A Random Projection Approach to Hypothesis Tests in High-Dimensional Single-Index Models

Abstract In this article, we consider the problem of hypothesis testing in high-dimensional single-index models. First, we study the feasibility of applying the classical F-test to a single-index model when the dimension of covariate vector and sample size are of the same order, and derive its asymptotic null distribution and asymptotic local power function. For the ultrahigh-dimensional single-index model, we construct F-statistics based on lower-dimensional random projections of the data, and establish the asymptotic null distribution and the asymptotic local power function of the proposed test statistics for the hypothesis testing of global and partial parameters. The new proposed test possesses the advantages of having a simple structure as well as being easy to compute. We compare the proposed test with other high-dimensional tests and provide sufficient conditions under which the proposed tests are more efficient. We conduct simulation studies to evaluate the finite-sample performances of the proposed tests and demonstrate that it has higher power than some existing methods in the models we consider. The application of real high-dimensional gene expression data is also provided to illustrate the effectiveness of the method. Supplementary materials for this article are available online.


Introduction
Modern data applications often lead the problem to a highdimensional regime, where the data dimension p can be larger than the sample size n.This phenomenon brings challenges to the classical statistical testing procedures, even in many basic settings.For example, the classical F-test for the linear model is not well-defined when p is larger than n.In fact, even when p < n, the F-test is adversely affected by the increasing dimension of the predictor and does not perform well when the dimension of the predictor is comparable to the sample size.Therefore, it is important to develop new testing procedures in highdimensional models.
In recent years, many efforts have been devoted to the problem of hypothesis testing in high-dimensional models.For the generalized linear models (GLM), a global test based on empirical Bayes method was proposed by Goeman, van Houwelingen, and Finos (2011), which allows p to be greater than n.Later, Guo and Chen (2016) proved that this test is feasible with diverging p and modified it to gain more power.Zhong and Chen (2011) considered a linear model and proposed a global test based on an U-statistic.This approach was further studied in Cui, Guo, and Zhong (2018) by implementing a new variance estimation method of Fan, Guo, and Hao (2012).Under the sparsity assumption, the tests based on the debiased method have been developed for many testing problems.In particular, statistical tests for single or low-dimensional components of the regression coefficients in linear models (Zhang and Zhang 2014), GLMs (Van de Geer et al. 2014) and general models (Ning and Liu 2017) were developed.For the high-dimensional global testing problem, Zhang and Cheng (2017) and Ma, Cai, and Li (2020) proposed tests in linear and logistic models, respectively.The technique of random projection was used in several highdimensional testing problems, including independence testing (Huang and Huo 2017), two-sample testing (Lopes, Jacob, and Wainwright 2011) and nonparametric testing (Liu, Shang, and Cheng 2018).An important advantage of the method based on random projection is its ability to preserve the significant information in data while reducing its dimension.
In this article, we consider the single-index model (SIM) where y is a response variable, x is a p × 1 covariate vector, β is a p × 1 vector of unknown coefficients, is a random error independent of x, and f is an unspecified link function.We are interested in testing the hypothesis Many commonly used parametric and semi-parametric models are included in SIM, such as linear, generalized linear and Cox models.However, the high flexibility of model assumption increases the difficulty to make statistical inference, especially in high-dimensional problems.In the classical settings, Li and Duan (1989) demonstrated that maximum likelihoodtype estimators are consistent for β up to a scalar, even though a misspecified link function f might be assumed in model (1.1).In addition, with the assumption that x had an elliptically symmetric distribution, Li and Duan (1989) showed that the Wald test as well as the likelihood ratio test were workable.Recently, this approach was considered in the usage of LASSO to the nonlinear model (1.1) by Neykov, Liu, and Cai (2016) and Thrampoulidis, Abbasi, and Hassibi (2015), where effective sparse recovery and explicit expressions for the mean-squared-error were obtained, respectively.In addition, Thrampoulidis, Abbasi, and Hassibi (2015) illustrated that the estimation performance of the generalized LASSO in the nonlinear model is asymptotically the same as that of the linear model.However, there is little systematic research on hypothesis testing for high-dimensional SIM.
We propose new statistical tests for hypothesis (1.2) in the high-dimensional single index model (1.1).First, the testing problem is investigated in a relatively high-dimensional regime, where p/n → ζ with ζ ∈ (0, 1) and x is assumed to follow the normal distribution.According to the results in Li and Duan (1989), the vector of coefficients in SIM (1.1) can be obtained by the least square up to a scalar.Motivated by this, we propose the F-statistic, whose asymptotic normality and asymptotic local power function are derived in high-dimensional SIM.While this approach seems to ignore the nonlinear link function f for simplification, the theoretical results are asymptotically the same as one working on the linear model y = c 0 x β + σ e, where e is the standard normal error, and c 0 , σ are constants depending on f .To investigate more high-dimensional settings, we study testing problem (1.2) in an ultrahigh-dimensional regime, where p can be much larger than n.The technique of random projection is used to reduce the data dimension, and the F-statistics is constructed based on the projected data which live in a lower-dimensional space.We prove that the proposed test statistic is asymptotically normal under the null hypothesis as (n, p) → ∞.We also derive the asymptotic local power function of the proposed test.With no extra sparsity assumption required, our proposed test has a wide application range in terms of general model assumption and mild conditions on the distribution.And the test is still applicable to misspecified models.In addition, it is simple in form and easy to compute.Finally, we extend the proposed testing procedures for global hypothesis (1.2) to the problem of testing partial coefficients and derive their asymptotic null distributions and asymptotic local power functions.
Testing whether a response depends on certain covariates is an important problem in regression analysis.In particular, testing the global null hypothesis can help form a holistic view about the relationship between the response and covariates.Once a significant relationship is confirmed, subsequent testing of the significance of a subset of covariates or individual covariates can proceed.For high-dimensional data, testing the significance of a large subset of covariates can reduce data dimension.This is because when a large subset of covariates has no significant relationship with the response, then the analysis can focus on the remaining covariates.Since a linear model assumption is restrictive, it is of great interest to develop tests for global null hypothesis under a nonlinear model such as the single-index model.
The rest of this article is organized as follows.In Section 2, we introduce the model and establish the theoretical foundation for the design of our proposed methods.In Section 3, we focus on a relatively high-dimensional regime, where we derive the asymptotic normality and the asymptotic local power function of the F-test.In Section 4, for an ultrahigh-dimensional regime, we establish the asymptotic null distribution of the randomprojection-based test statistics and derive the asymptotic local power functions.In particular, we compare the proposed test with other competing tests in Section 4.3.In Section 5.1, we conduct simulations to evaluate the finite-sample behaviors of the proposed test in terms of Type I error and empirical power, and compare it with the competing tests.We also illustrate its applications in high-dimensional gene expression data in Section 5.2.Finally, the conclusions are made in Section 6.The proofs of lemmas and theorems, additional theoretical results and numerical studies are given in the supplementary materials.

