Spearman's Rank Correlation

Learn how Spearman's rank correlation coefficient helps you to know the relationship between variables and make informed decisions in data analysis.

What is Spearman’s Correlation

Spearman's rank correlation is a statistical rule applied to uncover the degree and the direction of the association of the two ranked variables. Unlike other correlation techniques that require the data to be in the intervention or ratio scale, Spearman's rank correlation coefficient can be used for the ranked data but is not necessarily equidistant. This is why the ranking tool is considered very useful in many fields like psychology, sociology, economics, and biology, where the data is usually ranked.

The rank of a data observation according to its place in the data set is the main idea of Spearman's correlation. After the ranks have been obtained, they are used to find the correlation coefficient which is denoted by r (r)=(-1,1) and is a measure of the relationship between the two variables.

Importance in Statistics

1. Non-Parametric Analysis: Spearman's rank correlation, a non-parametric method, does not require a particular type of data distribution. It is highly resilient and adaptable to different kinds of data and their distributions.

2. Handles Non-Linear Relationships: Thus, unlike the Pearson correlation that is used for linear relationships, Spearman's rank correlation can measure non-linear relationships between variables. Hence, non-linear data can be analyzed, making it applicable to cases without a linear pattern.

3. Useful for Small Sample Sizes: Spearman's rank correlation is a great tool for small sample sizes where the normality is not yet shown, and thus the normality condition does not apply. It gave weak data that was nearly manageable, achieving a stable measure of the connection.

4. Interpretability: The Spearman's rank correlation coefficient is an easy-to-understand interpretation of the correlation between the ranked data. It is the method through which the relationship between the variables is outlined in terms of the rank, and therefore, it can be used by researchers and practitioners who are not familiar with the statistics.

5. Identifies Outliers: Spearman's rank coefficient is not affected by outliers as the Pearson correlation is. Therefore, it is simple to note the distinct observations that have a strong effect on the variable's correlation.

Formula for Spearman Rank Correlation and Calculation Steps

The formula for Spearman's rank correlation coefficient formula (ρ) can be expressed as:

Where:

• 𝜌 is the Spearman's rank correlation coefficient, in other words, the coefficient between the ranks of two variables.
• d is the shift that takes place in the order of the comparable variables.
• ∑ is the sign of the sum, that is, the total of the sums.
• 𝑛 is the count of data items.

The calculation steps for Spearman's rank correlation coefficient are as follows:

1. Assign the ranks to each value in the two variables; the smallest one will get the rank 1 first.
2. Determination of the distance between the orders of each of the same values.
3. The most suitable method is to square the different values.
4. In the last one, the total squared differences are found.
5. Give a go-to Spearman's rank correlation coefficient using the formula.

Spearman rank correlation formula for repeated ranks

The formula of Spearman's rank correlation coefficient for data with repeated ranks is the same as that for the data without repeated ranks. On the other hand, the computation of the squared differences in ranks (𝑑2d2) might need to be modified for the tied data.

Example of Spearman correlation Calculation

Let's consider a simple example to illustrate the calculation of Spearman's rank correlation coefficient:

We have two variables – X and Y, with the following data:

X: 10, 8, 4, 5, 6

Y: 5, 4, 2, 3, 1

1. Rank the data:

X: 4, 5, 6, 8, 10

Y: 2, 3, 1, 4, 5

2. Calculate difference between ranks for each pair:

d: 2, 2, 5, 4, 5

3. Square each difference:

d^2: 4, 4, 25, 16, 25

4. Sum up all the squared differences:

∑𝑑2=74

In this specific situation, the Spearman's rank correlation coefficient is -3. 44, hence it can be finally said that there is a solid negative monotonic relationship between the variables X and Y.

Assumptions and Limitations of Spearman's Rank Correlation

The Spearman’s rank correlation is a powerful statistical method that can be used in various situations. It is particularly useful in the following scenarios:

1. Outliers: Spearman’s rank correlation like other coefficients of correlation measures is also affected by outliers. Outliers significantly impact the ranking of observation, which is why they can influence the correlation coefficient.

2. Rank Differences: Spearman’s correlation merely deals with the rank of the data concerned and does not record the actual difference. This suggests that it may lack the ability to consider other factors in the relationship of these variables.

3. Linearity: Using Spearman correlation rankings, the outcomes are convenient, but they are different from the Pearson correlation since they do not suppose any kind of relationship, linear or otherwise. However, the technique might not work when identifying nonlinear interactions between variables.

4. Small Sample Sizes: The Spearman correlation is not highly affected by outliers and applies to studies with small data. Hence, it can be used in research with a limited amount of data.

5. Homoscedasticity: Like other correlation coefficients, Spearman’s correlation also presumes homoscedasticity, which presupposes that the variability of the differences between ranks should be equal to any point inside or outside a determined range. If this assumption is violated, then the value of the correlation coefficient could be inaccurate and misleading.

Factors Affecting Its Validity

1. Outliers: Although the Spearman correlation is not as much affected by outliers as the Pearson correlation, the extreme outliers can still be the main reason why this type of correlation is not valid. Considering the outliers and seeing their impact on the investigation is essential.

