Tschuprow's T

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

Definition

For an ''r'' × ''c'' contingency table with ''r'' rows and ''c'' columns, let $\pi_$ be the proportion of the population in cell $(i,j)$ and let
:$\pi_=\sum_^c\pi_$ and $\pi_=\sum_^r\pi_.$

Then the mean square contingency is given as

: $\phi^2 = \sum_^r\sum_^c\frac ,$

and Tschuprow's ''T'' as

: $T = \sqrt .$

Properties

''T'' equals zero if and only if independence holds in the table, i.e., if and only if $\pi_=\pi_\pi_$. ''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 $\pi_>0$ 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

: $\hat T = \sqrt ,$

where $p_=n_/n$ 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

: $\hat T = \sqrt .$

See also

Other measures of correlation for nominal data:
* Cramér's V
* Phi coefficient
* Uncertainty coefficient
* Lambda coefficient

Other related articles:
* Effect size

References

{{Reflist

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

Summary statistics for contingency tables