Spearman's Rank Correlation: Case of Tied Ranks | Correlation Coefficient
Quantitative Techniques in Management:
How to find the Coefficient of Correlation also called the Correlation Coefficient when the Ranks are given and there is a case of tie between two or more ranks?
Correlation is an Important topic of Statistics.
Spearman's Rank Correlation is a method finding the correlation between two variables when their ranks are mentioned instead of their score.
The other popular method of finding the correlation between two variables is the Karl Pearson's Coefficient of Correlation.
Case of Tied Ranks arise when two or more variables have the same rank.
Here we learn how to resolve this case and find the correlation between two variables.
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Well! Just take the numbers as they are and rank them from the highest to the lowest. In case of X there is a tie at the highest number i.e. at 16 so they will be ranked as per the average of their ranks if the ranks were going to be 1 and 2. so both will have rank 1.5. Then 15 will be assigned rank 3. Like wise there are ties at 12 for rank 4 & 5, so they both will have ranks 4.5. Similarly at 10, both the 10 will have ranks average of 6 & 7. Again a tie at 8 will have the ranks of both the 8s as 8.5. Last one remaining is no. 5 which will be ranked as 10. In the same manner the series Y will have the rank 1 for 28, 2 for 23, 3 for 22, now there is a tie at 20 and there are 3 numbers. So taking the average of 4,5 & 6 there prospective ranks had they been different, we get rank 5 for all the three. Then rank 7 for 18, 8 for 17 and again another tie at number 15 will have rank 9.5 for both. You can see the arrangement and working below.
The correction factor for a tie with 2 numbers will be 0.5 and for a tie with 3 numbers will be 2.We have 4 tie with 2 numbers in X and 1 tie with 2 numbers in Y. Also there is a tie of 3 numbers in Y. so total correction factor would be (0.5+0.5+0.5+0.5+0.5+2) = 4.5
The whole working is illustrated here.
Hope that makes it clear.
X : 5, 8, 12, 10, 15, 16,16,10,12,8
Y: 15, 17, 18,20,23,28,22,20,20,15
Rank X: 10, 8.5, 4.5, 6.5, 3, 1.5, 1.5, 6.5, 4.5, 8.5
Rank Y: 9.5, 8, 7, 5, 2, 1, 3, 5, 5, 9.5
d = Rank X - Rank Y = 0.5, 0.5, -2.5, 1.5, 1, 0.5, -1.5, 1.5, -0.5, -1
d^2 = 0.25, 0.25, 6.25, 2.25, 1, 0.25, 2.25, 2.25, 0.25, 1
Sum of (d^2) = 16
d^2 + cfs = 16+4.5 = 20.5
n = 10
n((n^2)-1) = 990
6(Sum(d^2)) = 123
Rho = 0.875757576
and the serial is also given for x and y values, i.e. for 1 x is given 5 and y is 15, for serial 2, x is 8 and y is 17. so how would I rank? I saw in another lecture that the rank, in case of tied, has given by average number of serial. so if I search for the rank 1 to highest number what are 16 and 16. then how to rank? if i rank both the 16 are 6.5 by average their serial number what will be the next rank? of course 3, is not it? as the first 2 numbers has been ranked. what will be third rank? 15? and the fourth? there is again tie of 12 whose serial number are 3 and 9. if I average the serial number the rank will be 6 for both the fourth and fifth rank? is not it?
help me to assign ranks?
Do you want me to solve it or just assign ranks ?
If I have to teach this, I'll have to do a new video.
Anyways, let me know....
The solution to this is 0.875757...
Try out with the help from the video, very similar steps. In case you need any more steps, let me know.
Thanks a lot for this video. I have my exam tomorrow and I was really confused with the case of more than two variables having same rank. Thank you soo much for clearing the doubt. Keep up the good work..!! :-)
Just one question, i did the calculation on the graphics calculators and used the x and y values you got, yet managed to get a slightly varied answer of 0.974789916 instead of the 0.967... you calculated. Is it possible my calculation is wrong, or have you calculated an error? Many thanks.
The value of r = 0.9844... and the value of rho = 0.966666666666667. I do not know how you got 0.9747899. In the video, it has been done twice by two methods, first manually and then on excel and you the answer is the same. Again here it is explained, so there is no chance of error.
Mobasshir Quamar I used the x and y values, calculated to the same as yours, but also put these values in the calculator( stats mode, mode 2) in the table it gives, and used the table to find an r value of 0.9844... . Using your method exactly, got another result which is the same as yours. I was just wondering if either result could be used in an exam.
Yeah Dear, sure. Well...!
You have 1-((6*4)/(9*80)) you can cancel (6/9) to get (2/3) and cancel out (4/80) to get (1/20). Again you can cancel (2/20) to get (1/10). Now the equation would look like 1-(1/30), which can be written as (30-1)/30 =29/30 =0.966666666666667. This can be round off to 0.967 and that is what we have got.
So, the calculation for this method is correct. But if you change the method the answer would be different. Like if you calculate the Karl Pearson Coefficient of Correlation you will get 0.984425072.
I do not know what method you have used to get the value 0.9747899, please check or let me know elaborately as to how you managed to get that value.
In case of tied ranks, I believe as per mu intuitive understanding, coefficient of correlation decreases.for example, in a particular sample of observations, 3 observations of x variable are the same while the corresponding 3 observations of the y variable increases steadily. So, for these 3 observations coefficient of correlation, I suppose, is 0 and therefore it is bringing down the coefficient of correlation of the entire sample in turn. In this connection, I am not understanding the the concept of the correction factor. What is it actually correcting? what is the derivation of the formula of the correction factor?
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thank you but there's something I really don't get - why aren't you first sorting the values from the lowest to the highest? It seems like the ranking is meaningless if it's not really sorted.. Am I wrong?
thank you very much! "In the example the ranks are allotted in the order from the highest data to to the lowest data" - but according to what values? values that are not displayed but were used to sort it according to them beforehand?
Well dear, it can be either sorted from "the lowest to the highest" or from "the highest to the lowest". Since the data given here is already in the order from highest to lowest, we are not sorting it further.
In the example the ranks are allotted in the order from the highest data to to the lowest data. Since the order is taken care of while allotting the ranks, it already got sorted. You can even first sort the data and then allot the ranks. You will get the same ranks and the same answer.
Hope that makes things clear.
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Hey, thanks for the great video.
I have a question (im not really strong in math).
If i have more than one number tied at "series a", and more than one number tied at "series b", do i then calculate cf for each tie and then add them to find di2 like this: di2 = di2 + cf1 + cf2 + cf3 + cf4 etc..?
Hope you understand?
Yes, in case if you have more than 1 number tied in both the series A and B then you need to calculate the cf for each of those ties and find the "Sum of di square (new)" = "Sum of di square (old) + cf1 + cf2 + cf3 + cf4 ... and so on till you add all the cf's for all the ties separately.
Hope your doubt is clear.
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