An introduction to log-linear analysis and implementing the Newton-Raphson algorithm in APL2

ABSTRACT

This paper introduces the method of log-linear analysis for performing hypothesis testing on contingency tables. The reader will see a brief development of this topic beginning with Pearson's classic chi-square test and examples of analyses on two and three dimensional tables. The idea of hierarchical models and backward elimination are discussed. Finally, the APL2 implementation of the Newton-Raphson algorithm is described. Newton-Raphson is an iterative procedure for finding the roots of a function. It is used in log-linear analysis to find the maximum-likelihood estimation of expected frequencies that cannot be calculated from expressions in closed form.

REFERENCES

• 1: Bishop, Y. M., Fienberg, S. E., & Holland, P. W. (1975). Discrete Multivariate Analysis: Theory_ and Practice. Cambridge, MA: The MIT Press.
• 2: Haberman, S. (1978). Analysis of Oualitative Data. 2 vols. New York: Academic Press.
• 3: Kennedy, J. J. (1992). Analyzing Oualitative Data: Lo~-Linear Analysis for Behavioral Re~ach. (2nd edition). New York: Praeger.
• 4: Bakeman, R., & Robinson, B. F., (In Prep). Understandin~ Lo~-Linear Ana!vsis with ILOG: An Interactive Aooroach.
• 5: Pearson, K., (1900). On a criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can reasonably be supposed to have arisen from random sampling. Philosophical Magazine, 50, 157-75.
• 6: Deming, W.E., & Stephan, F.F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Statist., 11, 427-444.

INDEX TERMS

Primary Classification:
  D. Software
  D.3 PROGRAMMING LANGUAGES
      D.3.2 Language Classifications

          Nouns: APL

Additional Classification:
  E. Data
  E.1 DATA STRUCTURES
      Subjects: Tables**; Arrays

General Terms:
Algorithms, Languages