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 |
|---|---|
| Pages (from-to) | 2075-2082 |
| Number of pages | 8 |
| Journal | Communications in Statistics - Theory and Methods |
| Volume | 27 |
| Issue number | 8 |
| DOIs | |
| Publication status | Published - 1998 Jan 1 |
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All Science Journal Classification (ASJC) codes
- Statistics and Probability
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Limiting conditional distribution for tests of independence in the two way table. / Kang, Seung Ho; Klotz, Jerome.
In: Communications in Statistics - Theory and Methods, Vol. 27, No. 8, 01.01.1998, p. 2075-2082.Research output: Contribution to journal › Article
TY - JOUR
T1 - Limiting conditional distribution for tests of independence in the two way table
AU - Kang, Seung Ho
AU - Klotz, Jerome
PY - 1998/1/1
Y1 - 1998/1/1
N2 - 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).
AB - 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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U2 - 10.1080/03610929808832210
DO - 10.1080/03610929808832210
M3 - Article
VL - 27
SP - 2075
EP - 2082
JO - Communications in Statistics - Theory and Methods
JF - Communications in Statistics - Theory and Methods
SN - 0361-0926
IS - 8
ER -