1 Introduction
Consider the following scenario: your research team received a small grant to study the relationship between anatomical brain features and a set of clinical phenotypes. For the past several months, you’ve been painstakingly enrolling patients in the study and collecting neuroimaging data on them. The subjectmatter expert on your team has already grouped the millions of voxels into regions of interest. A given region can now be represented by a dataset of dimension , for patients, and the clinical phenotypes by a dataset of dimension . Prior to the study, you had identified Canonical Correlation Analysis (CCA) as a potential analytical tool; it would allow the extraction of maximally correlated components from both datasets, and the overall relationship between and could be summarised with a series of canonical correlations. In the classical lowdimensional setting (), you could also test for significance of these canonical correlations using Rao’s statistic (Mardia et al., 1979). However, despite your colleague’s excellent work, most brain regions still contain more features or measurements than your overall sample size (
); in other words, you are no longer in the classical inference setting, but in a highdimensional setting. Building on your knowledge of linear algebra and matrix decomposition, you know that you can still extract the canonical components and canonical correlations using a truncated eigenvalue decomposition (EVD). But one last obstacle remains: the null distribution of the first canonical correlation is unknown in this highdimensional setting. Although you could rely on a permutation strategy to obtain a pvalue, you are also aware that such a procedure is very computationally expensive if you want to reach a high level of precision for the estimated pvalue, or if you want to analyze multiple regions in the brain.
In this article, we provide a fast computational approach for estimating the null distribution of the first canonical correlation in such high dimensional setting.
However, the contribution of this paper extends well beyond this CCA example provided above. More generally, a cursory look at the table of contents of recent volumes in both neuroimaging and genomics journals reveals a strong bias towards multivariate analysis methods such as Principal Component Analysis (PCA), Multivariate Analysis of Variance (MANOVA), Canonical Correlation Analysis (CCA), Principal Components of Explained Variance (PCEV), and Linear Discriminant Analysis (LDA); for a small yet broad sample, see
Park et al. (2017); Zhao et al. (2017); Hao et al. (2017); Pesonen et al. (2017); Gossmann et al. (2018); Fraiman and Fraiman (2018); Happ and Greven (2018); Yang et al. (2018); Turgeon et al. (2018). This is hardly surprising, given the evolving nature of technological capabilities data and the complex underlying biological processes that are now measurable. As described in our scenario above, using truncated matrix decomposition, we can often perform dimension reduction even in high dimensions. But beyond dimension reduction, many classical multivariate approaches also aim at summarizing the relationship between two datasets and; in the list above, CCA, MANOVA and PCEV all have this common goal. Furthermore, this subset of methods also provide a unified way of performing null hypothesis significance testing: they all rely on the largest root of a
double Wishart problem. Specifically, the strength of the association between and is measured in terms of the magnitude of the largest solution to the following determinantal equation:(1) 
where and are two independent random matrices, both following a Wishart distribution with the same scale matrix. The definition of and is formulated under the null hypothesis and is methodspecific. Several specific examples will be given later in this paper. From the distribution of this largest root, we can then compute a pvalue for the null hypothesis of interest.
More formally, in highdimensional settings, when the sample size is smaller than the number of measurements, both matrices and can have singular distributions. This singularity leads to both computational challenges for estimation and theoretical challenges for inference. On the one hand, common estimators in the nonsingular case can be illconditioned (or even undefined) for singular problems; on the other hand, classical asymptotic convergence results rely on large sample sizes and therefore may not directly apply to highdimensional settings.
In this work, we are interested in multivariate analysis involving two datasets, and , such that the dimension of one or both matrices may be much larger than the sample size . We posit that proper highdimensional inference in several multivariate statistical methods such as CCA, MANOVA and PCEV, can be attained by studying the singular double Wishart problem described above. Our main contribution is an empirical estimator of the distribution of the largest root that is applicable to the analysis of highdimensional data. This estimate provides valid pvalues by fitting a locationscale family of distributions to a small number of permutations of the original data. By a theorem of Johnstone (2008), this family of distributions is known to provide an excellent approximation in the nonsingular case, and we provide empirical evidence that this good performance extends to the singular case.
