I have data $$X_1,\ldots,X_T \qquad Y_1,\ldots,Y_T$$ where $X_i \sim \text{Categorical}(a_1,\ldots,a_k)$ and $Y_i \sim \text{Categorical}(b_1,\ldots,b_k)$, but the $X_i$ are not independent of one another (neither are $Y_i$). The $X_i$ ARE independent of $Y_i$ though.

I want to test whether $(a_1,\ldots,a_k)=(b_1,\ldots,b_k)$. Now if the $X_i$ and $Y_i$ were indepdnent then I would simply do Pearson Chi-square Test. What options do I have? What would you do?

  • $\begingroup$ What data are they? Can you be more specific with an example, if there is one? What caused the dependence? Is it due to MCMC or time series? $\endgroup$ – mac Dec 15 '15 at 23:15
  • $\begingroup$ dependence is due to the fact that the data come from MCMC (which you can view as a time series). $\endgroup$ – bdeonovic Dec 15 '15 at 23:23
  • $\begingroup$ What about adjusting the rows (X) and columns (Y) of the table by the effective sample size of X and the effective sample size of Y? $\endgroup$ – mac Jan 11 '16 at 1:23
  • $\begingroup$ thats actually what I ended up doing! $\endgroup$ – bdeonovic Jan 11 '16 at 1:26
  • $\begingroup$ Ha! Is that the correct way to do it? How did you get the effective sample sizes? By fitting AR(p)? $\endgroup$ – mac Jan 11 '16 at 1:48

Actually ended up writing a paper to answer this question. One of the approaches in the paper is similar to what @JunchiGuo suggested in the comments (adjusting by effective sample size) where the ESS was calculated assuming the data follow an NDARMA model.

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