The Spearman Rank Correlation Coefficient
In today's newspaper, a large investment firm published a list of their top 10 analysts in terms of total returns generated by the firm on the basis of their recommendations. If you get curious as to why they performed the way they did, you might decide to look at a correlation between years in the industry and the underlying performance measurement.
So you contact the firm and ask for the performance metric used. They laugh at you and hang up since you are working for a competitor. While you really can't blame them, you're left with a decision.
The last step in obtaining an estimate for the Spearman Rank Correlation Coefficient is to use these _d_ values in the following calculation:
$$\displaystyle r_s = 1 - \frac{6 \displaystyle \sum_{i=1}^{n}d^2_i}{n(n^2-1)} $$
Using the _d_ values you calculated, what is the Spearman Rank Correlation Coefficient in this example?
No, it's possible to obtain this answer by omitting the "1 -" at the front of the calculation.
Exactly!
This is calculated as:
$$\displaystyle r_s = 1 - \frac{6[0+4+0+2.25+9+1+2.25+4+49+1]}{10(99)}$$
> $$\displaystyle = 1 - \frac{435}{990} \approx 0.56 $$
This calculation is nested within the same process as always of determining a hypothesis and then finding an appropriate test statistic. For small-sample problems using this coefficient, special tables are typically used.
No, it's possible to obtain this answer by omitting the "6" in front of the summation.
Then is it significant? As is often the case, there's a t-test for that. For small samples like this n=10 example, there is a table that you can find; you'll see that the critical value for this sample size and an alpha of 0.05 is 0.6364 in absolute value, and 0.5606 isn't beyond that. So no rejection of the null. For larger samples, you would use this:
$$\displaystyle t = \frac{r_s \sqrt{n-2}}{\sqrt{1 - r_s^2}} $$
Perfect!
Now you have the two variables in ranks and a _d_ vector of the differences:
| Analyst | A | B | C | D | E | F | G | H | I | J |
|---|---|---|---|---|---|---|---|---|---|---|
| Performance Ranking | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th | 9th | 10th |
| Experience Ranking | 1st | 4th | 3rd | 5.5th | 8th | 7th | 5.5th | 10th | 2nd | 9th |
| d vector | 0 | -2 | 0 | -1.5 | -3 | -1 | 1.5 | -2 | 7 | 1 |
That's right.
No, this is not the difference between the two rankings. Notice that there is no difference between the rankings of the first analyst.
To summarize:
[[summary]]
What do you suppose would happen if you simply correlated the ranking of the manager with the number of years that the manager was in the industry?
No, you could. You can grab any set of numbers and run a correlation between them. It just wouldn't give you something particularly useful.
Incorrect. Statistically, this wouldn't be appropriate for analysis.
Exactly!
Rankings just don't work like ratio variables of degree.
The t-test for a correlation coefficient has some strict assumptions. If one or both of the variables is already in ranks, it's best to convert both data series to ranks and use the __Spearman Rank Correlation Coefficient__. The first step is to convert the following listing of how many years of experience each manager has in the industry into ranks:
| Analyst | A | B | C | D | E | F | G | H | I | J |
|---|---|---|---|---|---|---|---|---|---|---|
| Performance Ranking | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th | 9th | 10th |
| Years in the Industry | 32 | 14 | 19 | 10 | 8 | 9 | 10 | 4 | 20 | 7 |
In terms of experience, how would you rank these analysts?
No, this is their performance ranking order but not the ranking of years of experience.
Now, with this "experience ranking" in mind, you could augment the original table to include this information as shown:
| Analyst | A | B | C | D | E | F | G | H | I | J |
|---|---|---|---|---|---|---|---|---|---|---|
| Performance Ranking | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th | 9th | 10th |
| Years in the Industry | 32 | 14 | 19 | 10 | 8 | 9 | 10 | 4 | 20 | 7 |
| Experience Ranking | 1st | 4th | 3rd | 5th/6th | 8th | 7th | 5th/6th | 10th | 2nd | 9th |
Sometimes you have to deal with the issue of ties, such as in the case of the experience ranking for analysts D and G. Note that they should each have rankings 5 or 6, so best practice is to simply take the average of these two numbers, which is 5.5. Then you have a complete table for the ranking of the two variables:
| Analyst | A | B | C | D | E | F | G | H | I | J |
|---|---|---|---|---|---|---|---|---|---|---|
| Performance Ranking | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th | 9th | 10th |
| Experience Ranking | 1st | 4th | 3rd | 5.5th | 8th | 7th | 5.5th | 10th | 2nd | 9th |
Now that you have both series in ranks, the second step is to calculate the difference between each ranking. Call this third variable _d_. It doesn't matter which series you subtract from since you'll be squaring these _d_ values in a minute.
What is this _d_ vector?
No, this is not the difference between the two rankings. Notice that there is not a difference in rankings for the third analyst.
0.44
0.56
0.93
You would get a correlation that could be used for analysis
You would get a correlation which could be statistically indefensible
You couldn't correlate those two things
A, I, C, B, D&G, F, E, J, H
A, B, C, D, E, F, G, H, I, J
| d vector | 0 | 2 | 1 | 2.5 | 2 | 1 | 2.5 | 2 | 2 | 1 |
|---|---|---|---|---|---|---|---|---|---|---|
| d vector | 0 | -2 | 0 | -1.5 | -3 | -1 | 1.5 | -2 | 7 | 1 |
|---|---|---|---|---|---|---|---|---|---|---|
| d vector | 1 | 3 | 3 | 5 | 6.5 | 6.5 | 6 | 9 | 5.5 | 9.5 |
|---|---|---|---|---|---|---|---|---|---|---|
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