Sverdrup balance

In statistics, a rank correlation is any of several statistics that measure the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the labels "first", "second", "third", etc. to different observations of a particular variable. A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them.

Context

If, for example, one variable is the identity of a college basketball program and another variable is the identity of a college football program, one could test for a relationship between the poll rankings of the two types of program: do colleges with a higher-ranked basketball program tend to have a higher-ranked football program? A rank correlation coefficient can measure that relationship, and the measure of significance of the rank correlation coefficient can show whether the measured relationship is small enough to likely be a coincidence.

If there is only one variable, the identity of a college football program, but it is subject to two different poll rankings (say, one by coaches and one by sportswriters), then the similarity of the two different polls' rankings can be measured with a rank correlation coefficient.

Correlation coefficients

Some of the more popular rank correlation statistics include

1. Spearman's ρ
2. Kendall's τ
3. Goodman and Kruskal's γ

An increasing rank correlation coefficient implies increasing agreement between rankings. The coefficient is inside the interval [−1, 1] and assumes the value:

• 1 if the agreement between the two rankings is perfect; the two rankings are the same.
• 0 if the rankings are completely independent.
• −1 if the disagreement between the two rankings is perfect; one ranking is the reverse of the other.

Following Template:Harvtxt, a ranking can be seen as a permutation of a set of objects. Thus we can look at observed rankings as data obtained when the sample space is (identified with) a symmetric group. We can then introduce a metric, making the symmetric group into a metric space. Different metrics will correspond to different rank correlations.

General Correlation Coefficient

Template:Harvtxt showed that his tau and Spearman's rho are particular cases of a general correlation coefficient.

Suppose we have a set of $n$ objects, which are being considered in relation to two properties, represented by $x$ and $y$, forming the sets of values $\{x_{i}\}_{i\leq n}$ and $\{y_{i}\}_{i\leq n}$. To any pair of individuals, say the $i$-th and the $j$-th we assign a $x$-score, denoted by $a_{ij}$, and a $y$-score, denoted by $b_{ij}$. The only requirement made to this functions is anti-symmetry, so $a_{ij}=-a_{ji}$ and $b_{ij}=-b_{ji}$. Then the generalised correlation coefficient $\Gamma$ is defined by

$\Gamma ={\frac {\sum _{i,j=1}^{n}a_{ij}b_{ij}}{\sqrt {\sum _{i,j=1}^{n}a_{ij}^{2}\sum _{i,j=1}^{n}b_{ij}^{2}}}}$

Kendall's $\tau$ as a particular case

If $r_{i}$ is the rank of the $i$-member according to the $x$-quality, we can define

$a_{ij}=\operatorname {sgn}(r_{j}-r_{i})$

and similarly for $b$. The sum $\sum a_{ij}b_{ij}$ is twice the amount of concordant pairs minus the discordant pairs (see Kendall tau rank correlation coefficient). The sum $\sum a_{ij}^{2}$ is just the number of terms $a_{ij}$, equal to $n(n-1)$, and so for $\sum b_{ij}^{2}$. It follows that $\Gamma$ is equal to the Kendall's $\tau$ coefficient.

Spearman's $\rho$ as a particular case

If $r_{i}$, $s_{i}$ are the ranks of the $i$-member according to the $x$ and the $y$-quality respectively, we can simply define

$a_{ij}=r_{j}-r_{i}$

$b_{ij}=s_{j}-s_{i}$

The sums $\sum a_{ij}^{2}$ and $\sum b_{ij}^{2}$ are equal, since both $r_{i}$ and $s_{i}$ range from $1$ to $n$. Then we have:

$\Gamma ={\frac {\sum (r_{j}-r_{i})(s_{j}-s_{i})}{\sum (r_{j}-r_{i})^{2}}}$

now

$\sum _{i,j=1}^{n}(r_{j}-r_{i})(s_{j}-s_{i})=\sum _{i=1}^{n}\sum _{j=1}^{n}r_{i}s_{i}+\sum _{i=1}^{n}\sum _{j=1}^{n}r_{j}s_{j}-\sum _{i=1}^{n}\sum _{j=1}^{n}(r_{i}s_{j}+r_{j}s_{i})$

$=2n\sum _{i=1}^{n}r_{i}s_{i}-2\sum _{i=1}^{n}r_{i}\sum _{j=1}^{n}s_{j}$

$=2n\sum _{i=1}^{n}r_{i}s_{i}-{\frac {1}{2}}n^{2}(n+1)^{2}$

since $\sum r_{i}$ and $\sum s_{j}$ are both equal to the sum of the first $n$ natural numbers, namely ${\frac {1}{2}}n(n+1)$.

We also have

$S=\sum _{i=1}^{n}(r_{i}-s_{i})^{2}=2\sum r_{i}^{2}-2\sum r_{i}s_{i}$

and hence

$\sum (r_{j}-r_{i})(s_{j}-s_{i})=2n\sum r_{i}^{2}-{\frac {1}{2}}n^{2}(n+1)^{2}-nS$

$\sum r_{i}^{2}$ being the sum of squares of the first $n$ naturals equals ${\frac {1}{6}}n(n+1)(2n+1)$. Thus, the last equation reduces to

$\sum (r_{j}-r_{i})(s_{j}-s_{i})={\frac {1}{6}}n^{2}(n^{2}-1)-nS$

Further

$\sum (r_{j}-r_{i})^{2}=2n\sum r_{i}^{2}-2\sum r_{i}r_{j}$

$=2n\sum r_{i}^{2}-2(\sum r_{i})^{2}={\frac {1}{6}}n^{2}(n^{2}-1)$

and thus, substituting into the original formula these results we get

$\Gamma R=1-{\frac {6*SIGMAD2}{n^{3}-n}}$

which is exactly the Spearman's rank correlation coefficient $\rho$.

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010