Preliminaries
In this section, we introduce the target model and establish the theoretical foundation for the design of our proposed methods, which can be applied as the theoretical premise for our new tests both in relatively and extremely high-dimensional regimes.
For SIM in (1.1), the response variable y is generated from x based on a linear combination x β through an unknown link function.This model includes several important special cases, such as the GLM and the Cox model.When the information obtained from data is insufficient for specifying a particular link function, the high flexibility of (1.1) makes it a reasonable choice.However, with the introduction of an unknown link function in (1.1), statistical inference becomes significantly more difficult.In this article, we focus on the hypothesis testing problem (1.2) in SIM.
To motivate the proposed test, we first consider a risk function for the estimation problem, where L(θ , y) is a loss function, and u is a k-dimensional random variable with k ≤ min{n, p}.When u = x, this criterion is often used for estimating SIM in the classical settings, where the estimator of β can be obtained by solving a minimization problem of an empirical version of (2.1).When u satisfies a linear relationship with x, Lemma 2.1 suggests a way to develop a test procedure based on u for (1.2) in the p > n settings.Therefore, the selection of u is an important ingredient in our proposed method and will be concretely discussed in Sections 3 and 4.
The following lemma is obtained from the proof in Li and Duan (1989, Theorem 2.2).In the lemma, u satisfies a linear relationship with x according to the property of the normal distribution.The uniqueness of solution to the minimization problem is guaranteed when L(θ , y) is a strictly convex function of θ .The result of the lemma indicates that the direction of η can be estimated correctly even if the model is misspecified, which is of great significance in practice, since the underlying true model is unknown in prior for most of the cases.
In this article, we focus on the square loss function L(θ , y) = (y − θ) 2 .When u = x, the minimization problem of (2.1) becomes the least square problem, for which the solution is unique and has a closed form.From Lemma 2.1, the least square solution b ls is proportional to β, expressed as (2.2) When the response y and the covariate x satisfies E(y|x) = g(x β) for a differentiable function g, it can be shown that c 0 = E{g (x β)}, which is obtained from the normality assumption and Stein's lemma.From the property of the least squares and (2.2), model (1.1) can be written in a linear form (2.3) where the residual e satisfies E(e) = 0 and E(xe) = 0.The scalar coefficient c 0 is set to be 0 when β = 0.It is noted that the linear coefficients in the linear form are c 0 β, which brings out implicit information of β contained in the unknown link function f .When c 0 = 0, this makes it possible for us to do statistical inference about β without estimating the unknown link function.Motivated by this, we study the F-statistic in SIM.And its feasibility is investigated in Section 3.

