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Related Concept Videos

Introduction to Test of Independence01:21

Introduction to Test of Independence

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In statistics, the term independence means that one can directly obtain the probability of any event involving both variables by multiplying their individual probabilities. Tests of independence are chi-square tests involving the use of a contingency table of observed (data) values.
The test statistic for a test of independence is similar to that of a goodness-of-fit test:
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Friedman Two-way Analysis of Variance by Ranks01:21

Friedman Two-way Analysis of Variance by Ranks

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Friedman's Two-Way Analysis of Variance by Ranks is a nonparametric test designed to identify differences across multiple test attempts when traditional assumptions of normality and equal variances do not apply. Unlike conventional ANOVA, which requires normally distributed data with equal variances, Friedman's test is ideal for ordinal or non-normally distributed data, making it particularly useful for analyzing dependent samples, such as matched subjects over time or repeated measures...
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Hypothesis Test for Test of Independence01:16

Hypothesis Test for Test of Independence

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The test of independence is a chi-square-based test used to determine whether two variables or factors are independent or dependent. This hypothesis test is used to examine the independence of the variables. One can construct two qualitative survey questions or experiments based on the variables in a contingency table. The goal is to see if the two variables are unrelated (independent) or related (dependent). The null and alternative hypotheses for this test are:
H0: The two variables (factors)...
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Expected Frequencies in Goodness-of-Fit Tests01:19

Expected Frequencies in Goodness-of-Fit Tests

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A goodness-of-fit test is conducted to determine whether the observed frequency values are statistically similar to the frequencies expected for the dataset. Suppose the expected frequencies for a dataset are equal such as when predicting the frequency of any number appearing when casting a die. In that case, the expected frequency is the ratio of the total number of observations (n)  to the number of categories (k).
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Ranks01:02

Ranks

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Unlike parametric methods, nonparametric statistics are ideal for nominal and ordinal data, requiring fewer assumptions about the population's nature or distribution. This makes nonparametric methods easier to apply and interpret, as they do not depend on parameters like mean or standard deviation. One common approach in nonparametric analysis is to sort data according to a specific criterion. For instance, we might arrange weather data from hottest to coldest days in a month or rank cities...
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Test for Homogeneity01:23

Test for Homogeneity

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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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RANK-BASED INDICES FOR TESTING INDEPENDENCE BETWEEN TWO HIGH-DIMENSIONAL VECTORS.

Yeqing Zhou1, Kai Xu2, Liping Zhu3,4

  • 1School of Mathematical Sciences, Tongji University.

Annals of Statistics
|May 6, 2024
PubMed
Summary
This summary is machine-generated.

We introduce three novel rank-based tests for independence between high-dimensional random vectors. These distribution-free tests demonstrate superior performance over classic methods, especially when vector components have differing scales.

Keywords:
Bergsma-Dassios-Yanagimoto’s τ*Blum-Kiefer-Rosenblatt’s RDegenerate U-statisticsHoeffding’s DPrimary 62G10secondary 62G20

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Area of Science:

  • Statistics
  • High-Dimensional Data Analysis
  • Statistical Independence Testing

Background:

  • Testing independence is crucial in multivariate statistics.
  • Existing methods struggle with high-dimensional data and heavy tails.
  • Need for robust, distribution-free independence tests.

Purpose of the Study:

  • Propose novel rank-based independence tests for high-dimensional random vectors.
  • Analyze the asymptotic properties and power of these new tests.
  • Compare their efficiency against established distance covariance/correlation methods.

Main Methods:

  • Utilize rank-based indices from Hoeffding's, Blum-Kiefer-Rosenblatt's, and Bergsma-Dassios-Yanagimoto's statistics.
  • Establish asymptotic normality of test statistics under diverging dimensions.
  • Derive explicit rates of convergence and analyze local power.

Main Results:

  • Demonstrate asymptotic normality and provide convergence rates for the proposed tests.
  • Show these tests are distribution-free and applicable to heavy-tailed data.
  • Establish relationships between rank-based indices and Pearson's correlation.
  • Identify conditions where proposed tests outperform distance covariance/correlation tests.

Conclusions:

  • The proposed rank-based tests offer a robust alternative for high-dimensional independence testing.
  • These tests are particularly advantageous for data with heterogeneous component scales.
  • The study provides theoretical underpinnings for their application and efficiency.