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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:

Contingency Table

Contingency Table

A contingency table provides a way of portraying data that can facilitate calculating probabilities. It is a method of displaying a frequency distribution as a table with rows and columns to show how two variables may be dependent (contingent) upon each other; The table helps determine conditional probabilities quite quickly and can help systematically organize, analyze and quantify data. The table displays sample values concerning two variables that may be dependent or contingent on one...

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)...

McNemar's Test

McNemar's Test

McNemar's Test is a nonparametric statistical test used to determine if there is a significant difference in proportions between two related groups when the outcome is binary (e.g., yes/no, success/failure). It is beneficial when we have paired data, such as pre-test/post-test designs, where the same subjects are measured under two different conditions. The test is named after the statistician Quinn McNemar, who introduced it in 1947. It is commonly used in situations where subjects are...

Fisher's Exact Test

Fisher's Exact Test

Fisher's exact test is a statistical significance test widely used to analyze 2x2 contingency tables, particularly in situations where sample sizes are small. Unlike the chi-squared test, which approximates P-values and assumes minimum expected frequencies of at least five in each cell, Fisher's exact test calculates the exact probability (P-value) of observing the data or more extreme results under the null hypothesis. This feature makes it especially valuable when the assumptions of the...

Determination of Expected Frequency

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Suppose one wants to test independence between the two variables of a contingency table. The values in the table constitute the observed frequencies of the dataset. But how does one determine the expected frequency of the dataset? One of the important assumptions is that the two variables are independent, which means the variables do not influence each other. For independent variables, the statistical probability of any event involving both variables is calculated by multiplying the individual...

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Analysing 2 × 2 contingency tables: which test is best?

John Ludbrook¹

¹Department of Surgery, The University of Melbourne, Melbourne, Vic., Australia. ludbrook@bigpond.net.au

Clinical and Experimental Pharmacology & Physiology

|January 9, 2013

Summary

Biomedical researchers should avoid using Pearson's Chi-squared and Fisher's exact tests for 2x2 frequency tables. These statistical methods are often inappropriate for common study designs, unlike exact tests on the odds ratio or proportions.

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Area of Science:

Biostatistics
Medical Research Methodology

Background:

A survey of physiology and pharmacology journals revealed frequent use of Pearson's Chi-squared and Fisher's exact tests for analyzing 2x2 frequency tables.
Both Pearson's and Fisher's tests have stringent assumptions regarding fixed marginal totals that are often unmet in typical biomedical study designs.

Purpose of the Study:

To critically evaluate the appropriateness of Pearson's Chi-squared and Fisher's exact tests for analyzing 2x2 frequency tables in biomedical research.
To propose alternative statistical methods suitable for singly conditioned 2x2 tables common in biomedical studies.

Main Methods:

Analysis of the assumptions underlying Pearson's Chi-squared and Fisher's exact tests.
Comparison of these tests with methods appropriate for singly conditioned 2x2 tables, where only column marginal totals are fixed.
Discussion of exact tests on the odds ratio (OR=1) and proportions (relative risk, RR=1; difference in proportions=0) as suitable alternatives.

Main Results:

Pearson's test is inappropriate as it assumes random sampling from defined populations with unfixed marginal totals.
Fisher's test is inappropriate as it requires both row and column marginal totals to be fixed, a rare condition in practice.
Singly conditioned 2x2 tables, common in biomedical research, are best analyzed using exact tests on the odds ratio or proportions.

Conclusions:

Pearson's Chi-squared and Fisher's exact tests are frequently misapplied in biomedical research for analyzing 2x2 frequency tables.
Exact tests on the odds ratio or proportions are more appropriate for singly conditioned 2x2 tables, offering specific hypothesis testing.
Biomedical researchers should adopt exact permutation-based methods for analyzing singly conditioned 2x2 tables to ensure statistical validity.