A fast small-sample kernel independence test for microbiome community-level association analysis

Summary

To fully understand the role of microbiome in human health and diseases, researchers are increasingly interested in assessing the relationship between microbiome composition and host genomic data. The dimensionality of the data as well as complex relationships between microbiota and host genomics pose considerable challenges for analysis. In this paper, we apply a kernel RV coefficient (KRV) test to evaluate the overall association between host gene expression and microbiome composition. The KRV statistic can capture non-linear correlations and complex relationships among the individual data types and between gene expression and microbiome composition through measuring general dependency. Testing proceeds via a similar route as existing tests of the generalized RV coefficients and allows for rapid p-value calculation. Strategies to allow adjustment for confounding effects, which is crucial for avoiding misleading results, and to alleviate the problem of selecting the most favorable kernel are considered. Simulation studies show that KRV is useful in testing statistical independence with finite samples given the kernels are appropriately chosen, and can powerfully identify existing associations between microbiome composition and host genomic data while protecting type I error. We apply the KRV to a microbiome study examining the relationship between host transcriptome and microbiome composition within the context of inflammatory bowel disease and are able to derive new biological insights and provide formal inference on prior qualitative observations.

1. Introduction

The human body is inhabited by many complex communities of microorganisms and their composition (defined as the microbiome) have been increasingly recognized to play an important role in many human disease conditions, including obesity (Turnbaugh et al., 2009), type 2 diabetes (Qin et al., 2012), and inflammatory bowel disease (Morgan et al., 2015). Recent advances in next-generation sequencing technologies now allow investigators to quantify the composition of the microbiome using direct DNA sequencing of the 16S ribosomal RNA gene (Lasken, 2012). Based on their sequence similarity, the raw 16S sequence reads are often clustered into Operational Taxonomic Units (OTUs), which is a commonly used microbial diversity unit and can be considered as surrogate of a bacterial taxon when clustered at 97% similarity level (Stackebrandt and Goebel, 1994). Many downstream analyses are performed based on the OTU abundances, among which a powerful mode of analysis is the community level analysis, wherein overall microbiome composition of multiple OTUs is assessed for identifying overall shifts among different conditions (Li, 2015). Community level analysis can be more powerful than examination of individual taxa when there are systematic, modest changes in abundance but individual taxa do not have a strong effect (Zhao et al., 2015; Plantinga et al., 2017).

Recently, there is considerable interest in understanding the relationship between overall microbiome composition and profiles of other types of genomic data. For example, Morgan et al. (2015) was interested in determining whether host gene expression profiles, overall and within specific candidate pathways, are globally related to microbiome composition in patients with inflammatory bowel disease. Unfortunately, how to systematically examine the relationship between high-dimensional microbiome compositional profiles and other high-dimensional gene expression data remains unclear. The authors resorted to associating individual gene expression and individual OTUs by using the top principal components, as well as making qualitative observations regarding relationships, in which no formal inference was conducted. It would be of considerable practical interest to devise a means for formal inference of hypothesis testing and for conducting more systematic association analysis.

Assessing overall association relationships between two sets of variables can be accomplished using a range of different methods. For example, the RV coefficient (Escoufier, 1973) provides insight into the global correlation between the two random vectors (e.g., a vector of microbiome profiles and a vector of gene expression values). However, as a generalization of the Pearson correlation coefficient, RV coefficient can only measure linear dependency. The high dimensionality of the data, the complexity of the relationships between data types, and inherent structure (e.g., phylogenetic relationships) among the taxa pose grand challenges for the RV coefficient. To accommodate general dependency patterns beyond linearity, one strategy is to incorporate distance metrics as in the GRV statistic (Minas, Curry, and Montana, 2013). Motivated by GRV, we map the original vector spaces to reproducing kernel Hilbert spaces (RKHSs) and consider kernel RV (KRV) coefficient as the RV coefficient between the RKHS-images of the two random vectors. It turns out that this KRV statistic is closely related to existing statistics that measure multivariate statistical independence, including the Hilbert-Schmidt independence criterion (HSIC) (Gretton et al., 2005, 2008) and distance covariance (Székely, Rizzo, and Bakirov, 2007).

Despite the correspondences of KRV with many existing multivariate dependency metrics, the testing design of these existing statistics do not fit the current microbiome association analysis. This is because current microbiome studies often have a relatively small sample size, while most existing multivariate dependency tests depend on asymptotic results (e.g., the HSIC test). Thus, a more accurate finite-sample null distribution is desired for a microbiome association test (Chen et al., 2016; Plantinga et al., 2017). To evaluate significance based on the KRV statistic, we adopt the GRV testing strategy (Minas et al., 2013), which approximates the empirical null distribution of all KRV permutations to a Pearson type III distribution by matching the first three moments. Since the empirical moments of the null KRV permutation distribution are easy to calculate based on previous results on RV-type statistics (Kazi-Aoual et al., 1995; Josse, Pagè, and Husson, 2008), parameters of the Pearson type III distribution can be explicitly expressed in closed form. Finally, the p-value of a KRV test can be calculated analytically using this approximated Pearson type III density. The new test design is well-suited for small-sample microbiome studies without using any asymptotic results.

