abstract

Decompositions of symmetry using models based on
f -divergence for square contingency tables
Yusuke Saigusa
(Tokyo University of Science)
Kouji Tahata
(Tokyo University of Science)
Sadao Tomizawa (Tokyo University of Science)
1. ͸͡Ίʹ
ɹߦͱྻ͕ॱংͷ͋Δಉ͡෼ྨ͔ΒͳΔ r ʷ r ਖ਼ํ෼ׂදΛߟ͑ɼ(i, j) ηϧ֬཰Λ pij
ͱ͢Δ (i = 1, . . . , r; j = 1, . . . , r)ɽ͜ͷͱ͖ɼରশ (S) Ϟσϧ͸࣍ͷΑ͏ʹఆٛ͞ΕΔ
(Bowker [3])ɿ
pij = ψij
(i = 1, . . . , r; j = 1, . . . , r),
ψij = ψji .
ͨͩ͠
S Ϟσϧ͸ओର֯ઢʹؔ͢Δηϧ֬཰ͷରশߏ଄Λࣔ͢ɽS Ϟσϧ͕੒Γཱͨͳ͍ͱ͖ɼ
͍͔ͭ͘ͷ֦ுͨ͠Ϟσϧ͕ఏҊ͞Ε͍ͯΔɽͨͱ͑͹ɼ४ରশ (QS) Ϟσϧ͸࣍ͷ
Α͏ʹఆٛ͞ΕΔ (Caussinus [4])ɿ
pij = αi βj ψij
(i = 1, . . . , r; j = 1, . . . , r),
ͨͩ͠
ψij = ψji .
QS Ϟσϧ͸Φοζൺ θ(i<j;s<t) = (pis pjt )/(pjs pit ) Λ༻͍ͯ࣍ͷΑ͏ʹ΋දͤΔɿ
θ(i<j;s<t) = θ(s<t;i<j)
(1 ≤ i < j ≤ r; 1 ≤ s < t ≤ r).
QS Ϟσϧ͸ओର֯ઢʹؔ͢ΔΦοζൺͷରশߏ଄Λࣔ͢ɽ·ͨɼपลಉ౳ (MH) Ϟσ
ϧ͸࣍ͷΑ͏ʹఆٛ͞ΕΔ (Stuart [7])ɿ
pi· = p·i
(i = 1, . . . , r),
ͨͩ͠
pi· =
r
X
pit ,
p·i =
t=1
r
X
psi .
s=1
MH Ϟσϧ͸ɼߦม਺ͱྻม਺ͷपล෼෍͕ಉ౳Ͱ͋Δ͜ͱΛ͍ࣔͯ͠Δɽ
Caussinus [4] ͸࣍ͷఆཧΛ༩͑ͨɿ
ఆཧ 1ɽS Ϟσϧ͕੒ΓཱͭͨΊͷඞཁे෼৚݅͸ɼQS Ϟσϧͱ MH Ϟσϧͷ྆ํ
͕੒Γཱͭ͜ͱͰ͋Δɽ
{us } Λॱংͷ͋Δ‫ط‬஌ͷείΞͱ͢Δ (s = 1, . . . , r; u1 ≤ u2 ≤ · · · ≤ ur ; u1 < ur )ɽ
͜ͷͱ͖ɼॱং४ରশ (OQS) Ϟσϧ͸࣍ͷΑ͏ʹఆٛ͞ΕΔ (Agresti [1], p.429)ɿ
pij = αui β uj ψij
(i = 1, . . . , r; j = 1, . . . , r),
ͨͩ͠
ψij = ψji .
OQS Ϟσϧ͸ QS Ϟσϧͷಛผͳ৔߹Ͱ͋Δɽ
f Λ 2 ֊ඍ෼Մೳͳ‫ڱ‬ٛತؔ਺ɼF Λ f ͷ 1 ࣍ಋؔ਺ͱ͢Δɽf -μΠόʔδΣϯεʹ
‫ͮ͘ج‬४ରশ (QS[f ]) Ϟσϧ͸࣍ͷΑ͏ʹఆٛ͞ΕΔ (Kateri and Papaioannou [6])ɿ
pij = pSij F −1 (αi + γij )
(i = 1, . . . , r; j = 1, . . . , r),
ͨͩ͠ɼ
pSij =
pij + pji
,
2
γij = γji .
