For three random variables X1 , X2 and X3, the pairwise correlation coefficients are r12 = r13 = r23 = r. The multiple correlation coefficient \(\rm R_{1.2.3}^2\) is
Calculate Multiple Correlation Coefficient \(R_{1.2.3}^2\)
The question asks for the multiple correlation coefficient \(R_{1.2.3}^2\) for three random variables X1, X2, and X3, given their pairwise correlation coefficients \(r_{12}\), \(r_{13}\), and \(r_{23}\). We are specifically given that \(r_{12} = r_{13} = r_{23} = r\).
Understanding Multiple Correlation Coefficient
The multiple correlation coefficient \(R_{1.2.3}\) measures the linear relationship between a variable (X1 in this case) and a set of other variables (X2 and X3). The squared multiple correlation coefficient, \(R_{1.2.3}^2\), represents the proportion of the variance in X1 that can be explained by the linear combination of X2 and X3.
Method using Correlation Matrix
We can calculate the squared multiple correlation coefficient \(R_{1.2.3}^2\) using the correlation matrix of the variables. The correlation matrix R for X1, X2, and X3 is:
Given \(r_{12} = r_{13} = r_{23} = r\), the matrix becomes:
The formula for the squared multiple correlation coefficient \(R_{1.(2,3)}^2\) (which is \(R_{1.2.3}^2\)) using the correlation matrix is:
$$ R_{1.2.3}^2 = 1 - \frac{|R|}{C_{11}} $$where \(|R|\) is the determinant of the correlation matrix R, and \(C_{11}\) is the cofactor of the element in the first row and first column of R (which is the determinant of the submatrix obtained by removing the first row and first column).
Calculating the Determinant \(|R|\)
Let's calculate the determinant of R:
$$ |R| = \det \begin{pmatrix} 1 & r & r \\ r & 1 & r \\ r & r & 1 \end{pmatrix} $$ $$ |R| = 1 \cdot (1 \cdot 1 - r \cdot r) - r \cdot (r \cdot 1 - r \cdot r) + r \cdot (r \cdot r - 1 \cdot r) $$ $$ |R| = 1 \cdot (1 - r^2) - r \cdot (r - r^2) + r \cdot (r^2 - r) $$ $$ |R| = (1 - r^2) - r^2 + r^3 + r^3 - r^2 $$ $$ |R| = 1 - r^2 - 2r^2 + 2r^3 $$ $$ |R| = 1 - 3r^2 + 2r^3 $$Calculating the Cofactor \(C_{11}\)
The cofactor \(C_{11}\) is the determinant of the matrix obtained by removing the first row and first column of R:
$$ C_{11} = \det \begin{pmatrix} 1 & r \\ r & 1 \end{pmatrix} $$ $$ C_{11} = 1 \cdot 1 - r \cdot r $$ $$ C_{11} = 1 - r^2 $$Calculating \(R_{1.2.3}^2\)
Now, substitute the values of \(|R|\) and \(C_{11}\) into the formula for \(R_{1.2.3}^2\):
$$ R_{1.2.3}^2 = 1 - \frac{|R|}{C_{11}} $$ $$ R_{1.2.3}^2 = 1 - \frac{1 - 3r^2 + 2r^3}{1 - r^2} $$To simplify, find a common denominator:
$$ R_{1.2.3}^2 = \frac{(1 - r^2) - (1 - 3r^2 + 2r^3)}{1 - r^2} $$ $$ R_{1.2.3}^2 = \frac{1 - r^2 - 1 + 3r^2 - 2r^3}{1 - r^2} $$ $$ R_{1.2.3}^2 = \frac{2r^2 - 2r^3}{1 - r^2} $$Factor out \(2r^2\) from the numerator:
$$ R_{1.2.3}^2 = \frac{2r^2(1 - r)}{1 - r^2} $$Factor the denominator using the difference of squares formula (\(a^2 - b^2 = (a-b)(a+b)\)):
$$ R_{1.2.3}^2 = \frac{2r^2(1 - r)}{(1 - r)(1 + r)} $$Assuming \(1 - r \neq 0\) (i.e., \(r \neq 1\)), we can cancel the \((1 - r)\) term:
$$ R_{1.2.3}^2 = \frac{2r^2}{1 + r} $$Alternative Formula Check
Another common formula for \(R_{1.2.3}^2\) is:
$$ R_{1.2.3}^2 = \frac{r_{12}^2 + r_{13}^2 - 2 r_{12} r_{13} r_{23}}{1 - r_{23}^2} $$Substituting \(r_{12} = r_{13} = r_{23} = r\):
$$ R_{1.2.3}^2 = \frac{r^2 + r^2 - 2 (r)(r)(r)}{1 - r^2} $$ $$ R_{1.2.3}^2 = \frac{2r^2 - 2r^3}{1 - r^2} $$ $$ R_{1.2.3}^2 = \frac{2r^2(1-r)}{(1-r)(1+r)} = \frac{2r^2}{1+r} $$Both methods yield the same result for the multiple correlation coefficient \(R_{1.2.3}^2\) when all pairwise correlations are equal to \(r\).