The F-Test in Relatively High Dimensions
In this section, we focus on a relatively high-dimensional regime where p and n satisfy p/n → ζ ∈ (0, 1).In this setting, we derive the asymptotic null distribution and the asymptotic local power functions of the F-statistic.We also investigate the problem of testing partial coefficients and show the results in Appendix C.1 of the supplementary materials.For simplicity, we assume E(x) = 0 and E(y) = 0. Suppose that (x 1 , y 1 ), . . ., (x n , y n ) are iid copies of (x, y) from SIM (1.1).Let x i be the ith row of the design matrix X = (x 1 , . . ., x n ) and y = (y 1 , . . ., y n ) .The F-statistic is defined as where the hat matrix H = X(X X) −1 X .The test statistic T n is well-defined, since the matrix X X is invertible with the probability 1 when p < n and x follows the normal distribution.

Asymptotic Normality
First, we study the properties of the F-statistic (3.1) under H 0 .We make the following assumptions.
Assumption L1.The covariate x ∼ N (0, I) and x is independent of .
The asymptotic normality of the test statistic T n is established in the following theorem.
Theorem 3.1.Under H 0 and Assumptions L1-L3, as n → ∞, we have This asymptotic normality result justifies the following testing procedure.Given an α-level of significance, the test rejects where z α is the upper α-quantile of N (0, 1).

Asymptotic Local Power Function
We derive the asymptotic local power function of T n under H 1 .
We need the following additional assumption.
This is known as a local alternative, which is commonly used to study the asymptotic properties of a statistical test.Detailed discussions can be found in van der Vaart (1998, sec.14.1).To derive the asymptotic local power function, our analysis method is based on the linear form (2.3) and the analysis method of the F-test in the linear model.Because the residual e in (2.3) does not satisfy the conditions in a linear model, where the residual is often assumed to be conditionally independent of x.The method is further modified to adapt to our nonlinear highdimensional settings.The normality of x makes it possible to establish the independence between e and a new hat matrix, which is derived from a decomposition of the hat matrix H.The scalar c 0 = E(β xy)/||β|| 2 2 .The following theorem gives the asymptotic local power function.
This result shows that the power of the test becomes stronger as c 2 0 ||β|| 2 2 increases or ζ decreases.As shown in the linear form (2.3), c 2 0 ||β|| 2 2 is related to the level of linearity between y and x.Hence, it is reasonable to gain more testing power with larger c 2 0 ||β|| 2 2 .We note that an increase in the value of ζ leads to a decrease of testing power.Therefore, the F-test is adversely affected by the effect of high dimensionality and becomes powerless when the limit ζ of the ratio p/n is close to 1.

