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Chi-square Analysis

Chi-square Analysis

The chi-square test is a statistical hypothesis test. It is used to check whether there is a significant difference between an expected value and an observed value. In the context of genetics, it enables us to either accept or reject a hypothesis, based on how much the observed values deviate from the expected values.
The chi-square test was developed by Pearson in 1990.
The first step of performing a Chi-square analysis is to establish a null hypothesis, which assumes that there is no real...

Goodness-of-Fit Test

Goodness-of-Fit Test

The goodness-of-fit test is a type of hypothesis test which determines whether the data "fits" a particular distribution. For example, one may suspect that some anonymous data may fit a binomial distribution. A chi-square test (meaning the distribution for the hypothesis test is chi-square) can be used to determine if there is a fit. The null and alternative hypotheses may be written in sentences or stated as equations or inequalities. The test statistic for a goodness-of-fit test is given as...

Chi-square Distribution

Chi-square Distribution

How does one determine if bingo numbers are evenly distributed or if some numbers occurred with a greater frequency? Or if the types of movies people preferred were different across different age groups or if a coffee machine dispensed approximately the same amount of coffee each time. These questions can be addressed by conducting a hypothesis test. One distribution that can be used to find answers to such questions is known as the chi-square distribution. The chi-square distribution has...

Test for Homogeneity

Test for Homogeneity

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 be stated as...

Finding Critical Values for Chi-Square

Finding Critical Values for Chi-Square

Consider a curve representing sample data drawn randomly from a normally distributed population. One must construct confidence intervals to estimate or to test a claim regarding the population standard deviation. For example, a 95% confidence interval covers 95% of the area under the curve, and the remaining 5% is equally distributed on either side of the curve. To achieve such confidence intervals, one must determine the critical values. The critical values are simply the values separating the...

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:

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Related Experiment Video

Updated: Jun 14, 2026

The Adjuvant Efficacy of Angong Niuhuang Pill in the Treatment of Viral Encephalitis: A Meta-Analysis of Randomized Controlled Trials

The Adjuvant Efficacy of Angong Niuhuang Pill in the Treatment of Viral Encephalitis: A Meta-Analysis of Randomized Controlled Trials

Published on: April 19, 2024

A generalized formula for converting chi-square tests to effect sizes for meta-analysis.

Michael S Rosenberg¹

¹Center for Evolutionary Medicine and Informatics, The Biodesign Institute and School of Life Sciences, Arizona State University, Tempe, Arizona, United States of America. msr@asu.edu

|April 13, 2010

Summary

A common formula for converting chi-square tests to correlation coefficients for meta-analysis may overestimate effect sizes. This study provides a corrected formula to address a hidden assumption violation, ensuring more accurate effect size estimation.

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The Adjuvant Efficacy of Angong Niuhuang Pill in the Treatment of Viral Encephalitis: A Meta-Analysis of Randomized Controlled Trials

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Meta-analysis of Voxel-Based Neuroimaging Studies using Seed-based d Mapping with Permutation of Subject Images (SDM-PSI)

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Published on: November 27, 2019

Area of Science:

Statistics
Psychometrics
Meta-analysis

Background:

The chi-square test is frequently used in meta-analysis.
Converting chi-square test results to correlation coefficients is a common practice for effect size calculation.
Existing conversion formulas may contain hidden assumptions.

Purpose of the Study:

To identify and address a hidden assumption in the common formula for converting chi-square tests to correlation coefficients.
To provide a corrected formula for more accurate effect size estimation in meta-analysis.

Main Methods:

Analysis of the mathematical derivation of the standard chi-square to correlation conversion formula.
Identification of the specific condition under which the formula's assumption is violated.
Development and validation of a corrected formula.

Main Results:

The standard formula assumes equal cell frequencies or specific distribution patterns, which are not always met.
Violation of this assumption can lead to a significant overestimation of the effect size.
The proposed corrected formula accounts for variations in cell frequencies, yielding more accurate correlation coefficients.

Conclusions:

The corrected formula offers a more reliable method for calculating effect sizes from chi-square tests in meta-analysis.
Researchers should be aware of the assumption in the common formula to avoid biased effect size estimates.
Accurate effect size estimation is crucial for valid meta-analytic conclusions.