Tests Of Independence For Nominal And Ordinal Data; Comparison And Application
Author: Shakeel Shahzad


This study aims to analyze the performance of tests of independence for categorical data which may further be classified as nominal and ordinal data. Tests of independence are one of the most frequently used statistical tools in econometrics. Researchers are often interested in the independence of variables summarized in Contingency Tables (CTs). Many tests are available in the literature to test independence in CTs. However, there is no clarity about the choice of tests that are incapable to provide a comparison of a large number of tests.

A central problem and question facing researchers is to decide which tests of independence are most stringent for the data in hand. Most of the studies make pairwise comparisons of tests and such studies are unable to guide optimal tests among a wide set of tests. Furthermore, such studies used different conventional statistical techniques to find an optimal test of independence for nominal and ordinal data.

This study used Monte Carlo Simulations (MCS) to evaluate the performance of a large number of tests of independence for nominal and ordinal data.

The study compares eleven tests of independence for nominal data namely, Pearson’s Chi-Square ( χ 2 ) test, Log Likelihood Ratio (G 2 ) test, Fisher Exact Test (FES), Freeman and Tuckey Test (FTS), Cressie and Read Test (CRS), Kulber and Liaber test (KLS), Neyman Modified Chi-Square Test (NMCS), Modular Test (MDS), D Square (D 2 ), BP Test, and Logarithmic Minimum Square Test (LMS). We were able to calculate the most stringent test and it turned out that Logarithmic Minimum Square (LMS) is the most stringent test for nominal data in w × k CTs.

Similarly, seven popular tests of independence for ordinal data are compared namely, Spearman 𝜌 coefficient of correlation, Kendall’s𝜏 − 𝑎, Kendall’s𝜏 − 𝑏, Kendall’s 𝜏 − 𝑐 coefficient, Goodman and Kruskal γ, Sumer’s D and Novel Phi_k (ϕ𝑘). Since the likelihood function is not found in the literature; for ordinal data, the stringency criteria cannot be applied to compare tests. Therefore, the comparison was made based on power and our MCS concludes based on solid estimations using the power criteria that the most powerful test is Novel ϕ𝑘 in w × k CTs for ordinal data.

Meta Data

JEL Classification: C1, C12, C14, C15, C46, C63
Keywords : Contingency Table, Nominal Data, Ordinal data, Power of test, Size of test, Stringency Criteria, Tests of Independence
Supervisor: Saud Ahmed Khan
Cosupervisor: Atiq Ur Rehman

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