ಛʹɼf (x) = xlogxɼx > 0ɼͱͨ͠ QS[f ] Ϟσϧ͸ QS ϞσϧͰ͋Δɽ·ͨɼf (x) = (1−
x)2 ͱͨ͠ QS[f ] Ϟσϧ͸ Pearsonian QS Ϟσϧͱ‫ݺ‬͹ΕΔ (Kateri and Papaioannou
[6])ɽ
Kateri and Papaioannou [6] ͸࣍ͷఆཧΛࣔͨ͠ɿ
ఆཧ 2ɽS Ϟσϧ͕੒ΓཱͭͨΊͷඞཁे෼৚݅͸ɼQS[f ] Ϟσϧͱ MH Ϟσϧͷ྆
ํ͕੒Γཱͭ͜ͱͰ͋Δɽ
f -μΠόʔδΣϯεʹ‫ॱͮ͘ج‬ং४ରশ (OQS[f ]) Ϟσϧ͸࣍ͷΑ͏ʹఆٛ͞ΕΔ
(Kateri and Agresti [5])ɿ
pij = pSij F −1 (αui + γij )
(i = 1, . . . , r; j = 1, . . . , r),
ͨͩ͠ɼ
pij + pji
, γij = γji .
2
OQS[f ] Ϟσϧ͸ QS[f ] Ϟσϧͷಛผͳ৔߹Ͱ͋Δɽಛʹɼf (x) = xlogxɼx > 0ɼͱ͠
ͨ OQS[f ] Ϟσϧ͸ OQS ϞσϧͰ͋Δɽ·ͨɼf (x) = (1 − x)2 ͱͨ͠ OQS[f ] Ϟσϧ
͸ Pearsonian OQS Ϟσϧͱ‫ݺ‬͹ΕΔ (Kateri and Agresti [5])ɽ
Kateri and Agresti [5] ͸࣍ͷఆཧΛࣔͨ͠ɿ
pSij =
ఆཧ 3ɽS Ϟσϧ͕੒ΓཱͭͨΊͷඞཁे෼৚݅͸ɼOQS[f ] Ϟσϧͱ MH Ϟσϧͷ
྆ํ͕੒Γཱͭ͜ͱͰ͋Δɽ
S Ϟσϧ͸੍໿ͷ‫͍ڧ‬ϞσϧͰ͋ΔͨΊɼ࣮ࡍͷσʔλղੳʹ͓͍ͯ͸ద߹౓͕ѱ
͍͜ͱ͕ଟ͍ɽຊ‫Ͱڀݚ‬͸ɼS ϞσϧΛɼΑΓऑ੍͍໿Λ΋͍͔ͭͭ͘ͷϞσϧʹ෼͚
ΔɽͦͷΑ͏ͳ S Ϟσϧͷ෼ղ͸ɼS Ϟσϧͷσʔλʹର͢Δ౰ͯ͸·Γ͕ѱ͍ͱ͖ɼ
ͦͷ‫ݪ‬ҼΛߟ͑Δ্Ͱ༗༻Ͱ͋Δɽ
ද 1 ʹ r ʷ r දʹద༻֤ͨ͠Ϟσϧͷద߹౓‫ݕ‬ఆͷͨΊͷ໬౓ൺΧΠೋ৐‫ݕ‬ఆ౷‫ྔܭ‬
ͷࣗ༝౓Λ‫͢ه‬ɽද 1 ΑΓɼOQS[f ] Ϟσϧͱ MH Ϟσϧͷࣗ༝౓ͷ࿨͸ S Ϟσϧͷࣗ
༝౓ΑΓ΋େ͖͍ɽ͜͜ʹɼࣗ༝౓͸ r ʷ r දʹ͓͚Δ֤Ϟσϧͷ੍໿਺ʹ౳͍͜͠ͱ
ʹ஫ҙ͢ΔɽΑͬͯɼOQS[f ] Ϟσϧͱ MH Ϟσϧ͕ಉ࣌ʹ੒Γཱͭߏ଄͸ S Ϟσϧͷ
ࣔ͢ߏ଄ΑΓ΋੍໿͕‫͍͑ͱ͍ڧ‬Δɽ
ຊઅͰ͸ɼϞσϧͷ੍໿਺Λߟྀͨ͠ S Ϟσϧͷ෼ղΛ༩͑Δɽ
ද 1. r ʷ r දʹద༻֤ͨ͠Ϟσϧͷద߹౓‫ݕ‬ఆͷͨΊͷ໬౓ൺΧΠೋ৐౷‫ྔܭ‬ͷࣗ
༝౓
Ϟσϧ
S
OQS[f ]
MH
ࣗ༝౓
r(r − 1)/2
r(r − 1)/2 − 1
r−1
2. Ϟσϧͷ෼ղ
2.1. ରশੑͷ෼ղ
ɹߦͱྻͷείΞͷฏ‫ۉ‬Ұக (ME) ߏ଄Λߟ͑Δɿ
µ1 = µ2
ͨͩ͠
µ1 =
r
X
ui pi· ,
µ2 =
i=1
r
X
ui p·i .
