Limiting conditional distribution for tests of independence in the two way table

Seung Ho Kang, Jerome Klotz

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

For testing the hypothesis of independence in the r x c two way contingency table, we consider the likelihood ratio statistic -2 ln(λ) and the χ2 statistic conditionally given the sufficient statistic of row and column margin totals. We use Stirling's formula in the conditional characteristic function of normalized cell counts to give a direct proof that the limiting null conditional distribution is multivariate normal Z : script Nrc(0, Λ). We then establish the two statistics have a limiting conditional distribution equivalent to ZT Λ-Z, which has a central χv2, distribution with degrees of freedom v = (r - 1)(c - 1) = rank(Λ).

Original language English 2075-2082 8 Communications in Statistics - Theory and Methods 27 8 https://doi.org/10.1080/03610929808832210 Published - 1998 Jan 1

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Test of Independence
Conditional Distribution
Limiting Distribution
Table
Stirling's formula
Sufficient Statistics
Likelihood Ratio Statistic
Multivariate Normal
Null Distribution
Contingency Table
Characteristic Function
Margin
Statistic
Count
Limiting
Degree of freedom
Statistics
Testing
Cell
Independence

All Science Journal Classification (ASJC) codes

• Statistics and Probability

Cite this

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abstract = "For testing the hypothesis of independence in the r x c two way contingency table, we consider the likelihood ratio statistic -2 ln(λ) and the χ2 statistic conditionally given the sufficient statistic of row and column margin totals. We use Stirling's formula in the conditional characteristic function of normalized cell counts to give a direct proof that the limiting null conditional distribution is multivariate normal Z : script Nrc(0, Λ). We then establish the two statistics have a limiting conditional distribution equivalent to ZT Λ-Z, which has a central χv2, distribution with degrees of freedom v = (r - 1)(c - 1) = rank(Λ).",
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In: Communications in Statistics - Theory and Methods, Vol. 27, No. 8, 01.01.1998, p. 2075-2082.

Research output: Contribution to journalArticle

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AU - Klotz, Jerome

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