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Related Concept Videos

Contingency Table01:29

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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...
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Determination of Expected Frequency01:08

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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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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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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.
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Fisher's Exact Test01:08

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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...
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Expected Frequencies in Goodness-of-Fit Tests01:19

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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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Maximum Augmented Empirical Likelihood Estimation of Categorical Marginal Models for Large Sparse Contingency Tables.

L Andries van der Ark1, Wicher P Bergsma2, Letty Koopman3

  • 1Research Institute of Child Development and Education, University of Amsterdam, P.O. Box 15776, 1001, NG, Amsterdam, The Netherlands. L.A.vanderArk@uva.nl.

Psychometrika
|September 26, 2023
PubMed
Summary

Maximum augmented empirical likelihood (MAEL) estimation offers a solution for analyzing large, sparse categorical data. This new method overcomes limitations of maximum empirical likelihood (MEL) for complex models.

Keywords:
Cronbach’s alphacategorical marginal modellarge categorical data setsmarginal homogeneitymaximum empirical likelihood estimationmaximum likelihood estimationscalability coefficients

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Area of Science:

  • Statistics
  • Computational Statistics
  • Data Analysis

Background:

  • Categorical marginal models (CMMs) are effective for dependent categorical data when dependencies are not the primary focus.
  • Maximum likelihood (ML) estimation for CMMs becomes computationally infeasible with an increasing number of variables due to exponentially growing contingency tables.
  • Maximum empirical likelihood (MEL) estimation offers an alternative with optimal asymptotic efficiency but struggles with large, sparse tables.

Purpose of the Study:

  • To address the breakdown of maximum empirical likelihood (MEL) estimation in large, sparse contingency tables.
  • To introduce a novel estimation method for categorical marginal models (CMMs) that is computationally feasible for large datasets.
  • To provide a robust statistical tool for analyzing complex categorical data structures.

Main Methods:

  • Development of maximum augmented empirical likelihood (MAEL) estimation.
  • MAEL involves augmenting the empirical likelihood support with carefully selected cells.
  • Simulation studies were conducted to evaluate the performance of MAEL.

Main Results:

  • Maximum empirical likelihood (MEL) estimation was shown to be unreliable for large, sparse contingency tables.
  • The proposed maximum augmented empirical likelihood (MAEL) method demonstrates good finite sample performance.
  • MAEL is effective even for very large contingency tables, overcoming the limitations of previous methods.

Conclusions:

  • Maximum augmented empirical likelihood (MAEL) estimation provides a viable and robust alternative to maximum likelihood (ML) and maximum empirical likelihood (MEL) for CMMs.
  • MAEL is particularly well-suited for analyzing large and sparse categorical datasets where traditional methods fail.
  • The proposed method enhances the applicability of CMMs in complex statistical modeling scenarios.