# What are the Goodman-Kruskal statistics?

The Goodman-Kruskal statistics are measures of association between categorical variables. You can get Goodman-Kruskal tau and Goodman-Kruskal lambda statistics by choosing Stat > Tables > Cross Tabulation and Chi-Square and clicking Other Stats. You also get Goodman-Kruskal gamma when you perform Stat > Regression > Binary Logistic Regression or Stat > Regression > Ordinal Logistic Regression.

## What is Goodman-Kruskal tau?

Goodman-Kruskal tau measures association for cross tabulations of nominal level variables.

Goodman-Kruskal tau is based on random category assignment. It measures the percentage improvement in predictability of the dependent variable (column or row variable) given the value of other variables (row or column variables). Goodman-Kruskal tau is the same as Goodman-Kruskal lambda except the calculations of the tau statistic are based on assignment probabilities specified by marginal or conditional proportions.

Misclassification probabilities are based on random category assignment with probabilities specified by marginal or conditional proportion.

## What is Goodman-Kruskal lambda?

Goodman-Kruskal lambda measures association for cross tabulations of nominal level variables.

Goodman-Kruskal lambda is based on modal probabilities. It measures the percentage improvement in probability of the dependent variable (column or row variable) given the value of other variables (row or column variables).

Misclassification probabilities are calculated based on assignment to the category with the highest probability. Lambda with X (row variable) as the dependent variable is calculated as:
• Let S denote the sum of the highest cell count for each row.
• Let R denote the highest row total.
• Let N denote the total of all the cell counts.
• Lambda is the ratio of (S – R) / (N – R).

To calculate lambda with Y (column variable) as the dependent variable, follow the previous steps and replace S with the highest cell count for each column and replace R with the highest column total.

## What is Goodman-Kruskal gamma?

Goodman-Kruskal gamma (γ) shows how many more concordant than discordant pairs exist divided by the total number of pairs excluding ties. Use the Goodman-Kruskal gamma to measure the association between the ordinal variables.

Perfect association exists when |γ| = 1. In ordinal and binary logistic regression, if X and Y are independent, then γ = 0.

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