Methods of Combining P-values for Multiple Endpoints of Various Data Types

(1)

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j 3© e©r+«>\"f u«Ñ´òõ #Q ìøÍ6£xÃº(endpoints)Ð 8£¤&ñ|¨c M:, s[þt ìøÍ6£x

Ãº @/1px > ×æכ¹ # ÅÒכ¹ ìøÍ6£xÃº(primary endpoint)\¦ ×þ½+É Ãº \OH ©S!s µ

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ÏÒqt½+É Ãº e. O⁰Brien (1984)Ér s[þt ìøÍ6£xÃº ¸¿º\¦ 7áx½+Ë # u«Ñ´òõ\ @/ôÇ éß8£¤

&ñ(one-tailed testing) :x>|¾ÓÜ¼Ð"f ìøÍ6£xÃº 5Åq+þA(continuous) «ÑÐ 8£¤&ñ÷&%3

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¦ M: Ordinary Least Square(OLS)ü< Generalized Least Square(GLS) &ñ :x>|¾Ó`¦ ]

jr %i. Pocock 1px (1987)Ér #Q +þAI, 7£¤ 5Åq+þA, síß+þA(binary), Òqt>r(survival)

«Ñ_ ìøÍ6£xÃº\¦ <Êa ìr$3½+É Ãº e6£§`¦ /åL ¦ eÜ¼ z´]jÐ sü< °ú s #Q +þA I

_ ìøÍ6£xÃº #î½+Ë\ @/ôÇ ëH]j&h`¦ [O"î ½¨^&hÜ¼Ð ¸_z´+«>Ü¼Ð"f sQôÇ â Ä

º_ OLSü< GLS :x>|¾Ó_ ´òÖ¦$í`¦ ·úÐtH ·ú§¤. :r 7HëH\"fH :£¤y #Q +þA I

_ ìøÍ6£xÃº\¦ 7áx½+Ë # u«Ñ´òõ\ @/ôÇ :r`¦ ?/oHX< P °úכ_ #î½+Ë :x>|¾Ó`¦ ]j î

ß

9, sM: y ìøÍ6£xÃº_ u«Ñ´òõ\ @/ôÇ &ñ õ P °úכÉr "fÐ ©'a$ís >rF

H P °úכs. OLS x9 GLS &ñ :x>|¾ÓÐ ©&h`¦ t P °úכ_ #î½+Ë~½ÓZO ×æ, ~½ÓZO Fü<

GH ]j 17áx ¸ÀÓ Ä»_ÃºïrÐ &"f &ñ_ :rs ¸ú3lw ?/9|9 Ãº eH âÄº e¦

~

½

ÓZO BH ]j 17áx_ ¸ÀÓ ¸ú :x]j÷&¦ ¢¸ôÇ ´òÖ¦$ís Z}Ér כ Ü¼Ð z¤.

Å

Òכ¹6 x#Q: ×æ ìøÍ6£xÃº, 1lqwn)a P °úכ_ #î½+Ë, 'a)a P °úכ_ #î½+Ë, OLS, GLS.

1. " V´ * 0

1 l

qwn)a ¿º çH_ u«Ñ´òõ\¦ q§ H e©r+«>_ &³©\"f y ¨8\>"f Ñüt s©_ ìøÍ 6

£

xÃº 8£¤&ñ÷&#Q ×æ ìøÍ6£xÃº(multivariate endpoints)\¦ ìr$3K H âÄº ¥u ·ú§

>

µ1ÏÒqtôÇ. \V\¦ [þt#Q"f, p²DG d'õA(U.S. FDA)s wn q@/7£x(prostatic hyperplasia)

¨ 8

\ @/ôÇ u«ÑÐ 0A(placebo)õ q§ôÇ &h&ñ|¾Ó 5mg_ finasteride\¦ 5pxÙþ¡~ e©r+«>

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¦ ¶ú(RÐ, 8úx C¹7£x©&hÃº(urinary symptom score), כ¹5Åq(urinary-flow rate test), w

n

^ÂÒx(total prostate volume), ïß¹|¾Ó(residual volume), wn :£¤s½Ó"é¶(prostate specific antigen) 1px_ ìøÍ6£xÃºÐ u«Ñ´òõ\¦ ¨î %i. :£¤y, %6£§ [j ìøÍ6£xÃº_ :x>

&

h

Ä»_$í`¦ HÐ Õª ´òõ\¦ 7£x"î %i(Chowü< Liu, 2003). sQôÇ ×æ ìøÍ6£xÃº\ 'a

1) (137-701) "fÖ¦r "fí½¨ ìøÍí1lx 505, d¦aË:@/<Æ§ _<Æ:x><Æõ, $3õ&ñ. E-mail: kimsuyoung@catholic.ac.kr

2) (137-701) §$. "fÖ¦r "fí½¨ ìøÍí1lx 505, d¦aË:@/<Æ§ _<Æ:x><Æõ, §Ãº.

