A Powerful Bayesian Test for Equality of Means in High Dimensions

Abstract

We develop a Bayes factor based testing procedure for comparing two population means in high dimensional settings. In ‘large-p-small-n’ settings, Bayes factors based on proper priors require eliciting a large and complex p×p covariance matrix, whereas Bayes factors based on Jeffrey’s prior suffer the same impediment as the classical Hotelling T² test statistic as they involve inversion of ill-formed sample covariance matrices. To circumvent this limitation, we propose that the Bayes factor be based on lower dimensional random projections of the high dimensional data vectors. We choose the prior under the alternative to maximize the power of the test for a fixed threshold level, yielding a restricted most powerful Bayesian test (RMPBT). The final test statistic is based on the ensemble of Bayes factors corresponding to multiple replications of randomly projected data. We show that the test is unbiased and, under mild conditions, is also locally consistent. We demonstrate the efficacy of the approach through simulated and real data examples.

1 Introduction

High dimensional population mean testing is common in many application areas including genomics, where gene-set testing is often of more interest than individual gene tests (Ein-Dor et al., 2006; Subramanian et al., 2005). A natural high dimensional test is based on the distance between the sample mean vectors weighted by the inverse sample covariance matrix, also known as the Mahalanobis distance (Johnson and Wichern, 1992). However, the weight becomes undetermined when the dimension of the population mean vectors is larger than the total sample size minus 2.

To circumvent these limitations, two major approaches have emerged. The first approach is centered around constructing tests that eliminate the need to invert ill-formed covariance matrices. Bai and Saranadasa (1996) replaced the sample covariance matrix by a diagonal covariance matrix, for which the inverse exists. Srivastava (2007) substituted the inverse covariance matrix by its Moore-Penrose inverse, under the assumption that the groups have the same covariances. Wu et al. (2006) and Gregory et al. (2014) proposed tests based on the pooled squared univariate t-tests, eliminating the need to invert non-positive definitive matrices.

The latter approach centers around transforming the data, instead of the test statistics, so that existing tests could be applied to the transformed data. Random projection (RP) is one such method that works by projecting high dimensional data into lower dimensions while only slightly distorting the distances between the original vectors. See, for example, Dasgupta and Gupta (2003). RP has become a popular tool used extensively in machine learning literature where texts documents, imaging and MRI data are often high dimensional. Dasgupta (2000), for example, used RP to uncover the components of high dimensional mixtures of Gaussians. Fern and Brodley (2003) showed the improvement in clustering high dimensional data using RP over other standard approaches. Recently, Guhaniyogi and Dunson (2015) proposed a Bayesian compression regression approach in n ≪ p scenarios, where RP is used to reduce the covariate space. RP has entered the frequentist hypothesis testing literature where the T² statistics are based on the projected version of the data in ‘large-p-small-n’ setting. See, for example, Lopes et al. (2011) and Srivastava et al. (2016).

However, to our knowledge, no work has been done to extend Bayesian machineries to hypothesis testing in high dimensional group means testing. Bayesian hypothesis testing differs from its frequentist counterpart in that the decision to reject or accept a null hypothesis is based on the Bayes factor and a chosen evidence threshold (Jeffreys, 1961; Kass and Raftery, 1995). More precisely, the Bayes factor in favor of the alternative hypothesis H₁, denoted by BF₁₀, is defined as

$$B{F}_{10}=\frac{m(\mathit{Y}\mid H_1)}{m(\mathit{Y}\mid H_0)}=\frac{\int f(\mathit{Y}\mid \mathbf{\Theta })\pi (\mathbf{\Theta }\mid H_1)d\mathbf{\Theta }}{\int f(\mathit{Y}\mid \mathbf{\Theta })\pi (\mathbf{\Theta }\mid H_0)d\mathbf{\Theta }}$$ (1)

where m(Y|H_i) denotes the marginal distribution of Y under H_i; π(Θ|H_i) denotes the prior distribution of Θ under H_i, for i = 0, 1.

Equation (1) involves high dimensional integrals and the choice of π(Θ|H_i) often focuses on distributions that lead to closed form expressions for the Bayes factor.

In this paper, we use the random projection approach to develop a Bayes factor based restricted most powerful Bayesian test (RMPBT) (Goddard and Johnson, 2016) for the high dimensional group means testing problem. In an RMPBT, the prior distribution under the alternative is chosen by maximizing the power of the test with respect to a restricted class of priors. The evidence threshold is selected to match the rejection region of its non-Bayesian counterpart. We show that our proposed test is unbiased and consistent.

The paper is organized as follows. In Section 2, we derive RMPBT for testing differences between two mean vectors. We establish some asymptotic properties of the test in Section 3. Section 4 provides a simulation study investigating the power of the proposed test. We apply the proposed test to the analysis of some real data sets in Section 5. Section 6 concludes with a discussion.

2 Bayes factor in high dimensions

Let N_p(μ,Σ) denote a p-dimensional multivariate normal density with mean vector μ and covariance matrix Σ. Let X₁, · · ·, X_n1 ∈ ℝ^p and Y₁, · · ·, Y_n2 ∈ ℝ^p be independent random draws from N_p(μ₁,Σ) and N_p(μ₂,Σ), respectively. Also, let X_n1×p = (X₁, · · ·, X_n1)^T and Y_n2×p = (Y₁, · · ·, Y_n2)^T.

The minimal sufficient statistics are D = Ȳ − X̄, A = (n₁Ȳ +n₂X̄)/(n₁ +n₂) and $\mathit{S}=\{(n_1-1){\sum }_{i=1}^{n_1}({\mathit{X}}_i-\overline{\mathit{X}}){({\mathit{X}}_i-\overline{\mathit{X}})}^{\mathrm{T}}+(n_2-1){\sum }_{i=1}^{n_2}({\mathit{Y}}_i-\overline{\mathit{Y}}){({\mathit{Y}}_i-\overline{\mathit{Y}})}^{\mathrm{T}}\}/(n_1+n_2-2)$. Also, $\mathit{D}~{\mathit{N}}_p(\mathit{\delta },n_0^{-1}\mathbf{\sum })$, A ~ N_p(μ, n⁻¹Σ) and (n − 2)S ~ W_p(n − 2,Σ), independently, where δ = μ₂ − μ₁, μ = (n₁μ₁ + n₂μ₂)/n, $n_0^{-1}=n_1^{-1}+n_2^{-1}$, n = n₁ + n₂ and W_p(n,Σ) denotes a Wishart distribution on the space of p×p dimensional positive definite matrices with degrees of freedom n and mean nΣ.

The problem is to test H₀ : μ₁ = μ₂ against H₁ : μ₁ ≠ μ₂. We will work with the reparametrization in terms of δ, μ and Σ, which parametrize the distribution of the minimal sufficient statistics. The hypotheses of interest can accordingly be reformulated as

$$H_0:\mathit{\delta }=\mathbf{0}\text{against}{H}_1:\mathit{\delta }\ne \mathbf{0}.$$

The generic form of the prior that we consider is given by π(δ,μ,Σ |H_i) = π(μ,Σ) π(δ|Σ,H_i). For π(μ,Σ), we consider the Jeffrey’s prior given by

$$\pi (\mathit{\mu },\mathbf{\sum })\propto {\mid \mathbf{\sum }\mid }^{-(p+1)/2}.$$ (2)

The choice π(δ|Σ,H₀) = 1{δ = 0} is trivially dictated by H₀. For π(δ|Σ,H₁), we consider the prior

$$\pi (\mathit{\delta }\mid \mathbf{\sum },{\tau }_0,H_1)~{\mathit{N}}_p(\mathbf{0},\mathbf{\sum }/{\tau }_0).$$ (3)

The choice of the hyper-parameter τ₀ ∈ (0,∞) is crucial and is discussed in Section 2.1. Throughout the paper, we assume the same prior weight for each hypothesis, that is, P(H₀) = P(H₁) = 0.5.

When 1 < p < n−2, the Bayes factor admits a closed form expression under the assumed priors, as shown by the following result.

Lemma 1

With 1 < p < n− 2 and under the priors (2) and (3), we have

$$B{F}_{10}(\mathit{X},\mathit{Y})={(1+\eta )}^{-p/2}{\left\{\frac{1+\frac{pf}{(1+\eta )(n-p-1)}}{1+\frac{pf}{(n-p-1)}}\right\}}^{-(n-1)/2},$$ (4)

where η = n₀/τ₀ and $f=\frac{(n-p-1)}{(n-2)p}{n}_0{(\overline{\mathit{Y}}-\overline{\mathit{X}})}^{\mathrm{T}}{\mathit{S}}^{-1}(\overline{\mathit{Y}}-\overline{\mathit{X}})$.

