Permutation-based Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
Authors: Xinshuai Dong
Ignavier Ng
Boyang Sun
Haoyue Dai
Guang-Yuan Hao
Shunxing Fan
Peter Spirtes
Yumou Qiu
Kun Zhang
Venue: ICML 2025 poster (2025)
Keywords: Rank Test
Discretization
Causal Discovery
Primary area: general_machine_learning->causality
TL;DR: Permutation-based Rank test for Mixed data

Abstract: Recent advances have shown that statistical tests for the rank of cross-covariance matrices play an important role in causal discovery. These rank tests include partial correlation tests as special cases and provide further graphical information about latent variables. Existing rank tests typically assume that all the continuous variables can be perfectly measured, and yet, in practice many variables can only be measured after discretization. For example, in psychometric studies, the continuous level of certain personality dimensions of a person can only be measured after being discretized into order-preserving options such as disagree, neutral, and agree. Motivated by this, we propose Mixed data Permutation-based Rank Test (MPRT), which properly controls the statistical errors even when some or all variables are discretized. Theoretically, we establish the exchangeability and estimate the asymptotic null distribution by permutations; as a consequence, MPRT can effectively control the Type I error in the presence of discretization while previous methods cannot. Empirically, our method is validated by extensive experiments on synthetic data and real-world data to demonstrate its effectiveness as well as applicability in causal discovery (code will be available at https://github.com/dongxinshuai/scm-identify).

Permutation-Based Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
Xinshuai Dong' Ignavier Ng' Boyang Sun? Haoyue Dai Guang-Yuan Yuan Hao Shunxing Fan? Peter Spirtes 1 Yumou Qiu 3 Kun Zhang" 12
Abstract
(Sullivant et al., 2010; Spirtes, 2013). From one perspec- tive, Independence and Conditional Independence (CI) are crucial concepts in causal discovery and Bayesian network learning (Pearl et al., 2000; Spirtes et al., 2000; Koller Friedman, 2009) due to its relation to d-separations (Pearl, 1988), and it has been shown that rank tests take those linear CI tests as special cases (Sullivant et al., 2010; Di, 2009; Dong et al., 2024a). From another point of view, rank of cross-covariance matrix corresponds to t-separations in a graph (Sullivant et al., 2010), which contain graphical infor- mation that can be used to identify latent variables (Huang et al., 2022; Dong al., 2024a). more detailed discussion about related work can be found Appendix D.
Recent advances have shown that statistical tests for the rank of cross-covariance matrices play an important role in causal discovery. These rank tests include partial correlation tests as special cases and provide further graphical information about latent variables. Existing rank tests typ- ically assume that all the continuous variables can be perfectly measured, and yet, in practice many variables can only be measured after dis- cretization. For example, in psychometric studies, the continuous level of certain personality dimen- sions of a person only be measured after being discretized into order-preserving options such as disagree, neutral, and agree. Motivated by this, we propose Mixed data iornnutttioo-bbaeee Rank Test (MPRT), which properly controls the statis- tical errors even when some or all variables are discretized. Theoretically, we establish the ex- changeability and estimate the asymptotic null distribution by permutations; MPRT can effectively control the Type error in the presence of discretization while previous methods cannot. Empirically, our method vali- dated by extensive experiments on synthetic data and real-world data to demonstrate its effective- ness as well as applicability in causal discovery (code will be available at https: ///g//tthh////// com/dongx
Existing statistical rank tests (Anderson, 1984) are often built upon Canonical Correlation Analysis (CCA) (Jordan, 1875; Hotelling, 1992). with likelihood ratio based test statistics. Despite their effectiveness, existing methods rely on the strong assumption that all the variables concerned can be perfectly measured. However, in many fields, is often the case that the best available data are just discretized approximations of some underlying continuous variable (formally defined in Eq. 1). For example, in mental health, anxiety levels are often categorized into levels such as mild, moderate, or severe, according to some latent thresholds (Johnson et al., 2019). Examples can be found in multiple fields such as finance (Changsheng & Yongfeng, 2012), psychology (Lord & Novick, 2008), biometries (Finney, 1952) and econometries (Nerlove & Press, 1973). where continuous variables are often assumed to be observed as discretized values.
consequence,
ssmm identi fy).
When discretization is present, existing rank tests can hardly work. The main reason lies that the discretized values only reflect the order of the data, leading to cross- cross-covariance esti- mates that may differ significantly from the underlying cross- covariance matrix (also illustrated in Figure 1). Furthermore, even though the true underlying cross-covariance matrix can be estimated by maximum likelihood- baased methods such as polychoric and polyserial correlations (Olsson et al., 1982; Olsson, 1979), they cannot be directly plugged into existing rank tests. This is because the involved discretization and maximum likelihood processes change the distribution of
1. Introduction and Related Work
Recent advances have shown that the rank of cross- covariance matrix and its statistical test play essential roles in multiple fields of statistics especially in causal discovery
'Carnegie Mellon University MMhamedd bin Zayed University of Artificial Intelligence 'Peking University. Correspondence to: Kun Zhang <kunzl @ cmu.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 20255 by the author(s).

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Permutation-Bant Based Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
test statistics to considerable extent and thus the p-values cannot be correctly calculated. As consequence, Type errors of existing methods cannot be effectively controlled. Both of these points are elaborated in Section 2.2.
only have order-preserving discretized observations, respec- tively. Assume that we have N i.i.d., observations of these variables. The underlying true data matrix is D E IR RNNM while we only have access to D, where some columns are discretized. Specifically, for j E Cv, D., = D.j. while for those j E Dv, the observations are discretized in the following fashion:
To properly address the issue of discretization, in this paper, we propose a novel statistic rank test based on permuta- tion, i.e. Mixed data Permutation- based Rank Test (MPRT) that can accommodate continuous, partially discretized, or fully discretized observations. Specifically, in the presence of discretization, the underlying cross-covariance can be estimated by maximum likelihood estimator, but the infor- mation loss resulting from discretization and the additional estimation steps make the derivation of the null distribution highly non-trivial, To this end, we start with the continuous case and establish exchangeability of linear projections of concerned variables (Theorem 4), based on which the null distribution can be empirically estimated by permutations. When some observations are discretized, the exchangeabil- ity still holds but we do not have direct access to permutable data. Fortunately, we show that the concerned statistic distri- bution can still be consistently estimated by properly using permuted discretized observations (Theorem 5). We sum- marize our key contributions follows.
Dij= if < D S T4t1 for i E N3,t E (l,..,Cyh
(I)
where Cj is the cardinality of the domain of the discretized observation of Vj. T. refers to the I-th threshold for variable V,. Tÿ - -o,, and TV C,+1 8o.
We are interested in the rank of the population cross- covariance matrix over certain combinations of variables, c.g., Σx,y, where X C V and Y S V (X and Y are not necessarily disjoint). The rank information is crucial to causal discovery (Spirtes al., 2000) and will be detailed in Section 2.2. Ideally, would expect that we have infinite datapoints and there is no discretization; in this case, the sample covariance Σx, would be exactly the same as the population covariance, and the rank can be easily calculated by linear algebra. However, in practice we only have finite datapoints and for some of the variables we only have discretized observations. Thus, to consider the following problem: in the finite sample case and in the presence of discretization, we only have access to D instead of D, how to build a valid statistic test that properly controls the Type I error for testing the rank of a cooss-oovarianc matrix Σx,y?
To our best knowledge, we propose the first statistic rank test i.e., Mixed data Permutation-based Rank Test (MPRT), that properly deals with the problem of dis- cretization. Rank test takes partial correlation CI test as special case and thus the problem is crucial to many scientific fields such as psychology. biometrics, and econo- metrics, where discretizations are ubiquitous.
crucial
Theoretically, we estimate the asymptotic null distribution by effectively making use of data permutations, and thus properly controls the Type error. The setting considered is rather general: for the test of rank(Ex, both X and Y are allowed to be either fully continuous, partially discretized, or fully discretized. Thus, our method also includes the fully-comtin coly--oontinous rank test as special case.
2.2. Why this Problem is Important?
In this section we will briefly discuss why rank test is im- portant in the context of causal discovery as well as why it is crucial to deal with discretization.
Rank Test Takses Linear CI Test as a Special Case
Empirically, we validate our novel rank test under multiple synthetic settings where our method is shown to control Type I error properly and Type II error effectively, while existing methods cannot. We also use a real-world dataset to show the practicability of the proposed rank test and illustrate its application in causal discovery.
In causal discovery, we aim to find the underlying causal graph among variables given observational data. The most classical approach is to use conditional independence (CI) relationships to identify d-separations in a graph; see, e.g.. the PC algorithm (Spirtes et al., 2000). This idea is captured by the following theorem.
Theorem 1 (Conditional Independence and D-separation (Pearl, 1988)). Under the Markov and faithfulness assump- tion, for disjoint sets of variables A. B and C, d- separates A and B in graph 9, iff A 11 B holds for every distribution in the graphical model associated G.
2. Preliminaries
2.1. Problem Setting
Suppose that we have a set of M observed random variables 1 V = [V that are jointly Gaussian. However, for some of these variables, direct observations are unavailable. We use Cy and By to denote the index set of those variables in V that we have direct observations and that of those we
In practice, we often consider linear causal models where the CI test can be done by e.g., F (Fisher et al., 1921). It has been shown that, for linear causal models,
2

