Correlation is a method used to measure the relationship between two variables. It shows whether an increase in one variable corresponds to an increase or decrease in another, and it helps quantify the strength of this relationship. For example, suppose we want to determine whether taller fathers tend to have taller sons. In that case, we can apply correlation techniques, such as Spearman’s Rank-Order Correlation, to assess both the strength and direction of this relationship. By ranking the data and analyzing the consistency of their order, we can evaluate the extent of the correlation without assuming any specific distribution of the data.

Types of Correlation:

There are two main ways to calculate correlation, depending on the type of data and its properties:

1. Parametric Correlation (Pearson’s Correlation Coefficient, denoted as r)
   • What it does: Measures the strength and direction of a straight-line (linear) relationship between two variables, like height and weight.
   • When to use it: It works best for numerical data that follows a normal distribution (a common bell-shaped curve).
   • Example: Comparing the heights of fathers and sons. If taller fathers generally have taller sons, Pearson’s r will show a positive value, close to +1. If the relationship is weak, r will be closer to 0.
2. Non-Parametric Correlation (Kendall’s Tau and Spearman’s Rho)
   • What they do: Measure the relationship based on the order or rank of the data, rather than the actual values.
   • When to use them: They are ideal for categorical data (data grouped into categories) or when the data doesn’t meet the assumptions of a normal distribution.
   • Example: Ranking students in a class based on their grades and checking if their rank in sports performance matches their academic rank.
     • Spearman’s rho is commonly used for ranking numerical data.
     • Kendall’s tau is more specific when dealing with pairwise rankings.

What is Spearman’s Rank Correlation?

Spearman’s Rank Correlation is a statistical method used to measure the strength and direction of a relationship between two variables. It works by ranking the values of the two variables and then comparing those ranks. The method checks if the relationship between the two variables is monotonic, meaning as one variable increases, the other either consistently increases or decreases.

This correlation is represented by the symbol “rho” (ρ) and can range from -1 to +1:

• A positive rho (close to +1) means the two variables increase together.
• A negative rho (close to -1) means as one variable increases, the other decreases.
• A rho of 0 means there’s no association between the two variables.

Formula for Spearman’s Correlation

r = 1 −

\frac{n(n^2 − 1)}{6 \sum d_i^2}

Explanation of the terms:

• ρ (rho): The Spearman’s Correlation coefficient, which tells us the strength and direction of the relationship.
• Rank: The position of each value when the data is sorted in order.
• d_i​: The difference between the ranks of the two variables for each data point.
• n: The total number of observations in the dataset.

Compute Spearman’s Rank Correlation

To calculate Spearman’s Rank Correlation, we need to follow these steps, breaking down the process in simple terms:

Converting Data into Ranks

First, we assign a rank to each number in the dataset. The smallest number gets a rank of 1, the next smallest gets a rank of 2, and so on.

Data:

Ranking the Values for X1

• Start by sorting the values of X1 in ascending order: 2, 3, 4, 5, 6, 7, 7, 8, 9, 10
• Now, assign ranks based on the order of the sorted values:
  • 2 → Rank 1
  • 3 → Rank 2
  • 4 → Rank 3
  • 5 → Rank 4
  • 6 → Rank 5
  • 7 → Rank 6.5 (since there are two 7s, they share the rank, so their average rank of 6.5 is assigned)
  • 7 → Rank 6.5
  • 8 → Rank 8
  • 9 → Rank 9
  • 10 → Rank 10

Ranking the Values for Y1

• Similarly, sort the values of Y1 in ascending order: 1, 2, 4, 5, 6, 7, 8, 8, 9, 10
• Assign ranks:
  • 1 → Rank 1
  • 2 → Rank 2
  • 4 → Rank 3
  • 5 → Rank 4
  • 6 → Rank 5
  • 7 → Rank 6
  • 8 → Rank 7.5 (since there are two 8s, they share the rank, so their average rank of 7.5 is assigned)
  • 8 → Rank 7.5
  • 9 → Rank 9
  • 10 → Rank 10

Assigning the Ranks for X1 and Y1

Now, let’s match the ranks to each data point:

Spearman’s Correlation Calculations 

Spearman’s Rank Correlation is used to measure the strength and direction of the relationship between two variables by converting their original data into ranks. The goal is to assess the monotonic relationship without depending on the actual numerical values.

Let’s consider 10 data points for variables X1 and Y1. The first step is to arrange the values in ascending order and assign ranks. The smallest value gets a rank of 1, the second smallest gets 2, and so on. After assigning ranks to both X1 and Y1, we calculate the difference in ranks for each data point, then square these differences to get the d² values.

For example, for the first data point, the difference in ranks is 2, and squaring it gives a d² of 4. Similarly, we calculate the differences and squares for all data points. Once we have the d² values, we sum them up and apply the following formula to compute the Spearman correlation coefficient: ρ=1−6∑di2n(n2−1)\rho = 1 – \frac{6\sum d_{i}^{2}}{n(n^2-1)}ρ=1−n(n2−1)6∑di2​​

By plugging in the sum of d² and the number of data points (n), we get the Spearman correlation coefficient, which in this case is approximately 0.88, indicating a strong positive relationship.

Properties of Spearman’s Correlation:

• The coefficient ρ\rhoρ ranges from -1 (perfect negative correlation) to 1 (perfect positive correlation).
• A value of ρ=0\rho = 0ρ=0 means there is no association.
• It works even when the relationship between variables is not linear.
• It is suitable for ordinal data (ranked data).

Monotonic and non-monotonic relationships

A monotonic relationship is a type of relationship between two variables where the direction (whether increasing or decreasing) stays the same throughout. In other words, as one variable changes, the other variable consistently either increases or decreases.

