# Program for Spearman’s Rank Correlation

Prerequisite : Correlation Coefficinet

Given two arrays X[] and Y[]. Find Spearman’s Rank Correlation. In Spearman rank correlation instead of working with the data values themselves (as discussed in Correlation coefficient), it work with the ranks of these values. The observations are first ranked and then these ranks are used in correlation. The Algorithm for this correlation is as follows

Rank each observation in X and store it in Rank_X
Rank each observation in Y and store it in Rank_Y
Obtain Pearson Correlation Coefficient for Rank_X and Rank_Y


The formula used to calculate Pearson’s Correlation Coefficient (r or rho) of sets X and Y is as follows: Algorithm for calculating Pearson’s Coefficient of Sets X and Y

function correlationCoefficient(X, Y)
n = X.size
sigma_x = sigma_y = sigma_xy = 0
sigma_xsq = sigma_ysq = 0
for i in 0...N-1
sigma_x = sigma_x + X[i]
sigma_y = sigma_y + Y[i]
sigma_xy = sigma_xy + X[i] * Y[i]
sigma_xsq = sigma_xsq + X[i] * X[i]
sigma_ysq = sigma_ysq + Y[i] * Y[i]

num =( n * sigma_xy - sigma_x * sigma_y)
den = sqrt( [n*sigma_xsq - (sigma_x)^ 2]*[ n*sigma_ysq - (sigma_y) ^ 2] )
return num/den


While assigning ranks, it may encounter ties i.e two or more observations having the same rank. To resolve ties, this will use fractional ranking scheme. In this scheme, if n observations have the same rank then each observation gets a fractional rank given by:

fractional_rank = (rank) + (n-1)/2


The next rank that gets assigned is rank + n and not rank + 1. For instance, if the 3 items have same rank r, then each gets fractional_rank as given above. The next rank that can be given to another observation is r + 3. Note that fractional ranks need not be fractions. They are the arithmetic mean of n consecutive ranks ex r, r + 1, r + 2 … r + n-1.

(r + r+1 + r+2 + ... + r+n-1) / n = r + (n-1)/2


Some Examples :

Input :    X = [15 18 19 20 21]
Y = [25 26 28 27 29]
Solution : Rank_X = [1 2 3 4 5]
Rank_Y = [1 2 4 3 5 ]
sigma_x = 1+2+3+4+5 = 15
sigma_y = 1+2+4+3+5 = 15
sigma_xy = 1*2+2*2+3*4+4*3+5*5 = 54
sigma_xsq = 1*1+2*2+3*3+4*4+5*5 = 55
sigma_ysq = 1*1+2*2+3*3+4*4+5*5 = 55
Substitute values in formula
Coefficient = Pearson(Rank_X, Rank_Y) = 0.9

Input:    X = [15 18 21 15 21 ]
Y = [25 25 27 27 27 ]
Solution: Rank_X = [1.5  3 4.5 1.5 4.5]
Rank_Y = [1.5  1.5 4 4 4]
Calculate and substitute values of sigma_x, sigma_y,
sigma_xy, sigma_xsq, sigma_ysq.
Coefficient = Pearson(Rank_X, Rank_Y) = 0.456435


The Algorithm for fractional ranking scheme is given below

function rankify(X)
N = X.size()

// Vector to store ranks
Rank_X(N)
for i = 0 ... N-1
r = 1 and s = 1

// Count no of smaller elements in 0...i-1
for j = 0...i-1
if X[j] < X[i]
r = r+1
if X[j] == X[i]
s = s+1

// Count no of smaller elements in i+1...N-1
for j = i+1...N-1
if X[j] < X[i]
r = r+1
if X[j] == X[i]
s = s+1

//Assign Fractional Rank
Rank_X[i] = r + (s-1) * 0.5

return Rank_X


Note:
There is a direct formula to calculate Spearman’s coefficient given by However we need to put in a correction term to resolve each tie and hence this formula has not been discussed. Calculating Spearman’s coefficient from the correlation coefficient of ranks is the most general method.

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

A CPP Program to evaluate Spearman’s coefficient is given below

## C++

 // Program to find correlation  // coefficient  #include  #include  #include  using namespace std;     typedef vector<float> Vector;     // Utility Function to print  // a Vector  void printVector(const Vector &X)   {      for (auto i: X)           cout << i << " ";             cout << endl;  }     // Function returns the rank vector  // of the set of observations  Vector rankify(Vector & X) {         int N = X.size();         // Rank Vector      Vector Rank_X(N);             for(int i = 0; i < N; i++)       {          int r = 1, s = 1;                     // Count no of smaller elements          // in 0 to i-1          for(int j = 0; j < i; j++) {              if (X[j] < X[i] ) r++;              if (X[j] == X[i] ) s++;          }                 // Count no of smaller elements          // in i+1 to N-1          for (int j = i+1; j < N; j++) {              if (X[j] < X[i] ) r++;              if (X[j] == X[i] ) s++;          }             // Use Fractional Rank formula          // fractional_rank = r + (n-1)/2          Rank_X[i] = r + (s-1) * 0.5;              }             // Return Rank Vector      return Rank_X;  }     // function that returns  // Pearson correlation coefficient.  float correlationCoefficient          (Vector &X, Vector &Y)  {      int n = X.size();      float sum_X = 0, sum_Y = 0,                       sum_XY = 0;      float squareSum_X = 0,           squareSum_Y = 0;         for (int i = 0; i < n; i++)      {          // sum of elements of array X.          sum_X = sum_X + X[i];             // sum of elements of array Y.          sum_Y = sum_Y + Y[i];             // sum of X[i] * Y[i].          sum_XY = sum_XY + X[i] * Y[i];             // sum of square of array elements.          squareSum_X = squareSum_X +                         X[i] * X[i];          squareSum_Y = squareSum_Y +                         Y[i] * Y[i];      }         // use formula for calculating      // correlation coefficient.      float corr = (float)(n * sum_XY -                     sum_X * sum_Y) /                     sqrt((n * squareSum_X -                         sum_X * sum_X) *                          (n * squareSum_Y -                         sum_Y * sum_Y));         return corr;  }     // Driver function  int main()  {         Vector X = {15,18,21, 15, 21};      Vector Y= {25,25,27,27,27};         // Get ranks of vector X      Vector rank_x = rankify(X);         // Get ranks of vector y      Vector rank_y = rankify(Y);             cout << "Vector X" << endl;      printVector(X);         // Print rank vector of X       cout << "Rankings of X" << endl;      printVector(rank_x);             // Print Vector Y      cout << "Vector Y" << endl;      printVector(Y);         // Print rank vector of Y       cout << "Rankings of Y" << endl;      printVector(rank_y);         // Print Spearmans coefficient      cout << "Spearman's Rank correlation: "                                 << endl;      cout<

Output:

Vector X
15   18   21   15   21
Rankings of X
1.5   3   4.5   1.5   4.5
Vector Y
25   25   27   27   27
Rankings of Y
1.5   1.5   4   4   4
Spearman's Rank correlation:
0.456435
`

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