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Permutation-Basee Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
d-separations between variables can also be uncovered by rank tests, which is summarized in the following theorem.
In the context of rank test, what should we do deal with such ubiquitous discretization problem? One naive way is to just treat these ordinal values continuous ones and test the rank of a cross-con crvos-covare matrix as usual, and yet it cannot work. The reason lies in that the observed values of these discretized variables just represent the ordering and the values can be rather arbitrary. For example, assume that the original continuous observations are discretized into three levels represented by { 3) respectively; one can alternatively uses {1, 2.13 or {1, 2, 1046 to represent the three levels. If we directly use the ordinal values, the resulting cross-covariance matrix can be very different from the ground truth one, leading to meaningless results. An example can be found in Figure 1, where (a) shows the population cross-covariance and (b) shows the counterpart calculated by using discretized observations. Even with infinite samples, the two matrices are totally different, and the rank of the matrix in (a) is 1 while rank of that in (b) is 3. Next, we will show that, even if we can use maximum likelihood to estimate the correlation first, the problem is still highly non-trivial.
Theorem 2 (D-separation by Rank Test (Dong al., 2024a)). Suppose linear causal model with graph G and assume rank faithfulness (Spirtes, 2013). For disjoint vari- able sets A, B, and C, we have C d-separates A and B in graph G. if and only if rank( (AUC,EUU = |C|.
The above Theorem 2 says that d-separations can also be inferred from rank of a cross-covariance matrix, and thus for causal discovery of linear causal models, partial correlation test/ / linear CI test can be substituted by rank test.
Rank Relates to T-sep that Indicates Latent Variables
Next, we show that rank of cross-covariance informs something beyond d-separations. Specifically, t-separations (Sullivant et al., 2010) can be inferred from rank, and t-separations can be used to identify latent variables. The relation between rank and t-separations is given as follows.
Theorem 3 (Rank and T-separation (Sullivant et al., 2010)). Given two sets of variables A and B from linear model with graph G and assume rank faitfulnessss. We have:
2.3. Classical Rank Test with Estimated Correlation
rank(2AA.B) = min[|CAl + [CB]: : (CA, CB) I-separates A from B in G},
(2)
We have shown that the naive solution of directly using the ordinal values cannot work. Thus, one may wonder another straightforward one - estimate the correlations first (which can be done by maximizing likelihood, detailed in Section 3.3), and then plug the estimated correlations into standard CCA rank test. In this section we will show that this straightforward solution cannot work either; more specifi- cally, the Type-I errors cannot be effectively controlled.
where AA is the cross-covariance over A and B.
The left-hand side of Equation 2 is about properties of the observational distribution, while the right-hand side describes properties of the graph. An example highlighting the greater informativeness of rank compared to CI is as follows. Consider the graph G in Figure 5, where (X1, X2) and (X3, } are d-separated by L,. but we can never infer that from any CI test, i.e., can never check whether {x) IL [X3, X4) holds, L1 is not observed. In contrast, using rank information, we can infer that = 1, which implies (X1, X2) and (Xa, } are t-separated by one latent variable. The ratio- nale behind is that the t-separation of two set of variables A. B by (CA, CB) can be inferred through rank, without actually observing any element in (CA, CB). A more detailed discussion can be found in (Dong et al., 2024a).
We start with a brief introduction to the classical rank test, which is based on Canonical Correlation Analysis (CCA) (Jordan, 1875; Hotelling, 1992). The key design of test typically is to find suitable statistic and to derive its distri- bution under the null hypothesis. As for rank test of cross- covariance Σx,y, statistics based on CCA scores between X and Y are found to be very effective. For | = P, Y = Q, and K = min(P, Q), the CCA problem is as follows:
AER "RRPXXX,BRXXXX BERQXK tr(A ( Σx,y
max
(3)
Discretization is Ubiquitous and Needs to be Handled
s.t., A - II
Discretization is ubiquitous in many scientific fields. For instance, is common to come across concepts that cannot be measured directly, such as depression, anxiety, attitude, and the observations of such variables are often the result of coarse-grained measurement of the underlying continuous ones. More examples can be found in fields like psychology (Lord & Novick, 2008), biometrics (Finney, 1952) and econometrics (Nerlove & Press, 1973), where is widely accepted to assume a continuous variable underlies a dichotomous or polychotomous observed one.
Assume that the solution to Eq. 3 leads to CCA scores between X and Y as {r:jk With the null hypothesis that rank < k referred to HR. would expect that the top-k CCA scores are non-zero and the rest ones are all zero. This leads to a (Anderson, 1984) under HJ as follows.
test statistics
A 1--(N P P+Q+3
(NN
In In(1l
r2)), (4)
2
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Permutation-Basee Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
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(a) Population cross cross-covariance matrix over(b) Cross-covariance matrix using discretized(e) Distributions of p-values of CCART-C continuous variables.
data with N + 0o.
and SCART-DE.
Figure I. Subfigures (a) and (b) together show we cannot directly take the discrete values the calculation of rank the covariance. Subfigure (c) shows that directly plugging estimated cross-covariance into rank test does not work Type cannot be controlled.
which has been shown to approximately follow chi-square distribution with degree of freedom (P- k () - k 1). To perform the rank test, one only has to calculate A and the related chi-square distribution to get the p-value.
ces are D RXX and D RN respectively. The first crucial step is to solve the CCA problem defined in Eq 3, by Singular Value Decomposition (SVD) as follows.
USV 3 and B
In Eq 3, Êx,y refers to the sample covariance D DY N-1
(5)
A = 5xx
S TT
the presence of discretization, we only have access x and DY but we can still estimate the cross-correlation by maximizing the likelihood (detailed in Section 3,3), and take the estimation into Eq. 3 to calculate the CCA scores and thus the test statistics. However, due to the information loss introduced by discretization and the additional maximum likelihood steps, the distribution of the statistics changed considerable extent. An example is shown in Figure (c), where CCART-C refers to CCA rank test using the original continuous observations and CCART-DE refers to first estimating the correlations by maximum likelihood using discrete data and then plugging into the CCA rank test. As shown, the p-values of CCART-C are uniformly distributed while the p-values of CCART-DE are clearly not; most of them near zero and thus the test tends to reject everything, leading to unacceptably large Type errors (also validated in Section 4.2 and Figure 2).
where A and B are two linear projection matrices and the two CCA variables are Cx = A and Cy = BT Cx and Cy have two good properties: (i) Ecx = Ec = and E. By a diagonal matrix; (ii) under null hypothesis H6 rank(Exx,v)
k. only the top-k diagonal entries of Ecx,C Cy are nonzero and the rest of the diagonal entries should be zero. Taking these two into consideration, have the exchangeability between CX and CY which is formalized in the following Theorem 4 (proof of which can be found Appendix).
Theorem (Exchangeability of CX.: and CY Given a set of variables V that are jointly gaussian, under null hy- pothesis HJ raan((xx,x) S where X, Y S V, random vectors CXk: and Cy are asymptotically independent with each other.
Based on the exchangeability between CX and CY we can permute the data matrix of CX and CY in order to get resampling of CX and CY Specifically, given a random permutation matrix P. PD Lkk: and D together serve as N i.i.d. resamplings from the joint distribution of CXa: and CY Further, the statisties in Eq. 4 only depends the k-th to K-th CCA scores between X and Y, which can be equivalently calculated by the first to (K - k)- th CCA scores between CX and CYk:, formally captured by the following Lemma
Ideally, we would expect to derive the updated distribution of the statistics, and yet the involved likelihood maximiza- tion steps make it very difficult. Therefore, we aim to solve this problem by estimating the empirical edf of the null distribution using permutations, detailed in what follows.
3. Mixed Data Permutation-basee Rank Test
In this section, we propose MPRT. A brief introduction to permutation test can be found in Appendix C.I. We start with the all continuous case.
Lemma 1 (Alternative Way to Calculate Statistic in Eg. 4).
Let the CCA score between Cxk: and Cy be {f:)
3.1. All Continuous Case
The statistic defined Eq. can also be formulated as:
Assume that we are interested in the rank of Σx,y, where X = P and Y = Q and their corresponding data matri-
d = P P+ +++3
In K- (1- (1-f?)). (6)
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Permutation-Basee Based Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
By Lemma 1, we know that the test statistics only depends on Cx.: and Cy Further, Cxa: and Cy k: can be resam- pled by permutations. Taking these two into consideration, we can make use of permutation to estimate the empirical CDF of the null distribution, and thus correctly calculate the p-value. Below we give a detailed description of the procedure to do the permutation and consequently calculate the p-value, Given A and B. we have the observed data matrix of the two canonical variables as DC DX A and DC D B (where DCx, DC E BRN*K) For each random N x N permutation matrix P. we use PD. :,k: and c to calculate the test statistics under permutation P as X following Eq. 6, and the p-value is obtained as:
Remark (Remark on Theorem 5). Theorem 5 implies that we can consistently estimate Xf by making use of randomly permuted data D and p" DY Note that although here the transpose of permutation applies to DY, the correctness of the process still relies the exchangeability between Cxi and Cy and does not need the exchangeability between X and Y. In words, doing permutation on D* A will meet the problem of applying linear transformation to data that might contain ordinal values, and Theorem 5 provides a way to bypass the problem by permuting DY instead.