The rest of the article is structured as follows: in Section 2, we provide some detailed examples of double Wishart problems; these examples are later used to illustrate our approach. In Section 3, we describe our approach to approximating the distribution function of its largest root using an empirical estimator. Then, in Section 4, we investigate the numerical accuracy of the approximation and show how it leads to valid pvalues. We further illustrate our approach through an analysis of the association between DNA methylation and disease type in patients with systemic autoimmune rheumatic diseases. Finally, in Section 6, we explain how our empirical estimator can be extended to accommodate linear shrinkage covariance estimators within the double Wishart setting. Our approach has already been implemented in two R packages: pcev and covequal; both packages are available on CRAN.
Notation
In what follows, and will denote and matrices, respectively. We also write when is Wishartdistributed with parameters and scale matrix . Recall that this is equivalent to having an matrix where each row is independently drawn from a multivariate normal and such that .
2 Examples of double Wishart problems
As stated above, the developments in our paper build on theory associated with double Wishart problems. Therefore, we start here by giving four examples of well known multivariate tests that are each double Wishart problems in order to emphasize the number of different applications of the theoretical results to follow. In each case, if the two matrices and are singular or illconditioned, then the theoretical results developed for double Wishart problems no longer apply. The four examples are oneway MANOVA, a test of equality of covariance matrices, CCA, and Principal Component of Explained Variance (PCEV). The first example is given for its simplicity and historical importance; the other three examples are used later to illustrate our approach.
2.1 Manova
Suppose that we have a set of independent observations , where denotes the th observation of the th group, with , . In MANOVA, we are interested in the null hypothesis of equality of means . First, for each group, we can form the sample mean and covariance matrix . We then compute two basic quantities: 1) the withingroup sum of squares ; and 2) the betweengroup sum of squares , where is the overall mean. Under , the matrices and are independent and Wishartdistributed, with and . The unionintersection test of uses the largest root of as a test statistic or, equivalently, that of (Mardia et al., 1979, Chapter 12). In the notation of Equation 1, we therefore can express the unionintersection test as a double Wishart problem with and . This test statistic is also known as Roy’s largest root, and it is one of the standard tests in classical MANOVA.
2.2 Test of covariance equality
Suppose that independent samples
from two multivariate normal distributions
and lead to covariance estimates which are independent and Wishart distributed on degrees of freedom, respectively. For example, we could take to be the maximum likelihood estimate of the covariance matrix based on observations. We are interested in testing the null hypothesis ; this can done using the largest root of the determinantal equation 1 as a test statistic for which we set and (Anderson, 2003, Chapter 10).2.3 Cca
Suppose we have two datasets of dimension and , respectively. Recall that CCA seeks to find linear combinations of and that are maximally correlated with each other. Assume each row of
, where has the formFor simplicity, we first assume (which implies that ). Under our normality assumption, a hypothesis test for independence between and is equivalent to a hypothesis test for .
Write and let . Using these quantities, we can define
Under our assumptions, we have and . We can test the null hypothesis using as a test statistic the largest root of the double Wishart problem corresponding to the pair of matrices above (Mardia et al., 1979, Chapter 10). We note that this largest root corresponds to the square of the first canonical correlation between and .
The computation above can be generalized to the case when (and ) and with nonzero mean; for details, see Johnstone (2009).
2.4 Pcev
Similar to CCA, PCEV can also be used to simultaneously perform dimension reduction and test for association between two multivariate samples of dimension and , respectively. However, whereas CCA seeks to maximise the correlation between linear combinations of and , PCEV seeks the linear combination of whose proportion of variance explained by is maximised. Specifically, we assume that the relationship between and can be represented via a linear model:
where is a matrix of regression coefficients for the covariates of interest, and
is a vector of residual errors. This model assumption allows us to decompose the total variance of
as the sum of variance explained by the model and residual variance:where . PCEV seeks a linear combination of outcomes, , that maximises the proportion of variance being explained by the covariates :
where
(2) 
Then testing the null hypothesis is performed using as a test statistic the largest root of the determinantal equation 1 for which we set and . Turgeon et al. (2018) have further details on this dimensionreduction technique.