Conclusion
The calculated value for the multiple correlation coefficient \(R_{1.2.3}^2\) is \(\frac{2r^2}{1+r}\).
Comparing this result with the given options:
- Option 1: \(\rm \frac{2r}{1+r}\)
- Option 2: \(\rm \frac{r}{1+r}\)
- Option 3: \(\rm \frac{2r^2}{1-r}\)
- Option 4: \(\rm \frac{2r^2}{1+r}\)
The result matches Option 4.
Revision Table: Key Concepts
| Concept | Description | Formula (for 3 variables) |
|---|---|---|
| Pairwise Correlation (\(r_{ij}\)) | Measures linear association between two variables \(X_i\) and \(X_j\). Ranges from -1 to +1. | \(r_{ij} = \frac{\text{Cov}(X_i, X_j)}{\sigma_{X_i} \sigma_{X_j}}\) |
| Multiple Correlation Coefficient (\(R_{i.(j,k)}\)) | Measures linear association between one variable \(X_i\) and the best linear prediction of \(X_i\) using variables \(X_j\) and \(X_k\). Ranges from 0 to 1. | Not a simple direct formula; often derived from the squared coefficient. |
| Squared Multiple Correlation Coefficient (\(R_{i.(j,k)}^2\)) | Proportion of variance in \(X_i\) explained by \(X_j\) and \(X_k\). Ranges from 0 to 1. | \(R_{1.2.3}^2 = \frac{r_{12}^2 + r_{13}^2 - 2 r_{12} r_{13} r_{23}}{1 - r_{23}^2}\) or \(R_{1.2.3}^2 = 1 - \frac{|R|}{C_{11}}\) |
Additional Information: Properties of Correlation
- Correlation measures only linear relationships. Non-linear relationships might exist even if the correlation is zero.
- A correlation coefficient of +1 indicates a perfect positive linear relationship.
- A correlation coefficient of -1 indicates a perfect negative linear relationship.
- A correlation coefficient of 0 indicates no linear relationship.
- Correlation does not imply causation.
- For \(R_{1.2.3}^2\) to be defined, the variables X2 and X3 must not be perfectly linearly correlated themselves (i.e., \(|r_{23}| \neq 1\), so \(1 - r^2 \neq 0\)), and X1 should not be perfectly linearly predictable from X2 and X3. Also, the correlation matrix must be positive semi-definite. For the formula \(R_{1.2.3}^2 = \frac{2r^2}{1+r}\) to be valid when \(r_{12}=r_{13}=r_{23}=r\), we generally require \(r \neq 1\) and \(r \neq -1\). If \(r = 1\), all variables are perfectly correlated and the formula denominator is 2. If \(r = -1\), the determinant \(1 - 3(-1)^2 + 2(-1)^3 = 1 - 3 - 2 = -4\), and \(C_{11} = 1 - (-1)^2 = 0\), which leads to division by zero. The case \(r=-1\) needs careful consideration; when all pairwise correlations are -1, it's impossible for only three variables unless two are linearly dependent. If \(r = -0.5\), the determinant is \(1 - 3(0.25) + 2(-0.125) = 1 - 0.75 - 0.25 = 0\), meaning the variables are linearly dependent (e.g., X1 = X2 + X3). In general, for a valid correlation matrix, \(|R|\) must be non-negative. For three variables with equal pairwise correlation \(r\), \(|R| = (1-r)^2(1+2r)\). For \(|R| \ge 0\), we need \(1+2r \ge 0\), so \(r \ge -0.5\). Thus, the formula \(\frac{2r^2}{1+r}\) is typically valid for \(r \in [-0.5, 1)\).
Similar Questions
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Two variates, x and y, are uncorrelated and have standard deviations σ xand σ yrespectively. What is the correlation coefficient between x + y and x – y?
If the covariance between x and y is 30, variance of x is 25 and variance of y is 144, then what is the correlation coefficient?
The coefficient of correlation when coefficients of regression are 0.2 and 1.8 is
