On the Small Sample Distribution and its Consistency with the Large Sample Distribution of the Chi-Squared Test Statistic for a Two-Way Contigency Table with Fixed Margins

주변값이 주어진 이원분할표에 대한 카이제곱 검정통계량의 소표본 분포 및 대표본 분포와의 일치성 연구

  • Published : 2000.04.30


The chi-squared test statistic is usually employed for testing independence of two categorical variables in a two-way contingency table. It is well known that, under independence, the test statistic has an asymptotic chi-squared distribution under multinomial or product-multinomial models. For the case where both margins fixed, the sampling model of the contingency table is a multiple hypergeometric distribution and the chi-squared test statistic follows the same limiting distribution. In this paper, we study the difference between the small sample and large sample distributions of the chi-squared test statistic for the case with fixed margins. For a few small sample cases, the exact small sample distribution of the test statistic is directly computed. For a few large sample sizes, the small sample distribution of the statistic is generated via a Monte Carlo algorithm, and then is compared with the large sample distribution via chi-squared probability plots and Kolmogorov-Smirnov tests.


  1. Biometrika v.38 Note an Exact treatment of Contingency, Goodness of Fit and Other Problems of Significance Freeman G. H.;Halton, J. H.
  2. 응용통계연구 v.10 2원 분할표의 소표본 검증법 허명희
  3. Communications Statistics -Theory and Methods v.16 The Chi-Squared Test with Both Margins Fixed Alalouf, I. S.
  4. Journal of American Statistical Association v.72 Counting the Number of Contingency Tables With Fixed Margins Gail, M. H.;Mantel, N.
  5. StatXact: Statistical Software for Exact Nonparametric Inference(Version 2) StatXact.
  6. Journal of American Statistical Association v.78 A Network Algorithm for Performing Fisher's Exact Test in r × c contingency tables Mehta, C. R.;Patel, N. R.
  7. Journal of Royal Statistical Society, Series v.C30 An efficient method of generating random R × C tables with given row and column totals Patefield
  8. Communications Statistics-Theory and Methods v.24 Some Remarks on the Chi-Squared Test with Both Margins Fixed Park, C.