As I understand, Cramer's V method of counting correlation between categorical variables may find correlations between symmetrical data. I searched a ton of sites, but the only thing I found was that it may lead to loss of information. But why is that? Also, may the danger of symmetrical data lead us to choose Theil's U method of finding correlations? Does it perhaps mean that Theil's U method is more safe: we definitely won't have any loss and it's better compared to Cramer's V?

So, in the end, it leads me to the question when is it better to use each method? If both are OK, what must I look for when, for example, comparing heatmaps via Cramer and Theil? What will be the difference between them?

I don't want to dive in the mathematical part of the problem yet and only understand the logic behind it. Also, please, when answering consider describing an example of the data (maybe fictional).

  • $\begingroup$ I suspect that discussions about Cramer's V mention "symmetric" in the sense that the statistic is the same if you are considering the how, e.g. a) the rows of the table are correlated to the columns of the table, or b) the columns of the table are correlated to the rows of the table. In this sense, the statistic is symmetric. This is different than, for example Goodman Kruskal lambda, with which you have decide whether the rows or the columns are essentially the independent and dependent variable. $\endgroup$ – Sal Mangiafico Jan 8 at 20:00
  • $\begingroup$ It would be helpful, too, if you could cite specifically the language and the source you are looking at. $\endgroup$ – Sal Mangiafico Jan 8 at 20:02

Typically when summarizing an distribution with a single number there is loss of information, after all you cannot recover the distribution from just that single number. The question is not if there is loss of information but whether the summary is sensible.

The uncertainty coefficient (Theil's U) is a conditional measure: given one variable how well can we predict the other? On the other hand, Cramer's V is a symmetric measure giving an overall measure of "strength of association", but is difficult to interpret. If you can use the uncertainty coefficient, I think it is usually to be preferred due to its interpretability.

  • $\begingroup$ Ok, I understand about the summary, but can you please tell me why is it summarised in such a way in the first place?Are you saying that Cramer's V doesn't do a sensible summary(gives just a 'strength of assosiaction')?Can you please describe more clearly the difference?Because to me it seems like similar things('how can we predict the other' and the 'strength of assosiaction'). $\endgroup$ – EndOfTheGlory Jan 8 at 18:21
  • $\begingroup$ The interpretation of both is admittedly not straightforward, illustrating that for nominal categorical data it is quite hard to describe association in a meaningful way. Best to look online or in books for detailed explanations of the interpretation of the coefficients, I think eg Agresti's book on Categorical Data Analysis has some comments on the coefficients, but Wikipedia gives some information as well. $\endgroup$ – Wicher Jan 8 at 18:49

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