We show the derivation of BF₁₀ in Appendix A. Here, f is the scaled Hotelling’s T² statistic with f ~ F_p,n−p−1 under H₀, where F_ν1,ν2 denotes a central F distribution with ν₁ and ν₂ degrees of freedom. When p ≥ n − 2, S is no longer positive definite, hence its inverse is not unique and (4) can not be employed to test H₀ against H₁. To handle the dimensionality problem, we project the data vectors to a lower dimensional subspace using a random projection matrix R_p×m satisfying R^TR = I_m×m where 1 < m < n−2. The projected data for group 1 are then obtained as ${\mathit{X}}_i^{★}={\mathit{R}}^{\mathrm{T}}{\mathit{X}}_i$, i = 1, · · ·, n₁. Likewise, the projected data for group 2 are ${\mathit{Y}}_i^{★}={\mathit{R}}^{\mathrm{T}}{\mathit{Y}}_i$, i = 1, · · ·, n₂. For a given projection matrix R_p×m, under the priors (2) and (3), using Lemma 1 and basic properties of multivariate normal distributions, BF₁₀ based on the projected data is given by

$$B{F}_{10}({\mathit{X}}^{★},{\mathit{Y}}^{★})={(1+\eta )}^{-m/2}{\left\{\frac{1+\frac{m{f}^{★}}{(1+\eta )(n-m-1)}}{1+\frac{m{f}^{★}}{(n-m-1)}}\right\}}^{-(n-1)/2},$$ (5)

where $f^{★}=\frac{(n-m-1)}{(n-2)m}{n}_0{(\overline{\mathit{Y}}-\overline{\mathit{X}})}^{\mathrm{T}}\mathit{R}{({\mathit{R}}^{\mathrm{T}}\mathit{SR})}^{-1}{\mathit{R}}^{\mathrm{T}}(\overline{\mathit{Y}}-\overline{\mathit{X}})$, 1 < m < n− 2, p ≥ n − 2, and n = n₁ + n₂. Also, f^★ ~ F_m,n−m−1 under H₀. The following result establishes some desirable asymptotic properties of BF₁₀(X^★,Y^★) when τ₀ is fixed and also when τ₀ is allowed to depend on n.

Theorem 1

Let n_min = min{n₁, n₂} → ∞ and m→∞ with lim_nmin→∞ m/n = θ ∈ (0, 1).

1. If τ₀ is fixed, under H₀, $\mathit{log}\{B{F}_{10}({\mathit{X}}^{★},{\mathit{Y}}^{★})\}\stackrel{p}{\to }-\infty$, and, under H₁, $\mathit{log}\{B{F}_{10}({\mathit{X}}^{★},{\mathit{Y}}^{★})\}\stackrel{p}{\to }\infty$.

2. If n₀/τ₀ → 0 and mn₀/τ₀ → ∞, under H₀, log{BF₁₀(X^★,Y^★)} = 𝒪_p(1). For the corresponding sequence of $H_1^n,\mathit{log}\{B{F}_{10}({\mathit{X}}^{★},{\mathit{Y}}^{★})\}\stackrel{p}{\to }\infty$, with $f^{★}\stackrel{p}{\to }\infty$ as n_min →∞.

The proof of Theorem 1 is deferred to Appendix B. Part (a) of Theorem 1 states that for fixed τ₀, the Bayes factors is consistent under H₀ and a fixed alternative H₁. However, if τ₀ is allowed to depend on n so that n₀/τ₀ → 0 at a slower rate than 1/n, BF₁₀(X^★,Y^★) is not consistent under H₀, but is consistent for that sequence of local alternatives, provided that the F-statistic f^★ is unbounded as n_min →∞. Although the lack of consistency of the Bayes factors in part (b) of Theorem 1 under H₀ seems unsettling at first, this property is similar to that of frequentist tests, where, for a chosen significance level, the null hypothesis has a non-zero probability of being rejected regardless of the sample size when the null is actually true. We show below that the construction of the restricted most powerful Bayesian test satisfies the conditions enumerated above.

2.1 Restricted most powerful Bayesian tests

Recently, Johnson (2013b) introduced the idea of the uniformly most powerful Bayesian tests (UMPBTs) in the context of point hypothesis testing, providing a Bayesian parallel to the idea of uniformly most powerful tests (UMPTs) proposed by Neyman and Pearson (1928, 1933). UMPTs are defined as tests with the highest power among all possible tests of a given size. For a fixed size, UMPTs have the rejection region with the highest probability under the alternative. The rejection region refers to the range of values of the test statistics that leads to a rejection of the null.

In Bayesian hypothesis testing, the decision to reject the null hypothesis is based on the Bayes factor or evidence (log Bayes factor) for a given fixed alternative. Johnson (2013b) defines a UMPBT for testing a null against a fixed alternative as the test corresponding to the prior under the alternative that maximizes the probability of deciding in favor of the alternative for a fixed evidence level γ, among all possible data generating parameters. More precisely, we have a UMPBT for a given evidence threshold γ if the Bayes factor in favor of an alternative hypothesis H₁ against a fixed null hypothesis H₀ satisfies

$$P_{\mathit{\theta }}(B{F}_{10}>\gamma )\ge P_{\mathit{\theta }}(B{F}_{20}>\gamma )$$

for all possible values of the data generating parameter θ and all alternative hypotheses H₂. However, as noted in Johnson (2013b), UMPBTs exist in a limited number of relatively simple testing scenarios. Finding a UMPBT in our setting is also a daunting task. Recently, Goddard and Johnson (2016) introduced the idea of restricted most powerful Bayesian tests (RMPBTs). RMPBTs are obtained by restricting the choice of the alternatives to a smaller family of distributions. Here, we focus the search of alternatives to a narrow class of distributions, preferably to priors that lead to Bayes factors with closed forms, like the prior considered in (3). We can subsequently choose the hyper-parameter τ₀ using the idea of RMPBT by maximizing the probability of deciding in favor of the alternative for a fixed evidence level γ. In other words, we choose τ₀ = τ^★ so that

$$P_{\mathit{\theta }}\{B{F}_{10}({\mathit{X}}^{★},{\mathit{Y}}^{★},{\tau }^{★})>\gamma \}\ge P_{\mathit{\theta }}\{B{F}_{10}({\mathit{X}}^{★},{\mathit{Y}}^{★},{\tau }_0)>\gamma \},$$

for a chosen value of the evidence threshold γ, all possible values of τ₀ and all data generating model parameters θ = (δ,μ,Σ). That is, we choose τ₀ so as to maximize the following probability

$$P_{\mathit{\theta }}\left\{\frac{m{f}^{★}}{m{f}^{★}+n-m-1}>\left(\frac{1+n_0/{\tau }_0}{n_0/{\tau }_0}\right)[1-{\{\gamma {(1+n_0/{\tau }_0)}^{m/2}\}}^{-2/(n-1)}]\right\},$$

which is at its maximum when the quantity on the right-hand side of the inequality is at its minimum. The RMPBT is thus obtained with

$${\tau }_0=argmin\left(\frac{1+n_0/{\tau }_0}{n_0/{\tau }_0}\right)[1-{\{\gamma {(1+n_0/{\tau }_0)}^{m/2}\}}^{-2/(n-1)}].$$ (6)

The optimization in (6) requires a value of γ which can be chosen according to the evidence threshold suggested in Kass and Raftery (1995). Alternatively, we can choose γ by equating the rejection region of the Bayes factor to that of the classical F statistic. In the non-Bayesian setting, a level α test would reject H₀ if f^★ > F_{α,m,n−m−1}, where F_{α,m,n−m−1} is the upper α quantile of an F distribution with m and n − m − 1 degrees of freedom. The rejection region based on the Bayes factor in favor of the alternative can be expressed as

$$\left\{(\mathit{X},\mathit{Y}):f^{★}>\frac{C_n(n-m-1)}{m(1-C_n)}\right\},$$ (7)

where C_n = {(1 + n₀/τ₀)/(n₀/τ₀)}[1 − {γ(1 + n₀/τ₀)^m/2}^−2/(n−1)]. Setting

$$F_{\alpha ,m,n-m-1}=\{C_n(n-m-1)\}/(m(1-C_n)),$$ (8)

we can then solve for γ_α. This way, under H₀, we have P{BF₁₀(X^★,Y^★) > γ_α} = P(f^★ > F_{α,m,n−m−1}) = α. We obtain an exact form for τ₀, denoted τ_α(n), that satisfies (6) given (8) as

$${\tau }_{\alpha }(n)=n_0\frac{m}{(n-1)}\frac{(1-C_n)}{(C_n-\frac{m}{n-1})}=\frac{n_0}{F_{\alpha ,m,n-m-1}-1},$$ (9)

where, from (8), C_n = (mF_{α,m,n−m−1})/(mF_{α,m,n−m−1} + n − m − 1).

We can then obtain the equivalent value of γ, denoted γ_α(n), as

$${\gamma }_{\alpha }(n)={\{1+n_0/{\tau }_{\alpha }(n)\}}^{-m/2}{\left[1-\frac{n_0/{\tau }_{\alpha }(n)}{\{1+n_0/{\tau }_{\alpha }(n)\}}{C}_n\right]}^{-(n-1)/2}.$$ (10)

The plot of the evidence threshold γ_α(n) along with the associated τ_α(n) value for various values of α is shown in Figure 1 for two different cases. In both cases, we note that the values of γ_α(n) above 20 (strong evidence) are associated with very small significance level α < 0.007. This is consistent with the findings that the evidence reflected in Bayes factors often requires very strong evidence in classical settings (Johnson, 2013a).

Figure 1.

Plot of the estimated Bayes factor threshold γ_α(n) and the value of τ_α(n) for various values of α, the significance level. The values of Bayes factor above the horizontal line at 20 denotes the (γ_α, τ_α, α) triplet that represents strong evidence against the null hypothesis according to Kass and Raftery (1995).

2.2 Choice of R and m

We discuss the choice of R and m here. We make no attempt to find an optimal projection matrix but are primarily motivated by practical convenience.

Intuitively, however, the projection matrix R should be selected so to only slightly perturb all pairwise distances between the sample vectors (Li et al., 2006). One possible way to achieve this is to sample the entries of R from a distribution with mean zero and variance one. Since our test statistics involves the inversion of R^TSR, which is positive definite if R^TR = I_m (see Lemma 1 of Srivastava et al., 2016), we further restrict our choices to the family of semi-orthogonal matrices. We consider two constructions of the projection matrix. The first one, denoted R₁, is similar to the one permutation + one random projection considered in Srivastava et al. (2016) and yields a sparse matrix with only p non-zero elements. It is constructed as follows.

