Tschuprow's T Explained

In statistics, Tschuprow's T is a measure of association between two nominal variables, giving a value between 0 and 1 (inclusive). It is closely related to Cramér's V, coinciding with it for square contingency tables.It was published by Alexander Tschuprow (alternative spelling: Chuprov) in 1939.^[1]

Definition

For an r × c contingency table with r rows and c columns, let

\pi _{ij} be the proportion of the population in cell (i,j) and let\pi _{i+} = \sum ^{c\pi} _{ij} and \pi _{+j} = \sum ^{r\pi} _{ij} .

Then the mean square contingency is given as

\phi ^{2} = \sum ^{c ((\pi _{ij} - \pi _{i+} \pi _{+j} ) ^{2})/(\pi _{i+} \pi _{+j})} _{j =1} ,

and Tschuprow's T as

T = \sqrt{ (\phi ^{2})/(\sqrt{(r -1 )(c -1 )) }} .

Properties

T equals zero if and only if independence holds in the table, i.e., if and only if

\pi _{ij} = \pi _{i+} \pi _{+j} . T equals one if and only there is perfect dependence in the table, i.e., if and only if for each i there is only one j such that \pi _{ij} > 0 and vice versa. Hence, it can only equal 1 for square tables. In this it differs from Cramér's V, which can be equal to 1 for any rectangular table.

Estimation

If we have a multinomial sample of size n, the usual way to estimate T from the data is via the formula

\hat T = \sqrt{ (\sum ^{c ((p _{ij} - p _{i+} p _{+j} ) ^{2})/(p _{i+} p _{+j})} _{j =1})/(\sqrt{(r -1 )(c -1 )) } },

where

p _{ij} = n _{ij} /n is the proportion of the sample in cell (i,j) . This is the empirical value of T. With \chi ^{2} the Pearson chi-square statistic, this formula can also be written as\hat T = \sqrt{ (\chi ^{2/n})/(\sqrt{(r -1 )(c -1 )) } } .

See also

Other measures of correlation for nominal data:

• Cramér's V
• Phi coefficient
• Uncertainty coefficient
• Lambda coefficient

Other related articles:

• Effect size

References

• Liebetrau, A. (1983). Measures of Association (Quantitative Applications in the Social Sciences). Sage Publications

Notes and References

1. Tschuprow, A. A. (1939) Principles of the Mathematical Theory of Correlation; translated by M. Kantorowitsch. W. Hodge & Co.