Till now, the remaining problem is how to consistently esti- mate cross cros-coovrriance matrices in the presence of discretiza- tion, and it will be detailed in what follows.
Pk = E 1 1222
3.3. Correlation Estimation with Discretization
where the expectation is taken over random permutations.
Assume that we concern the rank of Σxx, Y. where some of the variables are discretized and X and Y are not necessar- ily disjoint. As mentioned, for those variables that we only have discretized observations, their variance can never be determined. Further, the rank of a cross-covariance matrix is equivalent to the rank of the corresponding cross-correlation matrix. Without loss of generality, we can assume all vari- ables have unit variance and zero mean. Thus, some- times use correlation and covariance interchangeably. The remaining crucial step is to estimate the correlation matrix for V = X U Y, i.c., R, by data D E IR N As some elements of V are discrete, we use Cv and By to denote the index set of continuous variables and discrete variables in V respectively.
3.2. Mixed Case in the Presence of Discretization
Here we discuss the case where some columns of the data matrices xx and DY are discretized. Under such a scenario, one can still estimate Σx, Σx and Σy by max- imizing likelihood, which will be detailed in Section 3.3. After that, A and B can still be estimated following Eq. 5, and the exchangeability between Cxx: and CY still holds.
However, to get the resampling of Cxk: and Cr by permutation, one has to apply linear transformation A and B to get DCx = D* A and Dcr = D B, respectively. In the all continuous case, it is straightforward, but in the presence of discretization, it makes no sense to apply a linear transformation A Dx when some columns of D are just ordinal values. As a consequence, we cannot make use of Theorem 4 to get resampling of Cx.: and Cy to calculate the statistic A and estimate the p-value anymore.
We first introduce the overall objective function for correla- tion estimation as follows.
I = arg min RERMXM L(D, R), logPi, (D... Rag),,
C R) =
Σ ISKSSM
(10)
Fortunately, it can be shown that to calculate Xe_ one does not have to really get the exact resampling from CX:: and CY ki Instead, for each random permutation P. we can get a consistent estimation of {f:)k K- K- and consequently calcu- late Xf. This is formalized by the following Theorem 5.
where the optimization objective is minimizing pair-wise negative log- ligeliiklihood, also referred to as pseudo likeli- hood, instead of the real joint liglliilldood over all the observed variables (Dong et al., 2024b). The reason lies in that optimizing over the joint log-likelihood is very compu- tationally expensive and the pseudo likelihood is tractable while also serves as a consistent estimator (Besag, 1974; Gourieroux et al., 1984; Gouriéroux et al., 2017; Fan et al., 2017).
Theorem (Consistent Estimation of {f:)K K under Per- mutation P). Under permutation P. the empirical CCA
scores between CXk: and Cyk: i.e., {f.)k-* are the singular values of c- Eox, ΣCxE, EYr Coy which can be consistently estimated by:
((ATEXA) ((AT DXT pT DY
Next, we specify the pair-wise log-likelihood in three see- narios between two continuous variables, between con- tinuous and discrete, and between two discrete variables.
N-1 B)bck)
(8)
(B B)),))
(i) Likelihood for Two Continuous Variables
where DxTPpT can be consistently estimated by using N-1 Dx and PT D and assuming unit variance of variables.
If both i,j E Cy. the log-likelihood function log Pig (D...s Rag) is just the joint gaussian pdf
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Permutation-Basee Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
parametrized by R given as follows:
Algorithm Mixed data Permutation-based Rank Test
1. R R 1
1, R
)
)
1, Ri. Ri,j 1
I: Input:
Sample D* DY indexes of discretized columns, null hypothesis Hy raank S k, and significant level a;
(tr
log det
)
R
(11)
Output: True (fail to reject 7(/) or False (reject Hj);
P - Q = -|X], and K min(P, (P, Q)
where Ra,j is the empirical correlation matrix that can directly calculated from data D..ú-
Get Σx. Σx.y and Ey as submatrices of R by Eq. 17 (unit variance assumed)
Calculate A and B following Eq. 5.
(ii) Likelihood for a Continuous and a Discrete Variable
6: Let P I (no permutation), calculate {A,)] K- fol- lowing Eq. 8 and then the statistic AE following Eq. 6
i i Cv and j E Dy, then the log-likelihood (also known as polyserial correlation estimation (Olsson et al., 1982))
7: for each random permutation P do
log Pig (D.ij; Ro.j) can be factorized as follows.
8: Calculate [f:]] K under P following Eq. 8 and then
the statistic under P. i.e., Xf, following Eq. 4
N Σ N
end for
1
log p(V P(V = DR.)p(V, = DR.j V; = DE.s. Riz),
10: Calculate p-value pa by Eq.
11: return pk 2
(12)
where p(V DR.e) is a standard gaussian pdf. For a specific value of Da.j. say, t, we have that:
3.4. Parameterization Trick for Rank Test
We note that the optimization problem in Eq. 9 does not constrain the space to be a pseudo-correlation matrix a matrix that is PSD with unit diagonal elements. we only care about the maximum likelihood estimator, the pseudo- correlation requirement might be unnecessary. However, as we rely on SVD for CCA and rank test, the require- ment of being pseudo-correlation matrix is crucial. A clas- sical way to solve this problem is by projected gradient descent: projecting the current solution to the space of pseudo-correlation matrices after each step of gradient de- scent. Yet, in practice we found this solution less effective. as the projection cannot be analytically solved and requires additional optimization step.
pVV = DD, V = DR Rijj)
=p(Ty <V, S 1V -k,,i, Ri.j)
(13)
= Φ T++1 - Rs, $ T² R DR.A R2,,)
(1 R2
where & is the standard gaussian edf. We note that the thresholds 7 are unknown, thus it could be taken as free pa- rameters during optimization. In practice, is more efficient to estimate the thresholds first by using inverse gaussian edf:
TV = Φb-1 Ei= - L0. 50
(14)
To this end, we directly parameterize the space of pseudo-correlation matrices in a geometric way following (Rousseeuw & Molenberghs, 1993), given as follows.
N
(iii) Likelihood for Two Discrete Variables
If both j E My, then the log-likelihood (also known as polychoric correlation estimation (Olsson, 1979; Jöreskog, 1994)) log Pig (D..s; Ri.) is as follows.
R -UUU,
cos 0:00ijj1,h on jsi 0.
U3S =
(16)
j j>b i'
N
log( 2 (Té
D + T
Da. - Rig))
s.t., 0,, 0, Vi.
1=1
T + (T)) D... Dr.." Ri.))
Therefore, we have alternative way to parameterize the correlation matrix, which gives rise to the following new formulation of our objective function (instead of Eq. 9):
(15)
(T) + 72 DK.3
Rii))
(TT Dr.,+il R())),
T
Å = arg ming L(D, R).
(17)
where (,(..,,7)) is the joint edf of two standard gaussian variables with correlation r and the thresholds for each variable can also be estimated by using Eq. 14.
We summarize the overall testing procedure of our proposed MPRT in Algorithm 1.
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1250 Permulation-Basee Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
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(a) The probability Type errors with a 0.05.
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Figure 2. The probability of Type and Type II errors with mixed data, by different rank test methods, under different sample sizes.
Type erroor aith 0.05
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(a) The probability Type errors with a = 0.05.
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Figure 3. The probability of Type and Type II errors with continuous data, by different rank test methods, under different sample sizes.
4. Experiments
small as possible. We consider different sample sizes, and for each comparison, we consider 3000 random trials. For MPRT, randomly generated 200 permutations to calcu- late the p-value. The ground truth covariance matrices are randomly generated. For the mixed scenario, we uniformly generate two thresholds from [-1.5, 1.5) for each variable that should be discretized, and use the thresholds together with -o and co to discretize the continuous observations into three categories {1, 2,
4.1. Experimental Setting
To empirically validate the proposed Mixed data Permutation-basee Rank Test (MPRT), we apply our method to synthetic data and compare it with the following methods. (i) CCART-C: CCA-based Rank Test (Anderson, 1984) that the original continuous observation input; as it has access to the original observations, its performance is taken as the best possible performance that we can achieve. (ii) CCART-D: CCA-based Rank Test with Discrete data; directly takes the ordinal values as input. (iii) CCART-DE: CCA-based Rank Test with Discrete data Estimating covariance; it takes the estimated correlation matrix as input (following Eq. 17).
We also apply the proposed MPRT method with mixed data to the classical causal discovery method PC algorithm (Spirtes et al., 2000) and see whether our test method can better test CI relations compared to the classical Fisher-Z test (Fisher et al., 1921), in the presence of discretiza- tion. Fisher-Z is only compared by the result of PC and cannot be not compared in the previous setting, linear CI relations can only correspond to part of the rank informa- tion. Finally, we employ real-life dataset to illustrate the applicability of the proposed method in real-life scenarios.
We consider two scenarios: mixed data scenario where data are partially discretized, and all continuous scenario where all the original observations are available. The first scenario to illustrate how well can we handle discretization while the second show that our method can serve general also work well when there is concern both
rank test method discretization. In terms of performance, Type errors and Type II errors. Specifically, we expect a good test can properly control the Type errors given a significance level a, while the Type errors should be as
4.2. Analysis on Type and Type Il Errors under Different Sample Sizes
In this section we analyze the performance of each method in terms of Type I and Type II errors under different sample sizes. For the mixed data scenario, the result shown in

