A Kernel-based Test of Independence for Cluster-correlated Data

1 Introduction . We are often interested in studying the dependence between two multivariate variables . For example , in genetic studies , we may want to assess the association between multiple genetic variants within a gene and a group of traits that likely share a common genetic mechanism [ 1 , 2 ] . In microbiome studies , we may wish to investigate the association between the overall composition of human microbiota , including hundreds of microbial taxa , and multiple host metabolites from a particular metabolic pathway [ 3 , 4 ] . Such multivariate analyses aggregate information across variables and are often more powerful than univariate analyses . Meanwhile , correlated observations arise in many practical situations . Family-based designs are common in genetic studies [ 5 ] , where multiple family members are recruited together into the study . Longitudinal data are common in epidemiological studies [ 6 ] , where variables of interest are measured on the subjects repeatedly over time . Such study designs introduce clustered dependence among the observations and require proper accommodation . In this work , we aim to develop an approach for assessing the dependence between two multivariate variables , based on cluster-correlated data . A variety of parametric and semi-parametric methods has been proposed to study the association between one or multiple exposure variables and a multivariate longitudinal ( or other cluster-correlated ) outcome . These methods often extend upon existing tools for univariate longitudinal data . For example , many studies stack the multivariate outcome into a single response vector and then apply the 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . usual methods that account for clustered data , such as generalized estimating equations ( GEE ) [ 7 , 8 ] and random effects models [ 9 , 10 ] . Such approaches generally apply to a low-dimensional setting [ 11 ] and are subject to limitations typical of parametric and semi-parametric methods . Randomeffect-based methods require assumptions on the distribution of the multivariate outcome . GEE-based methods rely on good estimation of the correlation structure within clusters as well as across different outcome variables to achieve a high efficiency . Finally , both approaches assume parametric ( often linear ) relationships between the exposure and the outcome . Therefore , they can only evaluate a limited number of dependence patterns , and are not sufficient as independence tests . Here we base our approach on the Hilbert-Schmidt Independence Criterion ( HSIC ) , a non-parametric kernel-based measure for assessing the generalized dependence between two multivariate , and potentially high-dimensional , variables [ 12 ] . By mapping the two variables into reproducing kernel Hilbert spaces ( RKHS ’ s ) , the population HSIC can be viewed as a measure of maximized covariance between functions in the two RKHS ’ s . When the RKHS ’ s being used are characteristic [ 13 ] , the population HSIC is zero if and only if the two variables are independent . This measure makes no assumption on the distributions of the variables or the nature of the dependence . The original HSIC-based independence test [ 14 ] applies to independent and identically distributed ( i.i.d . ) observations . Several extensions have been made to accommodate non-i.i.d . data , but none of the tests , to our knowledge , directly applies to clustered data at an observation level . Zhang et al . ( 2008 ) [ 15 ] extended the HSIC to certain sequence data , such as XOR sequence and Gaussian process , by specifying the correlation structure of the data as a graphical model and deriving the test statistic based on the maximal cliques . Chwialkowski et al . [ 16 ] and Wang et al . [ 17 ] developed HSIC-based tests to evaluate the dependence between two time series or random processes in general . Flaxman et al . [ 18 ] considered spatial and temporal data ; they proposed to first use Gaussian process regression to remove dependence on space and time from the raw variables , and then perform the HSIC test on the resulting de-correlated residuals . However , their approach generally applies to independence testing between two univariate variables . A study with an aim closest to ours is by Rudra et al . [ 19 ] : They analyzed the association between multiple genetic variants and a multivariate longitudinal outcome . Rudra et al . concatenated the outcome measurements from different time points at the subject level , and then applied the HSIC test to the subject-level data . Although this is a straightforward approach to deal with clustered correlation , there could be a loss of statistical power by analyzing data at the subject/cluster level rather than observation level . In this work , we present the first HSIC-based independence test for cluster-correlated data . Using the empirical HSIC [ 14 ] as our test statistic , we derive its asymptotic distribution under the null hypothesis of independence between the two variables but in the presence of clustered correlation among observations . We also examine the behavior of the test statistic under the alternative hypothesis and establish the consistency of our test . Furthermore , we provide a way to approximate the asymptotic null distribution of the test statistic and allow for statistical testing in practice . In simulation studies , our