Although we follow the GRV testing framework to examine the association between two vectors, there are key differences. The most important difference is that the proposed KRV test has been applied to a different domain. GRV tests for association between SNPs and gene expressions, where specific distance metrics for SNPs and gene expressions have been explored. In this paper, our major focus is kernel metrics for microbiome composition data. Beyond that, the KRV test also extends the GRV test in the following two aspects. First, the KRV test allows adjustment of confounding effect. Environmental exposures, clinical outcomes and treatment groups (all termed as covariates) are important in assessing the association between microbiome composition and host gene expression. It is possible that some covariates affect both the microbiome composition and gene expression. Under such a scenario, failure to account for these covariates can produce misleading bias of association or affect the testing power. Second, we propose an omnibus KRV test which can accommodate multiple candidate kernels, which is much more efficient than the permutation and meta analysis-based approach used in GRV to accommodate multiple distances. The choice of kernels in KRV is crucial for the success of the test. The optimal kernels with powerful KRV tests depends on both the specific data structures and the underlying association patterns, which however, are often unknown in practice. Without hacking p-values by selecting the most favorable kernels, we incorporate an omnibus procedure in KRV to accommodate multiple candidate kernels. The KRV test with this omnibus kernel is more robust in that it can always have adequate power under different scenarios. Finally, by approaching the problem from the perspective of kernels rather the distances, we are able to related the KRV to existing metrics of generalized statistical dependence to better understand properties.

The rest of the paper is organized as follows. In Section 2, we first introduce the KRV statistic and explore its connection with many existing statistics for multivariate association analysis. Then, we utilize existing testing strategy in RV-type statistics to evaluate significance based on KRV statistic. Next, we carefully adapt the KRV test to microbiome association analysis by enabling covariates adjustment as well as accommodating multiple OTU kernels in Section 3. The finite sample performance of the proposed KRV test both in testing statistical independence and microbiome association is assessed through numerical studies in Section 4. In Section 5, we apply the KRV test to the dataset of Morgan et al. (2015) examining the relation between host transcriptome and microbiome composition in samples taken from inflammatory bowel disease patients. Our analysis is able to provide additional insights. The paper concludes with a brief discussion in Section 6.

2. A KRV-based Fast Small-sample Kernel Independence Test

RV coefficient (Escoufier, 1973) was developed as a measure of linear correlation between sets of multivariate measurements collected on the same individuals. In particular, let X be an n × p matrix (of variables X¹, . . . , X^p) and Y be an n × q matrix (of variables Y¹, . . . , Y^q), corresponding to two sets of variables, such as gene expression values and OTU counts observed from the same n individuals. Then, RV coefficient between X and Y is defined as

$$RV(X,Y):=\frac{tr(S_{XY}{S}_{YX})}{\sqrt{tr(S_{XX}^2)tr(S_{YY}^2)}},$$ (1)

where S_XX = X′X/(n − 1), S_YY = Y′Y/(n − 1), S_XY = X′Y/(n − 1), S_YX = Y′X/(n − 1) are sample covariance matrices, given that X and Y are centered by columns.

A notable feature of RV coefficient is that it is only able to capture the linear dependency between two random vectors and does not accommodate nonlinearity or other more general dependencies (Robert and Escoufier, 1976). In practice, complex data such as microbiome and host genome data, often require general methods to detect more general dependencies that are of interest. Motivated by this, we propose the KRV coefficient to measure more general relationship between microbiome composition and host genome expression. Specifically, we kernelize the RV coefficient by embedding the original spaces 𝒳 and 𝒴 to some functional spaces spanned by kernels (Hofmann, Schölkopf, and Smola, 2008). Let k(·, ·) : 𝒳 × 𝒳 ↦ R and l(·, ·) : 𝒴 × 𝒴 ↦ R. be two kernel functions. Then, the KRV coefficient is proposed as

$$\mathit{KRV}(X,Y):=\frac{tr(\stackrel{\sim }{K}\stackrel{\sim }{L})}{\sqrt{tr(\stackrel{\sim }{K}\stackrel{\sim }{K})}\sqrt{tr(\stackrel{\sim }{L}\stackrel{\sim }{L})}},$$ (2)

where K̃ = HKH and L̃ = HLH. K and L are two n × n kernel matrices, where K_ij = k(X_i, X_j), L_ij = l(Y_i, Y_j), i, j = 1, . . . , n, H = I − 11′/n is a centering matrix, I is an identity matrix of order n, and 1 is a n × 1 vector of all ones. A sketch of calculating the KRV coefficient is included in Section A.1 of Web Appendix A.

If the kernel matrices are selected as K = XX′ and L = YY′, then the KRV coefficient reduces to the RV coefficient. If we replace the two kernel matrices K and L by two distance matrices, then KRV reduces to a GRV coefficient. Beyond its close connection with RV-type statistics, KRV is also similar to some other statistics. In particular, the numerator of KRV is simply the HSIC statistic tr(K̃L̃) (Gretton et al., 2005, 2008), which has been widely used to characterize statistical independence. Thus, given the kernels being appropriately chosen (Gretton et al., 2005), the KRV statistic can also be used to characterize independence. Such a property, however, has never been studied for other RV-type statistics (Josse et al., 2008; Minas et al., 2013). Similar to the HSIC statistic, distance covariance/correlation (Székely et al., 2007) is also widely used for measuring and testing independence between two groups of variables. It has been shown that distance covariance is equivalent to HSIC (Sejdinovic et al., 2013). In this spirit, KRV is equivalent to distance correlation.

Besides the HSIC statistic and distance covariance statistic, many other statistics have been proposed to measure generalized dependency. Readers are referred to Josse and Holmes (2013) and references therein for further details. Finally, it turns out that our KRV statistic coincides with some existing statistics including the RV for kernels (Purdom, 2006) and the centered kernel alignment statistic (Cortes, Mohri, and Rostamizadeh, 2012). However, the RV for kernels is used for kernel principal component analysis and kernel canonical correlation analysis, and the centered kernel alignment statistic is used to develop algorithms for learning kernels for classification and regression. Both RV for kernels and centered kernel alignment statistic have not been used for hypothesis testing, which is the focus of the current paper.