i=1
͜ͷͱ͖ɼ࣍ͷఆཧΛಘΔɽ
ఆཧ 4ɽS Ϟσϧ͕੒ΓཱͭͨΊͷඞཁे෼৚݅͸ɼOQS[f ] Ϟσϧͱ ME Ϟσϧͷ
྆ํ͕੒Γཱͭ͜ͱͰ͋Δɽ
ಛʹ f (x) = xlogxɼx > 0ɼͱͨ͠ʢ͢ͳΘͪ OQS[f ] Ϟσϧ͕ OQS ϞσϧͱͳΔͱ
͖ͷʣఆཧ 4 ͸ Tahata, Yamamoto and Tomizawa [8] ʹΑͬͯ༩͑ΒΕ͍ͯΔɽ
2.2. Ԡ༻ྫ
ɹද 2 ͸࿈ଓ͢Δ 2 ೥ͷ༽ͷग़ੜ਺ʹؔ͢ΔσʔλͰɼॱংΧςΰϦ 4 ʷ 4 ෼ׂදͰ͋
Δ (Tallis [9])ɽ
ද 3 ͸ɼද 2 ͷσʔλʹద༻֤ͨ͠Ϟσϧʹର͢Δద߹౓ΧΠೋ৐౷‫ྔܭ‬ͷ஋Λࣔ͢ɽ
ද 2. ࿈ଓ͢Δ 2 ೥ͷ༽ͷग़ੜ਺ʹ‫ͮ͘ج‬σʔλ (Tallis [9])
1952 ೥
1953 ೥ 0
1
2 ‫ܭ‬
0
58 52 1 111
1
26 58 3 87
2
8 12 9 29
‫ܭ‬
92 122 13 227
ද 3. ද 2 ͷσʔλʹରͯ͠ద༻֤ͨ͠Ϟσϧʹର͢Δ໬౓ൺΧΠೋ৐౷‫ ྔܭ‬G2 ஋
Ϟσϧ
S
OQS
Pearsonian OQS
ME
QS
Pearsonian QS
MH
ࣗ༝౓
3
2
2
1
1
1
2
G2 ஋
20.81*
20.74*
20.75*
0.07
1.35
2.16
18.65*
*ҹ͸ 5% ༗ҙΛࣔ͢
ද 3 ΑΓɼද 2 ͷσʔλʹରͯ͠ S Ϟσϧͷద߹౓͸ѱ͘ɼOQS Ϟσϧʢ͋Δ͍͸
Pearsonian OQS Ϟσϧʣͷద߹౓΋ѱ͍ɽ͔͠͠ɼME Ϟσϧͷద߹౓͸ඇৗʹྑ͍ɽ
ఆཧ 4 ΑΓɼME ϞσϧΑΓ΋ OQS Ϟσϧʢ͋Δ͍͸ Pearsonian OQS Ϟσϧʣͷࣔ͢
֬཰ߏ଄่͕Ε͍ͯΔ͜ͱ͕ɼS Ϟσϧͷద߹౓ͷѱ͍‫ݪ‬ҼͰ͋ΔͱਪଌͰ͖Δɽ
3. Ϟσϧͷ௚ަੑ
ɹϞσϧ M ͷద߹౓Λ‫ݕ‬ఆ͢ΔͨΊͷ໬౓ൺΧΠೋ৐౷‫ྔܭ‬Λ G2 (M ) ͱ͢Δɽఆཧ 1
ʹରͯ͠ɼTomizawa and Tahata [10] ͸‫ݕ‬ఆ౷‫ؔ͢ʹྔܭ‬Δ࣍ͷఆཧΛ༩͑ͨɿ
ఆཧ 5ɽG2 (S) ͸ɼG2 (QS) ͱ G2 (M H) ͷ࿨ʹ઴ۙతʹಉ౳Ͱ͋Δɽ
Ұൠʹɼ
ʮϞσϧ M3 ͕੒ΓཱͭͨΊͷඞཁे෼৚݅͸ɼϞσϧ M1 ͱϞσϧ M2 ͷ྆
ํ͕੒Γཱͭ͜ͱͰ͋Δʯͱ͢ΔɽG2 (M3 ) ͕ G2 (M1 ) ͱ G2 (M2 ) ͷ࿨ʹ઴ۙతʹಉ౳
ͷͱ͖ɼ༗ҙਫ४ α ͰɼϞσϧ M1 ͱϞσϧ M2 ͕ͦΕͧΕߴ͍֬཰Ͱ࠾୒͞ΕΔͳΒ
͹ɼϞσϧ M3 ΋࠾୒͞ΕΔ܏޲ʹ͋ΔɽҰํͰɼG2 (M3 ) ͕ G2 (M1 ) ͱ G2 (M2 ) ͷ࿨ʹ
઴ۙతʹಉ౳Ͱͳ͍ͱ͖ɼϞσϧ M1 ͱϞσϧ M2 ͕ͦΕͧΕߴ͍֬཰Ͱ࠾୒͞ΕΔʹ
΋ؔΘΒͣɼϞσϧ M3 ͕‫غ‬٫͞ΕΔͱ͍͏ໃ६ͨ͠ঢ়‫͜ى͕گ‬Γ͏Δ (Aitchison [2])ɽ
ຊ‫Ͱڀݚ‬͸ɼ࣍ͷఆཧΛಘΔɽ
ఆཧ 6ɽS ϞσϧͷԼͰɼG2 (S) ͸ɼG2 (QS[f ]) ͱ G2 (M H) ͷ࿨ʹ઴ۙతʹಉ౳Ͱ
͋Δɽ
ఆཧ 7ɽS ϞσϧͷԼͰɼG2 (S) ͸ɼG2 (OQS[f ]) ͱ G2 (M E) ͷ࿨ʹ઴ۙతʹಉ౳Ͱ
͋Δɽ
ఆཧ 6 ͱఆཧ 7 ͔Βɼఆཧ 2 ͱఆཧ 4 Ͱ༩͑ΒΕΔϞσϧͷ෼ղʹ͓͍ͯ͸ɼ্‫ه‬ͷ
Α͏ͳໃ६ͨ͠ঢ়‫گ‬͸‫͜ى‬Γʹ͍͘ͱ͍͑Δɽ
ࢀߟจ‫ݙ‬