1. Start with a p × m matrix of zero in each entry.

2. Simulate {r₁, r₂, · · ·, r_p} independently from a standard normal.

3. For each of the m columns, iteratively select ⌊p/m⌋ elements from r = {r₁, r₂, · · ·, r_p} without replacement and assign them respectively to the positions 1 to ⌊p/m⌋ for column 1 vector, ⌊p/m⌋+1 to 2⌊p/m⌋ for column 2 vector, and elements (m−1)⌊p/m⌋ to m⌊p/m⌋ for column m vector. Finally, assign the remaining elements of r, if any, one per column and per row in the remaining rows. Each row of R₁ should now have exactly one non-zero element.

4. Randomly permute the row vectors of R₁.

5. Finally, standardize the columns vectors so that they have length 1.

The second approach obtains R₂ as the Q matrix of the QR decomposition of a p × m matrix with entries simulated independently from a standard normal distribution.

QR decomposition of a large matrix is computationally intensive.

Note, however, that any matrix U ∈ ℝ^p×m admits a QR decomposition U = RB, where R ∈ ℝ^p×m is an orthonormal matrix, that is, R^TR = I_m and B ∈ ℝ^m×m is an upper triangular matrix with positive entries on the diagonal. This implies U(U^TSU)⁻¹U^T = RB(B^TR^TSRB)⁻¹B^TR^T = R(R^TSR)⁻¹R^T.

This suggests that we could simply replace R by U in the equation for f^★ to speed up the computation.

As one reviewer pointed out, the matrix R could also be obtained from singular value decomposition (SVD) of the sample covariance matrix by ignoring the eigenvectors associated with small eigenvalues. However, since such a construction involves the data, the projected data would have a more complex distribution, adding another layer of complexity to the test. Simulation experiments, where we approximated the null distribution of the test statistic using Monte Carlo simulations, also suggest that this approach does not perform well in practice. The results are summarized in Section S.2 of the Supplementary Material.

Now, we discuss how to choose m. Intuitively, small values of m will tend to ignore dependence in the data and the value m = 1 completely ignore any correlation. Large values of m close to n₁ +n₂ −2, on the other hand, will lead to tests with low power as the sample covariance matrix is getting closer to a degenerate matrix with small eigenvalues as noted by Bai and Saranadasa (1996). It is expected that the best value of m will tend to depend on the form of the true unknown covariance matrix - while smaller values of m may perform well when the true covariance matrix is diagonal, larger values will be appealing in more complex cases. Here, we present a heuristic approach to obtain m in general settings.

For a significance level α and a random projection matrix R(m), we have P_m[BF₁₀{X^★(m),Y^★(m),m} > γ_α | H₁,R(m)] = P_m{f^★(m) > F_{α,m,n−m−1} | H₁,R(m)}. Ideally, we would want to choose a value of m that maximizes this probability and hence optimizes the power. This, however, is a difficult problem since m is also involved in the construction of f^★ itself and in its distribution as well. We obtain an approximate solution instead by minimizing the value of the threshold F_{α,m,n−m−1} with respect to m. The values of m hence obtained are similar to the values of m obtained by Srivastava et al. (2016) using a similar argument. We show in the proof of Theorem 2 in Appendix C that such a choice of m satisfies the assumptions of Theorem 1.

Numerical experiments also suggest the empirical power of the test based on such a choice of m to be very close to the optimal power.

3 Test based on Bayes factor and random projections

3.1 Single random projection

We derived the Bayes factor in Section 2 after we apply a single random projection R to the data (see Equation 5). Given the sample sizes n₁ and n₂ and a choice of α, we choose m(n), τ_α(n), and γ_α(n) as discussed in Sections 2.1 and 2.2. A test based on the resulting Bayes factor is then obtained as

$$\varphi (\mathit{R})=\{\begin{array}{ll}1\hfill & \text{if}B{F}_{10}({\mathit{X}}^{★},{\mathit{Y}}^{★})>{\gamma }_{\alpha }(\eta ),\hfill \\ 0\hfill & \text{Otherwise},\hfill \end{array}$$ (11)

where ϕ(R) = 1 signifies rejection of H₀ in favor of H₁, and ϕ(R) = 0 accept H₀. We make the following observations about the test in (11).

Theorem 2

For a given significance level α ∈ (0, 1), we have

1. Under H₀, for fixed n₁, n₂, and m(n), E{ϕ(R) | H₀} = α.

2. Under H₁, for fixed n₁, n₂, and m(n), E{ϕ(R) | H₁} ≥ α.

3. Let the assumptions in Theorem 1 part (b) be satisfied such that n₁, n₂, p → ∞, with m(n), τ_α(n), γ_α(n) chosen as described in Section 2.2. Then, ${lim}_{n_{min}\to \infty }E\{\varphi (\mathit{R})\mid H_1^n\}=1$, where m(n)/n → θ ∈ (0, 1), n₀m(n)/τ_α(n)→∞.

We show the proof of Theorem 2 in Appendix C. Note that (a) shows that the test described in (11) has size α; (b) shows that the test is unbiased; finally (c) shows that the power converges to 1 with increasing sample size. In part (c) of Theorem 2, we impose that m(n)/n → θ ∈ (0, 1) and n₀m(n)/τ_α(n) → ∞ which are satisfied by our construction suggested in Section 2.2.

3.2 Multiple random projections

A test based on a Bayes factor obtained from a single random projection may lead to different decisions for two different random projection matrices. To avoid that, we consider a multitude of Bayes factors computed using many different random projections. Subsequently, we define our test statistic based on the ensemble of Bayes factors and study its power.

Let R₁, · · ·, R_N be a collection of independently and identically distributed random projection matrices. For a choice of n₁, n₂, and α, the values of m(n), τ_α(n), and γ_α(n) are obtained as discussed in Sections 2.1 and 2.2. We then define ϕ̄(N) as

$$\overline{\varphi }(N)=\frac{1}{N}\sum _{i=1}^N\varphi ({\mathit{R}}_i)=\frac{1}{N}\sum _{i=1}^N\mathbf{1}\{B{F}_{10}({\mathit{R}}_i)>{\gamma }_{\alpha }(n)\},$$ (12)

where 1{A} = 1 if A is true and 0 otherwise. Clearly, BF₁₀(R_i) depends on τ_α(n). Note that ϕ̄(N) represents the proportion of Bayes factors based on projected data that yield Bayes factor larger than the specified evidence threshold γ_α(n), for a choice of α and m(n). We then define RMPBT as

$$\{\begin{array}{ll}\text{Reject}{H}_0\hfill & \text{if}\overline{\varphi }(N)>{\varphi }_{\alpha }^0,\hfill \\ \text{Accept}{H}_0\hfill & \text{Otherwise},\hfill \end{array}$$ (13)

where ${\varphi }_{\alpha }^0$ is the upper α quantile of the distribution of ϕ̄(N) under H₀, which depends on m(n), n₁, n₂, p and α.

Theorem 3

Suppose the assumptions of Theorems 1 and 2 hold. Given a collection R₁, · · ·, R_N of random projections matrices, where ${\mathit{R}}_i^{\mathrm{T}}{\mathit{R}}_i=\mathit{I}$ for all i = 1, · · ·, N, then ${lim}_{n_{min}\to \infty }P\{\overline{\varphi }(N)>{\varphi }_{\alpha }^0\}=1$ under the sequence $H_1^n$ of alternatives.

We show the proof of Theorem 3 in Appendix D. For fixed m(n), p, n₁, n₂ and α, RMPBT in (13) requires that we first compute ${\varphi }_{\alpha }^0$. Under H₀, δ = 0, but μ₁ = μ₂ = μ and Σ are unknown. Fortunately, the asymptotic null distribution of ϕ̄(N) is independent of the nuisance parameters μ and Σ, providing a simple way of finding ${\varphi }_{\alpha }^0$. The result is formalized in the following theorem.

Theorem 4

Under H₀, the distribution of ϕ̄(N) as N →∞ is independent of μ and Σ for any fixed n₁, n₂, m ∈ (1, n₁ + n₂ − 2) and p ≥ n₁ + n₂ − 2.

We show the proof of Theorem 4 in Appendix E. Theorem 4 suggests that for large values of N we can approximate the null distribution of ϕ̄(N) by simulating data assuming Σ = I and μ₁ = μ₂ = 0.

4 Simulation study

We designed a simulation study aimed at investigating the power of the test proposed in (13) with respect to the proportion of true elements of δ that are actually zero for various choices of covariance matrices and different scenarios or conditions. We consider two simulation settings. In the first, we assume p = 200 and n₁ = n₂ = 50. Using the approach described in Section 2.2, we find m = 43. In the second setting, p = 1000, n₁ = n₂ = 70 and we get m = 62. We denote by p₀ the proportion of entries of the vector δ that are exactly zero. We choose p₀ = 0.5, .75, .80, 0.95, 0.99, 1.00. In each setting, the values of τ_α and γ_α are chosen according to our discussion in Section 2.1. We consider two types of random projections matrices, R₁ and R₂, as described in Section 2.2.

We consider the following choices for Σ = ((σ_ij)).

1. Σ₁ = I_p×p is the identity matrix.

2. Σ₂ is a diagonal matrix where the first 20 elements are set at 1 to 20 and the remaining are set exactly 1.