Despite the correspondences of KRV with HSIC and distance covariance, the design of the HSIC test (based on asymptotic results) and the distance covariance test (permutations) are often limited. In particular, asymptotic null distribution-based HSIC test is not appropriate for studies with small sample size, such as the micriobiome study considered in this paper. On the other hand, a permutation test of distance covariance can be computationally expensive when the nominal significance level is stringent. Thus, a new fast small-sample independence test based on the KRV statistic is necessary.

The distribution of the KRV statistic is generally unknown due to its complex form. A reasonable strategy is to use permutations. Unlike the permutation-based distance covariance test, we utilize permutations differently. To avoid the computational burden of explicitly resampling and recalculating permuted KRV statistics, we follow testing strategy of existing RV-type statistics (Josse et al., 2008; Minas et al., 2013), to approximate the empirical null distribution of KRV permutations by moment-matching. Specifically, let Q_i, i = 1, . . . , n! denote the KRV statistics calculated from all n! potential permutations by shuffling rows and columns of one kernel matrix simultaneously. The first three sample moments of {Q₁, . . . , Q_n!} are calculated and a Pearson type III density with the same first three moments is obtained. The final p-value is calculated from this approximated Pearson type III density. More details of the Pearson type III approximation can be found in Section A.2 and A.3 of Web Appendix A.

3. Adapting KRV for Microbiome Association Analysis

In this section, we tailor the KRV framework to facilitate the microbiome association analysis with host gene expression data mainly considered in this paper.

3.1 Kernel Choice

To evaluate the association between microbiome composition and host gene expressions via the KRV test, we first need to select kernels in KRV for both microbiome composition data and gene expression data. In many kernel-based genetic association tests, kernels are used as similarity measures, and concordance between genotype similarity and phenotype similarity is suggestive of association (Wu et al., 2011; Broadaway et al., 2016). Similarly, we treat K_ij and L_ij in KRV as similarity measures of sample i and j in terms of their microbiome composition profiles and host genomic expression profiles, respectively. The KRV statistic tends to be large if one similarity matrix resembles to the other. That is, concordance in microbiome similarity and host genome similarity is suggestive of association.

More rigorously, kernel matrices K and L need to be positive semi-definite so that the KRV statistic (2) is well-defined. Constructing positive semi-definite kernels for association analysis is a common practice for many different omics data types (Wu et al., 2011; Zhan et al., 2015; Zhao et al., 2015; Zhan et al., 2016). For the microbiome composition data considered in this paper, the UniFrac kernels are ecologically meaningful similarity metrics and can accommodate important features of OTU data, e.g. the phylogenetic structure (Lozupone and Knight, 2005; Lozupone et al., 2007; Chen et al., 2012). The UniFrac-type kernels quantify the similarity of two OTU profiles by incorporating both their abundance (or presence/absence) information and phylogenetic relationship. Besides the UniFrac kernels, the Bray-Curtis kernel is also widely used, which quantifies similarity of two microbial communities based on the OTU counts and can be useful when the phylogenetic tree information is unavailable and unreliable. For host gene expression data, some popular choice are the Gaussian kernel (K_ij = k(x_i, x_j) = exp(−||x_i − x_j||²/σ²)) and linear kernel (K = XX′) (Liu, Lin, and Ghosh, 2007). To account for correlation among gene expressions, the weighted linear kernel ( $K=X{\mathrm{\sum }}_{XX}^{-1}{X}^{\prime }$) can also be used (Broadaway et al., 2016).

3.2 Accommodating Multiple Kernels

The choice of kernels in KRV is crucial for the success of the test. Different kernels measure different aspects of data nature and assume different association patterns. Unfortunately, selecting the most powerful OTU (or gene expression) kernel requires both knowledge of the microbiome community structure and how the microbiome influences gene expression. Without such prior knowledge, it is necessary to develop an omnibus test which incorporates multiple candidate kernels. In GRV (Minas et al., 2013), a similar multiple candidate distances issue is solved by meta-analysis for different combinations of distances. P-values from all possible distance combinations are used to calculate the Fisher summary statistic, and permutations are used to establish the significance based on the Fisher summary statistic. The adjustment of multiple distances in GRV is often computationally inefficient due to the need of extra datasets for meta-analysis and also permutations for final p-value calculation.

To avoid potential limitations of GRV, we propose to combine the multiple candidates at the kernel level in KRV rather than the test p-value level as in GRV. Without loss of generality, suppose k_i, i = 1, . . . , m are candidate OTU kernels, with corresponding kernel matrices K_i, i = 1, . . . , m, and we fix the gene expression kernel l or L. The same omnibus OTU kernel strategy can be applied to accommodate multiple gene expression kernels. Motivated by existing literature in multiple kernel learning (Cortes et al., 2012) and genetic association studies (Wu et al., 2013), we propose to use an omnibus OTU kernel of the form $K_{om}={\sum }_{i=1}^m{\omega }_i{K}_i$ with ω_i ≥ 0 and ${\sum }_{i=1}^m{\omega }_i=C$. Since the KRV statistic is scale invariant, constant C in the constraint ${\sum }_{i=1}^m{\omega }_i=C$ does not make a real difference. There are many methods to determine the weights ω_i, i = 1, . . . , m. The simplest strategy is to use unsupervised weights such as $K_{om1}={\sum }_{i=1}^m{K}_i/m$ and $K_{om2}={\sum }_{i=1}^m{K}_i/tr(K_i)$. An advantage of K_om1 and K_om2 is that a direct KRV test between K_om and L can be used to establish the final significance. Another more complicated way to select the weights in a supervised way. For example, Cortes et al. (2012) suggest to select the weights that maximize the KRV statistic between the omnibus OTU kernel and gene expression kernel:

