Introduction

Goodman and Kruskal’s gamma, symbolized as G or γ, is a non-parametric measure used to estimate both the strength and the direction of the association between two variables measured on an ordinal scale. This method is particularly important when the data include many ties, since in such cases the use of other techniques, such as Spearman’s rank-order correlation or Kendall’s tau-b, may lead to less reliable results. Therefore, gamma is considered more appropriate and is often recommended by researchers analyzing data where rank order matters more than the absolute numerical difference.

Practical Applications

A characteristic example of gamma’s application is the investigation of the relationship between restaurant ratings, expressed with stars on a scale from one to five, and price category, which may be cheap, moderate, or expensive. In this case, both variables are ordinal, and the analysis can reveal whether higher star ratings are associated with placement in a higher price category. Another example is the study of the relationship between the level of anxiety a student feels before an exam and the duration of the exam. Anxiety may be categorized as low, moderate, or high, while exam duration may range from one to four hours. Using gamma, the analysis can show whether there is a systematic relationship, for instance, whether longer exams are associated with greater anxiety.

Special Case: Yule’s Q

Goodman and Kruskal’s gamma can also be applied when the two variables each have only two categories. For example, it may be used to examine the relationship between passing or failing an exam and the level of anxiety, which may be high or low. In such cases, however, it is often preferable to use Yule’s Q, which is considered a special case of gamma. Yule’s Q is particularly useful when the analysis focuses on dichotomous variables, such as gender, which can be categorized as male or female. In this way, Yule’s Q complements gamma and extends its usefulness to more limited scenarios.

Assumptions of Application

The correct use of gamma requires that certain assumptions be met in order for the results to be valid. First, both variables must be measured on an ordinal scale. This means their categories must have a specific order of ranking, without necessarily having equal distances between the categories. An example is a Likert scale, where participants are asked to express their level of agreement on a scale ranging from “strongly disagree” to “strongly agree.” The second assumption is the presence of a monotonic relationship between the two variables. A monotonic relationship exists when the variables move in the same direction—meaning as one increases, the other also increases—or when they move in opposite directions—meaning as one increases, the other decreases. Although this assumption cannot always be verified with certainty before conducting the analysis, it is considered essential for the validity of the results.

Procedure in SPSS Statistics

The calculation of Goodman and Kruskal’s gamma in SPSS is performed through the Crosstabs procedure. The user creates a cross-tabulation table by placing the two ordinal variables in their respective rows and columns and then requests the display of association measures. At this stage, the gamma option is activated, and the software calculates the value of the measure, its sign, which shows the direction of the relationship, and its significance level. Although technically straightforward, this procedure requires careful data entry and appropriate variable selection in order for the results to accurately reflect the true association.

Conclusions

Goodman and Kruskal’s gamma is a powerful measure for analyzing relationships between ordinal variables, especially when a large number of ties exist. It allows the researcher to understand not only the strength of the association but also its direction, leading to more complete conclusions. Its implementation through SPSS greatly facilitates the analysis, making it accessible even to researchers with limited statistical background. However, the validity of the results depends on meeting the required assumptions, namely that the variables are measured on an ordinal scale and that a monotonic relationship exists. With proper preparation and application, Goodman and Kruskal’s gamma stands out as a reliable and flexible method that can be widely used in the social and behavioral sciences, as well as in other fields where the analysis of ordinal data is critical.