---

Coimutaaiio-Basee Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
Table Fl score and SHD the PC algorithm, with different test methods (1 the bigger the better while + smaller the better).
Fl score for skeleton 1
SHD for skeleton +
Cl test method
N = 500
N
1000
N
2000
N
500
N
1000
N
2000
MPRT
0.84
0.9
0.96
0.80
0.60
0.20
Fisher-Z
0.81
0.80
0,78
1.20
1.20
1,40
KCI
0.81
0.88
0.86
1.00
0.80
0.93
CCART-D
0.75
0.79
0.77
1.60
1.60
1.80
CCART-DE
0.80
0.85
0.83
1.40
1.30
1.60
L1
Neuroticism
L1
[07]
[O2] have dificclty
[01] vocabulary.
unferstand things
understanding ebstract idea.
L2
[N10] roften feel buue
[N8] hawe frequent
L2
[04] ideas.
[08]
Openness
[N1] easily.
[N3]
[N2] most the time.
[N7] mind lot.
[010] lam falless
[05] have excellent ideas.
about sbout things.
(a) Discovered personality substructure for Openness.
(b) Discovered substructure for Neuroticism.
Figure 4. Application of MPRT in causal discovery using real-life Big Five human personality data.
Figure 2. Specifically, one can see that both our proposed MPRT and CCART-C can properly control the Type errors as the Type I errors of them are both very close to the signif- 0.05; in contrast, CCART-D and CCART- DE totally failed control the Type errors. As for Type II errors, it can be found that the Type II errors of MPRT are quite small, and decreases with the increase of sample size N, while CCART-D and CCART-DE cannot benefit from the increase in sample size. We note that it is very natural that MPRT cannot beat CCART-C as CCART-C takes the original continuous observation as input while MPRT takes mixed data as input. We show the performance of CCART-C just in order to show the minimal possible Type II errors that one can achieve in the presence of discretization.
4.3. Application in Causal Discovery
In this section we validate our test using the PC algorithm (Spirtes et al., 2000). Specifically, we consider linear causal models with gaussian noises V, = Σv, EPa(V, at V, + EV,, where the edge coefficients and the variance of the noises are randomly generated. We consider the scenario where data are partially discretized and compare MPRT with Fisher-Z to see which one works better with PC. We employ F1 score F1 = and Structural Hamming Distance (SHD) for skeleton (the smaller the better) to evaluate the performance. As shown in Table 1, MPRT achieves the best performance in terms of both F1 and SHD, under all sample sizes. This validates the claim that MPRT can serve as a powerful CI test for causal discovery in the presence of discretization.
icance level
E-RRcall-Precision for skeleton (the bigger the better)
Recall + Precision
We also show the performance when both CCART-C and MPRT have access to the original continuous observations, as in Figure 3. Specifically, both methods properly control the Type errors as in the subfigure (a). For the Type II er- rors, the performance of CCART-C and MPRT is almost the same. This is expected, as in this scenario both methods use exactly the same test statistics except that CCART-C uses the analytically derived null distribution to get the p- value while MPRT uses the empirical CDF to calculate the p-value; the two results are expected to be exactly the same asymptotically.
4.4. Real-world Causal Discovery Application
In this section, we further validate our proposed MPRT method using real-world Big Five Personality dataset ht /ttts::/oppnnee wwwpppppsscchtterrppppm 50 personality indicators and close to 20,000 data points. Each Big Five personality dimension, namely, Openness, Conscientiousness, Extraversion, Agreeableness, and Neu- roticism (O-C-E-A-N), are designed to be measured with their own 10 indicators and the values of each variable ordinal: Disagree, slightly disagree, Neutral, Slightly agree, and Agree We employ RLCD (Dong et al., 2024a), re- cently proposed rank based causal discovery method with our MPRT method. We choose items from openness and items from neuroticism to verify our method.
org/. It consists of
Taking the performance under these scenarios together into consideration it can be argued that MPRT is very general and valid rank test as it can handle all continous data, partially discretized data, and all discretized data and the Type I are properly controlled while the power is also good.
The results are shown in Figure 4. Specifically, for open-
8