$$\mathit{KRV}(K_{om},L)=\frac{{\sum }_{i=1}^{m}tr({\omega }_i{K}_{i}L)}{\sqrt{{\sum }_{i=1}^m{\sum }_{j=1}^{m}tr({\omega }_i{K}_i{\omega }_j{K}_j)}\sqrt{tr(L^2)}},$$ (3)

subjected to ω_i ≥ 0 and ${\sum }_{i=1}^m{\omega }_i=1$. The optimal weights ${\mathit{\omega }}^{\ast }={({\omega }_1^{\ast },\dots ,{\omega }_m^{\ast })}^{\prime }$ can be calculated by a Quadratic Programming (QP) algorithm (Cortes et al., 2012). As a consequence of supervised weights learning, p-value of the test KRV (K_om3, L), where $K_{om3}={\sum }_{i=1}^m{\omega }_i^{\ast }{K}_i$, is no longer a genuine p-value. Permutations are needed for establish the significance of the test based on K_om3. Finally, Wu et al. (2013) suggest to select the individual kernel with the minimum p-value. That is, K_om4 = K_i, where K_i has the smallest KRV p-value among K₁, . . . K_m. Like K_om3, a permutation-based procedure is needed to establish the significance between K_om4 and L. More details on K_om3 and K_om4 including the permutation-based p-value calculation procedures along with comprehensive numerical studies comparing K_om1, K_om2, K_om3 and K_om4 are presented in Section B.1 of Web Appendix B. Based on our numerical studies, it turns out that the omnibus kernel K_om2 with unsupervised weights ω_i = 1/tr(K_i) tends to have the best overall performance under most scenarios, and thus is used as the omnibus kernel in the rest of this paper.

3.3 Adjusting for Confounders

It is important to adjust for the effect of confounding variables when testing association. Let Y and Z denote host gene expression and microbiome composition respectively, X denote some covariates, such as age, gender, smoking status and other clinical or environmental variables, which may influence both host gene expression and microbial community diversity. Without adjusting for covariate effects, the association testing results between Y and Z can be misleading, sometimes leads to excessive false positive discoveries. To adjust for the potential confounding effects of X in KRV framework, we utilize the residual-based strategy as widely used in many kernel machine association tests (Liu et al., 2007; Hua and Ghosh, 2015; Broadaway et al., 2016). Let P_X = X(X′X)⁻¹X′ denote the projection matrix of the column space of X, and denote the residuals Ỹ = (I − P_X)Y. Then we can calculate the residual kernel as ${\stackrel{\sim }{L}}_{ij}^r=l({\stackrel{\sim }{Y}}_i,{\stackrel{\sim }{Y}}_j)$. Finally, we replace L̃ in equation (2) by L̃^r to calculate the statistic and conduct the test after adjusting for X. In the univariate scenario (dim(Y)=1) of kernel machine regression, the above procedure is equivalent to testing the association using a restricted maximum likelihood (REML)-based score test (Liu et al., 2007).

4. Simulation Studies

4.1 Statistical Independence Simulation

We first conducted simulations to evaluate the performance of the proposed KRV test in testing statistical independence. We compared our KRV test to the HSIC test and distance covariance (dcov) test, both of which have been widely used for testing statistical independence between two random vectors. As a benchmark, we also compared the GRV test, which has the same test design as the KRV test but uses distance metrics rather than kernels. The setup of this simulation was exactly the same as that in the dcov test paper (Székely et al., 2007). Two continuous random vectors X and Y were simulated, where p = dim(X) = dim(Y ) = 5, and the marginal distribution of each dimension of X and Y was standard normal. The following four scenarios (A) – (D) were used to simulate the data:

1. Cov(X_i, Y_j) = 0, for i, j = 1, . . . , p, and Cov(X_i, X_j) = 0, Cov(Y_i, Y_j) = 0 for any i ∉ j.

2. Cov(X_i, Y_j) = 0.1, for i, j = 1, . . . , p, and Cov(X_i, X_j) = 0.1, Cov(Y_i, Y_j) = 0.1 for any i ∉ j.

3. Y_ij = X_ijε_ij, i = 1, . . . n; j = 1, . . . p, where ε_ij are independent standard normal random variables independent of X.

4. $Y_{ij}=log(X_{ij}^2)$, i = 1, . . . n; j = 1, . . . p.

The empirical type I error rates were evaluated when generating data under scenario (A), and the empirical powers were assessed under scenarios (B), (C) and (D). Under each scenario, N = 10000 datasets were simulated with varied sample sizes n = {20, 40, 60, 80, 100}. For the KRV test and HSIC test, we applied the Gaussian kernel to both X and Y to test independence (Gretton et al., 2008). That is, k(X₁, X₂) = l(X₁, X₂) = exp{−||X₁ − X₂||²/σ²}, where ||X₁ − X₂||² is the Euclidean distance between X₁ and X₂, σ² is the shape parameter which was selected as the median of the Euclidean distance between each sample pair. The design of the HSIC test is different from the KRV test. The asymptotic null distribution of HSIC statistic is characterized as ${\sum }_{i=1}^n{\sum }_{j=1}^n{\lambda }_i{\mu }_j{\chi }_1^2$, where λ_i, μ_j are eigenvalues of kernel matrices K and L respectively. More details can be found in Sejdinovic et al. (2013). For the GRV test, Euclidean, Manhattan and Mahalanobis distance have been proposed for continuous variables (Minas et al., 2013). For simplicity, we selected both Euclidean distances for X and Y in GRV test (GRV results with Manhattan and Mahalanobis distance are qualitatively similar). Finally, B = 10000 permutations were used in the dcov test (Székely et al., 2007). The nominal significance level was set at α = 0.05 and the testing results are reported in Figure 1.

