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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:

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...

Contingency Table

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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...

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...

McNemar's Test

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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...

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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Effect sizes for 2×2 contingency tables.

Jake Olivier¹, Melanie L Bell

¹School of Mathematics and Statistics, University of New South Wales, Sydney, Australia. j.olivier@unsw.edu.au

|March 19, 2013

Summary

This summary is machine-generated.

Researchers often struggle with choosing appropriate effect sizes for sample size calculations. This study provides new, objective recommendations for the odds ratio in 2x2 tables, improving statistical analysis and research resource allocation.

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Area of Science:

Biostatistics
Statistical Methods
Research Methodology

Background:

Sample size calculations are crucial for ethical and efficient research.
Effect sizes are essential for sample size calculations but often unknown.
Existing effect size recommendations for 2x2 tables (e.g., phi coefficient) have limitations.

Purpose of the Study:

To propose evidence-based recommendations for effect sizes in 2x2 table analyses.
To provide guidance on selecting appropriate odds ratios for sample size calculations.
To address the limitations of existing effect size measures for binary data.

Main Methods:

Anchoring odds ratio recommendations to the phi coefficient (φ) for fixed marginal probabilities.
Developing a framework for odds ratio effect size recommendations.
Demonstrating the generalizability of the proposed method by relaxing marginal assumptions.

Main Results:

New, objective recommendations for odds ratio effect sizes are presented.
The proposed method offers a more statistically sound alternative to the phi coefficient for 2x2 tables.
The recommendations are robust and applicable even when marginal probabilities are unknown.

Conclusions:

The proposed odds ratio recommendations enhance the accuracy of sample size calculations for 2x2 tables.
This work provides researchers with practical tools for determining meaningful effect sizes.
Improved effect size guidance contributes to more efficient and ethical research practices.