Spearman Rank Order Correlation

For ordinal-level data, the Spearman rank order correlation is one of the most common methods to measure the direction and strength of the association between two variables. First put forth by British psychologist Charles E. Spearman in a 1904 paper, the nonparametric (i.e., not based on a standard distribution) statistic is computed from the sequential arrangement of the data rather than the actual data values themselves. The Spearman rank order correlation is a specialized case of the Pearson product-moment correlation that is adjusted for data in ranked form (i.e., ordinal level) rather than interval or ratio scale. It is most suitable for data that do not meet the criteria for the Pearson product-moment correlation coefficient (or Pearson's r), such as variables with a non-normal distribution (e.g., highly skewed) or that demonstrate a somewhat nonlinear tendency.

Although Pearson's r indicates the strength of the linear relationship between two variables, the Spearman rank order correlation shows the strength of the monotone associations. A monotonic relationship is one in which all x and y variables are arranged in ascending order and are compared for their differences in ranks. In other words, the statistic determines to what degree one data set influences another data set. An increasingly positive monotonic relationship would be one in which as x increases in rank (and value), y also increases (or stays the same) in rank, and the data pairs are concordant with one another. In contrast, an increasingly negative monotonic relationship exists when as x increases in rank (and value), y decreases (or stays the same) in rank. In this case, the data pairs are discordant with one another and exhibit large differences between their respective ranks. In contrast to Pearson's r, the Spearman rank order correlation does not differentiate between linear and monotonic associations because of the ordinal level scale of the data. This property enables the Spearman rank order correlation to be a suitable nonparametric alternative to the Pearson's r when assumptions regarding linearity cannot be met or are unknown.

This entry discusses several aspects of the Spearman rank order correlation, including methods for computing, the influence of tied rankings, and statistical significance and significance testing.

Computation

The Spearman rank order correlation measures the degree of association of ordinal-level data by examining the ratio of the sum of the squared differences in the ranks of the paired data values to the number of variable pairs. Computationally, the Spearman rank correlation coefficient (rs) is defined by the formula

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where d is the difference in statistical ranks between the paired variables x and y, ∑d2 is the sum of the squared differences between the ranks of the paired variables, and N is the number of paired data values. As with the Pearson's r, values of rs range from a minimum of −1.0 to a maximum of 1.0. Increasingly negative rs values indicates a negative monotonic relationship between the ordered pairs, and x and y are inversely related to one another (i.e., as one variable increases, the other tends to decrease). Conversely, increasingly positive rs values indicates a positive monotonic relationship, and x and y covary in the same direction (i.e., as one variable decreases, the other is apt to also decrease). The closer the Spearman rank correlation coefficient is to the extremes (1.0 or −1.0), the stronger the association between the variables. If no association exists between the variables, then rs is equal to or near 0. The statistical strength of the Spearman correlation has been demonstrated to be as robust as that of the parametric Pearson's r, especially for data sets with considerable range in values and reduced frequency of tied ranks.

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