Figure 1.

Empirical type I error/power of KRV, GRV, HSIC and dcov test. Scenario (A) is for type I error and Scenario (B)–(D) are for powers under different alternative models. Symbols ○, △, + and × represent KRV, GRV, HSIC and dcov respectively.

Under scenario (A), KRV, GRV and dcov test have correct type I error. The 95% CI of type I error is $0.05\pm 1.96\sqrt{0.05·0.95/10000}=[0.0457,0.0543]$, which are represented as dash lines in the top-left panel of Figure 1. Clearly, the HSIC test is outside this CI and is extremely conservative especially when sample size is small. This small-sample conservativeness has been observed for other kernel-based association test statistics (Chen et al., 2016). Under scenario (B), GRV and dcov are more powerful than KRV and HSIC. The dependence between X and Y under Scenario (B) is fully described by the Pearson correlation (Cov(X_i, Y_j) = 0.1, i, j = 1, . . . 5), and the Gaussian kernels as applied in KRV and HSIC are less sensitive to such a linear dependency pattern than the Euclidean distances implemented in GRV. The dependency between X and Y under scenario (C) is linear but with random coefficient. KRV and HSIC are more powerful than GRV under this scenario. Finally, there is a nonlinear dependency between X and Y under scenario (D). Since the dependency is purely deterministic, KRV, HSIC and docv is extremely powerful under this scenario. On the other hand, GRV with Euclidean distances fails to detect such a nonlinear dependency in the sense that it has a power close to the nominal type I error rate. GRV tests with other distances (such as Manhattan and Mahalanobis distance) can have improved power, which however, is still less powerful than KRV (data not shown).

To summarize, KRV test is powerful in detecting any kind of departure from statistical independence under each scenario given the kernels are appropriately chosen, such as Gaussian kernels (Gretton et al., 2008). Depending on the distances being used, GRV test can be powerful in detecting certain kind of dependency patterns. However, it is not clear, under what conditions/distances, GRV is able to capture any general dependency patterns among two random vectors. HSIC seems to be as powerful as KRV when the sample size is large. However, it is clear that HSIC is conservative when sample size is relatively small. The permutation-based dcov test tends to be slightly less powerful than KRV (except for Scenario (B)) and always has adequate power to detect any dependencies. However, the computational cost of dcov can be expensive if required number of permutations is large (e.g., for stringent significance levels).

4.2 Microbiome Association Simulation

We also conducted simulation studies to evaluate the performance of KRV in testing micro-biome association. We first generated the microbiome composition data which was reflective of real OTU counts in a upper-respiratory-tract microbiome dataset (Charlson et al., 2010). A total of 856 OTUs were simulated and were further partitioned into 20 clusters using the partitioning around medoids algorithm. Finally, we selected a relatively abundant cluster (denoted by 𝒜) as the one which affected the outcomes. After the OTU counts Z_ij, i = 1, . . . n, j = 1, . . . 856 were generated, we simulated q host gene expressions from

$$y_{it}=0.5X_{i1}+0.5X_{i2}+{\beta }_t·\mathit{scale}(\sum _{j\in \mathcal{A}}{Z}_{ij})+{\epsilon }_{it},i=1,\dots ,n,t=1,\dots ,q,$$ (4)

where X_i1, X_i2 are covariates such as age, gender and smoking status, which may also be related to the microbiome composition. In particular, two different ways of simulating covariates were considered. In the first scenario, the covariates were independent of OTUs, and simulated as X_i1 ~ Bernoulli(0.5), X_i2 ~ N (0, 1). In the second scenario, we simulated X_i2 as N (0, 1) + 0.4 · scale(Σ_j∈𝒜 Z_ij), which was related to the microbiome composition. The scale(·) function standardized the sum of OTU counts in cluster 𝒜. The error terms ε_ik are independent and identically distributed as normal with mean zero and covariance matrix Σ(ρ), where Σ(ρ) is compound symmetry covariance matrix with ρ = 0.2, 0.8 representing low and high correlation among gene expressions respectively. We simulated n = 200 samples and p = 30 gene expressions to mimic a mid-size pathway as analyzed in a real data example later in this paper. Under the null model, all β_t = 0 and 10000 datasets were simulated to evaluate type I error. Two different alternative models were considered. One was the sparse-association model, where only 20% of the gene expressions are related to OTUs. In particular, we set β_t = 0.5 for t = 1, . . . q^*(= 0.2q), and zero elsewhere. The other is the dense-association model, where β_t = 0.5 for t = 1, . . . q^*(= 0.5q), and zero elsewhere. Under both alternative models, we generated 1000 datasets to assess the power.

To test the association between the simulated microbiome composition and gene expressions data, six different methods were applied including KRV test, GRV test, Gene Association with Multiple Traits (GAMuT) test (Broadaway et al., 2016), Multi-trait Sequence Kernel Association Test (MSKAT) (Wu and Pankow, 2016), Multivariate MiRKAT (MMiRKAT) (Zhan et al., 2017) and the marginal MiRKAT (Zhao et al., 2015). GAMuT uses the same design of HSIC test in previous simulation (Broadaway et al., 2016). MSKAT combine multiple marginal score test statistic through the covariance matrix of all scores and also calculates its p-value asymptotically (Wu and Pankow, 2016). MMiRKAT incorporates a small-sample adjustment to a MSKAT-type test so that the test has a better finite-sample behavior (Zhan et al., 2017). Finally, the marginal MiRKAT tests the association between one gene expression and OTUs each time followed by Bonferroni correction to the minimum p-value, and we term it as minP for simplicity in the rest of this paper.