E-mail: hhsong@catholic.ac.kr

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ô

Ç ìr$3 r © /'îr ]XHÉr ÅÒ'a&h óøÍéß <ÊÉr ½¨3lq&h 1px\ qÆÒ#Q Ð ÅÒכ¹ ìøÍ6£x Ã

º(primary endpoint)ü< ÐØæ ìøÍ6£xÃº(subsidiary endpoint)\¦ Äº&hÜ¼Ð ×þ # y Ã

º\ @/K éß|¾Ó ìr$3(univariate analysis)`¦ z´r H כ s. ÕªQ Pocock 1px (1987)s

/åL %i1pws Ä»_ t ·ú§Ér ìøÍ6£xÃº\¦ ]jü@<ÊÜ¼Ð+ 0A&hÜ¼Ð Ä»_ôÇ :r`¦ %3H â Ä

º¸ µ1ÏÒqt 9, :£¤y «Ñ_ ÂÒìr&h :r`¦ ]jr HX< Õªu> ÷&H éß&hs e. sQôÇ Ð

o|ÃÌ\"f #Q ìøÍ6£xÃº @/1pxôÇ ×æכ¹$íÜ¼Ð ìr$3÷&H ~½ÓZOs ¦9÷&#Q ôÇ.

Hotelling_ T²:x>|¾ÓÉr ×æ ìøÍ6£xÃº ¸¿º\ HôÇ u«Ñ´òõ\¦ &ñ tëß, O⁰Brien (1984)õ Meier (1975) t&h %i1pws 8úx K>h_ ìøÍ6£xÃº_ y ìøÍ6£xÃº ´òõß¼l(effect sizes) δk= µ1k− µ2k 0õ Ér\ @/ôÇ @/wn[O_ &ñsÙ¼Ð u«Ñ´òõ\¦ _p H δk > 0÷rëß m u«Ñ Ä»KôÇ âÄº\¦ _p H δk < 0¸ <Êa ly # &ñ§4s ±ú`¦ Ã

º µ1Ú\ \O.

#

Q ìøÍ6£xÃº_ ´òõß¼l_ ª~½Ó¾Ó$í ëH]j\¦ x½+É Ãº eH ]XHÜ¼Ð"f éß8£¤ &ñ Êê Bonferroni Ãº&ñZOs e. K>h_ ìøÍ6£xÃº "fÐ 1lqwns &ñ½+É M:, y ìøÍ6£xÃº_ éß 8

£

¤ &ñ_ õÐ %3Ér P °úכ ×æ © Ér °úכ\ K\¦ YLôÇ °úכ`¦ Ä»_Ãºïr αü< q§ # &ñ

:r`¦ ?/oH ~½ÓZOs. &h6 xs çßéß ¦ sK l /'îr s ~½ÓZOÉr y ìøÍ6£xÃº\ @/K éß

|¾Ó ìr$3`¦ z´r > ÷&#Q #Q SÃº_ Ä»_$í &ñÜ¼Ð 7£x÷&H ]j 17áx ¸ÀÓ\¦ Ð&ñôÇ

H ©&hs etëß ìøÍ6£xÃº_ ©'a$í(ρ)s 7£x½+ÉÃº2¤, :£¤y ρ ≥ 0.5 âÄº\ Bonferroni Ã

º&ñZOs BÄº ÐÃº&he`¦ Pocock 1px (1987)Ér 7HëH_ ³ð 1\"f µ1ßy¦ e. sH Gupta (1963) ]jrôÇ |¾Ó &ñ½©(multivariate normal) ìrí_ ³ðïro)a &ñ½©¼# ×æ, þj@/°úכ_ S

X

Ò¦ ìrí\ HôÇ α⁰°úכõ Bonferroni Ãº&ñ\ _ôÇ α⁰⁰= α/K°úכ`¦ q§<ÊÜ¼Ð+ 7£x"î %i

. Bonferroni Ãº&ñZO_ ¢¸ Ér éß&hÉr #Q ìøÍ6£xÃº\ K{© H P °úכ ×æ © Ér P °úכ

\

H # :r`¦ ?/2;H כ s 9, "f K>h_ ìøÍ6£xÃº ×æ #Q" \"f Ìº§Â ôÇ u

«

Ñ´òõ >rFôÇH @/wn[O \"f Bonferroni Ãº&ñZOÉr &ñ§4s Z}tëß, ¸H ìøÍ6£x Ã

º\"f Ìº§Â tH ·ú§Ü¼ {9Ò¦&hÜ¼Ð #QÖ¼ &ñ¸_ u«Ñ´òõ >rF H @/wn[O \"f

H &ñ§4s Z}t ·ú§.

