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Let $X \in \{0,1\}^{n \times m}$ be a matrix with $n,m \geq 600$.

I want to calculate the correlations between the columns of the matrix $X$. To be more precise, I need the correlation matrix $C \in [-1,1]^{m \times m}$.

My question is, is a rank correlation coefficient like Spearman's $\rho$ or Kendall's $\tau_B$ suitable and if yes, which one more?

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    $\begingroup$ It looks like the "standard" Pearson correlation is out of the running, right? Of course, the Spearman correlation is simply the Pearson correlation of the ranked data... and for 0-1 data, the ranked data is just a linear rescaling of the data itself if you break ties by assigning the same rank (which is the only thing that makes sense), so the Spearman correlation should be exactly the same as the Pearson one. The Kendall correlation with its concordant vs. discordant pairs approach seems to make slightly more sense. $\endgroup$ – Stephan Kolassa Jan 7 '13 at 12:56
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    $\begingroup$ I don't think a correlation coefficient is a particularly useful measure of dependence for binary data. Does your measure have to be bounded between $-1$ and $1$? If not, I'd say it would be better to use something like an odds ratio to quantify the dependence. $\endgroup$ – Macro Jan 7 '13 at 13:30
  • $\begingroup$ @Marco The measure does not really have to be bounded by $-1$ and $1$ but it would be good. Unfortunately, the odds ratio is not appropriate, because I want to analyse the effects of the correlation I bring in. I do not change the amount of the symbols but their positions depending on the positions of the symbols in the previous columns. $\endgroup$ – Carl Jan 7 '13 at 15:38
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How about a tetrachoric correlation - a special case of polychoric correlation, where the variables are dichotomous? Polychoric correlations work by estimating the Pearson correlation of an underlying latent continuous variable that only presents as ordinal data.

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