We first selected the OTU kernels as used in all six tests. For a little abuse of notation, in this section, we simply use the term kernels for distances when the test is GRV. The weighted UniFrac kernel, unweighted UniFrac kernel, generalized UniFrac kernel with parameter θ = 0.5 and the Bray-Curtis kernel were considered (Zhao et al., 2015). We denote these kernels as K_w, K_u, K_0.5 and K_BC respectively. Then, the omnibus OTU kernel K_om = K_w/tr(K_w)+K_u/tr(K_u)+K_0.5/tr(K_0.5)+K_BC/tr(K_BC) was also calculated and applied in all six tests. For the gene expression data, the Gaussian kernel-based KRV/GAMuT is shown to be robust in the previous continuous variables simulation in Section 4.1. To capture the correlation among gene expression, the weighted linear kernel $L=Y{\widehat{\mathrm{\sum }}}_{YY}^{-1}{Y}^{\prime }$ is often shown to be useful (Broadaway et al., 2016). Based on the results of Section B.1 in Web Appendix B, we selected the gene expression kernel in KRV and GAMuT as G/tr(G) + L/tr(L). On the other hand, the Euclidean distance, Manhattan distance and Mahalanobis distance are recommended in the GRV test (Minas et al., 2013). The Mahalanobis distance tends to be powerful when outcome correlation is high while the other two distances are more powerful with weakly correlated outcomes. An omnibus distance to accommodate three distances was used. Since the trace of a distance matrix is zero, we simply used an average distance matrix of the three in the GRV test.

The empirical type I errors are reported in Table 1. Based on the table, KRV and GRV always have correct type I error under each scenario. GAMuT and MSKAT tend to be very conservative under each scenario, which is also observed in Section 4.1 and other studies (Zhan et al., 2017). This is because the asymptotic p-value calculation in GAMuT and MSKAT work for large-sample genetic association studies, and tends to be conservative with small samples due to estimation error in variance terms (Chen et al., 2016). The small-sample adjustment incorporated in MMiRKAT usually works well with low-dimensional outcomes (Zhan et al., 2017). However, MMiRKAT seems to be a little conservative in this simulation with p = 30 outcomes. Finally, minP has correct type I error when outcomes are weakly correlated (ρ = 0.2) and is very conservative when outcomes are highly correlated (ρ = 0.8). This is due to the conservativeness of the Bonferroni correction when individual tests are highly correlated. The type I errors of all tests with dependent (X,Z) scenario are similar and reported in Table S2 in Section B.2 of Web Appendix B.

Table 1.

Empirical type I error of KRV, GRV, GAMuT, MSKAT, MMiRKAT and minP at nominal level α = 0.05.

Test	ρ = 0.2					ρ = 0.8
	Kw	Ku	K0.5	KBC	Kom	Kw	Ku	K0.5	KBC	Kom
KRV	0.0493	0.0512	0.0499	0.0483	0.0500	0.0498	0.0504	0.0527	0.0510	0.0534
GRV	0.0455	0.0475	0.0476	0.0471	0.0452	0.0492	0.0497	0.0489	0.0545	0.0519
GAMuT	0.0271	0.0182	0.0158	0.0207	0.0130	0.0330	0.0200	0.0186	0.0229	0.0176
MSKAT	0.0349	0.0238	0.0254	0.0256	0.0258	0.0341	0.0227	0.0254	0.0278	0.0257
MMiRKAT	0.0383	0.0367	0.0381	0.0350	0.0360	0.0360	0.0367	0.0379	0.0390	0.0380
minP	0.0479	0.0454	0.0491	0.0434	0.0485	0.0188	0.0212	0.0220	0.0206	0.0201

The empirical powers are reported in Table 2. We first compare the performance of each test with different OTU kernels. Data generated in this simulation have two features. First, the simulated OTUs are phylogenetically related, and reflect a real upper-respiratory-tract microbiome data. Second, based on simulation model (4), the outcomes are affected by the abundance of OTUs (i.e. Z_ij), rather than the presence/absence of OTU (i.e. I[Z_ij > 0]). Given these facts, K_w and K_0.5 consider both phylogeny and abundance information, and hence are more powerful. On the other hand, K_u ignores the abundance information and K_BC ignores the phylogeny information, hence are less powerful. Finally, one can see that tests based on omnibus OTU kernel are quite robust. Under each scenario, the omnibus tests are slightly less powerful than the best test but much more powerful than the worst one.

Table 2.

Empirical power of KRV, GRV, GAMuT, MSKAT, MMiRKAT and minP at nominal level α = 0.05.

q*	Test	ρ = 0.2					ρ = 0.8
		Kw	Ku	K0.5	KBC	Kom	Kw	Ku	K0.5	KBC	Kom
6	KRV	0.784	0.084	0.718	0.345	0.677	0.856	0.066	0.804	0.403	0.759
	GRV	0.809	0.080	0.767	0.387	0.614	0.166	0.063	0.157	0.102	0.133
	GAMuT	0.706	0.032	0.532	0.203	0.475	0.803	0.031	0.637	0.275	0.550
	MSKAT	0.277	0.037	0.420	0.116	0.307	0.474	0.037	0.672	0.173	0.522
	MMiRKAT	0.546	0.063	0.458	0.185	0.424	0.799	0.061	0.688	0.333	0.651
	minP	0.834	0.068	0.913	0.381	0.828	0.610	0.036	0.684	0.234	0.601
15	KRV	0.978	0.086	0.946	0.579	0.935	0.969	0.096	0.951	0.603	0.925
	GRV	1.000	0.123	0.999	0.878	0.991	0.574	0.059	0.531	0.261	0.413
	GAMuT	0.963	0.038	0.886	0.439	0.817	0.960	0.034	0.870	0.441	0.827
	MSKAT	0.336	0.027	0.525	0.129	0.357	0.488	0.038	0.737	0.212	0.579
	MMiRKAT	0.662	0.054	0.548	0.250	0.510	0.844	0.066	0.772	0.357	0.729
	minP	0.971	0.081	0.991	0.593	0.971	0.697	0.033	0.772	0.313	0.684