3. Σ₃ is an AR(1) covariance matrix with σ_ij = σ²ρ^|i−j|. We chose σ² = 1 and ρ = 0.4.

4. Σ₄ is block diagonal matrix, with block B = 0. 85I_25×25 + 0.15J_25×25. J denotes a matrix with 1 in all of its entries.

5. Σ₅ is an ARIMA(1,1) covariance matrix with σ_ij = σ²γ^1{|i−j|>0}ρ^{|i−j|1{|i−j|≥2}}. We chose σ² = 1, γ = 0.5, ρ = 0.9.

We also consider two possible alternatives.

1. We simulate μ₂ ~ N_p(1, I), set p₀ randomly selected elements to zero, and scale μ₂ so that (μ₂ − μ₁)^TΣ⁻¹(μ₂ − μ₁) = 2.

2. We simulate μ₂ ~ N_p(1, I) and set p₀ of its elements to zero and rescale μ₂ so that $\frac{{\Vert {\mathit{\mu }}_2-{\mathit{\mu }}_1\Vert }^2}{\sqrt{tr\left({\mathbf{\sum }}^2\right)}}=0.1$.

μ₁ is chosen to be a vector of zeros. The two alternatives described above were also considered by Srivastava et al. (2016). We include in these comparisons the following competitors.

1. The approach of Srivastava et al. (2016) referred to as RAPTT.

2. The approach of Bai and Saranadasa (1996) referred to as BS96.

3. The approach of Srivastava and Du (2008) referred to as SD08.

4. The approach of Chen et al. (2010) referred to as CQ10.

To estimate the power of our test, we use N = 5000 random projections for each of the 1000 independently simulated data sets under each of the alternatives considered. Recall that when p₀ = 1.0, the null hypothesis is true and the power represents the type-I error rate estimate. When the true covariance matrix Σ is diagonal (Σ₁ and Σ₂) (see Table 1 and Table S.1 in the Supplemental Material), the tests that do not rely on random projections, namely BS96, SD08 and CQ10, perform slightly better compared to the tests based on random projections, namely RAPTT and the proposed RMPBT. However, RMPBT tended to have higher power when compared to RAPTT for Σ₁ and Σ₂. Also, for Σ₃, an AR(1) covariance matrix with ρ = 0.4, RMPBT performed better than all its competitors, especially for sparse alternatives (Table S.2 in the Supplemental Material). Also, the random projection based tests tended to be slightly more conservative when Σ = Σ₁, Σ₂, or Σ₃. For more complex covariance matrices (Σ₄, Σ₅), the non-random projection based approaches tended to have lower overall power (Table 2, Table 3), and, in some cases, significantly lower power than their random projection based competitors, especially when n₁ = n₂ = 50. However, random projection based approaches tended to have slightly higher or similar estimated type-I error rates for complex true covariance matrices.

Table 1.

Power analysis of 5 tests assuming the true covariance matrix is Σ₁ = I_p×p. We chose the significant level at α = 0.05. For the case n₁ = n₂ = 50, p = 200, we have m = 43, τ_α = 41.918, and γ_α = 3.841. For the case, n₁ = n₂ = 70, p = 1000, we have m = 62, τ_α = 72.318, and γ_α = 3.850. RMPBT is our approach. RAPTT is the approach of Srivastava et al. (2016). BS96 is the approach of Bai and Saranadasa (1996). SD08 is the approach of Srivastava and Du (2008). CQ10 is the approach of Chen et al. (2010).

	p0	RMPBT (R1)	RMPBT (R2)	RAPTT (R1)	RAPTT (R2)	BS96	SD08	CQ10	RMPBT (R1)	RMPBT (R2)	RAPTT (R1)	RAPTT (R2)	BS96	SD08	CQ10
		Alternative 1							Alternative 2
n1 = n2 = 50, p = 200	0.500	0.674	0.669	0.675	0.650	0.754	0.723	0.754	0.469	0.435	0.453	0.408	0.542	0.513	0.542
	0.750	0.684	0.663	0.669	0.654	0.769	0.726	0.769	0.474	0.463	0.455	0.414	0.535	0.503	0.535
	0.800	0.680	0.651	0.652	0.634	0.749	0.715	0.749	0.441	0.413	0.425	0.379	0.517	0.481	0.517
	0.950	0.716	0.698	0.649	0.622	0.756	0.720	0.756	0.480	0.439	0.425	0.384	0.525	0.482	0.525
	0.975	0.707	0.680	0.572	0.555	0.721	0.681	0.721	0.471	0.450	0.382	0.351	0.497	0.453	0.497
	0.990	0.793	0.771	0.561	0.553	0.748	0.726	0.748	0.553	0.525	0.367	0.326	0.554	0.506	0.554
	1.000	0.046	0.041	0.037	0.032	0.042	0.039	0.042	0.046	0.038	0.037	0.025	0.042	0.039	0.042
n1 = n2 = 70, p = 1000	0.500	0.372	0.343	0.347	0.311	0.473	0.422	0.473	0.677	0.588	0.644	0.559	0.767	0.727	0.767
	0.750	0.348	0.294	0.316	0.271	0.470	0.401	0.470	0.695	0.616	0.673	0.578	0.789	0.746	0.789
	0.800	0.337	0.304	0.314	0.267	0.448	0.389	0.448	0.660	0.581	0.634	0.552	0.762	0.717	0.762
	0.950	0.388	0.339	0.349	0.304	0.474	0.423	0.474	0.694	0.612	0.646	0.559	0.775	0.741	0.775
	0.975	0.332	0.309	0.298	0.253	0.450	0.384	0.450	0.685	0.612	0.619	0.532	0.761	0.722	0.761
	0.990	0.377	0.348	0.316	0.280	0.453	0.401	0.453	0.719	0.640	0.610	0.527	0.774	0.719	0.774
	1.000	0.031	0.030	0.028	0.023	0.063	0.040	0.063	0.031	0.022	0.028	0.020	0.063	0.040	0.063

Table 2.

Power analysis of 5 tests assuming the true covariance matrix is Σ₄. We chose the significant level at α = 0.05. For the case n₁ = n₂ = 50, p = 200, we have m = 43, τ_α = 41.918, and γ_α = 3.841. For the case, n₁ = n₂ = 70, p = 1000, we have m = 62, τ_α = 72.318, and γ_α = 3.850. RMPBT is our approach. RAPTT is the approach of Srivastava et al. (2016). BS96 is the approach of Bai and Saranadasa (1996). SD08 is the approach of Srivastava and Du (2008). CQ10 is the approach of Chen et al. (2010).

	p0	RMPBT (R1)	RMPBT (R2)	RAPTT (R1)	RAPTT (R2)	BS96	SD08	CQ10	RMPBT (R1)	RMPBT (R2)	RAPTT (R1)	RAPTT (R2)	BS96	SD08	CQ10
		Alternative 1							Alternative 2
n1 = n2 = 50, p = 200	0.500	0.668	0.653	0.682	0.669	0.644	0.602	0.644	0.534	0.504	0.539	0.489	0.508	0.465	0.508
	0.750	0.642	0.621	0.639	0.606	0.558	0.526	0.558	0.565	0.539	0.561	0.522	0.497	0.447	0.497
	0.800	0.658	0.649	0.645	0.617	0.576	0.535	0.576	0.591	0.569	0.577	0.532	0.528	0.485	0.528
	0.950	0.659	0.627	0.592	0.567	0.521	0.481	0.521	0.644	0.605	0.580	0.537	0.513	0.469	0.513
	0.975	0.688	0.667	0.564	0.538	0.531	0.478	0.531	0.678	0.656	0.556	0.521	0.523	0.473	0.523
	0.990	0.731	0.706	0.513	0.492	0.511	0.464	0.511	0.729	0.694	0.511	0.474	0.509	0.462	0.509
	1.000	0.049	0.049	0.049	0.048	0.057	0.044	0.057	0.049	0.046	0.049	0.042	0.057	0.044	0.057
n1 = n2 = 70, p = 1000	0.500	0.414	0.379	0.407	0.378	0.401	0.353	0.401	0.774	0.717	0.767	0.704	0.761	0.720	0.761
	0.750	0.327	0.294	0.323	0.291	0.331	0.278	0.331	0.790	0.727	0.783	0.711	0.764	0.718	0.764
	0.800	0.348	0.318	0.347	0.304	0.343	0.285	0.343	0.796	0.734	0.788	0.713	0.775	0.728	0.775
	0.950	0.335	0.307	0.319	0.287	0.311	0.270	0.311	0.827	0.776	0.806	0.742	0.782	0.730	0.782
	0.975	0.315	0.278	0.295	0.266	0.294	0.245	0.294	0.836	0.776	0.786	0.720	0.755	0.716	0.755
	0.990	0.335	0.308	0.298	0.261	0.294	0.248	0.294	0.871	0.819	0.769	0.714	0.775	0.738	0.775
	1.000	0.060	0.052	0.056	0.050	0.063	0.045	0.063	0.060	0.040	0.056	0.038	0.063	0.045	0.063

Table 3.