New Test in Ultrahigh Dimensions
In this section, we consider a higher-dimensional regime where the dimension p is much larger than the sample size n.In this case, the original F-statistic is not well-defined due to the singularity of the matrix X X.Therefore, a new high-dimensional test statistic is required to address the problem.Using the technique of random projection, we propose a new test statistic based on the F-statistic of the projected data.The new test has a less restrictive assumption on the distribution of x.We derive the asymptotic distribution under the null hypothesis and the asymptotic local power function of the proposed test.We also compare the properties of the proposed test with competing tests and provide sufficient conditions that guarantee its superior performance.
First, we concentrate on the problem of testing the global hypothesis With the normality assumption of x, when n > p, the feasibility of applying the F-test in SIM has been carefully investigated in Section 3.However, the F-test is inapplicable in the p > n settings.To solve the problem, we randomly project the highdimensional covariates into a lower-dimensional space, and then apply the F-test to the projected data.Specifically, for an integer 1 ≤ k < min{n, p}, let P k ∈ R p×k denote a random projection matrix with random entries, drawn independently of the data.Define u = P k x.We consider a model Note that the distribution of y in model (1.1) under hypothesis H 0 : β = 0 is the same as that in model (4.2) under hypothesis H 0 : η = 0.In addition, when η = 0 and P k has iid N (0, 1) entries, the probability of P k η = 0 is 1.We propose a test statistic where The test statistic T n,k can be well defined even when p > n, for the reason that the matrix U k U k is of full rank with probability 1, where P k has iid N (0, 1) entries.This is shown in the proof of Theorem 4.1.
One of the convenient ways to construct P k is to generate its iid entries from N (0, 1).Furthermore, Li, Hastie, and Church (2006) proposed that it is possible to generate other types of random projections P k , such as sparse random projections to obtain the same asymptotic performance as the normal random projections with a fast convergence speed.A sparse random projection is composed of entries p ij that are iid from distributions satisfying where the recommended value of l is √ p.In the theoretical analysis, we will focus on the random projection consisting of iid N (0, 1) entries.The results are also applicable to some nonnormal projections.The sparse random projections together with other types of random projections will be investigated in Appendix D.3 of the supplementary materials.

Asymptotic Normality
The first main result demonstrates the asymptotic normality of the proposed test under H 0 .We work under the following assumptions.
Assumption H1. x = μ + z, where is a p × m matrix with m ≥ p, μ is a p-dimensional vector and z = (z 1 , . . ., z m ) is an m-variate random vector with E(z) = 0, var(z) = I and var( z z m ) = O(m −1 ).For any nonnegative integers q 1 , . . ., q m , with m j=1 q j = 4, the mixed moments E m j=1 z q j j are bounded, and equal to 0 when at least one of the q j is odd.Assumption H2. is independent of x, and E(y 4 ) < ∞.

Assumption H3. p
n and there is a constant ρ ∈ (0, 1) such that k n → ρ.
As stated in Assumptions H1 and H3, there is no specific relationship between n and p, so that the dimension p, mean vector μ and covariance matrix = implicitly vary as n goes to infinity, making our test applicable to ultrahighdimensional problems.The covariate x is generated from a multivariate model, where the only restriction on is m ≥ p and the assumption on z is mild.Therefore, a rich collection of x can be generated, including the type of x consisting of continuous and discrete random variables and x generated from the elliptically distributions.Similar assumptions have been adopted in Bai and Saranadasa (1996), Zhong and Chen (2011), Guo and Chen (2016), and Cui, Guo, and Zhong (2018), where stricter conditions were imposed on each element of z.Suppose that the projection dimension k is asymptotically proportional to n with a coefficient ρ.The selection of ρ will be discussed in Section 5.
Since T n,k is invariant to the location shift of y and X, we assume that E(y) = 0 and μ = 0 in the rest of the article.
Under H 0 , the response y is independent of u, as the result of the independence between y and x.Therefore, it is sufficient to study the proposed test under the linear model.The asymptotic normality of the test statistic T n,k is established in the following theorem.
Theorem 4.1.Suppose that the random projection matrix P k consists of iid N (0, 1) entries.Under H 0 and Assumptions H1-H3, as n → ∞, we have This asymptotic normality result justifies the following test procedure.Given an α-level of significance, the proposed test rejects H 0 if where z α is the upper α-quantile of N (0, 1).

Asymptotic Local Power Function
In this section, we analyze the asymptotic local power function of the proposed test.We need the following additional assumption.

