A simple yet powerful test for assessing goodness-of-fit of high-dimensional linear models

Area of Science:

• Statistics
• Bioinformatics
• Genomics

Background:

• Linear models are widely used in statistical analysis.
• High-dimensional data, where the number of covariates (p) is large relative to the sample size (n), poses challenges for traditional statistical methods.
• Assessing the validity of statistical tests in high-dimensional settings is crucial for reliable data analysis.

Purpose of the Study:

• To evaluate the validity and performance of a projection-based test for linear models in the context of an increasing number of covariates.
• To analyze the consistency and asymptotic properties of the test using gene expression datasets.
• To demonstrate the dimension reduction capabilities and numerical performance of the proposed test.

Main Methods:

• A projection-based test for linear models was developed and evaluated.
• The test's consistency was analyzed as the number of covariates tends to infinity.
• Asymptotic distributions under null and alternative hypotheses were derived.
• The test was applied to two gene expression datasets.

Main Results:

• The projection-based test was found to be consistent even when the number of covariates approaches infinity.
• The asymptotic properties of the test were shown to be similar to those with a fixed number of covariates, provided p/n → 0 and mild assumptions are met.
• The test demonstrated significant dimension reduction capabilities.
• Remarkable numerical performance was observed in the analyzed gene expression datasets.

Conclusions:

• The projection-based test is a valid and robust method for analyzing linear models in high-dimensional settings.
• The test offers practical advantages in terms of dimension reduction and computational efficiency.
• The findings support the use of this test in fields like genomics where high-dimensional data is common.