Power analysis of 5 tests assuming the true covariance matrix is Σ₅. We chose the significant level at α = 0.05. For the case n₁ = n₂ = 50, p = 200, we have m = 43, τ_α = 41.918, and γ_α = 3.841. For the second case, n₁ = n₂ = 70, p = 1000, m = 62, τ_α = 72.318, and γ_α = 3.850. RMPBT is our approach. RAPTT is the approach of Srivastava et al. (2016). BS96 is the approach of Bai and Saranadasa (1996). SD08 is the approach of Srivastava and Du (2008). CQ10 is the approach of Chen et al. (2010).

	p0	RMPBT (R1)	RMPBT (R2)	RAPTT (R1)	RAPTT (R2)	BS96	SD08	CQ10	RMPBT (R1)	RMPBT (R2)	RAPTT (R1)	RAPTT (R2)	BS96	SD08	CQ10
		Alternative 1							Alternative 2
n1 = n2 = 50, p = 200	0.500	0.526	0.508	0.541	0.520	0.255	0.216	0.255	0.861	0.845	0.870	0.850	0.481	0.408	0.481
	0.750	0.490	0.468	0.504	0.489	0.214	0.180	0.214	0.921	0.914	0.916	0.900	0.480	0.417	0.480
	0.800	0.505	0.497	0.519	0.502	0.226	0.168	0.226	0.928	0.923	0.918	0.908	0.500	0.409	0.500
	0.950	0.491	0.469	0.468	0.445	0.172	0.128	0.172	0.969	0.957	0.929	0.909	0.466	0.390	0.466
	0.975	0.537	0.517	0.455	0.437	0.192	0.139	0.192	0.969	0.959	0.894	0.882	0.484	0.410	0.484
	0.990	0.587	0.559	0.426	0.415	0.203	0.155	0.203	0.990	0.983	0.841	0.835	0.525	0.452	0.525
	1.000	0.060	0.055	0.075	0.067	0.064	0.047	0.064	0.060	0.054	0.075	0.061	0.064	0.047	0.064
n1 = n2 = 70, p = 1000	0.500	0.295	0.273	0.326	0.304	0.175	0.132	0.175	0.950	0.930	0.954	0.934	0.763	0.695	0.763
	0.750	0.258	0.231	0.290	0.265	0.135	0.089	0.135	0.967	0.953	0.966	0.951	0.755	0.681	0.755
	0.800	0.273	0.242	0.288	0.264	0.133	0.097	0.133	0.966	0.950	0.969	0.954	0.772	0.708	0.772
	0.950	0.260	0.246	0.289	0.266	0.146	0.107	0.146	0.986	0.973	0.971	0.962	0.783	0.699	0.783
	0.975	0.229	0.207	0.248	0.225	0.109	0.078	0.109	0.986	0.978	0.970	0.956	0.767	0.692	0.767
	0.990	0.246	0.229	0.268	0.243	0.122	0.092	0.122	0.988	0.984	0.961	0.939	0.768	0.687	0.768
	1.000	0.096	0.078	0.112	0.103	0.059	0.034	0.059	0.096	0.071	0.112	0.089	0.059	0.034	0.059

This performance differences can be explained as follows. All three non-projection approaches attempt to get around the issue related to inverting an ill-formed sample covariance matrix. In doing so, BS96 based their test statistic on a quadratic norm of the vector sample means, ignoring any possible weighing for the vectors entries. CQ10 develop their test around the cross-product of the sample vectors, also ignoring possible weighing. SD08 also based their test statistic around the square differences of the group sample means, but used the diagonal sample covariance as weights. Consequently, all three approaches tended to only perform well for diagonal or near diagonal structure covariance matrices. On the other hand, because the random projection based approaches make no assumption about the data covariance matrix, they can use the additional information provided in the dependency relationships to improve power. Both random projection based approaches seem to perform similarly under less sparse alternatives, that is, smaller value of p₀.

However, RMPBT tended to have, in most cases, much higher power when compared to RAPTT for cases where the true mean differences are very sparse, especially in the scenario n₁ = n₂ = 50 and p = 200. Note that both RMPBT and RAPTT depend on the F-statistics. We think that the difference observed between the power of RMPBT and RAPTT are not due to the F-statistic and the choice of m per se. Instead, the differences found between RMPBT and RAPTT are due to the way each test quantifies the evidence contained in each F-statistic computed from each projected copies of a data set. For example, consider two arbitrary values, 20 and 1, for the F-statistic with 2 and 3 degrees of freedom. The RMPBT statistic is 0.5, and the equivalent test statistic for the RAPTT is 0.256. Suppose now that the values of the F-statistics are 10 and 0.01 instead. The test statistic for the RMPBT remains unchanged, but the test statistic for RAPTT become 0.518. Because in sparse alternatives scenarios and small sample setting, a large number of F-statistics are reported large (small p-values), and a smaller number is reported small (large p-values), the test statistic in RAPTT would tend to be affected by these few less significant p-values and causing non-rejection. RMPBT does not suffer this problem since the test statistic only relies on a 0–1 decision. In less sparse alternatives however, this discrepancy between RMPBT and RAPTT is less severe, which is reflected in very similar power reported by both approaches.

The Bayes factor based test proposed in this article assumed a non-informative prior for the nuisance parameters with a single scalar hyper-parameter. Other possibilities include proper priors, like a joint normal-inverse-Wishart prior for the nuisance parameters. Specifying such a prior, however, requires a practitioner to carefully choose high dimensional prior hyper-parameters, including a large p×p covariance matrix, a very difficult exercise for most applications. We found the power of the resulting test to be highly sensitive to the choice of the covariance matrix hyper-parameter of the normal-inverse-Wishart prior. Results are deferred to Section S.3 of the Supplemental Material.

5 Applications

5.1 Colon organoids data

Stem cells have some unique regenerative abilities and offer new potential to treating chronic diseases. But stem cells are often modulated by many factors, like the aryl hydrocarbon receptors (AhR), which are not well understood. A study was designed to examine the effect of the AhR on intestinal stem cells. Intestinal crypts were isolated from one mouse, plated, cultivated, separated in 4 sets of 3 plates and each set was treated with one of the following 4 treatments: TCDD only, Indole only, TCDD+Indole, DMSO (control). TCDD is a cancer inducing agent and has the effect of changing the expressions of many genes by activating the AhR. Indole has the role of modulating the effect of TCDD whereas DMSO has an anti-inflammatory property. Finally, RNA is isolated from each of the 12 organoids and sequenced. A gene-by-gene comparison between the TCDD only group versus TCDD+Indole group resulted in only 6 differentially expressed (DE) genes after adjusting for multiple comparisons (McCarthy et al., 2012). We use RMPBT to compare the expression of p = 2000 genes simultaneously between TCDD only and TCDD+Indole groups, with n₁ = 3 and n₂ = 3. Before we apply the tests, we take a log₂ transformation of the gene expressions (after adding one to avoid issues with log₂ of 0). In this set of 2000 genes, all the 6 DE genes previously found are included. For RMPBT, we use m = 2, N = 100000 random projections, ${\varphi }_{\alpha }^0=0.054$, τ_α = 0.175, and γ_α = 4.302. Based on the results in Table 4, RMPBT is the only test that reports significance, when the random projection matrix is R₁. This is consistent with the findings in the simulation where RMPBT showed high power for sparse alternative when the projection matrix was R₁. The next smallest p-value is also produced by RMPBT with the projection matrix R₂.

Table 4.

Summary of the analysis of 3 data sets: ORGNDS=organoids, BC=breast cancer, and SRBCT=small round blue cell tumors. In each case, we assume a significance level of α = 0.05 and report the probability of exceeding the test statistic obtained based on the data under the null (p-value). RMPBT is our approach. RAPTT is the approach of Srivastava et al. (2016). BS96 is the approach of Bai and Saranadasa (1996). SD08 is the approach of Srivastava and Du (2008). CQ10 is the approach of Chen et al. (2010).

Data Set	(n1, n2)	Subset	p	RMPBT (R1)	RMPBT (R2)	RAPTT (R1)	RAPTT (R2)	BS96	SD08	CQ10
ORGNDS	(3, 3)		2000	0.0058	0.212	0.1392	0.510	0.6309	-	0.6309
BC	(111, 57)	Chromosome 1	374	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.19	0.11	0.23
		Chromosome 2	233	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.012	0.043	0.022
		Chromosome 12	191	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.0004	0.026	0.003
	(37, 19)	Chromosome 1	374	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.416	0.371	0.466
		Chromosome 2	233	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.223	0.273	0.327
		Chromosome 12	191	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.135	0.266	0.304
SRBCT	(11, 18)		2308	< 0.0001	< 0.0001	< 0.0001	< 0.0001	< 0.0001	< 0.0001	< 0.0001

5.2 Breast cancer data

We apply the proposed method to the analysis of a breast cancer data set reported by Gravier, Eleonore et al. (2010). The study investigates the involvement of small, invasive ductal carcinoma without lymph (T1T2N0) in predicting metastasis of small node-negative breast carcinoma. Gene expression levels of around 2905 genes were reported for 168 patients over five years. Of the 168, n₁ = 111 patients with no event after diagnosis were labelled good and the remaining n₂ = 57 with early metastasis were labelled poor. We performed three gene-set comparisons between the two groups, good and poor. The gene-sets were similar to the sets compared in Thulin (2014). The first gene set had p = 374 genes located on Chromosome 1. The second set had p = 233 genes located on Chromosome 2 and the third set had p = 191 genes located on Chromosome 12. A restricted most powerful test 15 is obtained by choosing τ_α = 86.880 and evidence threshold γ_α = 3.852 with m = 75 and N = 10000 random projections. The values of ${\varphi }_{\alpha }^0$, the cutoff values for the test statistic in Chromosome 1, 2 and 12 are 0.127, 0.165, and 0.175 respectively. We also compared both groups using RAPTT, BS96, SD08 and CQ10. The results reported in Table 4 indicate that RMPBT and RAPTT found significance in each gene set, but the non-random projection based approaches failed to find significance for the genes on Chromosome 1.

