$$ \begin{bmatrix} & P & Q & R & S \\ P & 0 & 0 & 0 & 0 \\ Q & 0 & 3.73 & 1.85 & 3.23 \\ R & 0 & 1.85 & 0.93 & 1.61 \\ S & 0 & 3.23 & 1.61 & 2.79 \end{bmatrix} $$
Correlation Calculation from Variance-Covariance Matrix
To find the pair with the highest correlation, we use the correlation coefficient formula: $ \rho_{ij} = \frac{\sigma_{ij}}{\sigma_i \sigma_j} $ where $ \sigma_{ij} $ is the covariance between variables i and j, and $ \sigma_i $ and $ \sigma_j $ are their respective standard deviations.
Extracting Variances and Standard Deviations
From the diagonal elements of the variance-covariance matrix, we get the variances ($ \sigma^2 $). The standard deviations ($ \sigma $) are the square roots of the variances.
| Variable | Variance ($ \sigma^2 $) | Standard Deviation ($ \sigma $) |
|---|---|---|
| P | 0 | $ \sqrt{0} = 0 $ |
| Q | 3.73 | $ \sqrt{3.73} \approx 1.931 $ |
| R | 0.93 | $ \sqrt{0.93} \approx 0.964 $ |
| S | 2.79 | $ \sqrt{2.79} \approx 1.670 $ |
Extracting Covariances
The off-diagonal elements represent the covariances ($ \sigma_{ij} $).
| Covariance | Value |
|---|---|
| $ \sigma_{P,Q} $ | 0 |
| $ \sigma_{P,R} $ | 0 |
| $ \sigma_{Q,R} $ | 1.85 |
| $ \sigma_{Q,S} $ | 3.23 |
| $ \sigma_{R,S} $ | 1.61 |
Calculating Pairwise Correlations
Now, we calculate the correlation for each pair listed in the options:
- P-Q: Since $ \sigma_P = 0 $, the correlation $ \rho_{P,Q} $ is 0 (or undefined).
- P-R: Since $ \sigma_P = 0 $, the correlation $ \rho_{P,R} $ is 0 (or undefined).
- Q-R: $ \rho_{Q,R} = \frac{\sigma_{Q,R}}{\sigma_Q \sigma_R} = \frac{1.85}{\sqrt{3.73} \sqrt{0.93}} \approx \frac{1.85}{1.931 \times 0.964} \approx \frac{1.85}{1.862} \approx 0.9935 $
- Q-S: $ \rho_{Q,S} = \frac{\sigma_{Q,S}}{\sigma_Q \sigma_S} = \frac{3.23}{\sqrt{3.73} \sqrt{2.79}} \approx \frac{3.23}{1.931 \times 1.670} \approx \frac{3.23}{3.225} \approx 1.0013 $
- R-S: $ \rho_{R,S} = \frac{\sigma_{R,S}}{\sigma_R \sigma_S} = \frac{1.61}{\sqrt{0.93} \sqrt{2.79}} \approx \frac{1.61}{0.964 \times 1.670} \approx \frac{1.61}{1.611} \approx 0.9994 $
Identifying the Highest Correlation
Comparing the calculated correlation coefficients:
- $ \rho_{P,Q} = 0 $
- $ \rho_{P,R} = 0 $
- $ \rho_{Q,R} \approx 0.9935 $
- $ \rho_{Q,S} \approx 1.0013 $
- $ \rho_{R,S} \approx 0.9994 $
The pair Q-S has the highest correlation coefficient ($ \approx 1.0013 $).
Important Questions from Correlation Analysis
- Let $X$ be a random variable that follows $Uniform(-1, 1)$ distribution. The conditional distribution of the random variable $Y$ given $X = x$ is the $Uniform(x^2 - 0.1, x^2 + 0.1)$ distribution.
The value of $correlation(X, Y)$ is __________ . (Answer in integer) - Consider 62 pair of observations $(x_i, y_i)$, i = 1, …., 62, obtained from the distribution of $(X, Y)$. The Pearson correlation coefficient ($R$) of these observations is 0.27.
Which one of the following statements is CORRECT for a significance level $\alpha = 0.05$ and $t_{0.025,60} = 2.0$?
Hint: Test statistic $t = \frac{R\sqrt{n - 2}}{\sqrt{1 - R^2}}$ - Let $X = \cos \theta$ and $Y = \sin \theta$, where $\theta$ is a random variable uniformly distributed over $(0, 2\pi)$. Which of the following options is/are CORRECT?
Hint: $\text{cov}(X, Y) = E[(X - E(X)) (Y - E(Y))]$ - The multiple correlation ranges between :
- Arrange the following correlation coefficients in descending order in terms of their strength of relationship, as defined by common variance :
