Goodman and Kruskal's gamma

Goodman and Kruskal's gamma is a statistical measure used to assess the strength and direction of the association that exists between two variables measured on an ordinal scale. It is a non-parametric measure, meaning it does not assume a normal distribution of the variables. Goodman and Kruskal's gamma is particularly useful in situations where the data are ordinal but not necessarily continuous, making it a valuable tool in fields such as social sciences, psychology, and medicine where ordinal data are common.

Overview

Goodman and Kruskal's gamma (γ) is calculated based on the difference between the number of concordant pairs and discordant pairs divided by the total number of pairs, excluding ties. A concordant pair is a pair of observations that have the same order on both variables, whereas a discordant pair has a different order on the two variables. The value of gamma ranges from -1 to 1, where 1 indicates a perfect positive association, -1 indicates a perfect negative association, and 0 indicates no association.

Formula

The formula for Goodman and Kruskal's gamma is:

\[ \gamma = \frac{N_c - N_d}{N_c + N_d} \]

where \(N_c\) is the number of concordant pairs, and \(N_d\) is the number of discordant pairs.

Application

Goodman and Kruskal's gamma is widely used in various research fields to analyze ordinal data. For example, in medicine, it can be used to assess the relationship between the severity of a disease and the effectiveness of a treatment. In psychology, it might be applied to study the association between stress levels and coping mechanisms.

Advantages and Limitations

One of the main advantages of Goodman and Kruskal's gamma is its ability to provide a measure of association that is not influenced by the presence of tied observations. However, it is important to note that gamma can only be used with ordinal data and is not suitable for nominal or interval data. Additionally, while gamma indicates the strength and direction of an association, it does not imply causation.

Comparison with Other Measures

Goodman and Kruskal's gamma is often compared to other non-parametric measures of association such as Spearman's rank correlation coefficient and Kendall's tau. While all these measures assess the association between two ordinal variables, they differ in their calculation and interpretation. Choosing the appropriate measure depends on the specific characteristics of the data and the research question.

See Also

• Non-parametric statistics
• Ordinal data
• Spearman's rank correlation coefficient
• Kendall's tau