To investigate the impact of smaller sample sizes on the performance of each test, we compare again the two groups, now by only considering one-third of the total samples (n₁ = 37 and n₂ = 19). We run the tests on 100 independently sampled data sets from the original data set. For all three chromosomes, the median (over the 100 p-values) p-value for RMPBT (R₁), RMPBT (R₂), RAPTT (R₁), RAPTT (R₂) are all highly significant, whereas the p-values for BS96, SD08 and CQ10 are highly insignificant (see Table 4). We see that the approaches based on random projection are able to detect differences between both groups even for relatively smaller sample sizes. However, the other approaches fail to detect difference between both groups for smaller samples sizes.

5.3 SRBCT data

We finally apply our approach to the small round blue cell tumors (SRBCTs) data set which is available at http://www.biolab.si/supp/bi-cancer/projections/info/SRBCT.htm. SRBCT’s are comprised of 4 different childhood tumors. In this exercise, we would like to test for equality of expression mean of the genes between the neuroblastoma(NB) and the Burkitt’s lymphoma(BL) tumors group. The data contain p = 2308 gene expression for both NB and BL tumors with sample sizes 11 and 18 respectively. We estimate ${\varphi }_{\alpha }^0=0.0604$, τ_α = 4.836, γ_α = 3.720, and N = 10000. We report the p-values for each of these tests (see Table 4 SRBCT part). All the tests rejected the null hypothesis with high significance.

6 Conclusion

In this article, we proposed a Bayes factor based test for differences between group means in high dimensions. We transformed the data points to lower dimensional spaces using random projections. Using the transformed data, we obtained a closed form Bayes factor by carefully choosing the priors for the model parameters, involving a single scalar hyper-parameter. The hyper-parameter was chosen to obtain a restricted most powerful Bayesian test (RMPBT). Our final test was based on an ensemble of Bayes factors obtained from multiple projected copies. We showed unbiasedness and consistency of the proposed test under mild conditions. We illustrated the efficacy of the test in real and simulated examples.

An ongoing extension of the proposed test also considers non-local priors of Johnson and Rossell (2010) for the distribution of δ under the alternative and relaxes the assumption of equal covariance matrices between the two groups.

Figure 2.

Empirical distribution function of ϕ̄(N) under the null hypothesis for 5 different covariance matrices based on N = 50000 random projections and 1000 data sets.

Appendix

Appendix A Lemma 1 - Derivation of the Bayes factor

We recall that the minimal sufficient statistics are D = Ȳ−X̄, A = (n₁Y +n₂X)/(n₁+n₂) and $\mathit{S}=\{(n_1-1){\sum }_{i=1}^{n_1}({\mathit{X}}_i-\overline{\mathit{X}}){({\mathit{X}}_i-\overline{\mathit{X}})}^{\mathrm{T}}+(n_2-1){\sum }_{i=1}^{n_2}({\mathit{Y}}_i-\overline{\mathit{Y}}){({\mathit{Y}}_i-\overline{\mathit{Y}})}^{\mathrm{T}}\}/(n_1+n_2-2)$. Also, $\mathit{D}~{\mathit{N}}_p(\mathit{\delta },n_0^{-1}\mathbf{\sum })$, A ~ N_p(μ, n⁻¹Σ) and (n − 2)S ~ W_p(n − 2,Σ), independently, where δ = μ₂ − μ₁, μ = (n₁μ₁ + n₂μ₂)/n, $n_0^{-1}=n_1^{-1}+n_2^{-1}$, n = n₁ + n₂. Under our assumed framework, the joint distribution of the data in terms of the minimal sufficient statistics D, A and S is given by

$$P(\mathit{Data}\mid \mathit{\mu },\mathit{\delta },\mathbf{\sum })={\mathit{N}}_p(\mathit{D}\mid \mathit{\delta },n_0^{-1}\mathbf{\sum }){\mathit{N}}_p(\mathit{A}\mid \mathit{\mu },n^{-1}\mathbf{\sum }){\mathit{W}}_p\{(n-2)\mathit{S}\mid n-2,\mathbf{\sum }\}.$$

We assume the following joint prior for μ and Σ as π(μ,Σ) ∝ |Σ|^−(p+1)/2. Under H₁, we choose the following conditional prior for δ as δ | Σ ~ N_p(0,Σ/τ₀). Under H₁, we have

$$P(\mathit{D}\mid \mathbf{\sum })=\int {\mathit{N}}_p(\mathit{D}\mid \mathit{\delta },n_0^{-1}\mathbf{\sum }){\mathit{N}}_p(\mathit{\delta }\mid \mathbf{0},\mathbf{\sum }/{\tau }_0)d\mathit{\delta }={\mathit{N}}_p(\mathit{D}\mid \mathbf{0},n_{\tau }^{-1}\mathbf{\sum }),$$

where $n_{\tau }^{-1}={\tau }_0^{-1}+n_0^{-1}$. If we denote the marginal distribution of the data under H₁ by m₁(Data), then we have

$$\begin{gathered} m_1(\mathit{Data})\propto \int {\mathit{N}}_p(\mathit{D}\mid \mathbf{0},n_{\tau }^{-1}\mathbf{\sum }){\mathit{N}}_p(\mathit{A}\mid \mathit{\mu },n^{-1}\mathbf{\sum }){\mathit{W}}_p\{(n-2)\mathit{S}\mid n-2,\mathbf{\sum }\}{\mid \mathbf{\sum }\mid }^{-(p+1)/2}d\mathit{\mu }d\mathbf{\sum } \\ \propto \int {\mathit{N}}_p(\mathit{D}\mid \mathbf{0},n_{\tau }^{-1}\mathbf{\sum }){\mathit{W}}_p\{(n-2)\mathit{S}\mid n-2,\mathbf{\sum }\}{\mid \mathbf{\sum }\mid }^{-(p+1)/2}d\mathbf{\sum }, \end{gathered}$$

where the second line is obtained by taking the integral with respect to μ. This integral only involves N_p(μ | A, n⁻¹Σ) and evaluates to 1. With a little bit of algebra, we get that

$$m_1(\mathit{Data})=A_0{n}_{\tau }^{p/2}{\left(1+\frac{n_{\tau }}{n-2}{\mathit{D}}^{\mathrm{T}}{\mathit{S}}^{-1}\mathit{D}\right)}^{-(n-1)/2},$$

where A₀ represents a quantity independent of the data. Similarly, under H₀, δ = 0 and we can show that the marginal distribution of the data, denoted m₀(Data), is

$$m_0(\mathit{Data})=A_0{n}_0^{p/2}{\left(1+\frac{n_0}{n-2}{\mathit{D}}^{\mathrm{T}}{\mathit{S}}^{-1}\mathit{D}\right)}^{-(n-1)/2}.$$

Therefore, the Bayes factor in favor of the alternative is

$$B{F}_{10}={\left(\frac{n_{\tau }}{n_0}\right)}^{p/2}{\left(\frac{1+\frac{n_{\tau }}{n-2}{\mathit{D}}^{\mathrm{T}}{\mathit{S}}^{-1}\mathit{D}}{1+\frac{n_0}{n-2}{\mathit{D}}^{\mathrm{T}}{\mathit{S}}^{-1}\mathit{D}}\right)}^{-(n-1)/2}.$$

Setting η = n₀/τ₀ and $f=\frac{(n-p-1)}{(n-2)p}{n}_0{(\overline{\mathit{Y}}-\overline{\mathit{X}})}^{\mathrm{T}}{\mathit{S}}^{-1}(\overline{\mathit{Y}}-\overline{\mathit{X}})$, we get Equation (4).

Appendix B Proof of Theorem 1

• Part(a) For 1 < m < n− 2, we integrate out the parameters with respect to the conjugate priors to obtain the Bayes factor in favor of the alternative as

  $$B{F}_{10}({\mathit{X}}^{★},{\mathit{Y}}^{★})={(1+\eta )}^{-m/2}{\left\{1-\frac{m{f}^{★}\eta /(1+\eta )}{m{f}^{★}+n-m-1}\right\}}^{-(n-1)/2},$$
  where

  $$f^{★}=\frac{n-m-1}{(n-2)m}{n}_0{(\overline{\mathit{Y}}-\overline{\mathit{X}})}^{\mathrm{T}}\mathit{R}{(\mathit{RS}{\mathit{R}}^{\mathrm{T}})}^{-1}{\mathit{R}}^{\mathrm{T}}(\overline{\mathit{Y}}-\overline{\mathit{X}}).$$
  Recall that n = n₁+n₂, 1/n₀ = 1/n₁+1/n₂, η = n₀/τ₀, and n_min = min{n₁, n₂}. Since τ₀ is fixed, η → ∞ as n_min → ∞. For a randomly chosen R, under H₀, f^★ ~ F_m,n−m−1 with m and n − m − 1 degrees of freedom. Thus, f^★ = O_p(1). Also, from well-known properties of the F distribution, we have that

  $$U=\frac{m{f}^{★}/(n-m-1)}{\{m{f}^{★}/(n-m-1)+1\}}=\frac{m{f}^{★}}{(m{f}^{★}+n-m-1)}~\text{Beta}\{m/2,(n-m-1)/2\},$$
  where Beta(a, b) denotes a Beta distribution. Therefore, {η/(1+η)}U = O_p(1). Hence, log {1 − ηU/(1 + η)} = O_p(1) as n_min →∞. We then get

  $$-\frac{m}{2n}log(1+\eta )-\frac{(n-1)}{2n}log\{1-\eta U/(1+\eta )\}\stackrel{p}{\to }-\infty ,$$

  since log(1+η)→ ∞ as n_min → ∞ and lim_nmin→∞ m/n = θ ∈ (0, 1). We conclude that $log\{B{F}_{10}(\mathit{R})\}\stackrel{p}{\to }-\infty$ under the null hypothesis.