Assumption H4. E(yx
This is known as a local alternative.In the linear model, Assumption H4 is converted to β β = o(1).Considering a family of models where E(y|x) = g(x β) for a differentiable function g while x follows the normal distribution, Assumption H4 can be denoted as c 2 0,k β β = o(1) with c 0,k = E{g (x β)}.Specifically, the GLMs are special cases with specific forms of link functions.
In the study of the asymptotic local power function, the analysis method of the F-test proposed in Section 3 is considered.The empirical distribution of randomly projected data tends to be approximately normal (Diaconis and Freedman 1984;Hall and Li 1993;Leeb 2013;Steinberger and Leeb 2018).Therefore, it is expected that the result of the asymptotic local power function will remain valid even when the p-dimensional data are not generated from the normal distribution.This is supported by the simulation studies in Section 5. Let η = −1 1 P k β with 1 = P k P k , the scalar c 0,k = E(η uy)/(η 1 η) with P k considered as deterministic in the expectation, and ω 2 = η 1 η.The formal result is stated as follows.
Theorem 4.2.Suppose that Assumptions H1-H4 hold and z follows the standard normal distribution.Let RP n (β; P k ) denote the power function of the proposed test T n,k .Then RP where σ 2 = var(y) − c 2 0,k ω 2 , is the cumulative distribution function of the standard normal distribution, and z α is the upper α-quantile of .
The asymptotic local function relies on P k and is an increasing function of c 2 0,k ω 2 .When the vector β is in the space generated by P k , ω 2 can reach its upper bound β β.To reach this bound asymptotically, we give a sufficient condition.

Assumption H5 (Tail eigenvalue condition).
There is an integer s and a real number γ > 0 such that s < k and Assumption H5 is a tail eigenvalue condition, since it requires that the product of ||β|| 2 2 and the sum of tail eigenvalues of to be of order less than p/ √ n.
Lemma 4.3.Let P k ∈ R p×k be composed of entries from iid N (0, 1) entries.Suppose that Assumption H5 holds.Then we have for some ζ ∈ R k with probability tending to 1.
This lemma indicates that we can approximate β by P k ζ with a negligible approximation error.Consequently, the asymptotic local power function for the proposed test is shown as follows.
Corollary 4.1.Suppose that Assumptions H1-H5 hold and z follows the standard normal distribution.As n → ∞, we have where σ 2 = var(y)−c 2 0,k β β, is the cumulative distribution function of the standard normal distribution, and z α is the upper α-quantile of .
In the corollary, since ω 2 is sufficiently close to β β, it can be regarded as a deterministic value inside the asymptotic local power function, even when P k is randomly generated.In addition, when E(y|x) = g(x β) for a differentiable function g, we have c 0,k = E{g (x β)}, that is, c 0,k can be determined by the model.Therefore, it is proved that RP n (β; P k ) can be a nonrandom function in some certain conditions, and it is a decreasing function of ρ.To illustrate the forms of c 0,k and σ 2 , we give some examples in Appendix C.3 of the supplementary material.
For the problem of testing partial regression coefficients in an ultrahigh-dimensional regime, we propose a new testing procedure and derive its asymptotic null distribution and asymptotic local power function.The results are shown in Appendix C.2 of the supplementary material.

