Author Response
Thank you for the time dedicated to reviewing our paper, the insightful comments, and valuable feedback. Please see our point-by-point responses below.
**Q1:** It remains unclear how the proposed method would perform with non-normal variables.
**A1:** Thank you for your insightful question. The proposed method can still work with non-normal variables, as long as the parametric form is given. To be specific, if variables follow a linear non-gaussian causal model and our observation is discretized, we only need to modify the likelihood function according to the corresponding parametric form for correlation estimation and the proposed method can still work.
As a comparison, the traditional CCA rank test must assume normality to infer the chi-square null distribution.
On the other hand, if the parametric form is not given, which means we do not have any information about the shape of the distribution,
it may not be possible to consistently recover the underlying correlation (due to insufficient information), and thus the problem cannot be solved. Thank you again for the comment and we have added a related discussion to our revision to avoid any further confusion.
**Q2:** What happens when the number of categories increases? Additionally, what is the computational cost associated with the proposed method?
**A2:**
The asymptotic validity of the method is not influenced by the number of level of
discretization, but more levels are beneficial. Specifically, the proposed method can handle any
level of discretization as long as it is greater than 1, with Type-I errors properly controlled. Further,
more levels of discretization is beneficial because it leads to less information loss
during the discretization process, and thus the correlation matrix can be more efficiently estimated
for building the test.
Our method is also quite computationally efficient: as we employ pseudo-likelihoods, the estimation of each entry of the correlation matrix can be parallelized. In our implementation, a test with 100 random permutations can be done within 2 seconds. Another advantage of our method is that the computational cost increases only marginally as the sample size grows. This is because we only need to compute the probability mass of each category once (which depends on the sample size) and save it for future use; once this step is done, the remaining computations do not depend on the sample size.
**Q3:** How can one distinguish whether a variable is discrete or continuous in practice, particularly when the number of categories is large?
**A3:** In real-world causal discovery applications, it is often the case that we know the name and meaning of each variable, as well as how they are measured. This allows us to leverage domain knowledge to determine whether a variable should be treated as discrete or continuous. For example, in the finance field, discrete survey measures of investor sentiment are often used to provide insights into investor behavior, while variables such as stock prices and annual revenue are inherently continuous.
**Q4:** Does the estimation of correlations in different scenarios impact the asymptotic distribution of the test statistic? Any theoretical justification?
**A4:** Yes, different scenarios of correlation estimation will lead to different asymptotic null distributions (as different extent of information loss was introduced during discretization). However, we note that this does not influence asymptotic correctness the proposed permutation test. The reason lies in that given our established exchangeability, the asymptotic distribution of the test statistic can always be estimated by making use of data permutations (as in Thms 4 and 5, and thus properly control Type-I errors), no matter which scenarios of correlation estimation are considered.
We genuinely appreciate the reviewer's effort and hope that your concerns/questions are addressed.