proposed test controls type I error rates well and has a much higher statistical power than competing methods across a range of scenarios . In an application to a longitudinal microbiomemetabolite data set , compared to other approaches , our proposed test identifies a larger number of metabolic pathways significantly associated with the overall microbiome composition , highlighting the value of our test in scientific studies . The remaining sections are organized as following . In Section 2 , we provide our background assumption on clustered data and an overview of the HSIC statistic . In Section 3 , we study the asymptotic behavior of the HSIC statistic under null and alternative hypotheses , and construct a statistical test of independence for cluster-correlated data . In Section 4 , we demonstrate the performance of our proposed test on both simulated and real data . In Section 5 , we summarize our work , discuss the limitations of our proposed test and provide a conclusion . 2 Background . In this section , we introduce our assumption on cluster-correlated data and give an overview of the HSIC statistic . 2.1 General setting . Let PXY be a probability measure defined on a sample space X × Y , where both X and Y can be multi-dimensional . Let PX and PY be the marginal distributions on X and Y , respectively . The variables X and Y are statistically independent if PXY = PXPY ( equivalently , we can write X ⊥⊥ Y ) . We consider a sample of clustered data { ( Xj , Yj ) } nj=1 drawn from PXY , where the pattern of clustered correlation is balanced and complete : Assumption 2.1 . The observations ( X1 , Y1 ) , · · · , ( Xn , Yn ) are identically distributed according to PXY , and can be divided into m clusters of fixed size d ( i.e. , n = md ) . In particular , the m clusters { [ ( Xdi−d+1 , Ydi−d+1 ) , · · · , ( Xdi , Ydi ) ] } m i=1 are independent from one another while having identical within-cluster correlation structure . The specific correlation structure among the observations in each cluster can be arbitrary . We are interested in studying the dependence between X and Y based on the sample { ( Xj , Yj ) } nj=1 . 2.2 Hilbert-Schmidt Independence Criterion . We briefly review the Hilbert-Schmidt Independence Criterion ( HSIC ) proposed by Gretton et al . ( 2005a ) [ 12 ] . The HSIC measures the generalized dependence between two variables X and Y , by embeddingX and Y into reproducing kernel Hilbert spaces ( RKHS ’ s ) and maximizing the covariance between functions of X and Y in the RKHS ’ s . Let HX be an RKHS on X with associated kernel function ( i.e. , inner product in the RKHS ) kX : X × X → R , and letHY be an RKHS on Y with associated kernel function kY : Y × Y → R. Following Gretton et al . ( 2007 ) [ 14 ] , the cross-covariance operator CXY : HY → HX can be defined such that , for any f ∈ HX and g ∈ HY , 〈f , CXY g〉HX = EXY [ ( f ( X ) − EX [ f ( X ) ] ) ( g ( Y ) − EY [ g ( Y ) ] ) ] = Cov ( f ( X ) , g ( Y ) ) . As shown by Gretton et al . ( 2005b ) [ 20 ] , the operator norm ( i.e. , the largest singular value ) of CXY , defined by ‖CXY ‖ : = supf∈HX , g∈HY , ‖f‖∞≤1 , ‖g‖∞≤1 Cov ( f ( X ) , g ( Y ) ) , is zero if and only if X ⊥⊥ Y , given that the kernels kX and kY are universal ( see Definition 5 of [ 20 ] ) . In this sense , ‖CXY ‖ is a measure of independence between X and Y . The largest singular value of CXY becomes zero when the sum of all squared singular values , denoted as the squared Hilbert-Schmidt norm [ 12 ] , is zero . Therefore , the squared Hilbert-Schmidt norm of CXY , ‖CXY ‖2HS , is also an independence criterion . This measure is defined as the population HSIC , which can be expressed conveniently in terms of kernel functions : HSIC ( PXY ) : = ‖CXY ‖2HS = EXX′Y Y ′ [ kX ( X , X ′ ) kY ( Y , Y ′ ) ] + EXX′ [ kX ( X , X ′ ) ] EY Y ′ [ kY ( Y , Y ′ ) ] − 2EXY [ EX′ [ kX ( X , X ′ ) ] EY ′ [ kY ( Y , Y ′ ) ] ] , where X ′ is an independent copy of X . It is obvious that , if X is independent from Y , then we have HSIC ( PXY ) = 0 . Furthermore , for certain characteristic kX and kY [ 13 ] , HSIC ( PXY ) = 0 if and only if X ⊥⊥ Y . Example characteristic kernels include Gaussian kernels and Laplacian kernels [ 21 ] . To estimate the population HSIC from a sample { ( Xj , Yj ) } nj=1 , the empirical HSIC can be used : HSIC ( Pn ) : = 1 n2 n∑ i , j kX ( Xi , Xj ) kY ( Yi , Yj ) + 1 n4 n∑ i , j , q , r kX ( Xi , Xj ) kY ( Yq , Yr ) − 2 n3 n∑ i , j , q kX ( Xi , Xj ) kY ( Yi , Yq ) . Define the kernel matrices KX and KY such that the ( i , j ) -th element of KX is kX ( Xi , Xj ) and the ( i , j ) -th element ofKY is kY ( Yi , Yj ) . Then the empirical HSIC can also be written in terms of KX andKY : HSIC ( Pn ) = 1 n2 tr ( HKXHKY ) , whereH = I − 1n11 T is a centering matrix . Both Gretton et al . ( 2007 ) [ 14 ] and Zhang et al . ( 2012 ) [ 22 ] have derived the asymptotic distribution of HSIC ( Pn ) under the null hypothesis of independence between X and Y as a weighted sum of chi-square variables , when the observations are i.i.d . In Section 3 , we examine the asymptotic behavior of HSIC ( Pn ) based on cluster-correlated observations . It turns out that the null distribution in this case is still a weighted sum of chi-square variables , where the weights are now modified .

Summary: The authors propose a test of independence for cluster correlated data. The test statistic is still HSIC. However they derive a different asymptotic distribution under cluster correlation.