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Understanding Categorical Data Research: Goodness-of-Fit, Independence, and Homogeneity Tests

This presentation provides an overview of research involving categorical data, focusing on key statistical methods such as the Goodness-of-Fit test, χ² test of independence, and χ² test of homogeneity. It outlines the assumptions of these tests, their reporting standards, and follow-up analyses, including Cramér’s phi coefficient and the McNemar test for correlated samples. Expect to gain insights into how to fit data into categories, assess independence between variables, and test relationships in categorical research.

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Understanding Categorical Data Research: Goodness-of-Fit, Independence, and Homogeneity Tests

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Presentation Transcript

  1. Slides to accompany Weathington, Cunningham & Pittenger (2010), Chapter 16: Research with Categorical Data

  2. Objectives • Goodness-of-Fit test • χ2 test of Independence • χ2 test of Homogeneity • Reporting χ2 • Assumptions of χ2 • Follow-up tests for χ2 • McNemar Test

  3. Background • Sometimes we want to know how people fit into categories • Typically involves nominal and ordinal scales • Person only fits one classification • The DV in this type of research is a frequency or count

  4. Goodness-of-Fit Test • Do frequencies of different categories match (fit) what would be hypothesized in a broader population? • χ2 will be large if nonrandom difference between Oi and Ei • If χ2 < critical value, distributions match

  5. Figure 16.1

  6. Table 16.2

  7. Calculation Example

  8. Another Example – Table 16.4

  9. Goodness-of-Fit Test • χ2 is nondirectional (like F) • Assumptions: • Categories are mutually exclusive • Conditions are exhaustive • Observations are independent • N is large enough

  10. χ2 Test of Independence • Are two categorical variables independent of each other? • If so, Oij for one variable should have nothing to do with Eij for other variable and the difference between them will be 0.

  11. Table 16.5

  12. Table 16.6

  13. Computing χ2 Test Statistic

  14. Interpreting χ2 Test of Independence • Primary purpose is to identify independence • If Ho retained, then we cannot assume the two variables are related (independence) • If Ho rejected, the two variables are somehow related, but not necessarily cause-and-effect

  15. χ2 Test of Homogeneity • Can be used to test cause-effect relationships • Categories indicate level of change and χ2 statistic tests whether pattern of Oi deviates from chance levels • If significant χ2, can assume c-e relation

  16. χ2 Test of Homogeneity Example

  17. Reporting χ2 Results • Typical standard is to include the statistic, df, sample size, and significance levels at a minimum: χ2 (df, N = n)= #, p < α χ2(6, N = 240)= 23.46, p < .05

  18. Follow-up Tests to χ2 • Cramér’s coefficient phi (Φ) • Indicates degree of association between two variables analyzed with χ2 • Values between 0 and 1 • Does not assume linear relationship between the variables

  19. Post-Hoc Tests to χ2 • Standardized residual, e • Converts differences between Oi and Ei to a statistic • Shows relative difference between frequencies • Highlights which cells represent statistically significant differences and which show chance findings

  20. Follow-up Tests to χ2 • McNemar Test • For comparing correlated samples in a 2 x 2 table • Table 16.9 illustrates  special form of χ2 test • Ho: differences between groups are due to chance • Example presented in text and Table 16.10 provides an application

  21. What is Next? • **instructor to provide details

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