O⁰Brien (1984)Ér ¸H ìøÍ6£xÃº\¦ 7áx½+Ë # ¿º çH_ u«Ñ´òõ\¦ q§ H ~½ÓZOÜ¼Ð"f OLSü< GLS &ñ :x>|¾Ó`¦ ]jîß %i. s &ñ :x>|¾ÓÉr ~½Ó¾Ó$í`¦ °úH @/wn[O \"f

&ñ§4s Z}Ér ©&h`¦ tmH ìøÍ, $ 1pxÉr 5Åq+þA «Ñ\ @/ôÇ &h6 xëß`¦ ¦9 %i.

s

\ Pocock 1px (1987)Ér 5Åq+þA, síß+þA, Òqt>r «Ñü< °ú s #Q «Ñ +þAI_ ìøÍ6£xÃº

<

Êa ]jr)a âÄº\ GLS :x>|¾Ós 8¹¡¤ &h]X<Ê`¦ /åL %i. GLS :x>|¾ÓÉr ½¨^&hÜ¼Ð y

éß|¾Ó ìr$3\"f >íß)a &ñ :x>|¾Ó`¦ #Q ìøÍ6£xÃºçß_ /BNìríßÜ¼Ð ×ær& ½+Ëíß ô

Ç :x>|¾Ós. ÕªQ &³z´&hÜ¼Ð +þAI "fÐ Ér ìøÍ6£xÃº_ éß|¾Ó ìr$3`¦ :xK %3Ér

&ñ :x>|¾ÓÉr BÄº ª # #î½+Ë\ e#Q #Q9¹¡§s e. :r 7HëH\"fH #Q +þAI_

×

æ ìøÍ6£xÃº_ ìr$3~½ÓZOÜ¼Ð"f ·ú¡\"f /åLôÇ #Q éß&hõ #Q9¹¡§s \O¦ óøÍéß÷&H éß

|¾Ó ìr$3_ õ P °úכ`¦ #î½+Ë H ~½ÓZO`¦ ]jîßôÇ. e©r+«>_ :rs _ P °úכÜ¼Ð כ

¹Hds ×æכ¹ 9, #Q ìøÍ6£xÃº\ _ôÇ "fÐ ©'a$ís >rF H P °úכ_ #î½+Ë~½ÓZOÜ¼Ð"f

#

Q s:r\ HôÇ ~½ÓZOs ]jr|¨c Ãº e. ·ú¡\"f /åLôÇ OLSü< GLS_ ~½ÓZOÐ &ñ§4 s

Z}Ér P °úכ #î½+Ë~½ÓZO`¦ :r 7HëH\"f ]jîß½+É כ s.

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2. î |q @ è

O⁰Brien (1984)õ Pocock 1px (1987)_ OLSü< GLS :x>|¾Ó`¦ [O"î 9, 6£§Ü¼Ð ©'a

$ í

s eH P °úכ`¦ #î½+Ë H ~½ÓZO[þt`¦ ]jîßôÇ. sp 1lqwn&h P °úכ_ #î½+Ë~½ÓZOÜ¼Ð Fisher (1950)_ ~½ÓZO`¦ q2© # BÄº ´ú§Ér ~½ÓZO[þts ]jîß÷&%3Ü¼, ©'a$ís eH P °úכ_ #î½+Ë~½Ó Z

O

Ér Brown (1975)õ Zaykin 1px (2002)\ Ô¦õ . s[þt_ ~½ÓZOõ Ér, e©r+«>_ âÄº

\

½+Ë{©ôÇ P °úכ_ #î½+Ë~½ÓZO`¦ [O"î½+É כ s.