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crimuutation-asedd Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
ness we discovered two latent variables. L.2 corresponds to whether a person has lot of ideas while 1.1 corresponds to the general concept of openness. As for neuroticism, we also discovered two latent variables. L1 relates more to one's emotions while 1.2 relates to one's stress level. In contrast, we directly use the ordinal values to do the rank test, i.c., using CCART-D, all the p-values lend to be very small, and thus we have to use very small significance level (around 1e-10) in order to have some structures discovered; yet ing such an extremely small alpha value will induce a lot of Type II errors. This result illustrates the superiority of using MPRT in the presence of discretizations in real-life scenar- ios, and again empirically validate the proposed method.
Acknowledgment
We would like to acknowledge the support from NSF Award No. 2229881, Al Institute for Societal Decision Making (AI-SDM), the National Institutes of Health (NIH) under Contract R01HL159805, and grants from Quris AI, Florin Court Capital, and MBZUAI-WIS Joint Program. IN ac- knowledges the support of the Natural Sciences and Engi- neering Research Council of Canada (NSERC) Postgraduate Scholarships Doctoral program.
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In this paper, we propose novel permutation-biat based rank test that works in the presence of discretization. It is rather general as it can accommodate fully continuous data, par- tially discretized data, or fully discretized data as input. Extensive experiments empirically validate our method.
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Impact Statement
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Permutation-Basel Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
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Permutation-Basse Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
A. Proofs
A.1. Proof of Theorem 4
Theorem 4 (Exchangeability of CXk: and Cy Given a set of variables that are jointly gaussian, under null hypothesis Hb: rank(Exxx)) S k where X, Y S V, random vectors Cx2: and Cy k: are asymptotically independent with each other.
Proof of Theorem 4. First, Σx, Σy, and Σx,Y Y by pseudo-likelihood, converge in probability Ex. Ey, and Σx.Y. respectively (Besag, 1974; Gourieroux et al., 1984; Gouriéroux et al., 2017; Fan et al., 2017).
Plus, as we need to apply the continuous mapping theorem, we show the continuity and uniqueness of SVD in what follows, SVD is not continuous only when the input matrix has repeated singular values. Specifically, if a matrix A has distinct singular values, then SVD is continuous in the neighborhood of A, and unique only up to sign flip (chapter 2 section 5.3
of (Kato, 2013)). Thus, to make use of the continuous mapping theorem, we assume that >x Y does not have repeated singular values (the set of matrices with repeated singular values has Lebesgue measure zero (Lemma 1.4.2 in (Kunisky), also in (Bochnak et al., 2013)).). To further eliminate the sign indeterminacy, we can just follow scikit-learn to impose the largest coefficient of each column in U in absolute value is positive (svd flip in scikit-lear).
Given (Éx, Ex, εX,x) & (Ex, Σy, Ex,y). we aim to show the desired asymptotic independence. Specifically we want to show (i) CXa: L CXi: and CY 2 Cy and (ii) CXi:, Cy are independent under the null hypo. Here X, and Cy = B² where (A, B) and (A", B²) are produced by SVD using estimated covariance and population one respectively as follows.
Cx = x Cy = B"Y, Cx = A X
USV = -xx Ex A - U Ex |T VT. VT. V'V'==E Ex Ex A Ex AT U*, B" =Ey E V.T.
For (i): By continuous mapping theorem, under the assumption of repeated singular values, we have U 1 U As Ex is positive definite, the matrix inverse square root is continuous and thus 5 1r 2x xxx 1rr Given (U, U 2x
b7 2 (U", Exx"" )
we have - T U A 2 A === 1r U*. Similarly, we have B -, B Thus
x
((AT
T X. BB"-B-T YY) 3 (((AT~^**) x))x k: (B - B ())) 30= (Cxi:) CY 3 (Cx Cx
For (ii): Under the null hypo, the cross-covariance between Cxi: and Cr are all zeros. As Cx k Cv.: are jointly gaussian (linear mixing of X, Y), zero cross-covariance implies independence.
A.2. Proof of Lemma 1
Lemma (Alternative Way to Calculate Statistic in Eq. 4). Let the CCA score between Cx and CY be (fi) Kr The statistic defined in Eq. 4 can also be formulated as:
In [T P+Q+3
A =
N
-??))
(6)
2
Proof of Lemma 1. The CCA scores between CXk: and CY k: are just the diagonal entries of their cross-covariance matrix, which corresponds to the i to K CCA scores between X and Y. Thus we have f; = ri4k for = (1,..., K k), and thus
A+- -(N r+Q+3) ( = 1+1 (1 (1-r?) r))).
2
A.3. Proof of Theorem 5
Theorem (Consistent Estimation of (f:) K under Permutation P). Under permutation P, the empirical CCA scores
between Cx and CY ie., {f,) K-k are the singular values of E Ecxx, 5-3 Cov.' which can be consistently estimated by:
DXT T pT DY
((AT A) A); =1((AT
B) B))ck)))
N-1
(8)
(BBTEx B).,)))
12

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Permutation-Baset Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
where Dx pd DY N-1
can be consistently estimated by using Dx and pt DY and assuming unit variance of variables.
Proof of Theorem 5. We are interested Cx., Σcx.. Cr., Gyy Assume that we have access to the original data D
and D By the exchangeability, for each random P. we have (PD* A):.:: and (D) B) are the N samples from joint
distribution of Cxk: and CYR:. Then the S- Cx.."
Ecxi, and E Cy are as follows:
(PDx AT A).
Cxk
N (PD* A)) A)lx: )=
(18)
(19)
N-1
(A" D A) Dankey --3
(20)
N-1 -(ATEX
I.x A) A)k
(21)
((D) B).,.)' (D' BYB) B)ck
6 CY
(22)
N
(( B)" (DY B Ki.ks
(23)
N -1
(BT DY" DD DYB)es B)Ks
(24)
N-1
=
=(B ():.::
(25)
((PD* A (DD
(26)
--1
((PDx A) DY B)ack:
B)
(27)
N -
(A" DXT p' DY B) R)
(28)
N
(AT D
DY PT
B)ack
(29)
N-1
Further, x and PT DY can be taken sampled from the joint distribution of two independent gaussian random vectors. As each of them are marginally gaussian, they are also jointly gaussian. Thus, DxT p DY maximizing likeilhood as in Eq. 17.
can be consistently estimated by N-1
B. Other Definitions
B.1. T-separation
The definitions of trek and t-separation are follows.
Definition (Treks (Sullivant al., 2010)). In G, trek from X to Y is ordered pair of directed paths (P1, P2) where P1 has sink X, Pz has sink Y, and both P1 and Pg have the same source Z.
Definition 2 (T-separation (Sullivant et al., 2010)). Let A, B, CA. and CB be four subsets of Ve in graph & (not necessarilly disjoint). AC CB) t-separates A from B if for every trek (P1,P,) from a vertex in A to vertex in B, either P; contains a vertex in CA or Ps contains vertex in Cp.
Example 1. In Figure 5, there are multiple treks. For example, X4 + L + X3 is a trek between Xa and X3, X4 + L is trek between X4 and L,, and L + X3 a trek between L and X3. As for t-separations, we have (X1, X2) and (X3, X4 ) are I-separated by (0, (L,)).
13

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Permutation-Bast Based Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
X1
X3
4
X2
X
Figure 5. An illustrative example show that rank contains graphical information than CL When using CL. cannot deduce that
(2, and (X3, } are d-separated by L1 as L, is latent, while by using rank we can.
P(X, Y)
P(X)
Random peruntatiin on Y
Figure 6. An illustration of exchangeability and permutation test. The left figure refer to N i..dd samples from P(X, Y). After random permutation the permutated data can considered as random i.i.d. samples from P(X) and P(Y). ). If exchangeability holds, ie.. random vectors X and Y are independent, then we have P(X, Y) P(X) and thus the permuted data can serve as another N i.i.d. samples from P(X,Y: Y).
C. Discussion
C.1. Brief Introduction to Permutation Test
Permutation tests aim to empirically estimate the CDF the null distribution a test statistic. The core such an CDF estimation is the exchangeability, under which we can make use of permuted data to serve as additional samples from the same distribution.
Take Figure 6 as example. The left figure in Figure 6 refer to N i.i.d. samples from P(X, Y). After random permutation on Y, the permutated data can be considered as random i..dd samples from P(X) and P(Y). If the exchangeability holds under the null hypothesis, i.e., random vectors X and Y are independent, then we have P(X, Y) = ((X))(()) and thus the permuted data can serve as another i.i.d. samples from P(X, Y). Now we know how to generate additional N i.i.d. samples. a test statistic is just a deterministic function the N i.i.d., samples. For each randomly permuted data, we can calculate the value of the test statistic, and thus all these calculated test statistics can be considered as sampled from the distribution of the test statistic. Given these samples, consequently correctly calculate the p-value.
construct the empirical CDF of the null distribution, and
C.2. Number of Categories and Analysis of Type-I error and Power
The proposed method handle any level of discretization, long it is greater than 1, with Type-I errors properly controlled. At the same time, more levels are always 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.
Regarding Type-I errors, as we establish the exchangeability even in the discretized scenario, the asymptotic null distribution can estimated by random permutations. Consequently, Type-I errors can be properly controlled at any significance level. At the same time, we do not have theoretical result on the analysis of the power yet. To be specific, even without
14

