Spearman Rank Correlation

Definition

Spearman rank correlation is a statistical measure of the strength and direction of the monotonic relationship between two variables. It is a non-parametric alternative to the Pearson correlation coefficient, used when the assumptions for the Pearson correlation are not met or when the relationship between variables is not linear.

5 Must Know Facts For Your Next Test

1. Spearman rank correlation is calculated by first ranking the data for each variable and then computing the correlation between the ranked values.
2. The Spearman rank correlation coefficient, denoted as $\rho$ (rho), ranges from -1 to 1, where -1 indicates a perfect negative monotonic relationship, 0 indicates no relationship, and 1 indicates a perfect positive monotonic relationship.
3. Spearman rank correlation is less sensitive to outliers compared to Pearson correlation, making it more robust to non-normal distributions and non-linear relationships.
4. The formula for Spearman rank correlation is: $\rho = 1 - \frac{6\sum d^2}{n(n^2 - 1)}$, where $d$ is the difference between the ranks of corresponding values, and $n$ is the number of data points.
5. Spearman rank correlation is commonly used in fields such as psychology, social sciences, and business research, where the assumptions for Pearson correlation may not be met.

Review Questions

• Explain the key differences between Spearman rank correlation and Pearson correlation.
  • The primary difference between Spearman rank correlation and Pearson correlation is that Spearman correlation measures the strength and direction of a monotonic relationship between variables, while Pearson correlation measures the strength and direction of a linear relationship. Spearman correlation is a non-parametric test, meaning it does not require the data to follow a specific probability distribution, unlike Pearson correlation. Additionally, Spearman correlation is less sensitive to outliers and can be used when the relationship between variables is not linear.
• Describe the process of calculating the Spearman rank correlation coefficient.
  • To calculate the Spearman rank correlation coefficient, $\rho$, the following steps are followed: 1) Rank the data for each variable from lowest to highest, 2) Calculate the difference, $d$, between the ranks of corresponding values for each data point, 3) Compute the sum of the squared differences, $\sum d^2$, 4) Apply the formula: $\rho = 1 - \frac{6\sum d^2}{n(n^2 - 1)}$, where $n$ is the number of data points. The resulting coefficient, $\rho$, will range from -1 to 1, indicating the strength and direction of the monotonic relationship between the variables.
• Discuss the practical applications of Spearman rank correlation in business and social science research.
  • Spearman rank correlation is particularly useful in business and social science research when the assumptions for Pearson correlation are not met, such as when the relationship between variables is not linear or the data does not follow a normal distribution. For example, in market research, Spearman rank correlation can be used to analyze the relationship between customer satisfaction ratings and sales performance, even if the relationship is not perfectly linear. In psychology, Spearman rank correlation can be used to assess the correlation between personality traits and job performance, without making assumptions about the underlying distribution of the data. The robustness of Spearman rank correlation makes it a valuable tool for researchers in these fields.

Related terms

Monotonic Relationship: A relationship between two variables where as one variable increases, the other variable either increases (positive monotonic) or decreases (negative monotonic) consistently, without changing direction.

Non-parametric Test: A statistical test that does not rely on the data following a specific probability distribution, making it more robust to violations of assumptions compared to parametric tests.

Pearson Correlation Coefficient: A measure of the linear relationship between two variables, which assumes a normal distribution and a linear relationship between the variables.