Strength of the association in the crosstab (gamma, tau, Somers' d, lambda, uncertainty coefficient)

In brief

The chi-square test says whether two categorical characteristics are related; it does not say how strongly. Under "More options" of the question "Does something occur at different rates in different groups?" there is the section "Strength of the association". Statable suggests the measure that fits the level of measurement of the two columns; the others can be switched on. The result is a table with value, 95% interval, test statistic T and p, plus a sentence.

Which measure fits?

• Two ordered variables (grades, levels) with the same number of levels: Kendall's τb. It includes ties (equal values) and can reach ±1 only in square tables.
• Unequal numbers of rows and columns: Kendall's τc (Stuart) or Somers' d.
• One variable is the dependent one (for example exam by grade): Somers' d, directional. It asks how many more concordant than discordant pairs there are when looking at the dependent variable. Statable shows both directions.
• Gamma ignores pairs with the same value and so usually comes out higher; never the default. With two by two cells gamma is the same as Yule's Q.
• Nominal variables (groups without an order): Cramér's V is already there; in addition the uncertainty coefficient (Theil), which shows the share of uncertainty cleared up, and on request lambda.
• Two groups and an ordered answer: a rank test (Mann-Whitney) with the effect size d = 2 · AUC − 1 fits better.

Ordinal measures depend on the order

Statable reads the rows and columns as they stand in the table: from low to high. For numbers with labels this is the order of the numbers. For text Statable sorts alphabetically, so "high" would come before "medium" before "low". The levels should therefore have numbers with labels (describe columns) before ordinal measures are read. A warning appears when a text column counts as ordered.

Interval and test

• The interval is value ± 1.96 · standard error at the strength found (ASE1), limited to −1 to 1 (gamma, tau, d) or 0 to 1 (lambda, UC). The test uses T = value / standard error under "no association" (ASE0). The two standard errors differ; so the interval can contain 0 while p is below .05, especially with strong association.
• Gamma, tau-b, tau-c and Somers' d share the same T and thus the same p. Statable therefore shows the test once in the row "Test (ordinal measures)", as SPSS does. The rank correlation card calculates the p of τb with a different, equally valid variance; with strong association and many ties the two differ.
• The symmetric Somers' d has no interval: PSPP gives no standard error for it, and the approximation used by SPSS differs from the exact calculation. The directional d has one and is usually the more sensible measure.
• The p of the uncertainty coefficient comes from the likelihood ratio test (G²), as in SPSS (footnote "Likelihood ratio chi-square probability"); the T next to it does not belong to this p.

Lambda is not a measure of strength

Lambda shows by how much the errors in predicting the most frequent answer fall when the group is known. If the same answer is the most frequent in every group, λ = 0 - even with a clear association. Example: 90 to 10 in one group, 60 to 40 in the other: λ (answer from group) = 0, but V = .35, chi-square test p < .001, OR = 6. Statable then shows the value 0 without interval and without test and adds the sentence: "That does not mean there is no association." That is why lambda is never suggested.

Special cases

• Fewer than two rows or columns with cases: there is no measure. Statable says so instead of showing "not calculable".
• Perfect association: gamma and τb are 1, the interval is [1; 1] - without scatter it is not meaningful. τc is smaller (rectangular tables).
• Fewer than 30 cases or more than one cell in five with fewer than 5 expected cases: the interval is an approximation for large samples.
• Case weight: with a frequency weight Statable calculates the weighted table (N = sum of the weights). With a survey weight the section is blocked because standard errors and tests need the sampling design.

Reporting the result

τb = .41, 95% CI [.27, .56], p < .001; N = 112

Example: "The higher the level of education, the higher the level of satisfaction tends to be (Kendall's τb = .41, 95% CI [.27, .56], p < .001; N = 112)." Do not word it causally.

For specialists

PSPP calculates the measures (CROSSTABS with GAMMA, D, BTAU, CTAU, LAMBDA, UC); Statable reads the tables "Symmetric Measures" and "Directional Measures" by their labels and assigns the direction by the name of the dependent variable, never by order. The interval and the p are added where PSPP does not give them (gamma, tau). Cross-checked against the formulas of the IBM documentation (SPSS Algorithms, CROSSTABS), a numerical delta method and the printed examples of the manual "IBM SPSS Exact Tests" (smoking, party). DescTools deviates in the sixth place for τb and d and is not the reference for those two.

Sources

Goodman, L. A., & Kruskal, W. H. (1954). Measures of association for cross classifications. Journal of the American Statistical Association, 49, 732-764. Kendall, M. G. (1945). The treatment of ties in ranking problems. Biometrika, 33, 239-251. Stuart, A. (1953). The estimation and comparison of strengths of association in contingency tables. Biometrika, 40, 105-110. Somers, R. H. (1962). A new asymmetric measure of association for ordinal variables. American Sociological Review, 27, 799-811. Theil, H. (1970). On the estimation of relationships involving qualitative variables. American Journal of Sociology, 76, 103-154. Brown, M. B., & Benedetti, J. K. (1977). Sampling behavior of tests for correlation in two-way contingency tables. Journal of the American Statistical Association, 72, 309-315. Agresti, A. (2010). Analysis of Ordinal Categorical Data (2nd ed.). Wiley. IBM (2014). SPSS Statistics Algorithms: CROSSTABS. Mehta, C., & Patel, N. IBM SPSS Exact Tests, chapters 13 to 15.