3 Empirical Estimator of the Distribution of the Largest Root
As we can see from the examples above, many null hypothesis significance tests in multivariate analysis follow the same two steps 1) computing the largest root of equation 1
; and 2) from its distribution under the null hypothesis, compute a pvalue. For the first point, we may have to use a truncated version of singular value decomposition (SVD) (or EVD) when both
andare singular; this singularity is common in highdimensional datasets. In essence, these matrix decompositions are restricted to the subspace spanned by the eigenvectors corresponding to the nonzero eigenvalues; the mathematical details are reviewed in Section 1 of the Supplementary Materials. In this section, we focus on the second point. We show how the distribution of the largest root can be accurately approximated in the singular setting.
3.1 Known results in the nonsingular setting
In the nonsingular setting, the distribution of the largest root to the determinantal equation 1 is well studied (Mardia et al., 1979; Muirhead, 2009). To provide a theoretical underpinning to the remainder of the section, we highlight two approaches to computing this distribution: an exact approach, and an asymptotic result.
First, Chiani (2016)
described an explicit algorithm for computing the cumulative distribution function (CDF) of the largest root
. Building on his earlier work (Chiani, 2014), he proposed a new expression that relates the CDF to the Pfaffian of a skewsymmetric matrix. He also provided a set of recursive equations that provide a fast and efficient way to compute this matrix. However, this matrix quickly becomes very large when the parameters of the Wishart distribution increase, leading to both computational instability and numerical overflow problems. And yet, this matrix can only be computed in the nonsingular setting.
The second approach we wish to highlight uses results from random matrix theory to derive an approximation to the marginal distribution. Specifically,
Johnstone (2008) showed that after a suitable transformation of the largest root, its distribution can be approximated by the TracyWidom distribution of order 1. His result, using the notation of this manuscript, is given below:3.2 TracyWidom Empirical Estimator
Unfortunately, in the singular setting, the marginal distribution of the largest root is not as wellunderstood. Crucially, the results from both Johnstone (2008) and Chiani (2016) depend on the nonsingularity of the matrix , and therefore they cannot readily be applied to the singular setting.
As highlighted in the introduction, a common approach to computing pvalues when the null distribution is unknown is to use a permutation procedure. But high precision requires a large number of permutations, and therefore is computationally burdensome. In this section, we show that we can drastically reduce the number of permutations by using Johnstone’s theorem above as inspiration.
We suggest an empirical estimator for approximating the CDF of the largest root. Indeed, we propose to estimate the distribution using a locationscale family of the TracyWidom distribution of order 1 indexed by two parameters . Algorithm 1 below describes our approach.
A few comments are required:

As we will show in Section 4, the number of permutations can be chosen to be relatively small while retaining good performance.

The appropriate permutation procedure on Line 3 depends on the particular double Wishart problem being studied. For a test of association between and , we typically permute the rows of and keep those of fixed. The test of equality of covariance matrices requires a different strategy, and we give all the relevant details in Section 4.2.1 below.

Line 7 of Algorithm 1 refers to a fitting procedure. In Section 4.1, we investigate four different approaches: Method of moments; Maximum Likelihood Estimation; and Maximum GoodnessofFit estimation (Luceño, 2006) using the AndersonDarling statistic and a modified version that gives more weight to the right tail. Details about this latter approach, including computational formulas of these statistics, are given in Section 2 of the Supplementary Materials.
Therefore, we propose this locationscale transformation of the TracyWidom distribution as a suitable parametric family for estimating the distribution of the largest root in singular double Wishart problems.
4 Simulation results
To investigate the performance of our empirical estimator, we performed two simulation studies: 1) we compared the distribution obtained from our empirical estimator to an empirically generated cumulative distribution function (CDF), as described below, for different number of permutations and using four different fitting strategies; 2) we compared pvalues derived from the estimated distribution to those obtained through a permutation procedure. The first simulation study is aimed at assessing the performance of our empirical estimator over the whole range of the distribution; the second simulation study specifically looks at the upper tail of the distribution and therefore at the validity of the resulting pvalues.
4.1 Comparison to the true distribution
Since the distribution function of the largest root distribution is not available analytically, we cannot use an analytical function as the benchmark for the “true” CDF. Therefore, an estimate of the true distribution was derived computationally. Specifically, we started by generating 1000 pairs of singular Wishart variates as follows: each singular Wishart variate was generated by first generating a sample of multivariate normal variates , and then defining ; the Wishart variate was generated similarly. For each pair of Wishart variates, we then computed its corresponding largest root using truncated SVD (cf. §1 of the Supplementary Materials). From these 1000 largest roots, we calculated the empirical CDF for the marginal distribution: this estimate was considered our benchmark for assessing the performance of our approach.