Theoretical Comparison
From the analysis in Sections 4.1 and 4.2, our new test is applicable in ultrahigh-dimensional settings with mild assumptions about the model and the covariate.In this section, we will further consider the performance of the proposed test by comparing it with two existing tests.Specifically, we compare our test with the GLM test proposed in Guo and Chen (2016) and the test developed by Ma, Cai, and Li (2020), which focused on the logistic model in sparse settings.These tests are designed for high-dimensional testing problems (4.1) and have been demonstrated to have powerful testing performance.We denote these competing tests as GC test and MCL test, respectively.First, the GC test is adopted in a certain class of GLM for comparison.For theoretical analysis, the criterion asymptotic relative efficiency (ARE) is used, for which we give a sufficient condition to guarantee that the proposed test is asymptotically more powerful than the GC test.Since the MCL test has no closed form of the asymptotic power function, the comparison with the MCL test is conducted through simulation studies, shown in Section 5, where the GC test is also investigated.In this section, we compare the proposed test with the MCL test in terms of model assumptions to demonstrate the general application range for our test.
For the GLMs with a canonical link, the response variable y satisfies E(y|x) = g(x β) and var(y|x) = φg (x β), where g is a monotone differentiable function and φ is a dispersion parameter.When x ∼ N (0, ), the asymptotic power function of the GC test is denoted as where the matrices satisfy With a slight abuse of notation, we also denote the asymptotic power function of our random-projection-based (RP) test by where c 0,k = E{g (x β)}.
Since the term added to −z α inside function is the component controlling power, the ratio of such terms is defined as the ARE.More explicitly, we define (4.7) Whenever the ARE is larger than 1, our procedure is considered to have a greater asymptotic power than the competing test.For this purpose, an inequality for (4.7) is derived below.Let r 0 = y − g(0) and F(β) = 1/2 I − P 1/2 β 1/2 , where P 1/2 β denotes the projection matrix for 1/2 β.Then, Plugging this into (4.7),under the conditions in Guo and Chen (2016), where satisfies tr( 4 ) = o(tr 2 ( 2 )), we obtain where Clearly, when the right side in the last inequality in (4.8) is larger than 1, the proposed test has a higher local power.To derive this, using the method given in Theorem 2 of Lopes, Jacob, and Wainwright (2011), we obtain the following result.
Lemma 4.4.Suppose that δ follows a spherical distribution with P(δ = 0) = 0, and is independent of P k .There is a constant γ ∈ [0, 1) such that k/p → γ .And assume 1 √ k tr( ) 1).For a fixed value 1 > 0, let c( 1 ) be any constant strictly greater than holds for all large n, then When tr( ) 2 /tr( 2 ) has a slower growing speed compared with the sample size n, our proposed test will have an ARE greater than 1.
Under the conditions in Corollary 4.1, ω 2 can be regarded as a deterministic value, for which (4.8) leads to the following inequality.

ARE( RP
(4.9) A sufficient condition can be derived to ensure that the right-hand side in (4.9) is larger than 1.Specifically, when ), where ξ = β/||β|| 2 and γ is a sufficiently small positive constant, the right hand side in (4.9) goes to infinity as n → ∞.Therefore, the proposed test is asymptotically more powerful than the GC test.
Next, we compare the proposed test with the MCL test.Ma, Cai, and Li (2020) studied global testing problem (4.1) in a highdimensional logistic model and constructed the MCL test based on a bias-corrected estimator.The covariate x is assumed to be normal or follow a bounded design, with a strong assumption on the covariance matrix.In particular, the covariance matrix is assumed to have bounded eigenvalues and a sparse inverse.Therefore, our proposed test has a wider range of applications.In addition, Ma, Cai, and Li (2020) considered sparse alternatives, H 1 : β ∈ β ∈ R p : ||β|| ∞ ≥ γ , ||β|| 0 ≤ s for some γ > 0. A further study via simulation is given in Section 5 and Appendix D of the supplementary materials.

