If six hand-writings were ranked by two judges in a competition and the rankings are as follows:Hand Writing 1 2 3 4 5 6 Judge1 6 5 4 3 2 1 Judge2 1 2 3 4 5 6
Then the value of Spearman's rank correlation coefficient is
-1
Calculating Spearman's Rank Correlation Coefficient
Spearman's rank correlation coefficient, denoted by \( \rho \) (rho) or \( r_s \), is a non-parametric measure of the strength and direction of association between two ranked variables. It assesses how well the relationship between two variables can be described using a monotonic function.
The formula for Spearman's rank correlation coefficient when there are no tied ranks is:
\[ \rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)} \]
Where:
- \( n \) is the number of pairs of observations (in this case, the number of hand-writings).
- \( d_i \) is the difference between the ranks of the \( i \)-th observation for the two variables (the difference between the ranks given by the two judges for each hand-writing).
- \( \sum d_i^2 \) is the sum of the squared differences between the ranks.
Applying the Formula to the Hand-Writing Ranking Data
We are given the ranks of six hand-writings by two judges. So, the number of observations, \( n \), is 6.
The ranks are:
| Hand Writing | Judge 1 Rank (\( R_1 \)) | Judge 2 Rank (\( R_2 \)) |
|---|---|---|
| 1 | 6 | 1 |
| 2 | 5 | 2 |
| 3 | 4 | 3 |
| 4 | 3 | 4 |
| 5 | 2 | 5 |
| 6 | 1 | 6 |
Now, we calculate the difference (\( d_i = R_1 - R_2 \)) and the squared difference (\( d_i^2 \)) for each hand-writing:
| Hand Writing | Judge 1 Rank (\( R_1 \)) | Judge 2 Rank (\( R_2 \)) | Difference (\( d_i = R_1 - R_2 \)) | Squared Difference (\( d_i^2 \)) |
|---|---|---|---|---|
| 1 | 6 | 1 | \( 6 - 1 = 5 \) | \( 5^2 = 25 \) |
| 2 | 5 | 2 | \( 5 - 2 = 3 \) | \( 3^2 = 9 \) |
| 3 | 4 | 3 | \( 4 - 3 = 1 \) | \( 1^2 = 1 \) |
| 4 | 3 | 4 | \( 3 - 4 = -1 \) | \( (-1)^2 = 1 \) |
| 5 | 2 | 5 | \( 2 - 5 = -3 \) | \( (-3)^2 = 9 \) |
| 6 | 1 | 6 | \( 1 - 6 = -5 \) | \( (-5)^2 = 25 \) |
Next, we sum the squared differences (\( \sum d_i^2 \)):
\[ \sum d_i^2 = 25 + 9 + 1 + 1 + 9 + 25 = 70 \]
Now, substitute \( n = 6 \) and \( \sum d_i^2 = 70 \) into the formula for Spearman's rank correlation coefficient:
\[ \rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)} \]
\[ \rho = 1 - \frac{6 \times 70}{6(6^2 - 1)} \]
\[ \rho = 1 - \frac{420}{6(36 - 1)} \]
\[ \rho = 1 - \frac{420}{6(35)} \]
\[ \rho = 1 - \frac{420}{210} \]
\[ \rho = 1 - 2 \]
\[ \rho = -1 \]
The value of Spearman's rank correlation coefficient is -1.
A Spearman's correlation coefficient of -1 indicates a perfect negative monotonic relationship between the two variables. In this context, it means that as the rank given by Judge 1 increases, the rank given by Judge 2 decreases, and vice versa, in a perfectly consistent manner.
Revision Table: Spearman's Correlation Key Concepts
| Concept | Description | Formula (No Ties) |
|---|---|---|
| Spearman's Rho (\( \rho \)) | Measures the strength and direction of monotonic association between ranked variables. | \( \rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)} \) |
| Monotonic Relationship | A relationship that is consistently increasing or consistently decreasing, but not necessarily linear. | |
| \( n \) | Number of data pairs. | - |
| \( d_i \) | Difference in ranks for each pair. | \( R_{1i} - R_{2i} \) |
Additional Information: Correlation Coefficients and Their Interpretation
Correlation coefficients are statistical measures used to quantify the relationship between two variables. Different coefficients are used depending on the type of data and the nature of the relationship being assessed.
- Pearson Correlation Coefficient (\( r \)): Measures the strength and direction of a linear relationship between two continuous variables. It assumes the data are normally distributed and the relationship is linear. Values range from -1 to +1.
- Spearman Rank Correlation Coefficient (\( \rho \) or \( r_s \)): Measures the strength and direction of a monotonic relationship between two ranked variables or variables that can be ranked. It does not assume linearity or normality. Values also range from -1 to +1.
- Interpreting Correlation Values:
- \( +1 \): Perfect positive correlation. As one variable increases, the other increases proportionally (Pearson) or consistently (Spearman).
- \( -1 \): Perfect negative correlation. As one variable increases, the other decreases proportionally (Pearson) or consistently (Spearman).
- \( 0 \): No linear (Pearson) or monotonic (Spearman) relationship.
- Values between 0 and 1 (positive) or 0 and -1 (negative) indicate weaker relationships. The closer the value is to +1 or -1, the stronger the relationship.
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Important Questions from Correlation and Regression
- For testing the significance of overall regression the test to be used is
- Including relevant lagged values of the dependent variable on the right hand side of a regression equation could lead to which one of the following ?
- If OLS is used in the presence of auto-correlation, which of the following will be likely consequences ?
i. Coefficients estimates may be misleading.
ii. Hypothesis tests could reach the wrong conclusions.
iii. Forecasts made from the model could be biased.
iv. Standard errors may be inappropriate. - Regression analysis is of particular interest
- Which of the following can be linked with those providing method for estimating black money in India ?
I. Kaldor’s approach
II. NIPFP method
III. O.P. Chopra’s method
IV. Arjun Sengupta
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