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The goodness–of–fit test can be used to decide whether a population fits a given distribution, but it will not suffice to decide whether two populations follow the same unknown distribution. A different test, called the test for homogeneity, can be used to conclude whether two populations have the same distribution. To calculate the test statistic for a test for homogeneity, follow the same procedure as with the test of independence. The hypotheses for the test for homogeneity can...
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The Behrens-Fisher test is a statistical method designed to address the Behrens-Fisher problem, which arises when comparing the means of two normally distributed populations with unequal variances. Unlike the Student's t-test, which assumes equal variances, the Behrens-Fisher test allows for mean comparison without this restrictive assumption. This flexibility makes it particularly valuable in scenarios where two independent samples exhibit normality but lack variance homogeneity.
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Combined Immunofluorescence and DNA FISH on 3D-preserved Interphase Nuclei to Study Changes in 3D Nuclear Organization
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Distribution-free simultaneous tests for location-scale and Lehmann alternative in two-sample problem.

Wolfgang Kössler1, Amitava Mukherjee2

  • 1Department of Computer Science, Humboldt University of Berlin, Berlin, Germany.

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|October 22, 2019
PubMed
Summary

This study introduces two novel distribution-free tests for comparing two populations, addressing versatile alternatives where location, scale, or shape may change simultaneously. These tests demonstrate robust power across various distribution shifts in biomedical data.

Keywords:
Euclidean distanceLepage-type statisticMahalanobis distancebiomedical experimentsversatile alternative

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Area of Science:

  • Biostatistics
  • Statistical Inference
  • Hypothesis Testing

Background:

  • The classical two-sample testing problem is fundamental in biomedical research and case-control studies.
  • Existing tests often assume invariance in distribution shape, location, or scale, limiting their applicability.
  • Real-world data shifts can involve simultaneous changes in location, scale, and shape parameters.

Purpose of the Study:

  • To address the lack of statistical tests for comparing two populations against versatile alternatives (simultaneous changes in location, scale, and shape).
  • To introduce two novel distribution-free tests based on Euclidean and Mahalanobis distances for versatile alternatives.

Main Methods:

  • Development of two distribution-free test statistics utilizing Euclidean and Mahalanobis distances.
  • Derivation of asymptotic distributions and study of asymptotic power for the proposed tests.
  • Investigation of p-value approximation methods for small sample sizes.
  • Monte Carlo simulations to compare power performance against existing distribution-free tests.

Main Results:

  • The proposed tests exhibit strong power across a wide range of distribution shifts, outperforming existing methods in versatility.
  • Asymptotic distributions and power properties of the new test statistics were established.
  • The tests were illustrated with practical applications in biomedical experiments.

Conclusions:

  • The introduced Euclidean and Mahalanobis distance-based tests provide a powerful and versatile approach for two-sample comparisons.
  • These tests are suitable for situations where multiple distributional parameters (location, scale, shape) may change concurrently.
  • The proposed methods offer a valuable alternative to existing tests, which are often limited to specific types of distributional changes.