For our simulations, we fixed the degrees of freedom and . In the context of a oneway MANOVA, this would correspond to four distinct populations and a total sample size of 100; in the context of CCA, this would correspond to one set of variates of dimension 4 (the dimension of the other is ), and a total sample size of 99. We also fixed the scale matrix , but we varied the parameter .
To compute the TracyWidom empirical estimator, we sampled of these largest roots (with or ) and fitted the locationscale family as described in Algorithm 1. Finally, we looked at four different fitting strategies: the method of moments (MM); Maximum Likelihood Estimation (MLE); and Maximum GoodnessofFit estimation (Luceño, 2006) using the AndersonDarling statistic (AD) and a modified version that gives more weight to the right tail (ADR).
The simulation results appear in Figure 1. As we can see, with a larger number of samples , all four methods estimate the distribution of the largest root reasonably well; on the other hand, for a smaller value of , the method of moments clearly outperforms the other fitting strategies. For this reason, unless otherwise stated, we use the method of moments in the remainder of this article.
4.2 Comparison of pvalues
Next, we used our empirical estimator to compute pvalues for three double Wishart problems: a test of equality of covariances, CCA, and PCEV. In all settings, we performed 100 simulations, and we fixed the sample size at . We also used permutations to fit the TracyWidom empirical estimator using the method of moments. Finally, we compared our approach to a permutation procedure with 500 permutations. As a reference, we summarise the parameters for all simulation scenarios appear in Table 1.








Autoregressive  
CCA  Fixed at 20  for , otherwise  
PCEV  Fixed at 1  Linear model 
4.2.1 Test of equality of covariances
For the test of equality of covariances, we simulated two datasets and , both with observations. We selected an autocorrelation structure for the covariance matrix , with . We varied two parameters: 1) the dimension of both and ; and 2) the correlation parameter . Note that corresponds to the null hypothesis of the same covariance, and correspond to alternative hypotheses.
The permutation procedure for testing the equality of covariance matrices started by centring both and . Then, the observations were permuted between and : that is, a valid permutation would sample 50 observations from both and to create a permuted , and the remaining 50 observations from and were used to create a permuted . The results are summarised using QQplots (see Figure 2). The computations were performed using the R package covequal.
The simulation results appear in Figure 2. As we can for these QQplots, there is excellent agreement between the pvalues obtained from a permutation procedure and those obtained from the TracyWidom empirical estimator.
4.2.2 Cca
For CCA, we again simulated two datasets and , both with observations, and with fixed . We selected an exchangeable structure with parameter for the covariance matrix . We again varied two parameters: 1) the dimension of ; and 2) the correlation parameter . As above, the value corresponds to the null hypothesis of no association between and , and the values correspond to alternative hypotheses. Finally, the permutation procedure consisted in permuting the rows of and keeping those of fixed.
The simulation results appear in Figure 3. As above, there is excellent agreement between the pvalues obtained from a permutation procedure and those obtained from the TracyWidom empirical estimator.
4.2.3 Pcev
For PCEV, we looked at the following highdimensional simulation scenario: we fixed the number of observations and a balanced binary covariate
. We then varied the number of response variables
, and fixed the covariance structure of the error term . We induced an association between and the first 50 response variables in . This association was controlled by the parameter ; this parameter is related to the univariate regression coefficient through the following relationship:As above, we summarised the results using QQplots (Figure 4). The computations were performed using the R package pcev (Turgeon et al., 2018); the methodology presented here is part of the package starting with version 2.1.
The simulation results appear in Figure 4. Once again, there is excellent agreement between the pvalues obtained from a permutation procedure and those obtained from the TracyWidom empirical estimator.
As we can see, our TracyWidom empirical estimator yields valid pvalues in a variety of highdimensional scenarios that include both null and alternative hypotheses. For further simulation results, see Section 3 of the Supplementary Materials.
