Z, the square of the multiple correlation coefficient when predicting one variable (e.g., X) from the other two (Y and Z) is given by the formula:

$$\rm \mathop R\nolimits_{x.yz}^2 = \frac{\mathop r\nolimits_{xy}^2 + \mathop r\nolimits_{xz}^2 - 2 \mathop r\nolimits_{xy} \mathop r\nolimits_{xz} \mathop r\nolimits_{yz}}{1 - \mathop r\nolimits_{yz}^2}$$

Where:

• \(\rm \mathop R\nolimits_{x.yz}^2\) is the square of the multiple correlation coefficient between X and the combination of Y and Z.
• \(\rm \mathop r\nolimits_{xy}\) is the simple correlation coefficient between X and Y.
• \(\rm \mathop r\nolimits_{xz}\) is the simple correlation coefficient between X and Z.
• \(\rm \mathop r\nolimits_{yz}\) is the simple correlation coefficient between Y and Z.

The question provides the following simple correlation coefficients:

• \(\rm \mathop r\nolimits_{xy} = 0.80\)
• \(\rm \mathop r\nolimits_{xz} = 0.64\)
• \(\rm \mathop r\nolimits_{yz} = 0.79\)

We are asked to find \(\rm \mathop R\nolimits_{xyz}^2\), which we interpret as the square of the multiple correlation coefficient when predicting one variable from the other two. Let's assume we are predicting X from Y and Z, as this is a common application of the formula and will allow us to use the given simple correlation values directly in the standard formula \(\rm \mathop R\nolimits_{x.yz}^2\).

Let's calculate the components needed for the formula:

• \(\rm \mathop r\nolimits_{xy}^2 = (0.80)^2 = 0.64\)
• \(\rm \mathop r\nolimits_{xz}^2 = (0.64)^2 = 0.4096\)
• \(\rm \mathop r\nolimits_{yz}^2 = (0.79)^2 = 0.6241\)
• \(\rm 2 \mathop r\nolimits_{xy} \mathop r\nolimits_{xz} \mathop r\nolimits_{yz} = 2 \times 0.80 \times 0.64 \times 0.79\)
• \(\rm 2 \times 0.80 \times 0.64 \times 0.79 = 1.60 \times 0.64 \times 0.79\)
• \(\rm 1.60 \times 0.64 = 1.024\)
• \(\rm 1.024 \times 0.79 = 0.80896\)

Now, substitute these values into the formula for \(\rm \mathop R\nolimits_{x.yz}^2\):

$$\rm \mathop R\nolimits_{x.yz}^2 = \frac{0.64 + 0.4096 - 0.80896}{1 - 0.6241}$$

Calculate the numerator:

$$\rm Numerator = 0.64 + 0.4096 - 0.80896 = 1.0496 - 0.80896 = 0.24064$$

Calculate the denominator:

$$\rm Denominator = 1 - 0.6241 = 0.3759$$

Now, divide the numerator by the denominator to find \(\rm \mathop R\nolimits_{x.yz}^2\):

$$\rm \mathop R\nolimits_{x.yz}^2 = \frac{0.24064}{0.3759} \approx 0.640116$$

Rounding the result to two decimal places, we get 0.64. This value represents the square of the multiple correlation coefficient when predicting X from Y and Z.

Comparing Calculated Value with Options

Let's compare our calculated value with the given options:

Our calculated value, approximately 0.64, matches Option 4.

Revision Table: Multiple Correlation Concepts

Additional Information: Understanding \(R^2\)

The value of \(R^2\) provides a measure of the goodness of fit of a multiple regression model. For example, if we were predicting X based on Y and Z, an \(R^2\) of 0.64 means that 64% of the variance in X can be explained by the combined linear relationship with Y and Z. A higher \(R^2\) indicates a better fit of the model to the data, suggesting that the independent variables (Y and Z) are good predictors of the dependent variable (X).

It is important to note that while a high \(R^2\) suggests a strong predictive relationship, it does not imply causation. Also, adding more independent variables to a multiple regression model will always increase \(R^2\) (or keep it the same), but this does not necessarily mean the new model is better, especially if the added variables are not truly related to the dependent variable. Adjusted \(R^2\) is sometimes used to account for the number of predictors in the model.