Negative Correlation: Inverse Association between Two Variables

Negative Correlation occurs when one variable increases as the other decreases, represented by correlation coefficients less than 0.

Negative Correlation, also known as an inverse correlation, occurs when an increase in one variable is associated with a decrease in another variable. Mathematically, this relationship is represented by correlation coefficients less than 0.

Mathematical Representation

The correlation coefficient, usually denoted as \( r \), quantifies the degree to which two variables are linearly related. The value of \( r \) ranges from -1 to +1. For negative correlations:

$$ -1 \leq r < 0 $$

Example: Consider the correlation between hours of exercise per week and body fat percentage. Generally, as the hours of exercise increase, the body fat percentage decreases, demonstrating a negative correlation.

Types of Correlation

Pearson’s Correlation Coefficient

The most commonly used measure of correlation is the Pearson correlation coefficient, defined as:

$$ r = \frac{ \sum (x_i - \overline{x})(y_i - \overline{y}) }{ \sqrt{\sum (x_i - \overline{x})^2 \sum (y_i - \overline{y})^2} } $$

Spearman’s Rank Correlation

For non-linear associations, Spearman’s rank correlation coefficient can be used. It evaluates the monotonic relationship between two variables.

Special Considerations

• Causation vs. Correlation: A negative correlation does not imply causation. It merely indicates an association.
• Outliers: Extreme values can distort the correlation coefficient, giving a misleading interpretation.

Historical Context

Negative correlation concepts date back to the early 20th century with contributions from Karl Pearson, who developed methods to quantify linear relationships between variables.

Applicability

Negative correlations are widely applicable in domains such as:

• Finance: Inverse relationship between asset prices and interest rates.
• Health: Relationship between physical activity and risk factors for various diseases.
• Economics: Trade-off between unemployment rates and inflation as described in the Phillips Curve.

Comparisons

• Positive Correlation: Here, an increase in one variable is associated with an increase in another. Represented by \( 0 < r \leq 1 \).

Related Terms

• Positive Correlation: Correlation where variables move in the same direction.
• Correlation Coefficient: A measure of the strength and direction of a linear relationship between two variables.
• Causation: Relationship where one variable directly affects another.
• Outliers: Data points that are significantly different from others in the dataset.

FAQs

Can a negative correlation be 0?

No, a correlation of 0 indicates no linear relationship between the variables.

What does a correlation coefficient of -1 mean?

A correlation coefficient of -1 indicates a perfect negative linear relationship between the variables.

Are negative correlations always linear?

While the Pearson correlation measures linear relationships, other methods like Spearman’s correlation can measure non-linear monotonic relationships.

References

1. Pearson, K. (1896). Mathematical contributions to the theory of evolution. III. Regression, heredity, and panmixia. Philosophical Transactions of the Royal Society of London. Series A, 187, 253-318.
2. Spearman, C. (1904). The proof and measurement of association between two things. American Journal of Psychology, 15(1), 72-101.

Summary

Negative correlation demonstrates the inverse relationship between two variables. Represented by correlation coefficients less than zero, it’s a crucial concept in data analysis, helping to understand the dynamics between variables in various fields. Understanding its limitations, such as the distinction between correlation and causation, is essential for accurate interpretation of data.