On the other hand, a non-monotonic relationship is a relationship where the direction of the relationship between two variables is not consistent. The variables may increase and decrease at different points, and there’s no predictable pattern of continuous increase or decrease.

Advantages of Spearman’s Rank Correlation:

1. Simplicity: Spearman’s Rank Correlation is easy to understand and apply, making it user-friendly for those without advanced statistical knowledge.
2. Handles Qualitative Data: It is particularly useful when measuring subjective attributes, like intelligence or appearance, which don’t have exact numerical values.
3. Order of Preference: This method is ideal when you only have information about the order of preferences or rankings (like in surveys) but not the exact values of the variables.
4. Resistant to Outliers: Spearman’s rank correlation is not greatly affected by outliers, making it a reliable method in datasets with extreme values.
5. Monotonic Relationships: It’s great for capturing monotonic relationships, where one variable consistently increases or decreases with another, even if the relationship is not linear.
6. Non-Normal Data: Spearman’s rank correlation doesn’t assume the data follows a normal distribution, so it works well with skewed or non-normal data.

Disadvantages of Spearman’s Rank Correlation:

1. Not for Grouped Data: It cannot be applied to grouped data, where values are aggregated into ranges or categories.
2. Limited Data Handling: It becomes less reliable when dealing with very large datasets, as the complexity and computation increase with more data points.
3. Ignores Non-Monotonic Relationships: Spearman’s rank correlation can’t detect non-monotonic relationships, such as curvilinear or nonlinear associations.
4. Rank-Based: The method only considers the relative order of data points (ranks), disregarding the actual numerical differences between the values, which can lead to a loss of detailed information.
5. Information Loss: Transforming data into ranks means you’re losing information about the magnitude of the values. This can be problematic if the actual differences between the values are meaningful.
6. Limited to Pairwise Comparisons: Spearman’s rank correlation works best with pairs of variables. For more complex, multi-variable relationships, it may not be as effective.
7. Sensitivity to Tied Ranks: If many values are tied (i.e., they are the same), it can affect the correlation calculation, making the results less accurate.

Conclusion:

Spearman’s Rank Correlation is a valuable statistical tool for measuring the strength and direction of a monotonic relationship between two variables, particularly when the data is ordinal, non-normally distributed, or contains outliers. It is simple to understand and applies well to qualitative data or rankings, making it useful in many fields like social sciences, psychology, and education. However, it has its limitations, such as its inability to capture non-monotonic relationships or handle grouped data effectively. Additionally, it only considers ranks, which may lead to the loss of important information about the actual magnitude of differences between data points. Despite these drawbacks, Spearman’s Rank Correlation remains a reliable and robust method when working with data that fits its assumptions, offering insights into relationships that might not be apparent with other methods.

FAQs for Spearman’s Rank Correlation

1. What is Spearman’s Rank Correlation?
   Spearman’s Rank Correlation is a non-parametric measure used to assess the strength and direction of a monotonic relationship between two variables. It is based on the ranks of the data rather than the actual values.
2. When should I use Spearman’s Rank Correlation?
   Use Spearman’s Rank Correlation when:

• The data is ordinal (i.e., ranks or preferences).
• You need to measure the relationship between two variables without assuming a linear relationship.
• The data includes outliers or is non-normally distributed.
• You only have information about the order, not the exact values of the variables.

3. How is Spearman’s Rank Correlation calculated?
   Spearman’s Rank Correlation is calculated by ranking the data points for each variable and then calculating the difference in ranks for each pair of data points. The formula is:

ρ=1−6∑d2n(n2−1)\rho = 1 – \frac{6 \sum d^2}{n(n^2 – 1)}ρ=1−n(n2−1)6∑d2​

Where:

• ρ\rhoρ is the Spearman rank correlation coefficient.
• ddd is the difference between the ranks of each pair of data points.
• nnn is the number of data points.

4. What is the range of Spearman’s Rank Correlation?
   The Spearman’s Rank Correlation coefficient (ρ) ranges from -1 to +1:

• +1 indicates a perfect positive monotonic relationship.
• -1 indicates a perfect negative monotonic relationship.
• 0 indicates no monotonic relationship.

5. What is the difference between Pearson and Spearman correlation?
   Pearson’s correlation measures the linear relationship between two variables, assuming the data is normally distributed. Spearman’s Rank Correlation, on the other hand, measures the monotonic relationship (whether linear or non-linear) and can be used with ordinal, non-normal, or skewed data.
6. Can Spearman’s Rank Correlation be used with non-numerical data?
   Yes, Spearman’s Rank Correlation can be used with non-numerical data as long as the data can be ranked, such as preferences, ratings, or orderings.
7. Can Spearman’s Rank Correlation handle ties in the data?
   Yes, Spearman’s Rank Correlation can handle ties, but it adjusts the ranks for tied values. When multiple data points have the same value, the average rank is assigned to those tied values.
8. Is Spearman’s Rank Correlation the same as Kendall’s Tau?
   No, while both are non-parametric tests for measuring ordinal association, Kendall’s Tau is another method for calculating correlation that differs in its calculation approach. Spearman’s Rank Correlation is generally easier to compute but may not be as accurate in some cases as Kendall’s Tau.
9. Can Spearman’s Rank Correlation be used for more than two variables?
   No, Spearman’s Rank Correlation is specifically used to measure the relationship between two variables at a time. For more complex relationships involving multiple variables, other methods like Spearman’s rank partial correlation or multiple regression analysis may be more appropriate.
10. How does Spearman’s Rank Correlation handle outliers?
    Spearman’s Rank Correlation is relatively resistant to outliers because it focuses on the ranks of the data rather than their actual values, making it less sensitive to extreme values.