Goodness-of-Fit Statistics for Discrete Multivariate Data

1. Front Matter

   Pages i-xi

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2. Introduction to the Power-Divergence Statistic

   • Timothy R. C. Read, Noel A. C. Cressie

   Pages 1-4

3. Defining and Testing Models: Concepts and Examples

   • Timothy R. C. Read, Noel A. C. Cressie

   Pages 5-18

4. Modeling Cross-Classified Categorical Data

   • Timothy R. C. Read, Noel A. C. Cressie

   Pages 19-43

5. Testing the Models: Large-Sample Results

   • Timothy R. C. Read, Noel A. C. Cressie

   Pages 44-63

6. Improving the Accuracy of Tests with Small Sample Size

   • Timothy R. C. Read, Noel A. C. Cressie

   Pages 64-80

7. Comparing the Sensitivity of the Test Statistics

   • Timothy R. C. Read, Noel A. C. Cressie

   Pages 81-97

8. Links with Other Test Statistics and Measures of Divergence

   • Timothy R. C. Read, Noel A. C. Cressie

   Pages 98-113

9. Future Directions

   • Timothy R. C. Read, Noel A. C. Cressie

   Pages 114-132

10. Historical Perspective: Pearson’s X² and the Loglikelihood Ratio Statistic G²

    • Timothy R. C. Read, Noel A. C. Cressie

    Pages 133-153

11. Back Matter

    Pages 154-211

    PDF

The statistical analysis of discrete multivariate data has received a great deal of attention in the statistics literature over the past two decades. The develop­ ment ofappropriate models is the common theme of books such as Cox (1970), Haberman (1974, 1978, 1979), Bishop et al. (1975), Gokhale and Kullback (1978), Upton (1978), Fienberg (1980), Plackett (1981), Agresti (1984), Goodman (1984), and Freeman (1987). The objective of our book differs from those listed above. Rather than concentrating on model building, our intention is to describe and assess the goodness-of-fit statistics used in the model verification part of the inference process. Those books that emphasize model development tend to assume that the model can be tested with one of the traditional goodness-of-fit tests 2 2 (e.g., Pearson's X or the loglikelihood ratio G ) using a chi-squared critical value. However, it is well known that this can give a poor approximation in many circumstances. This book provides the reader with a unified analysis of the traditional goodness-of-fit tests, describing their behavior and relative merits as well as introducing some new test statistics. The power-divergence family of statistics (Cressie and Read, 1984) is used to link the traditional test statistics through a single real-valued parameter, and provides a way to consolidate and extend the current fragmented literature. As a by-product of our analysis, a new 2 2 statistic emerges "between" Pearson's X and the loglikelihood ratio G that has some valuable properties.

• Chi-squared distribution
• Estimator
• Likelihood
• Variance
• best fit
• information theory
• multinomial distribution

• Hewlett-Packard Company, Palo Alto, USA

  Timothy R. C. Read

• Department of Statistics, Iowa State University, Ames, USA

  Noel A. C. Cressie

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