Tschuprow's T

$T=\sqrt{\frac{{\varphi }^2}{\sqrt{(r-1)(c-1)}}}$
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.

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

${\pi }_{i+}=\sum _{j=1}^c{\pi }_{ij}$ and ${\pi }_{+j}=\sum _{i=1}^r{\pi }_{ij}.$

Then the mean square contingency is given as

${\varphi }^2=\sum _{i=1}^r\sum _{j=1}^c\frac{({\pi }_{ij}-{\pi }_{i+}{\pi }_{+j}{)}^2}{{\pi }_{i+}{\pi }_{+j}},$

and Tschuprow's T as

$T=\sqrt{\frac{{\varphi }^2}{\sqrt{(r-1)(c-1)}}}.$

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

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

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

$\hat{T}=\sqrt{\frac{{\chi }^2/n}{\sqrt{(r-1)(c-1)}}}.$

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