2. Sample Size: Although the Spearman correlation can be applied to a small sample size, a larger sample size will give a more exact correlation coefficient.

3. Rank Differences: If the data has many equal ranks, the Spearman correlation may be too small and may not show the actual relationship between the variables. The survey was sent to a small group of people, so it becomes imperative to be careful when reading the results.

4. Linearity: The opposite is also true. If there is a non-linear relationship, the Spearman correlation does not assume it, but it does assume a monotonic relationship. Additionally, the correlation between variables that are not continuous may not be best measured with Spearman correlation.

5. Homoscedasticity: The Spearman rank correlation test doesn't have the assumption of the equalities of the variances of the variables. If the variances of the two variables are very different, it will be difficult to analyze their relationship.

Comparison with Other Correlation Methods

Spearman's rank correlation is a nice and powerful method to study the relationship between variables, but it is not the only correlation technique that one can use. Here's a comparison with other popular correlation methods:

1. Pearson Correlation: Pearson rank correlation is a statistic that tells the extent of the linear relationship between two continuous variables. The sentence shows that the variables are normally distributed and have a linear connection. Spearman correlation is not based on the linearity assumption and applies to ordinal data, whereas Pearson correlation assumes a linearity assumption.

2. Kendall's Tau: Kendall's Tau is a non-parametric correlation coefficient generally used to determine the level of interdependence between two variables. The coherent and incoherent pairs of the observations are used in the method, which is akin to Spearman correlation but is based on these pairs rather than the actual ranks. Kendall's Tau is usually used in situations where there are tied ranks.

3. Point-Biserial Correlation: Point-biserial correlation is suitable for the case when one variable is dichotomic (binary), and the other variable is continuous. It is the level and the meaning of the link between the two entities. The Spearman correlation, nonetheless, can be used for the variables that are either ordinal or continuous.

4. Spearman vs. Pearson: The Spearman correlation is more robust in dealing with outliers and non-linear relationships than the Pearson correlation. The opposite is the case for Spearman correlation, which is the best for ordinal data, while Pearson correlation is the ideal for continuous data with a linear relationship.

Spearman Rank Correlation Test

Purpose and Procedure

The Spearman rank correlation test is employed to ascertain the degree and the way the monotone relationship between two variables exists. It is a non-parametric test that does not impose any assumptions on the data distribution. The procedure for conducting the Spearman rank correlation test is as follows:

1. Rank the Data: Listen to the values of each variable separately and put them in order from the lowest to the highest, giving the lowest value rank 1, the next lowest rank 2, and so on. If there are ties, the average rank of the tied values will be assigned.

2. Calculate the Difference in Ranks: For each particular pair of corresponding ranks, determine the difference in ranks (𝑑d).

3. Square the Differences: The squares of differences in ranks (𝑑2) are being calculated.

4. Sum the Squared Differences: The sum of all the squared differences in ranks is the conclusion.

5. Determine the Significance: Create a statistical table or use software to establish the correlation coefficient's significance. The null hypothesis is that there is no monotonic relation between the variables.

Conclusion

The Spearman rank correlation test is a useful tool to analyze the relationship between the variables when the criteria for other correlation methods are not fulfilled. It is a strong indicator of the correlation that can assist researchers in the data analysis process; on the other hand, it can be used as a tool to comprehend the hidden patterns of the data. Through the classification of the data and the computation of the correlation coefficient, researchers can examine if there is a monotonic relationship between the variables and the degree and importance of this relationship.

Frequently Asked Questions (FAQs)

Q. What is Spearman's rank correlation?

A. Spearman's rank correlation is a statistical approach that evaluates the strength and the direction of the correlation between two ranked variables.

Q. How is Spearman's rank correlation calculated?

A. The first ranking of the data determines Spearman's rank correlation, then the calculation of the differences in ranks, the squaring of these differences, the summing up of such differences, and the application of the formula to the calculation of the correlation coefficient.

Q. When is Spearman's rank correlation used?

A. Spearman's rank correlation is used when the data is of ordinal or ranked type and when the relationship between variables is monotonic rather than linear.

Q. What is the difference between Pearson and Spearman rank correlation?

A. Pearson correlation determines the linear relationships between the variables, while Spearman correlation does the monotonic ones, and both of them use ranks.

Q. Why do we use Spearman correlation?

A. The Spearman correlation is employed when the conditions for Pearson correlation are not satisfied, for instance, in the case of the non-normal distribution of data or outliers.

Q. What is an example of Spearman's correlation?

A. An example of Spearman's correlation is the correlation between the ranks of the students in a class and their exam scores.

Q. Should I use Spearman or Pearson?

A. In such cases, Spearman correlation should be used when the data is ordinal or when the relationship between the variables is monotonic.

Q. Can Spearman's rank be negative?

A. Spearman's rank correlation coefficient can be -1 to 1, where -1 signifies an absolute negative monotonic relationship, one stands for a perfect positive monotonic relationship, and 0 indicates no monotonic relationship.