In spite of its urgent importance in the era of big data, testing high-dimensional parameters in generalized linear models (GLMs) in the presence of high-dimensional nuisance
parameters has been largely under-studied, especially with regard to constructing powerful
tests for general (and unknown) alternatives. Most existing tests are powerful only against
certain alternatives and may yield incorrect Type I error rates under high-dimensional
nuisance parameter situations. In this paper, we propose the adaptive interaction sum of
powered score (aiSPU) test in the framework of penalized regression with a non-convex
penalty, called truncated Lasso penalty (TLP), which can maintain correct Type I error
rates while yielding high statistical power across a wide range of alternatives. To calculate its p-values analytically, we derive its asymptotic null distribution. Via simulations,
its superior finite-sample performance is demonstrated over several representative existing
methods. In addition, we apply it and other representative tests to an Alzheimer’s Disease
Neuroimaging Initiative (ADNI) data set, detecting possible gene-gender interactions for
Alzheimer’s disease. We also put R package “aispu” implementing the proposed test on
GitHub.
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High-dimensional general linear hypothesis tests via non-linear spectral shrinkage
We are interested in testing general linear hypotheses in a high-dimensional multivariate linear regression model. The framework includes many well-studied problems such as two-sample tests for equality of population means, MANOVA and others as special cases. A family of rotation-invariant tests is proposed that involves a flexible spectral shrinkage scheme applied to the sample error covariance matrix. The asymptotic normality of the test statistic under the null hypothesis is derived in the setting where dimensionality is comparable to sample sizes, assuming the existence of certain moments for the observations. The asymptotic power of the proposed test is studied under various local alternatives. The power characteristics are then utilized to propose a data-driven selection of the spectral shrinkage function. As an illustration of the general theory, we construct a family of tests involving ridge-type regularization and suggest possible extensions to more complex regularizers. A simulation study is carried out to examine the numerical performance of the proposed tests.
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- NSF-PAR ID:
- 10176773
- Date Published:
- Journal Name:
- Bernoulli
- ISSN:
- 1350-7265
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Summary Covariate-adaptive randomization is popular in clinical trials with sequentially arrived patients for balancing treatment assignments across prognostic factors that may have influence on the response. However, existing theory on tests for the treatment effect under covariate-adaptive randomization is limited to tests under linear or generalized linear models, although the covariate-adaptive randomization method has been used in survival analysis for a long time. Often, practitioners will simply adopt a conventional test to compare two treatments, which is controversial since tests derived under simple randomization may not be valid in terms of type I error under other randomization schemes. We derive the asymptotic distribution of the partial likelihood score function under covariate-adaptive randomization and a working model that is subject to possible model misspecification. Using this general result, we prove that the partial likelihood score test that is robust against model misspecification under simple randomization is no longer robust but conservative under covariate-adaptive randomization. We also show that the unstratified log-rank test is conservative and the stratified log-rank test remains valid under covariate-adaptive randomization. We propose a modification to variance estimation in the partial likelihood score test, which leads to a score test that is valid and robust against arbitrary model misspecification under a large family of covariate-adaptive randomization schemes including simple randomization. Furthermore, we show that the modified partial likelihood score test derived under a correctly specified model is more powerful than log-rank-type tests in terms of Pitman’s asymptotic relative efficiency. Simulation studies about the type I error and power of various tests are presented under several popular randomization schemes.
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Summary We consider testing regression coefficients in high dimensional generalized linear models. By modifying the test statistic of Goeman and his colleagues for large but fixed dimensional settings, we propose a new test, based on an asymptotic analysis, that is applicable for diverging dimensions and is robust to accommodate a wide range of link functions. The power properties of the tests are evaluated asymptotically under two families of alternative hypotheses. In addition, a test in the presence of nuisance parameters is also proposed. The tests can provide p-values for testing significance of multiple gene sets, whose application is demonstrated in a case-study on lung cancer.
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Summary A formal likelihood ratio hypothesis test for the validity of a parametric regression function is proposed, using a large dimensional, non-parametric double-cone alternative. For example, the test against a constant function uses the alternative of increasing or decreasing regression functions, and the test against a linear function uses the convex or concave alternative. The test proposed is exact and unbiased and the critical value is easily computed. The power of the test increases to 1 as the sample size increases, under very mild assumptions—even when the alternative is misspecified, i.e. the power of the test converges to 1 for any true regression function that deviates (in a non-degenerate way) from the parametric null hypothesis. We also formulate tests for the linear versus partial linear model and consider the special case of the additive model. Simulations show that our procedure behaves well consistently when compared with other methods. Although the alternative fit is non-parametric, no tuning parameters are involved. Supplementary materials with proofs and technical details are available on line.
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Summary In this paper, we develop a systematic theory for high-dimensional analysis of variance in multivariate linear regression, where the dimension and the number of coefficients can both grow with the sample size. We propose a new U-type statistic to test linear hypotheses and establish a high-dimensional Gaussian approximation result under fairly mild moment assumptions. Our general framework and theory can be used to deal with the classical one-way multivariate analysis of variance, and the nonparametric one-way multivariate analysis of variance in high dimensions. To implement the test procedure, we introduce a sample-splitting-based estimator of the second moment of the error covariance and discuss its properties. A simulation study shows that our proposed test outperforms some existing tests in various settings.more » « less
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Accepted Manuscript1.0
- Journal Article:
- The DOI is not currently available.
