Why does rank correlation coefficient differ from Pearsonian correlation coefficient?
The rank correlation coefficient differs from the Pearsonian correlation coefficient in several ways. First, the rank correlation coefficient is derived from the simple correlation coefficient where individual values are replaced by ranks. It measures the linear association between ranks assigned to units, not their actual values. Due to this, all the information concerning the data is not utilised, making it generally less accurate than the ordinary method.
Second, the rank correlation coefficient is not affected by extreme values, whereas the Pearsonian coefficient is sensitive to them. This makes Spearman's coefficient very useful when data contains extreme values. Third, rank correlation can be used in cases where the relation is non-linear but the direction is clear, while Karl Pearson's coefficient should only be used for linear relationships. If the first differences of values were constant, both coefficients would give identical results. Generally, the rank correlation coefficient is less than or equal to the Pearsonian coefficient.
Explanation
The textbook explains that rank correlation differs from Pearsonian correlation primarily because it uses ranks instead of actual values, losing some information in the process. It also highlights that rank correlation is robust against extreme values and can handle certain non-linear relationships, making it more versatile in specific situations. The mathematical relationship r_k r is also mentioned as an important distinction.