5 Data analysis
To showcase our ideas in the context of a real analysis of a highdimensional dataset, we decided to look at the association between DNA methylation and disease type in patients with four systemic autoimmune rheumatic diseases: Scleroderma, Rheumatoid arthritis, Systemic lupus erythematosus, and Myositis. DNA methylation is an epigenetic mark, meaning that it is a chemical modification of the DNA that does not alter the nucleotide sequence (Baylin, 2005). It is known to be associated with changes in RNA transcription, and it is therefore correlated with gene expression.
The DNA methylation used for this analysis was measured prior to treatment on Tcell samples from 28 patients using the Illumina 450k platform (Hudson et al., 2017). Baseline characteristics of the patients appear in Table 2.
Scleroderma  Other diseases  
(n = 14)  (n = 14)  
Age Mean (sd)  48 (16)  52 (14) 
Female (%)  50%  71% 
We opted to test for differential methylation between scleroderma and the other three diseases at the pathway level: from the Kyoto Encyclopedia of Genes and Genomes (KEGG) (Kanehisa et al., 2017), we extracted the list of genes included in their manually curated list of molecular pathways; these pathways correspond to networks of genes interacting and reacting together as part of a given biological process. For each of the 320 pathways, we then identified all CpG dinucleotides contained in at least one gene of this pathway. All CpG dinucleotides mapped to a given pathway were analysed jointly. The extraction of gene lists was performed using the R package KEGGREST (Tenenbaum, 2017).
For each pathway, we thus have two datasets: an matrix that contains the methylation values at all CpG dinucleotides (where ranges from 39 to 21,640 over the 320 pathways) and an matrix indicating whether an individual has scleroderma. Recall that , and therefore all pathways lead to a highdimensional dataset. The analysis performed had two steps: first, we did a test of equality for the covariance matrices between both disease groups; then we used PCEV to test for differential methylation between these two groups. The PCEV analysis included both age and sex as possible confounders (ElMaarri et al., 2007; Horvath, 2013). For both steps, the TracyWidom empirical estimator was computed using 50 permutations of the data.
Since we repeated the same analysis independently for all 320 pathways, we need to account for multiple comparison. However, since a given gene may appear in multiple pathways, the 320 hypothesis tests are not independent; therefore, a naive Bonferroni correction would be too conservative. To estimate the effective number of independent tests, we looked at the average number of pathways in which a given CpG dinucleotide appears. Overall, 134,941 CpG dinucleotides were successfully matched to at least one of 320 KEGG pathways. On average, each dinucleotide appeared in 4.5 pathways; this leads to effectively 70 independent hypothesis tests. For a nominal familywise error rate of , an appropriate Bonferronicorrected significance threshold is therefore given by .
We compared the pvalue obtained from our procedure above to that obtained from a permutation procedure; the latter is a common approach when the null distribution of a test statistic is unknown but has the disadvantage of being computationally expensive. Given our significance threshold, we determined that we needed to perform at least 10,000 permutations in order to be able to assess significance.
In Table 3, we present the five most significant pathways, with the top pathway achieving overall significance for the test of differential methylation. None of the covariance equality tests yielded a significant pvalue; to improve clarity, we omitted these results from the table.
KEGG code  Description  # CpGs  3cmTracyWidom  
Pvalue  3cmPermutation  
Pvalue  
path:hsa00120  Glutamatergic synapse 
225  
path:hsa03040  Ras signaling pathway  2119  
path:hsa03450  Circadian rhythm  267  
path:hsa00563  Histidine metabolism  394  
path:hsa04380  Pathogenic E. coli infection  2312 
For the most significant pathway (i.e. Glutamatergic synapse), we computed the Variable Importance Factors (VIF) and compared them to the pvalues obtained from a univariate approach where each CpG dinucleotide is tested individually against the disease outcome (Turgeon et al., 2018). The VIF is defined as the correlation between the individual response variables and the principal component maximising the proportion of variance. The comparison is presented in Figure 5. As previously showed in the literature (Turgeon et al., 2018), there is some degree of agreement between these two measures of association. Moreover, we can see evidence that the overall association between this pathway and our disease indicator is driven by a few CpG dinucleotides.
6 Extension to linear shrinkage covariance estimators
In Section 4, we presented graphical evidence that our proposed TracyWidom empirical estimator provides a good approximation of the distribution of the largest root of the determinantal equation 1. As discussed in Section 2, the estimates of the matrices appearing in this equation often involve highdimensional covariance matrices. However, a common problem with such highdimensional covariance matrices is their instability (Pourahmadi, 2013, Chapter 1). As a result, the power of statistical tests derived from the double Wishart problem decreases as the dimension of and increases. However, it would seem that the TracyWidom empirical estimator relies on the assumption that and are Wishartdistributed, and it is not clear a priori that this estimator can be applied with other, more efficient estimators of the underlying highdimensional covariances matrices.