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Permulation-Basee Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data
considering discretization, the analysis of power involves tools from advanced random matrix theories and is highly nontrivial. Furthermore, in our setting with discretized variables, the involved maximum likelihood step makes such an analysis even more challenging, To our best knowledge, there is not any existing result available for the analytic form of the power in our setting, and we plan to leave it for future exploration.
D. Related Work
Conditional independence and rank test. A line of conditional independence tests imposes simplifying assumptions on the distributions. For instance, when the variables have linear relations with additive Gaussian noise, the Fisher's classical z-test based partial correlations can be used (Fisher, 1924; Baba et al., 2004). Ramsey (2014) developed an approach that separately regresses X" and Y on Z, and further perform independence test on the corresponding residuals. Fukumizu et al. (2007) proposed a conditional independence test method based on Hilbert-Schmidt independence criterion (HSIC) (Gretton et al., 2007). Zhang et al. (2012) further provided kernel-based conditional test that yields pointwise asymptotic level control. Shah & Peters (2018) investigated the hardness of conditional independence test, and developed a method based on kernel-ridge regression and generalised covariance measure. On the other hand, existing statistical tests for rank of a cross-covariance matrix (Anderson, 1984) often rely on CCA (Jordan, 1875; Hotelling, 1992), with likelihood ratio based test statistics. Recently, Sun et al. (2025b) also establishes valid partial correlation test in the presence of discretization, with focus on the binary discretization scenario, and later Sun et al. (2025a) better solves this problem with general method of moment.
Permutation test. Research and applications related to permutation tests have addressed increased attention in recent years (David, 2008; Pesarin & Salmaso, 2010; Welch, 1990). These tests lead to valid inferences while requiring weak assumptions that are commonly satisfied, base on the exchangeability of observations under the null hypothesis. Recently, permutation-basete CI test was proposed (Doran al., 2014) and more recently permutation-base rank test (Winkler et al., 2020). However, they cannot deal with the discretization problem. In contrast, our MPRT can take all continuous, partially discretized, or all discretized data as input, and our Type errors can be properly controlled.
Constraint- causal discovery. Constraint-based methods leverage statistical tests, such as conditional independence tests, to estimate the causal structure. Spirtes & Glymour (1991) proposed the PC algorithm that estimates the skeleton and orient certain edges to identify the Markov equivalence class. FCI (Spirtes et al., 1995; Colombo et al., 2012) was developed allow for latent and selection variables, while the CCD algorithm (Richardson, 1996) can accommodate cycles. Furthermore, Huang et al. (2020) developed constraint-based method that allows for heterogeneity non-stationarity in the data distribution, while Silva et al. (2006); Huang et al. (2022); Dong et al. (2024a) proposed algorithms based rank test that recover the causal structure involving latent confounders.
15


