# Spearman’s Rank Correlation

**What is correlation test?**

The strength of the association between two variables is known as the correlation test. For instance, if we are interested to know whether there is a relationship between the heights of fathers and sons, a correlation coefficient can be calculated to answer this question.

For know more about correlation please refer this.

**Methods for correlation analysis:**

There are mainly two types of correlation:

**Parametric Correlation – Pearson correlation(r) :**It measures a linear dependence between two variables (x and y) is known as a parametric correlation test because it depends on the distribution of the data.**Non-Parametric Correlation – Kendall(tau)**and**Spearman(rho):**They are rank-based correlation coefficients, are known as non-parametric correlation.

**Spearman Correlation formula:**

where,

r= Spearman Correlation coefficient_{s}

d= the difference in the ranks given to the two variables values for each item of the data,_{i}

n= total number of observation

**Example**: In the Spearman’s rank correlation what we do is convert the data even if it is real value data to what we call ranks. Let’s consider taking 10 different data points in variable X_{1} and Y_{1}. And find out their respective ranks. Then find out the square of the difference in the ranks given to the two variables values for each item of the data.

Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

X_{1} |
7 | 6 | 4 | 5 | 8 | 7 | 10 | 3 | 9 | 2 |

Y_{1} |
5 | 4 | 5 | 6 | 10 | 7 | 9 | 2 | 8 | 1 |

Rank X_{1} |
6.5 | 5 | 3 | 4 | 8 | 6.5 | 10 | 2 | 9 | 1 |

Rank Y_{1} |
4.5 | 3 | 4.5 | 6 | 10 | 7 | 9 | 2 | 8 | 1 |

d^{2} |
4 | 4 | 2.25 | 4 | 4 | 0.25 | 1 | 0 | 1 | 0 |

**Step 1**: Finding Rank-

- Rank X
_{1}: So, what we have done is looked at all the individual values of X_{1}and assigned a rank to it. For example, the lowest value, in this case, is 2 and it is given a rank 1 the next highest value is 3 that is given a rank 2 and so on. So, we are ranked all of these points. Notice that the sixth and the first value both are tied. So, they get the rank of 6.5(the midway the half of it) because there is a tie. Similarly, if there are more than 2 values that are tied we take all these ranks and average them by the number of data points that have equal values, and correspondingly you have to give the rank. - Rank Y
_{1}: Similarly, you can give rank to Y_{1}data points in the same manner.

**Step 2**: Calculate d^{2}–

Once you have got the rank you compute the difference in the ranks. So, in this case, the difference in the rank for the first data point is 2 and we square it, similarly, we take the difference in the second data point in the ranks between X_{i} and Y_{i} which is 2 and square it and we get 4. So, like this, we make the difference in the ranks and by squaring it we get the final what we call the d squared values. We sum overall values and then we compute the Spearman coefficient by using this value in the above formula.

By putting the value of the overall sum of d^{2}and n value rho/r_{s}= 1 - ((6 x 20.5) / 990) = 1 - (123 / 990) = 1 - 0.1242 = 0.88

**Properties**:

- r
_{s}takes a value between -1(negative association) and 1(positive association). - r
_{s}= 0 means no association. - It can be used when association is non linear.
- It can be applied for ordinal variables.

**Spearman Correlation for Anscombe’s Data:**

Anscombe’s data also known as Anscombe’s quartet comprises of four datasets that have nearly identical simple statistical properties, yet appear very different when graphed. Each dataset consists of eleven (x, y) points. They were constructed in 1973 by the statistician Francis Anscombe to demonstrate both the importance of graphing data before analyzing it and the effect of outliers on statistical properties.

Those 4 sets of 11 data-points are given here. Please download the csv file here.

When we plot those points it looks like this. I am considering 3 sets of 11 data-points here.

**A brief explanation of the above diagram:**

So, if we apply Spearman correlation coefficient for each of these data sets we find that it is nearly identical, it does not matter whether you actually apply into a first data set (top left) or second data set (top right) or the third data set (bottom left). So, what it seems to indicate is that if we apply the Spearman correlation and we find the reasonably high correlation coefficient close to one in this first data set(top left) case. The key point is here we can’t conclude immediately that if the Spearman correlation coefficient is going to be high then there is a linear relationship between them, for example in the second data set(top right) this is a non-linear relationship and still gives rise to a reasonably high value.

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