Simulation Studies
We conducted simulation studies to evaluate the finite-sample performance of the proposed tests and compare it with the GC and MCL tests.
We test the global hypothesis Two SIMs were considered: logistic model and Poisson model.The covariate x was generated from 1/2 z, where each entry of z was iid from N (0, 1) or U(− √ 3, √ 3).The condition of with different category was studied, which includes the settings that is sparse and nonsparse.Specifically, was generated based on ODO , where O was an orthogonal matrix and D = diag(d 1 , . . ., d p ).Let s = [n 0.8 ] and L = n, where the function [x] takes the greatest integer less than or equal to the number x.The diagonal entries of D were set as d i = 1, for i ≤ s, and d i = (L − s)w i /W, for i = s + 1, . . ., p, where w i = (i − s) −4 and W = p i=s+1 w i .The design of O came from the block-wise diagonal matrix structure.Specifically, O = diag(O 1 , . . ., O B ) was a block-wise diagonal matrix with B blocks, and each block was independently and uniformly generated from the m × m orthogonal group, with p = Bm.To study different types of , two different settings were analyzed as follows: (i) 1 : B = 1.(ii) 2 : B = 100 for strong sparsity.To create regimes of high dimensionality, we considered (n, p) = (400, 1000) and (n, p) = (600, 3000) in the simulation.For the alternative hypothesis, both the sparse and nonsparse cases were considered.The vector of coefficients was generated by β = bδ/ δ δ, where b was a positive real number and δ was an p-dimensional vector determining the sparsity of β.In the simulation, two different types of δ were considered as follows: (i) δ 1 : δ 1 j = 1, for j ∈ S, where the set S was randomly selected over {1, . . ., p} and had size |S| = 10, otherwise, δ 1 j = 0. (ii) δ 2 : randomly selected from Span{u 1 , . . ., u 100 }, where u i was the ith column of O.The value of b 2 was selected in the range such that c 2 0,k b 2 ≤ 0.25.Specifically, for the logistic model, b 2 = 0.4, 0.8 were investigated in the simulation.For the Poisson model, b 2 = 0.1, 0.2 were considered in the simulation.
In Tables 1 and 2, we report the Type I errors and the empirical powers of RP test, GC test and MCL test in the experimented models based on 1000 simulations.The Type I errors for the RP and GC tests are close to 0.05, which supports the result in Theorem 4.1.For the MCL test, when the covariance matrix in the simulation model is 1 , the Type I errors might exceed 0.1.The results illustrate that the MCL test requires a strong sparsity assumption.Therefore, our proposed test has a wider range of applications as discussed in Section 4.3.
For the empirical powers of the tests, since the MCL test is designed for the binary response, a comparison with the MCL test was only considered in the logistic model.As shown in the tables, the empirical power increases as b 2 grows.For the RP test, higher powers are attained in the cases with ρ = 0.4.This may be the result of a higher probability of deriving a larger value of ω 2 in a larger projection space.However, for the settings with almost fixed ω 2 , that is, when Assumption H5 is satisfied by β and , larger empirical powers are achieved in the case with a smaller ρ, which is demonstrated in the simulation study in Appendix D.5 of the supplementary materials.According to Lopes, Jacob, and Wainwright (2011), the ratio tr( ) 2 /tr( 2 ) can be viewed as measuring the decay rate of the spectrum of , with the tail eigenvalue condition satisfied when tr( ) 2 /tr( 2 ) p.It indicates that ρ could be determined by an estimation of the ratio.Compared with the GC test, our proposed test performs better in all the simulation models and settings.In comparison with the MCL test, our proposed test has higher power in most settings.
Figure 1 shows the numerical comparisons between the RP test and the competing tests in the logistic model when the nominal significance level varies from 0.1 to 0.9.Figures 1(a) and (b) report the Type I errors.The blue lines are close to the diagonal black lines, indicating that the RP test has well control of the Type I error.In Figures 1(c)-(f), the empirical powers of the tests under different alternatives and covariance matrices of x are presented.As shown in the plots, the blue lines stay above the other lines.This shows that our proposed test has better performance in the models considered here.The results for the Poisson model are reported in Appendix D.1 of the supplementary materials.
In Appendix D of the supplementary material, we conducted additional simulation studies to give more illustrations on the performance of the proposed test.We also investigated multiplerandom-projection-based tests with different aggregation methods.In addition, we considered other classical tests constructed from the projected data and examined their performances and compared them with the proposed test.We also conducted simulation studies on the problem of testing part of the regression coefficients, and investigated the feasibility of testing the global null hypothesis by simultaneously testing each element of β.