Reviews:
[{"review_id": "2mqOi0teDb", "reviewer": "Reviewer_uits", "summary": "This paper introduces the Mixed Data Permutation-Based Rank Test (MPRT), an approach designed to address the challenge of discretization in rank tests. The proposed MPRT estimates the asymptotic null distribution by leveraging data permutation, which is grounded in the exchangeability condition of the linear projection of the relevant variables. The authors demonstrate the efficacy of the MPRT through comprehensive synthetic and real-world experiments.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful comments, which have greatly helped us refine the quality of our paper. We first provide responses to your concerns and then give an updated sketch of proof for Thm 4.\n\n**Q1:** The convergence of $\\hat{\\Sigma}\\_\\mathbf{X}$ by pseudo-maximum likelihood.\n\n**A1:** Yes, $\\hat{\\Sigma}\\_\\mathbf{X}$ in our method converges in probability to the population one. [1] is one of the earliest on pseudo-likelihood, and its consistency was later discussed in [2]. As a member of M-estimators, the  consistency can also be derived from the general theory for M-estimators [3]. Specifically in the presence of discretization, [6] derived the consistence of estimating precision matrix under mixed non-paranormal model; similar arguments can be applied for the consistency of $\\hat{\\Sigma}\\_\\mathbf{X}$ by pseudo-maximum likelihood in our paper.\n\n**Q2:** Continuity and uniqueness of SVD.\n\n**A2:** SVD is  not continuous only when the input matrix has repeated singular values. Specifically, if a matrix $A$ has distinct singular values, then SVD is continuous in the neighborhood of $A$, and unique only up to sign flip (chapter 2 section 5.3 of [4]).\nThus, to make use of the continuous mapping theorem, we  assume that $\\Sigma\\_\\mathbf{X}^{-\\frac{1}{2}}\\Sigma\\_{\\mathbf{X},\\mathbf{Y}}\\Sigma\\_\\mathbf{Y}^{-\\frac{1}{2}}$ does not have repeated singular values (the justification of which is discussed in Q3). To further eliminate the sign indeterminacy, we can follow scikit-learn to ensure the largest coefficient in each column, in terms of absolute value, is positive (svd flip in scikit-learn).\n\n**Q3:** The justification of assuming non-repeated singular values.\n\n**A3:** Faithfulness is one of the most important assumptions in causal discovery and is typically justified by that the set of parameters violating it is of Lebesgue measure zero [7]. \nSimilarly, it has been shown that the set of matrices with repeated singular values has Lebesgue measure zero (Lemma 1.4.2 in [8], also in [5]).\n\nAn updated sketch of proof for Thm 4.\n\n> Sketch of proof\n\\\nGiven $(\\hat{\\Sigma}\\_\\mathbf{X},\\hat{\\Sigma}\\_\\mathbf{Y},\\hat{\\Sigma}\\_{\\mathbf{X},\\mathbf{Y}})\\overset{p}{\\to}(\\Sigma\\_\\mathbf{X},\\Sigma\\_\\mathbf{Y},\\Sigma\\_{\\mathbf{X},\\mathbf{Y}})$,\nwe aim to show the desired asymptotic independence. Specifically we want to show (i) $\\mathbf{C_X}\\_{k:} \\overset{p}{\\to} \\mathbf{C_X}\\_{k:}^\\*$  and $\\mathbf{C_Y}\\_{k:} \\overset{p}{\\to} \\mathbf{C_Y}\\_{k:}^\\*$, and  (ii) $\\mathbf{C_X}\\_{k:}^\\*,\\mathbf{C_Y}\\_{k:}^\\*$ are independent under the null hypo.\n\\\nHere $\\mathbf{C_X}=A^T\\mathbf{X},\\mathbf{C_Y}=B^T\\mathbf{Y}$,\n$\\mathbf{C_X}^\\*={A^\\*}^T\\mathbf{X}$, and $\\mathbf{C_Y}^\\*={{B^\\*}^T}\\mathbf{Y}$, where $(A,B)$ and $(A^\\*,B^\\*)$  are produced by SVD using  estimated covariance and population one respectively as follows.\n$$USV=\\hat{\\Sigma}\\_\\mathbf{X}^{-\\frac{1}{2}}\\hat{\\Sigma}\\_{\\mathbf{X},\\mathbf{Y}}\\hat{\\Sigma}\\_\\mathbf{Y}^{-\\frac{1}{2}},A=\\hat{\\Sigma}\\_\\mathbf{X}^{-\\frac{1}{2}T}U,B=\\hat{\\Sigma}\\_\\mathbf{Y}^{-\\frac{1}{2}T}V^T,~~~U^\\*S^\\*V^\\*  =\\Sigma\\_\\mathbf{X}^{-\\frac{1}{2}}\\Sigma\\_{\\mathbf{X},\\mathbf{Y}} \\Sigma\\_\\mathbf{Y}^{-\\frac{1}{2}},A^\\*=\\Sigma\\_\\mathbf{X}^{-\\frac{1}{2}T}U^\\*,B^\\*=\\Sigma\\_\\mathbf{Y}^{-\\frac{1}{2}T}{V^\\*}^{T}.$$\nFor (i): By continuous mapping theorem, under the assumption of no repeated singular values, we have $U \\overset{p}{\\to} U^\\*$. As $\\Sigma\\_\\mathbf{X}$ is positive definite, the matrix inverse square root is continuous and thus \n$\\hat{\\Sigma}\\_\\mathbf{X}^{-\\frac{1}{2}T}\\overset{p}{\\to}\\Sigma\\_\\mathbf{X}^{-\\frac{1}{2}T}$. Given $(U,\\hat{\\Sigma}\\_\\mathbf{X}^{-\\frac{1}{2}T})\\overset{p}{\\to}(U^\\*,\\Sigma\\_\\mathbf{X}^{-\\frac{1}{2}T})$, we have $\\hat{\\Sigma}\\_\\mathbf{X}^{-\\frac{1}{2}T}U=A\\overset{p}{\\to}A^\\*=\\Sigma\\_\\mathbf{X}^{-\\frac{1}{2}T}U^\\*$. Similarly, we have $B\\overset{p}{\\to}B^\\*$.\n\\\nThus $$((A^T-{A^\\*}^T)\\mathbf{X},(B^T-{B^\\*}^T)\\mathbf{Y})\\overset{p}{\\to}0\\Rightarrow (((A^T-{A^\\*}^T)\\mathbf{X})\\_{k:},((B^T-{B^\\*}^T)\\mathbf{Y})\\_{k:})\\overset{p}{\\to}0\\Rightarrow(\\mathbf{C_X}\\_{k:},\\mathbf{C_Y}\\_{k:})\\overset{p}{\\to}(\\mathbf{C_X}\\_{k:}^\\*,\\mathbf{C_Y}\\_{k:}^\\*).$$\nFor (ii): Under the null hypo, the cross-covariance between $\\mathbf{C_X}\\_{k:}^\\*$ and $\\mathbf{C_Y}\\_{k:}^\\*$ are all zeros. \nAs $\\mathbf{C_X}\\_{k:}^\\*,\\mathbf{C_Y}\\_{k:}^\\*$ are jointly gaussian (linear mixing of $\\mathbf{X,Y}$),\nzero cross-covariance implies independence.\n\nPlease feel free to let us know if any part remains unclear or you have further questions. Thank you again for your valuable feedback!\n\n[1] Besag, Spatial interaction ... system. 1974.\n\n[2] Gourieroux, Pseudo maximum likelihood methods: Theory. 1984.\n\n[3]  Gourieroux, Consistent pseudo ... estimators. 2017.\n\n[4] Kato, Perturbation theory for linear operators. 2013.\n\n[5] Bochnak, Real algebraic geometry.2013.\n\n[6] Fan, High dimensional ... mixed data. 2017.\n\n[7] Spirtes, Causation, prediction, and search. 2000.\n\n[8] Kunisky, Lecture Notes on Random Matrix Theory. 2024.", "claims_and_evidence": "No. The correctness of Thm. 4, which is the key result that the proposed test relies on, is problematic.\n\nFirst, there is no clear definition of \"asymptotic independence\". There are various definitions in the literature [1,2], but it is unclear which definition the authors adopt. From the proof, I presume that the authors define two sequence of random vectors $X_n$ and $Y_n$ are asymptotically independent if there exists $X, Y$ such that $X \\perp Y$ and $X_n \\to X, Y_n \\to Y$ in distribution [1]. But this needs clarification.\n\nSecond, the asymptotic independence between $C_{X k:}$ and $C_{Y k:}$ is not discussed in a rigorous way. The authors should follow the definition and show the convergence of  $C_{X k:}, C_{Y k:}$ to some $C^\\prime_{X}, C^\\prime_{Y}$ such that $C^\\prime_{X} \\perp C^\\prime_{Y}$.\n\nFrom the proof, I conjecture that the authors may want to show this by first showing the convergence of $\\hat{\\Sigma}_X, \\hat{\\Sigma}_Y$ to $\\Sigma_X, \\Sigma_Y$, and then using the continous mapping theorem to show the convergence of C{X k:}, C{Y k:}, which are computed from $\\hat{\\Sigma}_X, \\hat{\\Sigma}_Y$ with SVD decomposition. \n\nBut this can be highly problematic:\n\n1. The convergence of $\\hat{\\Sigma}_X, \\hat{\\Sigma}_Y$ obtained from pseudo-maximum likelihood is unkown. I do not find results in the paper cited by the authors (Besag, 1974). \n\n2. The SVD is not unique and thus cannot be considered as a continuous function (see [3]). So the continous mapping theorem may not apply here.\n\nCan the authors provide further clarifications to my questions? I ask for particular rigor because the proposed test highly rely on this independence result. I am happy to increase my score if the authors can address these questions properly.\n\n[1] https://math.stackexchange.com/questions/1272661/is-there-a-concept-of-asymptotically-independent-random-variables\n\n[2] https://arxiv.org/pdf/1910.04243\n\n[3] https://math.stackexchange.com/questions/3389899/continuity-of-singular-value-decomposition", "methods_and_evaluation_criteria": "The permutation-based test relies on the exchangeability result (Thm. 4), which can be problematic (see Claims and Evidence). Therefore, the proposed test may not be valid.", "theoretical_claims": "Yes. The proof of Thm. 4 is problematic. See Claim and Evidence.", "experimental_designs_or_analyses": "Yes. The experiment is comprehensive.", "relation_to_broader_scientific_literature": "This paper extends the classifical rank test (Jordan, 1875; Hotelling, 1992) to cases with discretization.", "essential_references_not_discussed": "Yes.", "other_strengths_and_weaknesses": "1. The problem of discretization in rank-based test is important. \n\n2. The writting is clear.", "other_comments_or_suggestions": "See Claim and Evidence.", "questions_for_authors": "See Claim and Evidence.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rLIRdHgGdo", "reviewer": "Reviewer_vw91", "summary": "The authors consider the task of finding the rank of cross covariance matrices, when some variables have been discretised. The authors consider a permutation based rank test that is able to handle both discrete and continuous variables. They show that their method is able to perform well with both continuous and mixed data.\n\n## update after rebuttal\nI thank the authors for their response. I will keep my positive score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the time dedicated to reviewing our paper, the insightful comments, and valuable feedback. Please see our point-by-point responses below.\n\n**Q1:** About data generation for experiments?\n\n**A1:** \nWe assume that data is generated following a linear structural causal model $\\mathsf{V}\\_i=\\sum \\nolimits\\_{\\mathsf{V}\\_j \\in \\text{Pa}(\\mathsf{V}\\_i)} f\\_{j,i} \\mathsf{V}\\_j + \\epsilon\\_{\\mathsf{V}\\_i}$, where each edge coefficient $f_{j,i}$ is uniformly sampled from $[-2, 2]$ and the noise terms are Gaussian with variance uniformly from $[1,5]$. \nWithin this model, the population covariance can be directly specified once $f\\_{j,i}$ and the variance of $\\epsilon\\_{\\mathsf{V}\\_i}$ are specified.\n%\nInformation about the process of discretization can be found in line 369-373. We thank the reviewer and have revised our paper to include the above description.\n\n\n**Q2:** The metric used in Table 1?\n\n**A2:** As mentioned in line 425, the metrics used in Table 1 are SHD and F1 scores for the skeleton, which is also commonly used in the literature. In light of your suggestion, we conducted additional experiments to compare the CPDAGs in terms of SHD and F1, and our proposed MPRT still consistently outperforms other methods across different sample sizes. We thank the reviewer for the valuable comment and have revised our paper to include this additional result.\n\n**Q3:** Although they tackle the problem of discretisation, they only do this for rank tests. Causal discovery has frameworks that can handle both mixed and continuous data. For example [1] is tested on both continuous and discrete data?\n\n**A3:** \nThank you for mentioning this interesting work and we have revised our manuscript to add a related discussion.