Next, we compare the power of different tests. We first compare four kernel-based multivariate association tests: KRV, GAMuT, MSKAT and MMiRKAT. Both KRV and GAMuT gain additional power by utilizing an additional kernel to model the structures in gene expression data. Also, as observed in Table 1, GAMuT, MSKAT and MMiRKAT are more or less conservative under small sample size. These two facts explain that KRV is consistently more powerful than GAMuT, MSKAT and MMiRKAT in Table 2. Next, we compare KRV and GRV. Under ρ = 0.2, GRV is slightly more powerful than KRV. However, KRV is much more powerful than GRV under ρ = 0.8 especially when q^* = 6, where the power of KRV and GRV are 0.856 and 0.166 respectively. We also tried other GRV tests. For example, Mahalanobis distance-based GRV has improved power under ρ = 0.8 but has much lower power than KRV under ρ = 0.2. Similar to previous simulations in Section 4.1, the Gaussian kernel in KRV is often robust to capture general relationship while it is not clear which distance in GRV can achieve such goals. Finally, the comparison between KRV and minP is simple. Under low correlation and sparse signal, minP is slightly more powerful. However, under other scenarios, the association signal can be largely amplified by collectively analyzing multiple outcomes and thus KRV can be much more powerful than minP. The powers of all tests with dependent (X,Z) scenario are similar and reported in Table S3 in Section B.2 of Web Appendix B.

To conclude, there is no uniform most powerful multivariate association test in our simulations. Unlike other methods, which suffer from huge power loss under certain scenarios, the proposed KRV test is always one of the most powerful method in testing the association between OTUs and gene expressions, and always has an adequate power under each scenario.

5. Analysis of host transcriptome and microbiome data

We further applied the KRV test to a dataset from an inflammatory bowel disease (IBD) study (Morgan et al., 2015), which examines how host transcriptome interacts with mi-crobiome in the pathogenesis of IBD. Paired host transcriptome and microbial metagenome data were collected from 255 samples, among which 196 were pre-pouch ileum (PPI) samples and 59 were pouch samples. For each sample, 19908 host transcript expressions and 7000 OTU counts were measured by microarray and 16S rRNA analysis respectively (Morgan et al., 2015). Besides host gene expression and microbiome composition, three additional covariates are available: antibiotic use (yes/no), inflammation score (0–13), and disease outcome (familial adenomatous polyposis or not). Due to heterogeneity reasons, only the 196 PPI samples were used to test the association between host transcriptome and microbiome (Morgan et al., 2015). In particular, a linear model was applied to test the association between each individual transcript and each individual OTU after accounting for the covariates. To reduce multiple testing burden and improve statistical power, principal component analysis (PCA) was applied to the 19908 host transcripts and 7000 OTUs for dimension reduction. The top 9 host PCs (which explain 50% variance in host transcripts) and the top 9 clade PCs (which explain 50% variance in OTUs) were included in individual association analysis, where one host PC and one clade PC is tested for association each time. Finally, after multiple testing adjustment, significant associations between host PCs and clade PCs can be detected at a false discovery rate (FDR) of 0.25. The authors also noted enrichment of microbiome-associated host transcript patterns within the interleukin-12 (IL12) pathway, but no formal statistical testing results were reported (Morgan et al., 2015).

Alternatively to the individual PC based association analysis implemented in the original study, we jointly tested the association between host gene expressions (either the whole transcriptome or within a certain pathway as IL12) and all 7000 OTUs using all six methods as illustrated in simulation studies. Besides the whole transcriptome and IL12 pathway, we also analyzed two additional pathways. One is Inflammatory mediator regulation of TRP channels pathway (KEGG: hsa04750), and the other is IBD pathway (KEGG:hsa05321). These two pathways are either related to the underlying biological process or related to the disease itself, hence can be of interest. To be consistent with the original studies (Morgan et al., 2015), only the 196 PPI samples were used in our analysis.

For the OTU data, the Bray-Curtis kernel can be directly calculated from the counts, and the phylogenetic tree needs to be first trained for calculating UniFrac-type kernels. Specifically, PyNAST (Caporaso et al., 2010) was used to generate a multiple sequence alignment from the representative OTU sequences identified in the original study. Of the 7000 available OTU sequences, 1646 could not be aligned and were excluded from the phylogenetic tree. A phylogenetic tree relating the remaining 5354 OTUs was produced using FastTree (Price, Dehal, and DehalArkin, 2009). The unweighted, weighted, and generalized UniFrac distances/kernels were calculated using this tree. The same kernel/distance for gene expression data as in Section 4.2 were used in this real data application. For the whole transcriptome, which contains too many genes (p = 19908 > n = 196) such that Σ̂ is not invertible. Thus we simply used the Gaussian kernel in KRV, GRV and GAMuT, and Σ̂⁻¹-based MSKAT and MMiRKAT are not evaluated under this scenario.

