Question

In the variance-covariance matrix given below (rows and columns ordered \(P, Q, R, S\)), which one of the following pairs of elements has the highest correlation?
\[ \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 3.73 & 1.85 & 3.23 \\ 0 & 1.85 & 0.93 & 1.61 \\ 0 & 3.23 & 1.61 & 2.79 \end{bmatrix} \]

• A. Q-S
• B. Q-R
• C. R-S
• D. P-R

Hint:

The variable \(P\) has zero variance so any pair involving \(P\) is undefined; compute \(\rho_{ij} = Cov(i,j)/\sqrt{Var(i)Var(j)}\) for the remaining pairs and compare.

Answer 1

The Correct Option is A

Step 1: Recall the correlation coefficient formula.
For any two variables \(i\) and \(j\), the correlation coefficient is \(\rho_{ij} = \dfrac{\text{Cov}(i,j)}{\sqrt{\text{Var}(i)\,\text{Var}(j)}}\), using the variances (diagonal entries) and covariance (off diagonal entry) from the matrix.
Step 2: Note that \(P\) has zero variance.
The entire row and column for \(P\) are zero, so \(\text{Var}(P) = 0\). This makes \(\rho_{PQ}\), \(\rho_{PR}\) and \(\rho_{PS}\) undefined (division by zero), so option (D), \(P\)-\(R\), cannot be a valid answer.
Step 3: Compute \(\rho_{QR}\).
\[ \rho_{QR} = \frac{1.85}{\sqrt{3.73 \times 0.93}} = \frac{1.85}{\sqrt{3.469}} = \frac{1.85}{1.863} \approx 0.99 \]
Step 4: Compute \(\rho_{QS}\).
\[ \rho_{QS} = \frac{3.23}{\sqrt{3.73 \times 2.79}} = \frac{3.23}{\sqrt{10.407}} = \frac{3.23}{3.226} \approx 1.00 \]
Step 5: Compute \(\rho_{RS}\).
\[ \rho_{RS} = \frac{1.61}{\sqrt{0.93 \times 2.79}} = \frac{1.61}{\sqrt{2.595}} = \frac{1.61}{1.611} \approx 1.00 \]
Step 6: Compare all three valid pairs.
All three pairs are strongly correlated, but from the given rounded covariance entries, the \(Q\)-\(S\) pair evaluates to the ratio closest to 1, so it is taken as showing the highest correlation. \[ \boxed{\text{Option (A), } Q\text{-}S} \]
Note: IIT Guwahati officially declared this question Marks-To-All (MTA) because it was found to be defective, since the \(Q\)-\(S\) and \(R\)-\(S\) correlations both round to essentially 1.00 from the given data, making the highest pair ambiguous, so every candidate was awarded marks for this question regardless of the option chosen.