Breast Cancer Data Example
Traditional breast cancer treatment methods are guided by a classification based on the expression levels of estrogen receptor (ER), progesterone receptor (PR), and human epidermal growth factor receptor 2 (HER2).In the last decade, research on global gene expression analyses indicated a more complex breast cancer portrait and identified at least four intrinsic molecular subtypes, which have significant differences in terms of response to therapies.To identify the subtypes, a clinically gene expression-based test, named as prediction analysis of microarray 50 (PAM50) was introduced (Parker et al. 2009), which is based on the expression measurement of 50 genes and shows a high accuracy of identification.
We illustrate our proposed methods by analyzing a real dataset of breast cancer, which is available under the accession number GSE50948 in the Gene Expression Omnibus (GEO).In this dataset, the gene expression analysis was performed for 156 samples from the enrolled patients in the NOAH trail, which consisted of 114 patients with HER2+ locally advanced or inflammatory breast cancer and 42 patients with HER2-disease.According to completeness of the information, n = 152 samples were used in the study.The detail of the dataset was given in Prat et al. (2014).As multiple probes might represent the same gene, the measurement for each gene was from the probe with the highest interquartile range.Then, expression values of 20,592 genes were obtained.In addition, some standard clinicopathologic variables were also considered in the analysis, including age at diagnosis, histologic grade, tumor size, histology, and hormonal receptor status with values of 1 and 0 corresponding to positive and negative status, respectively.
First, we studied the overall association between the status of HER2, ER, PR with other standard clinicopathologic features and gene expression levels.To illustrate our analysis method, the case considering HER2 was given as an example.The status of HER2 was set as the response variable and the covariates were composed of the rest standard clinicopathologic features and gene expression levels.The testing problem considered a extremely high-dimensional setting where n = 152 and p = 20601.The calculation of our proposed testing procedure reported a significance association with a p-value< 0.001.And the same conclusion was also obtained by the GC test and the MCL test under the assumption that the data followed the logistic model.Hence, it showed that HER2 was associated with the clinicopathologic features and gene expression.As the dimension of the genes was extremely high, it was impractical to examine all of them in practice.Then, we proposed a conjecture that there was a representative subsets of genes to explain almost all the influence from the genes given in the first study.Specifically, the set of 50 genes from the PAM50 was considered.We divided the covariates by setting the clinicopathologic features and the set of genes as the nuisance variables, and the rest genes as the interested variables.
The result of our proposed methods are given in Table 3, which indicates a weak association between HER2 and the rest genes.This confirms our conjecture and demonstrates that HER2 can be explained by a much smaller amount of genes together with some clinicopathologic features.It also helps to understand the powerful classification ability of PAM50.Similarly, the cases where the response variable was the status of ER or PR were also considered.The results are reported in Table 3.

Conclusion
In this article, we have developed an approach for hypothesis testing in a high-dimensional nonlinear single-index model.An appealing feature of the proposed method is its ability to conduct statistical test about the regression coefficient β without estimating the link function.Specifically, we transform an SIM into a linear form and find the linear coefficient β up to a scalar c 0 = 0.This makes it possible to design simple and effective methods for testing the significance of β.In the highdimensional regime with p/n → ζ ∈ (0, 1) settings, we have provided a detailed analysis of the asymptotic null distribution of the F-statistic and its asymptotic local power.We have also studied the role of ζ in affecting the power of the test.Another contribution of our work is a new method that combines the technique of random projection and the F-test.We have shown that the proposed test is valid in an ultrahigh-dimensional setting under some mild conditions, where p can be much larger than n.Theoretical and simulation studies demonstrate that the proposed test is more powerful than the two existing methods considered over a wide range of alternatives and possesses certain advantages in the sparse cases.In addition, a sufficient condition is derived to guarantee that the proposed test has good statistical power.An important aspect of our proposed method is the use of random projection.It reduces the dimension of the data while preserves the main information simultaneously.Moreover, the property that the randomly projected data is asymptotically normal facilitates the theoretical analysis under more general distributions.The selection of ρ, the proportion of the dimension of the random projection data over the dimension p of the covariate, is analyzed in several different settings.
The proposed test is an effective approach to hypothesis testing in high-dimensional SIM and is computationally simple to implement.Therefore, it is a useful contribution to the literature on statistical inference in high-dimensional models.

Figure 1 .
Figure 1.Type I errors and empirical powers of RP, GC and MCL tests for the logistic model when ρ = 0.4 and (n, p) = (400, 1000).

Table 1 .
Type I errors and empirical powers of RP, GC and MCL tests at the significance level 0.05 in logistic models.

Table 2 .
Type I errors and empirical powers of RP and GC tests at the significance level 0.05 in Poisson models.

Table 3 .
The p-values of the proposed tests, CG test and MCL test in the global and partial testing problems.The p-values of the following tests in the global testing problem: