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


 * $$ \phi^2 = \sum_{i=1}^r\sum_{j=1}^c\frac{(\pi_{ij}-\pi_{i+}\pi_{+j})^2}{\pi_{i+}\pi_{+j}} ,$$

and Tschuprow's T as


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

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


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

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


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