One strategy for improving the stability of a covariance estimator is to use a shrinkage estimator. One such estimator for the covariance matrix was introduced by Ledoit and Wolf (2004). In this section, we show that by replacing the matrix by a linearly shrunk version in Equation 1, our TracyWidom empirical estimator still provides a good approximation of the distribution of the largest root.
Let , and be the identity matrix. Ledoit and Wolf (2004) look for an optimal linear combination to estimate the population covariance matrix; the optimality criterion is described in the following result:
Theorem 2
[Ledoit and Wolf (2004)] Let be the squared Frobenius norm, and let be its corresponding inner product. Consider the optimization problem:
where the coefficients are nonrandom. Its solution verifies:
where
Furthermore, Ledoit and Wolf (2004) provide consistent estimators for all quantities
under mild conditions that hold true for normal random variables, and therefore hold true in our setting. The resulting estimator for
is denoted , and we thus get a linearly shrunk .To assess the performance of our TracyWidom empirical estimator under this extended setting, we repeated the simulations from Section 4.1 but by substituting the matrix with its linearly shrunk equivalent . The results appear in Figure 6, and they are very similar to the earlier results; we get good agreement even with small values of , and the method of moments provides a better approximation than the other fitting procedures.
7 Discussion
In this work, we investigated the singular double Wishart problem, which arises in the multivariate analysis of highdimensional datasets. We presented an empirical estimator of the distribution of the largest root that is simple, efficient, and valid for highdimensional data. Through simulation studies, we showed how our approach leads to a good approximation of the true distribution, and we also showed that it leads to valid pvalues. Finally, using a pathwaybased approach, we analysed the relationship between DNA methylation and disease type in patients with systemic autoimmune rheumatic diseases. Our analysis used the empirical estimator in two settings: a test for the equality of covariance matrices, and with the dimensionreduction method known as PCEV.
The empirical estimator we presented fills a gap in highdimensional multivariate analysis. Many common methods, such as MANOVA and CCA, fit into the framework of double Wishart problems. However, classical hypothesis testing breaks down in high dimensions, and therefore analysts often rely on computationally intensive resampling techniques to perform valid inference. For example, genomic studies often require significance thresholds of the order of and lower in order to correct for multiple testing. In this context, a permutation procedure would require at least 1 million resamples. By relying on results from random matrix theory, we can drastically cut down the required number of permutations. Since we only need to estimate two parameters from a locationscale family, good approximation is achieved with less than 100 permutations. Critically, this number is independent of the number of tests performed, and therefore the computation time is reduced by several orders of magnitude.
We motivated our empirical estimator of the CDF using an approximation theorem of Johnstone (2008). Our approach is further motivated by several results from random matrix theory that suggest a centrallimittype theorem for random matrices, with the TracyWidom distribution replacing the normal distribution (Deift, 2007). More evidence in support of our empirical estimator is given from the study of the largest root of Wishart variates. On the one hand, Srivastava (2007) and Srivastava and Fujikoshi (2006) showed that the asymptotic distribution of is approximately equal to the distribution of the largest eigenvalue of a scaled Wishart matrix. On the other hand, several results analogous to Johnstone’s theorem were also derived for Wishart distributions. Indeed, it has been shown that if is the largest root of a Wishart variate, then there exists , both functions of the parameters of the Wishart distribution, such that the ratio converges in distribution to (Johansson, 2000; Johnstone, 2001; El Karoui, 2003; Tracy and Widom, 2009).
Finally, the results from Section 6 show that the domain of applicability of our empirical estimator extends beyond that of a strict double Wishart problem. Specifically, we showed that we can replace the matrix in Equation 1 by a linearly shrunk estimator based on earlier work by Ledoit and Wolf (2004). This approach can easily be applied to multivariate analysis approaches for which is explicitly the covariance matrix of a multivariate random variable. Examples include the test of equality of covariance matrices and PCEV. However, more care is required in order to use these results with CCA.
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