\nFrom one perspective, since conditional independence constraint (in the linear case) is a special case of rank constraint of covariance matrices [3], the proposed method can also be used to test CI in the presence of discretization, as empirically validated by the results in Table 1. From another perspective,\nas you mentioned, there are some existing works that can handle mixed data for CI, e.g., RDC [1] and KCI [2]. In light of your suggestion, we conducted additional experiments to compare with KCI in Table 1 (it is hard to compare to RDC as it can only measure independence rather than conditional independence). We observe that the proposed MPRT outperforms KCI by a clear margin, e.g., when $N=2000$, MPRT achieves a F1 score of 0.96 while KCI achieves only 0.86. This outcome is expected, as KCI (like other CI tests designed for mixed data) directly test the relations between observed variables, while the real objective is to identify the relations between the underlying continuous variables. Please kindly refer to our response to Q2 for reviewer QSUp for a related discussion.\n\n\n**Q4:** Regarding the arguement in L241 LHS, why can Theorem 4 not be used for discrete values?\n\n**A4:** There are two reasons. (i) Directly using the ordinal values cannot produce the correct linear transformation $A$ (does not converge to the true one). (ii) Even if we can produce the correct $A$, when some columns of data are just ordinal values, after the linear transformation the entries of $\\mathbf{C_X}_{k:}$ would be linear combination of some continuous and some ordinal values, which can be arbitrary (as the choice of ordinal values can take either $\\{1,2,3\\}$ or $\\{1,2,10^{10}\\}$) and does not have psychical meaning. \n\n**Q5:** Regarding Section 4.2, just to be clear, CCART-C when applied on the mixed data, is not actually fed mixed data, but works on the original continuous values? If so, it might also be interesting to see what happens when mixed data is provided to CCART-C.\n\n**A5:** Yes, CCART-C takes the original continuous values as input and thus in our experiments its performance serves as the upper-bound. When CCA-based rank test takes the mixed data as input, it is named as CCART-D in our experiments (as mentioned in section 4.1). As expected, CCART-D \ncannot properly control the type-I error and it does not benefit from the increase of the sample size, as shown in Figure 2. \n\n**Q6:** The ground truth for section 4.4?\n\n**A6:** The structure that underlies human personality remains an open research problem and thus there is no established ground truth for it yet. \n\nWe genuinely appreciate the reviewer's effort and hope that your concerns/questions are addressed.\n\n[1] Lopez-Paz, The randomized dependence coefficient. 2013.\n\n[2] Zhang, Kernel-based conditional independence test and application in causal discovery. 2012.\n\n[3] Sullivant, Trek separation for gaussian graphical models. 2010.", "claims_and_evidence": "The claims made in the paper are supported.", "methods_and_evaluation_criteria": "I can't find any information about the data generation for the experiments. It would be useful to see the performance under varying data generating assumptions and varying degrees of discretisation.", "theoretical_claims": "Theoretical claims seem correct.", "experimental_designs_or_analyses": "- The PC algorithm outputs the CPDAG, not a single DAG, but its Markov equivalence class. As such, the SHD and F1 scores might not make sense. The SHD between CPDAGs might be a better metric. If this is the actual metric that is calculated, it should be made clear.", "relation_to_broader_scientific_literature": "The main contribution is a permutation based rank test that can also handle discrete data.\n\nIt is not clear which how other parts are the contributions of the paper. Is the estimation of the correlation with discretisation novel (section 3.3)? It seems to follow known results.", "essential_references_not_discussed": "Not to my knowledge.", "other_strengths_and_weaknesses": "Strengths:\n- The paper tackles an important problem of discretisation.\n\nWeaknesses:\n- Lack of information on the experiments makes it hard to judge whether they were comprehensive.\n- Although they tackle the problem of discretisation, they only do this for rank tests. Causal discovery has frameworks that can handle both mixed and continuous data. For example [1] is tested on both continuous and discrete data.\n\n[1] Lopez-Paz, David, Philipp Hennig, and Bernhard Schölkopf. \"The randomized dependence coefficient.\" Advances in neural information processing systems 26 (2013).", "other_comments_or_suggestions": "N/A", "questions_for_authors": "- I don't understand the arguement in L241 LHS, why can Lemma 4 not be used for discrete values?\n- Section 4.2: Just to be clear, CCART-C when applied on the mixed data, is not actually fed mixed data, but works on the original continuous values? If so, it might also be interesting to see what happens when mixed data is provided to CCART-C. \n- What is the ground truth for section 4.4?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ka3L9b04gc", "reviewer": "Reviewer_QSUp", "summary": "This paper introduces the Mixed data Permutation-based Rank Test (MPRT) for testing the rank of cross-covariance matrices in the presence of discretized variables. The authors address a critical gap in existing rank tests, which assume continuous measurements and fail when variables are discretized. MPRT leverages permutation-based resampling and maximum likelihood correlation estimation to handle mixed data. Theoretical guarantees on Type I error control and empirical validation on synthetic and real-world datasets demonstrate its effectiveness. The method is also applied to causal discovery, showing improved performance over traditional conditional independence tests in discretized settings.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the time dedicated to reviewing our paper, the insightful comments, and valuable feedback. Please see our point-by-point responses below.\n\n\n**Q1:**  While Appendix C.2 briefly discusses non-Gaussian cases, robustness to severe non-Gaussianity or nonlinear relationships is unclear.\n\n**A1:** \nTo the best of our knowledge, under the linear Gaussian assumption,\nnone of the existing rank tests can properly handle the issue of  discretization and thus we believe our method is already a non-trivial contribution to this field. At the same time, we totally agree that non-Gaussiani or nonlinear cases can also be highly relevant in practical applications. Extending our framework to accommodate non-Gaussian cases is certainly feasible, as discussed in Appendix C.2. Addressing nonlinear cases is certainly more challenging because, even without discretization, the rank constraints (i.e., vanishing determinant constraints implied by trek separation) relies on linearity; extending this to arbitrary nonlinear relationships remains an open problem in the field of causal discovery. We thank the reviewer for the insightful comments and we plan to leave it for future exploration.\n\n\n**Q2:**  Can discretization create or break CI relationships? \n\n**A2:** Yes, discretization generally breaks CI relations regardless of the existence of non-Gaussianity or non-linearity (as long as we are not referring to the trivial case where variables are discretized into just one value, resulting in complete loss of information).\nFor example, consider continuous variables $\\mathsf{X},\\mathsf{Y},\\mathsf{Z}$ following a causal graph $\\mathsf{X}\\leftarrow\\mathsf{Z}\\rightarrow\\mathsf{Y}$.\nBy d-separation we know that $\\mathsf{X} \\perp  \\mathsf{Y} | \\mathsf{Z}$. However, if we can only observe a discretized version of $\\mathsf{Z}$, say $\\mathsf{Z}'$, the CI relation  $\\mathsf{X} \\perp  \\mathsf{Y} | \\mathsf{Z}'$ generally does not hold. The reason is that discretization from $\\mathsf{Z}$ to $\\mathsf{Z}'$ introduces information loss and thus $\\mathsf{Z}'$ no longer retains the complete information to explain the dependence between $\\mathsf{X}$ and $\\mathsf{Y}$. \n\nThe same reasoning also applies to the rank test. Roughly speaking, a lower rank of the cross-covariance indicates greater independence between two sets of variables.  The presence of discretization generally induces higher rank than the correct one, leading to a false indication of dependence and thus breaks the independence. Foe example, as shown in in Figure 1 of our submission,\n  panel (a) shows the population cross-covariance without discretization, where the rank is 1, while panel (b) shows the cross-covariance computed from discretized observations, resulting in a rank of 3.\n\n\n**Q3:**  It seems that Equation 14 assumes the underlying variables follow a standard Gaussian distribution. How might deviations from this assumption affect the validity of correlation estimates or rank test results?\n\n**A3:** Violation of this assumption, e.g., shifting and rescaling, does not affect the validity of the whole method, as long as all the variables are still jointly Gaussian.\nThis is because we care about the rank of the cross-covariance matrix, which is equal to  the rank of the cross-correlation matrix; the latter is clearly invariant to shift or rescaling of either some or all variables (also mentioned in line 229). Thus, we can simply assume that all variables are standardized for Equation 14.\n\n\nWe genuinely appreciate the reviewer's effort and hope that your concerns/questions are addressed.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "No.", "experimental_designs_or_analyses": "Yes, I checked the experiments in Section 4.", "relation_to_broader_scientific_literature": "This paper utilizes the permutation strategy to extend classical rank tests to discretized data.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strength: This paper proposes a valid rank test for discretized data, addressing limitations of classical CI tests that fail under discretization.\n\nWeakness:  The theoretical guarantees rely on joint Gaussianity. While Appendix C.2 briefly discusses non-Gaussian cases, robustness to severe non-Gaussianity or nonlinear relationships is unclear.", "other_comments_or_suggestions": "None.", "questions_for_authors": "1. Can discretization create or break CI relationships, especially when the data is non-Gaussian or exhibits nonlinear dependencies? Specifically, are there cases where CI holds before discretization but not after, or vice versa?\n\n2. It seems that the threshold estimation in Equation (14) assumes the underlying variables follow a standard Gaussian distribution. How might deviations from this assumption affect the validity of correlation estimates or rank test results?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes the Mixed Data Permutation-Based Rank Test (MPRT) to assess the rank of cross-covariance matrices when data includes both discrete and continuous variables. 


Pros:

+ Addresses a critical limitation in classical rank tests by effectively handling discretized and mixed data.

+ Clear writing and solid motivation; the application to causal discovery adds practical relevance.

Cons:

+ The key theoretical result (Theorem 4) needs clarification and refinement, especially regarding asymptotic independence and SVD continuity.

+ Experimental details (e.g., data generation, metrics, and assumptions) are insufficient in parts, making reproducibility and evaluation difficult.

+ Robustness to non-Gaussianity and nonlinear dependencies is underexplored, which may limit the method's applicability in real-world scenarios.
Decision: Accept (poster)