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[2] Aitchison, J. (1962). Large-sample restricted parametric tests. Journal of the Royal
Statistical Society, Series B, 24, 234-250.
[3] Bowker, A. H. (1948). A test for symmetry in contingency tables. Journal of the American Statistical Association, 43, 572-574.
[4] Caussinus, H. (1965). Contribution `a l’analyse statistique des tableaux de corr´elation.
Annales de la Facult´e des Sciences de l’Universit´e de Toulouse, 29, 77-182.
[5] Kateri, M. and Agresti, A. (2007). A class of ordinal quasi-symmetry models for square
contingency tables. Statistics and Probability Letters, 77, 598-603.
[6] Kateri, M. and Papaioannou, T. (1997). Asymmetry models for contingency tables.
Journal of the American Statistical Association, 92, 1124-1131.
[7] Stuart, A. (1955). A test for homogeneity of the marginal distributions in a two-way
classification. Biometrika, 42, 412-416.
[8] Tahata, K., Yamamoto, H. and Tomizawa, S. (2008). Orthogonality of decompositions
of symmetry into extended symmetry and marginal equimoment for multi-way tables
with ordered categories. Austrian Journal of Statistics, 37, 185-194.
[9] Tallis, G. M. (1962). The maximum likelihood estimation of correlation from contingency
tables. Biometrics, 18, 342-353.
[10] Tomizawa, S. and Tahata, K. (2007). The analysis of symmetry and asymmetry: Orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry
for multi-way tables. Journal de la Soci´et´e Fran¸caise de Statistique, 148, 3-36.

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