  Under the alternative, μ₁ ≠ μ₂ and δ ~ N_p(0,Σ/τ₀). Then, f^★ | λ ~ F_m,n−m−1(λ) with non-centrality λ = n₀δ^TR(R^TΣR)⁻¹R^Tδ. Since δ ~ N_p(0,Σ/τ₀), $\lambda ~n_0{\chi }_m^2/{\tau }_0$, where ${\chi }_m^2$ denotes a χ² distribution with m degrees of freedom. The non-centrality parameter depends on n through n₀. We can show that the unconditional distribution of f^★/(1+η) ~ F_m,n−m−1 (see Johnson, 2005, page 704). If we denote f⁰ = f^★/(1+η), we have f⁰ = O_p(1), and mf⁰/n = O_p(1), as n_min →∞. We have that

  $$\frac{\eta U}{(1+\eta )}=\frac{m{f}^0\eta }{m{f}^0(1+\eta )+n-m-1}$$
  From the above equation, we get

  $$-log\left\{1-\frac{\eta U}{(1+\eta )}\right\}=log\left\{\frac{m{f}^0(1+\eta )/(n-m-1)+1}{\{m{f}^0/(n-m-1)\}+1}\right\}.$$
  Since f⁰ = O_p(1), and m/(n − m − 1) converges, we have

  $$-log\left\{1-\frac{\eta U}{(1+\eta )}\right\}\stackrel{p}{\to }\infty .$$

  We conclude that $log\{B{F}_{10}(\mathit{R})\}\stackrel{p}{\to }\infty$, under the alternative hypothesis.

• Part(b) We now assume that η → 0 and mη → ∞. We have

  $$log\{B{F}_{10}({\mathit{X}}^{★},{\mathit{Y}}^{★})\}=\frac{n}{2}\left(1-\frac{m}{n}\right)log(1+\eta )-\frac{n}{2}log\{1+\eta (1-U)\}+\frac{1}{2}log\left\{1-\frac{\eta U}{1+\eta }\right\},$$

  where U ~ Beta {m/2, (n − m − 1)/2} under H₀. For large n, none of the terms with n dominates and their difference converges. The distribution log{BF₁₀(X^★,Y^★)} then depends on that of U, which is bounded in probability. Therefore, under H₀, log{BF₁₀(X^★,Y^★)} = 𝒪_p(1).

  Under $H_1^n$, again we have

  $$log\{B{F}_{10}({\mathit{X}}^{★},{\mathit{Y}}^{★})\}=-\frac{m}{2}log(1+\eta )-\frac{(n-1)}{2}log\left\{1-\frac{\eta U}{1+\eta }\right\},$$

  where $U\stackrel{p}{\to }1$ with $f^{\ast }\stackrel{p}{\to }\infty$. Since log(1 + η){(n − 1)/2 − m/2} → ∞, we conclude that $log\{B{F}_{10}({\mathit{X}}^{★},{\mathit{Y}}^{★})\}\stackrel{p}{\to }\infty$.

Appendix C Proof of Theorem 2

In this proof, we denote m(n), τ_α(n) and γ_α(n) simply as m_n, τ_n and γ_n, respectively. That the projection matrix R, the Bayes factor, the variable f^★, their distributions etc. all depend on m_n is implicitly understood and hence m_n is suppressed in their notation.

• Part(a) Under H₀, and some chosen values of α and m_n, we have

  $$E\{\varphi (\mathit{R})\}=E_{\mathit{R}}\left[P_{\mathit{X},\mathit{Y}}\{B{F}_{10}(\mathit{XR},\mathit{YR})>{\gamma }_n\mid \mathit{R}\}\right]=E_{\mathit{R}}(\alpha )=\alpha ,$$

  making use of the results in Section 2.1 and noting that P_X,Y {BF₁₀(XR,YR) > γ_n} is computed over the data generating model under the null hypothesis.

• Part(b) Under the alternative hypothesis, δ = δ₁ ≠ 0, and f^★ | λ ~ F_{mn,n−mn−1}(λ) distribution, with degrees of freedoms m_n and n − m_n − 1 and non-centrality parameter $\lambda =n_0{\mathit{\delta }}_1^{\mathrm{T}}\mathit{R}{({\mathit{R}}^{\mathrm{T}}\mathbf{\sum }\mathit{R})}^{-1}{\mathit{R}}^{\mathrm{T}}{\mathit{\delta }}_1$. Hence, under the alternative hypothesis P{BF₁₀(XR,YR) > γ_α} = P(f^★ > F_{α,mn,n−mn−1}) ≥ α, where F_{α,mn,n−mn−1} is the α upper quantile of a F_{mn,n−mn−1} distribution, as seen in Section 2.1. Marginalizing over δ₁ under the alternative, f^★/(1+η_n) ~ F_{mn,n−mn−1}, where η_n = n₀/τ_n. Therefore, α = P{f^★/(1 + η_n) > F_{α,mn,n−mn−1}} = P{f^★ > (1 + η_n)F_{α,mn,n−mn−1}} ≤ P(f^★ > F_{α,mn,n−mn−1}). We conclude E_R{ϕ(R) | H₁} ≥ α.

• Part(c) First, we show that our construction of m_n satisfies m_n/n → θ ∈ (0, 1). For chosen n, F_{α,m,n−m−1} is a convex function over the range of possible values of m, suggesting that m_n and n − m_n both diverge. See Figure 3.

  The mean μ_n and variance ${\sigma }_n^2$ of an F_{mn,n−mn−1} distribution are given by ${\mu }_n=\frac{n-m_n-1}{n-m_n-3}>1$ and ${\sigma }_n^2=\frac{2{(n-m_n-1)}^2(n-3)}{m_n(n-m_n-5){(n-m_n-3)}^2}$. For large m_n and n − m_n, we have F_{α,mn,n−mn−1} ≈ μ_n + Σ_nΦ⁻¹(α), where Φ⁻¹(α) is the upper α-quantile of the standard normal distribution. So F_{α,mn,n−mn−1} is at its minimum when Σ_n is at its minimum, which happens when $m_n\approx \frac{n-5}{2}$. Hence, m_n/n → θ = 1/2 ∈ (0, 1) as n_min → ∞. Second, we show that nη_n → ∞ as n_min → ∞. From equation (8), we have

  $$C_n=\frac{m_n{F}_{\alpha ,m_n,n-m_n-1}}{m_n{F}_{\alpha ,m_n,n-m_n-1}+n-m_n-1}.$$ (A.1)
  We have

  $$\begin{gathered} n{\eta }_n=\frac{n{n}_0}{{\tau }_n}=\frac{n(n-1)\{C_n-m_n/(n-1)\}}{m_n(1-C_n)}=n(n-1)\left\{\frac{C_n}{m_n(1-C_n)}-\frac{1}{(n-1)(1-C_n)}\right\} \\ =n(n-1)\left\{\frac{F_{\alpha ,m_n,n-m_n-1}}{(n-m_n-1)}-\frac{m_n{F}_{\alpha ,m_n,n-m_n-1}+n-m_n-1}{(n-1)(n-m_n-1)}\right\} \\ =\frac{n}{(n-m_n-1)}\{(n-m_n-1)F_{\alpha ,m_n,n-m_n-1}-(n-m_n-1)\} \\ =n(F_{\alpha ,m_n,n-m_n-1}-1). \end{gathered}$$
  The variance of a central F distribution with m_n and n−m_n −1 degrees of freedom is 2(n−m_n−1)²(n−3)/{m_n(n−m_n−3)²(n−m_n−5)} = O(1/m_n). The convergence of F_{α,mn,n−mn−1} is thus slower than $1/\sqrt{n}$. We conclude that nη_n and hence m_nη_n → ∞. For a chosen α and the sequence of alternatives $H_1^n$, we have

  $$\begin{gathered} \underset{n_{min}\to \infty }{lim}{E}_{\mathit{R}}\varphi (\mathit{R}\mid H_1^n)=E_{\mathit{R}}\left[\underset{n_{min}\to \infty }{lim}{P}_{\mathit{X},\mathit{Y}}\{B{F}_{10}(\mathit{XR},\mathit{YR})>{\gamma }_{\alpha }(n)\mid \mathit{R},H_1^n\}\right] \\ =E_{\mathit{R}}\left[\underset{n_{min}\to \infty }{lim}{P}_{\mathit{X},\mathit{Y}}\{f^{★}>F_{\alpha ,m_n,n-m_n-1}\mid \mathit{R},H_1^n\}\right]. \end{gathered}$$

  Since $f^{★}\stackrel{p}{\to }\infty$ and F_{α,mn,n−mn−1} → 1, we conclude that ${lim}_{n_{min}\to \infty }{E}_{\mathit{R}}\varphi (\mathit{R}\mid H_1^n)=1$.

Figure 3.

Plot of F_{α,m,n−m−1} against m = 1, 2, · · ·, n − 2 for different values of n = n₁ + n₂. The arrows point to values of m_n obtained by our method for different n.