The testing results are reported in Table 3. For the overall association between microbiome composition and all 19908 genes in the whole transcriptome, KRV, GRV and GAMuT are all highly significant while minP is not, probably due to the heavy multiple testing correction burden. Compared with the claimed significance at FDR=0.25 of the original individual analysis, our KRV test is much more powerful detecting associations since it can amplify the marginal association signal by analyzing both OTUs and gene expressions collectively.

Table 3.

P-values of different tests examining the host-microbiome association in the real data. The whole transcriptome contains all 19908 genes, IL12 pathway contains 21 genes, Inflammatory pathway contains 96 genes, and IBD pathway has 62 genes.

Pathway	Test	Kw	Ku	K0.5	KBC	Kom
Whole transcriptome	KRV	0.0011	0.0002	0.0003	0.0014	0.0002
	GRV	0.0055	0.0003	0.0015	0.0024	0.0012
	GAMuT	0.0015	0.0006	0.0026	0.0029	0.0005
	minP	1.0000	1.0000	1.0000	1.0000	1.0000
IL12	KRV	0.0010	0.0004	0.0004	0.0014	0.0003
	GRV	0.0040	0.0003	0.0011	0.0021	0.0009
	GAMuT	0.0017	0.0011	0.0009	0.0029	0.0007
	MSKAT	0.1931	0.5000	0.3024	0.1739	0.2105
	MMiRKAT	0.1744	0.4376	0.3295	0.1674	0.2184
	minP	0.0759	0.0060	0.0237	0.0448	0.0164
Inflammatory	KRV	0.0013	0.0003	0.0003	0.0015	0.0003
	GRV	0.0042	0.0002	0.0011	0.0020	0.0009
	GAMuT	0.0018	0.0008	0.0007	0.0029	0.0007
	MSKAT	0.6772	0.5859	0.8096	0.3337	0.6921
	MMiRKAT	0.6288	0.7016	0.7127	0.4383	0.6475
	minP	0.3236	0.0189	0.0974	0.1207	0.0602
IBD	KRV	0.0015	0.0002	0.0004	0.0016	0.0003
	GRV	0.0041	0.0002	0.0011	0.0021	0.0009
	GAMuT	0.0022	0.0007	0.0008	0.0032	0.0007
	MSKAT	0.8046	0.3658	0.6958	0.4789	0.6711
	MMiRKAT	0.7286	0.4402	0.6199	0.4788	0.6248
	minP	0.2090	0.0079	0.0502	0.0698	0.0357

For the IL12 pathway, KRV, GRV, GAMuT and minP (except for K_w) are significant at α = 0.05 level, which are consistent with findings of the original study stating that microbiome-associated host genome PCs were enriched in IL12 pathway (Morgan et al., 2015). Thus, formal statistical inference by KRV and other methods provides support for previous scientific observations. Compared with MSKAT and MMiRKAT, the additional gene expression kernel in KRV boosts its power of detecting associations. For the other two pathways (Inflammatory and IBD), KRV, GRV, and GAMuT are significant while MSKAT, MMiRKAT and minP mostly fail to detect any significance at α = 0.05 level except for K_u-based minP. Among all tests, KRV seems to be most powerful in that it always has the smallest p-value under each scenario.

To summarize, the association between individual host transcript and microbiome seems to be weak and complicated. KRV can amplify the association signal by collectively analyzing multiple OTUs and multiple genes, which is more powerful than the original PC-based individual association analysis. The usage of an additional kernel modeling structures and capturing general relationship, along with the fast and robust p-value calculation make KRV more powerful than other methods.

6. Discussion

In this paper, we consider the problem of associating overall microbiome composition with host genomics and propose the KRV test, which can both adjust for confounder effect and accommodate multiple candidate kernels reflecting different data structures or association patterns. As shown in the simulation studies, the proposed KRV test has correct size and can have substantially higher power than existing similar tests in many scenarios. Moreover, KRV testing results on the host-microbiome data not only provides formal statistical inference to support original conclusion (Morgan et al., 2015), but also is able to facilitate microbiome community level analysis and provide additional insights on some other related pathways.

One major contribution of this paper is that we largely adapted the existing GRV test in the microbiome association analysis framework, making it better suited to the host genome-microbiome association problem considered in this paper. KRV extends GRV in the following aspects. First, by applying kernels, KRV is able to capture both more complicated data structure (i.e., the phylogenetic structure inherent to microbiome data) and more general dependencies between two sets of variables. Second, we further extend the GRV test in a comprehensive association testing framework. KRV can adjust for confounder effect, which is important yet has never been discussed in the GRV test. Furthermore, we propose an omnibus KRV test based on a linear combinations of multiple candidate kernels, which is computationally much more efficient than the way GRV accommodates multiple distances. The omnibus KRV test is robust against the underlying data structures and association patterns. Due to these differences, we think that KRV not only can coexist with the existing GRV test but also can provide beneficial complements to GRV. Another contribution of this paper is that the KRV test provides an important complement to existing statistical independence tests (Székely et al., 2007; Gretton et al., 2008) by providing an efficient test design which neither relies on large samples nor requires permutations. The approximated Pearson type III distribution of the KRV statistic may also shed light on the finite-sample distribution of other statistics such as HSIC and distance covariance.

The proposed KRV in this paper is mainly aimed at microbiome association analysis, however, application of KRV can be beyond this aim. The proposed KRV test can also be useful in other domains due to the following reasons. First, KRV is extremely flexible. X or Y considered in KRV can be either a single variable or a high-dimensional vector. Moreover, its good finite-sample performance makes it an ideal tool for those studies with relatively small sample size, such as metabolomics and proteomics (Zhan et al., 2015). Second, the application of kernels enables KRV to capture structured data types, such as networks, shapes and images as long as appropriate kernels are designed. We leave these to future investigations.