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Introduction to Test of Independence

Introduction to Test of Independence

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:

Hypothesis Test for Test of Independence

Hypothesis Test for Test of Independence

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)...

Unusual Results

Unusual Results

Unusual results are those that have a very low chance of occurring. Unusual results can be identified using probabilities and the range rule of thumb. In problems involving probability, unusual results can be observed in 2 instances – an unusually high number of successes or an unusually low number of successes.
According to the range rule of thumb, any value above or below two standard deviations, 2σ from the mean, μ is considered unusual.
Maximum unusual value = μ + 2σ
Minimum unusual value...

One-Way ANOVA: Unequal Sample Sizes

One-Way ANOVA: Unequal Sample Sizes

One-way ANOVA can be performed on three or more samples of unequal sizes. However, calculations get complicated when sample sizes are not always the same. So, while performing ANOVA with unequal samples size, the following equation is used:

One-Way ANOVA

One-Way ANOVA

One-way ANOVA analyzes more than three samples categorized by one factor. For example, it can compare the average mileage of sports bikes. Here, the data is categorized by one factor - the company. However, one-way ANOVA cannot be used to simultaneously compare the sample mean of three or more samples categorized by two factors. An example of two factors would be sports bikes from different companies driven in different terrains, such as a desert or snowy landscape. Here, two-way ANOVA is used...

Law of Independent Assortment

Law of Independent Assortment

While Mendel’s Law of Segregation states that the two alleles for one gene are separated into different gametes, a different question of how different genes are inherited remains. For example, is the gene for tall plants inherited with the gene for green peas? Mendel asked this question by experimenting with a dihybrid cross; a cross in which both parents are homozygous for two distinct traits resulting in an F1 generation that are heterozygous for both traits.

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Related Experiment Video

Updated: Jun 26, 2026

Selecting Multiple Biomarker Subsets with Similarly Effective Binary Classification Performances

Selecting Multiple Biomarker Subsets with Similarly Effective Binary Classification Performances

Published on: October 11, 2018

High Dimensional Classification Using Features Annealed Independence Rules.

Jianqing Fan¹, Yingying Fan

¹Princeton University.

Annals of Statistics

|January 27, 2009

Summary

This summary is machine-generated.

High-dimensional classification can fail even with independence rules due to noise. Feature selection is crucial, leading to the new Features Annealed Independence Rules (FAIR) method for improved accuracy.

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Last Updated: Jun 26, 2026

Selecting Multiple Biomarker Subsets with Similarly Effective Binary Classification Performances

Selecting Multiple Biomarker Subsets with Similarly Effective Binary Classification Performances

Published on: October 11, 2018

Area of Science:

Statistical learning
Bioinformatics
Machine learning

Background:

High-dimensional data classification is common in fields like tumor classification.
The impact of high dimensionality on classification performance is not well understood.
Existing methods like Fisher's discriminant and independence rules can perform poorly due to noise accumulation and diverging spectra.

Purpose of the Study:

To investigate the limitations of existing classification rules in high-dimensional settings.
To propose a new feature selection-based classification method, Features Annealed Independence Rules (FAIR).
To establish conditions for selecting important features and determine the optimal number of features for classification.

Main Methods:

Demonstrated that using all features can lead to random guessing performance.
Introduced the Features Annealed Independence Rules (FAIR) procedure.
Established theoretical conditions for feature selection using the two-sample t-statistic.
Proposed a method for selecting the optimal number of features based on classification error bounds.

Main Results:

Showed that almost all linear discriminants can perform as poorly as random guessing in high dimensions.
Identified the critical importance of selecting a subset of relevant features.
Established theoretical guarantees for feature selection under specific conditions.
Validated the proposed FAIR method through simulations and real-world data analysis.

Conclusions:

Feature selection is paramount for effective high-dimensional classification.
The proposed FAIR method offers a robust approach to high-dimensional classification.
The study provides theoretical insights and practical validation for the new classification procedure.