Appendix D Proof of Theorem 3

The power of RMBPT is $P\{\overline{\varphi }(N)>{\varphi }_{\alpha }^0\mid H_1^n\}$. Henceforth, we make it explicit that ${\varphi }_{\alpha }^0$ depends on (n₁, n₂) and write ${\varphi }_{\alpha }^0(n_1,n_2)$ instead.

For given n₁, n₂ and α, we choose ${\varphi }_{\alpha }^0(n_1,n_2)$ so that $P\{\overline{\varphi }(N)>{\varphi }_{\alpha }^0(n_1,n_2)\mid H_0\}=\alpha$. Since $0<{\varphi }_{\alpha }^0(n_1,n_2)<1$, for 0 < α < 1, we have that $P\{{\sum }_{i=1}^N\varphi ({\mathit{R}}_i)\ge 0\mid H_1^n\}\ge P\{{\sum }_{i=1}^N\varphi ({\mathit{R}}_i)>N{\varphi }_{\alpha }^0(n_1,n_2)\mid H_1^n\}\ge P\{{\sum }_{i=1}^N\varphi ({\mathit{R}}_i)\ge N\mid H_1^n\}$.

We have that $P\{\varphi ({\mathit{R}}_i)=1\mid H_1^n\}\to 1$ as n_min → ∞, under the alternative for i = 1, · · ·, N. So, $P\{{\sum }_{i=1}^N\varphi ({\mathit{R}}_i)\ge 0\mid H_1^n]=1-{\prod }_{i=1}^{N}P\{\varphi ({\mathit{R}}_i)=0\mid H_1^n\}\to 1$. Additionally, $P\{{\sum }_{i=1}^N\varphi ({\mathit{R}}_i)\ge N\mid H_1^n\}=P\{{\sum }_{i=1}^N\varphi ({\mathit{R}}_i)=N\mid H_1^n\}={\prod }_{i=1}^{N}P\{\varphi ({\mathit{R}}_i)=1\mid H_1^n\}\to 1$ for fixed N as n_min →∞. We conclude that $P\{\overline{\varphi }(N)\ge {\varphi }_{\alpha }^0\mid H_1^n\}\to 1$ as n_min →∞.

Appendix E Proof of Theorem 4

The proof is similar to that of Theorem 2 of Srivastava et al. (2016) except that we do not rely on the cumulative distribution function of the F distribution.

Suppose R₁, R₂, · · ·, R_N is a collection independently sampled projection matrices. Let ϕ_i = ϕ(R_i) = 1{BF₁₀(R_i, τ_α) > γ_α} for i = 1, 2, · · ·, N. Recall that $\overline{\varphi }(N)={\sum }_{i=1}^N{\varphi }_i/N$. Evaluating the conditional probability that ϕ̄(N) < x over the distribution of the random projection matrices given X and Y and then taking the expectation over the data, we get

$$P\{\overline{\varphi }(N)<x\}=E_{\mathit{X},\mathit{Y}}[P_{\mathit{R}}\{\overline{\varphi }(N)<x\mid \mathit{X},\mathit{Y}\}].$$ (A.2)

We have

$$P_{\mathit{R}}\{\overline{\varphi }(N)<x\mid \mathit{X},\mathit{Y}\}=P_{\mathit{R}}\left\{\frac{\overline{\varphi }(N)-E_{\mathit{R}}({\varphi }_1\mid \mathit{X},\mathit{Y})}{\sqrt{{var}_{\mathit{R}}({\varphi }_1\mid \mathit{X},\mathit{Y})/N}}<\frac{x-E_{\mathit{R}}({\varphi }_1\mid \mathit{X},\mathit{Y})}{\sqrt{{var}_{\mathit{R}}({\varphi }_1\mid \mathit{X},\mathit{Y})/N}}\mid \mathit{X},\mathit{Y}\right\},$$

where E_R(ϕ₁ | X,Y ) and var_R(ϕ₁ | X,Y ) are respectively the conditional mean and variance of ϕ₁. Given X and Y, the binary variables ϕ_i, i = 1, · · ·, N, are independent and identically distributed with finite mean and variance. By the Central Limit Theorem,

$$\underset{N\to \infty }{lim}\left[P_{\mathit{R}}\{\overline{\varphi }(N)<x\mid \mathit{X},\mathit{Y}\}-\mathrm{\Phi }\left\{\frac{x-E_{\mathit{R}}({\varphi }_1\mid \mathit{X},\mathit{Y})}{\sqrt{{var}_{\mathit{R}}({\varphi }_1\mid \mathit{X},\mathit{Y})/N}}\right\}\right]=0$$ (A.3)

where Φ(a) is the standard normal cumulative distribution function evaluated at a. We need to show that both E_R(ϕ₁ | X,Y ) and var_R(ϕ₁ | X,Y ) have a distribution independent of μ₁, μ₂ and Σ under H₀. This is equivalent to showing that E_X,Y {E_R(ϕ₁ | X,Y )}^r is independent of μ₁, μ₂, and Σ, for r = 1, 2, · · ·. From (7), we have

$$E_{\mathit{R}}({\varphi }_1\mid \mathit{X},\mathit{Y})=P_{\mathit{R}}\{f({\mathit{R}}_1\mid \mathit{X},\mathit{Y})>F_{\alpha ,m_n,n-m_n-1}\},$$

where $f(\mathit{R}\mid \mathit{X},\mathit{Y})=\frac{n-m_n-1}{(n-2)m_n}{n}_0{(\overline{\mathit{Y}}-\overline{\mathit{X}})}^{\mathrm{T}}\mathit{R}{({\mathit{R}}^{\mathrm{T}}\mathit{SR})}^{-1}{\mathit{R}}^{\mathrm{T}}(\overline{\mathit{Y}}-\overline{\mathit{X}})$. P_R is a probability measure associated with the random projection matrices. Thus, the r^th moment of E_R{ϕ₁ | X,Y )} is expressed as

$$E_{\mathit{XY}}{\{E_{\mathit{R}}({\varphi }_1\mid \mathit{X},\mathit{Y})\}}^r=\int {\left[\int \mathbf{1}\{f(\mathit{R}\mid \mathit{X},\mathit{Y})>F_{\alpha ,m_n,n-m_n-1}\}d{P}_{\mathit{R}}\right]}^{r}d{P}_{\mathit{X},\mathit{Y}}.$$ (A.4)

Since 0 ≤ E_R(ϕ₁ | X,Y ) ≤ 1, we have

$$\int \left(\int {[\mathbf{1}\{f(\mathit{R}\mid \mathit{X},\mathit{Y})>F_{\alpha ,m_n,n-m_n-1}\}]}^{r}d{P}_{\mathit{R}}\right)d{P}_{\mathit{X},\mathit{Y}}=\int \left(\int {[\mathbf{1}\{f(\mathit{R}\mid \mathit{X},\mathit{Y})>F_{\alpha ,m_n,n-m_n-1}\}]}^{r}d{P}_{\mathit{X},\mathit{Y}}\right)d{P}_{\mathit{R}},$$ (A.5)

where we can safely interchange the order of integration by Fubini’s Theorem. Under H₀, f(R | X,Y ) ~ F_{mn,n−mn−1} and P_X,Y {f(R | X,Y ) > F_{α,mn,n−mn−1}} = α. We conclude that (A.5) is independent of μ₁, μ₂, and Σ for any positive integer r.

Next, from (A.4), we have

$$E_{\mathit{XY}}{\{E_{\mathit{R}}({\varphi }_1\mid \mathit{X},\mathit{Y})\}}^r=\int \cdots \int \left[\int \prod _{i=1}^r\mathbf{1}\{f({\mathit{R}}_i\mid \mathit{X},\mathit{Y})>F_{\alpha ,m_n,n-m_n-1}\}d{P}_{\mathit{X},\mathit{Y}}\right]\prod _{i=1}^{r}d{P}_{{\mathit{R}}_i},$$ (A.6)

where we can again safely exchange the order of integration using Fubini’s Theorem. In (A.6), since R_i, i = 1, · · ·, N, are identically and independently distributed with respect to the probability measure P_R, we get that

$$\int \prod _{i=1}^r\mathbf{1}\{f({\mathit{R}}_i\mid \mathit{X},\mathit{Y})>F_{\alpha ,m_n,n-m_n-1}\}d{P}_{\mathit{X},\mathit{Y}}$$

is also independent of μ₁, μ₂, and Σ based on the result obtained in (A.5).

Also, we have

$$\left|P_{\mathit{R}}\{\overline{\varphi }(N)<x\mid \mathit{X},\mathit{Y}\}-\mathrm{\Phi }\left\{\frac{x-E_{\mathit{R}}({\varphi }_1\mid \mathit{X},\mathit{Y})}{\sqrt{{var}_{\mathit{R}}({\varphi }_1\mid \mathit{X},\mathit{Y})/N}}\right\}\right|\le 2.$$ (A.7)

Using (A.2), (A.3), (A.7) and the bounded convergence theorem, we have

$$\underset{N\to \infty }{lim}\left(P\{\overline{\varphi }(N)<x\}-E_{\mathit{X},\mathit{Y}}\left[\mathrm{\Phi }\left\{\frac{x-E_{\mathit{R}}({\varphi }_1\mid \mathit{X},\mathit{Y})}{\sqrt{{var}_{\mathit{R}}({\varphi }_1\mid \mathit{X},\mathit{Y})/N}}\right\}\right]\right)=0$$ (A.8)

We conclude that for n₁, n₂, m_n, and α, the asymptotic distribution of ϕ̄(N) as N → ∞ does not depend on the true parameters μ₁, μ₂, and Σ under H₀.