2.1. OLSØþ GLS 5ùç=ê ïÕ65ï| 2.1.1. OLS

OLS :x>|¾Ó`¦ [O"îv 0AK 5Åq+þA «ÑÐ Ø¦µ1Ï 9 s 5Åq+þA «Ñ Yijk(i = 1, 2, j = 1, . . . , ni, k = 1, . . . , K)H iP: |9éß\ 5ÅqôÇ jP: >h^_ kP: ìøÍ6£xÃº\¦ ·p. "f

Ð Ér >h^_ «ÑH 1lqwnstëß 1lx{9 >h^_ #Q ìøÍ6£xÃºH 'a÷&#Q e¦ |¾Ó &ñ

½

= (Yij1, . . . , Yijk)^T, µ_i = (µi1, . . . , µik)^T (i = 1, 2)s¦ Σkk = σ_k²Ð"f y ìøÍ6£xÃº_ ìríß

É

r "fÐ ØÔ 9 Ãºçß /BNìríß Σkk⁰s >rF<Ê`¦ &ñôÇ. OLS :x>|¾ÓÉr y ìøÍ6£xÃº\¦ Äº

³ðïrorvH כ Ü¼Ð Ø¦µ1ÏôÇ. 7£¤, ³ðïro)a ìøÍ6£xÃºH 6£§õ °ú .

Y_ijk^∗ =Yijk− ¯Y..k

S..k

. (2.1)

#

l"f ¯Y..kü< S_..k² H y ìøÍ6£xÃº\"f_ ¨îçHõ ½+Ë#îìríß(pooled variance)_ ÆÒ&ñ°úכÜ¼Ð 6

£

§õ °ú .

Y¯..k = n1Y¯1.k+ n2Y¯2.k

n1+ n2 , (2.2)

S_..k² = (n1− 1)S_1.k² + (n2− 1)S_2.k²

n1+ n2− 2 . (2.3)

#

l"f ¯Yi.kH y çH_ ¨îçHÜ¼Ð"f ¯Yi.k = (1/ni)P_n_i

j=1YijkÐ &ñ_)a.

s

]j ³ðïro)a Y_ijk^∗ \ H # y ìøÍ6£xÃº\ @/K u«Ñ´òõ eH\ @/ôÇ t :x>

|

¾

Ó`¦ ½¨½+É Ãº eÜ¼ 9 Lehmacher 1px (1991)_ l ñ\¦ "f s\¦ ¼#_© ZÐ ³ð&³ôÇ.

Zk= Y¯_1.k^∗ − ¯Y_2.k^∗

p(1/n₁) + (1/n₂). (2.4)

#

l"f ¯Y_i.k^∗ = (1/ni)P_n_i

j=1Y_ijk^∗ s.

OLS :x>|¾ÓÉr sü< °ú s ½¨ôÇ y ìøÍ6£xÃº_ Zk\¦ +þA ½+Ëíßr& ½¨ôÇ כ s. s]j 0A

\

"f /åLôÇ K>h_ Zk[þtÐ sÀÒ#Q K × 1 7'\¦ Z = (Z1, . . . , Zk)^TÐ, Y_ijk^∗ ÐÂÒ' ÆÒ&ñ

)

a K × K _ ©'a'§>=(correlation matrix)`¦ ˆRÜ¼Ð, y "é¶è 1Ð sÀÒ#Q K × 1 7'

\

¦ J = (1, . . . , 1)^TÐ &ñ_ôÇ. OLS :x>|¾ÓÉr 6£§õ °ú .

T_OLS= J^TZ p

J^TRJˆ . (2.5)

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s

:x>|¾Ó TOLS_ ìr\¦ ¶ú(RÐ Zk\¦ éßíH ½+Ëíß(unweighted sums)rv¦ e#Q Êê\ [O

"

î

H GLS_ ×æ ½+Ëíß(weighted sums)õ ØÔ.

OLS :x>|¾Ó TOLSH y ìøÍ6£xÃºZ>Ð «ÑÐÂÒ' ÆÒ&ñôÇ ½+Ë#îìríßÜ¼Ð ³ðïror Y_ijk^∗ \ HôÇ Zk\¦ ½+ËíßrvÙ¼Ð ) Áº[O \"f¸ {9ìøÍ&hÜ¼Ð &ñSXy t ìrí t ·ú§ Ü

¼ 9, ¢¸ôÇ @/wn[O \"f¸ &ñSXy q×æd t ìrí t ·ú§H. ) Áº[O \"f :£¤Z>ôÇ

â

Äº\ ôÇ # TOLSH &ñSXy t ìrí HX< sQôÇ âÄºêøÍ ¸H ìøÍ6£xÃº 1lx{9 8£¤&ñ°úכ

# 3

0A(same scale)\¦ t"f 1lx{9 ìríß(7£¤, σk = σ, k = 1, . . . , K){9 M: Yijk\¦ S..k

©

Ãº S...Ð ¾º#Q ³ðïrorvH âÄºs. O⁰Brien (1984)\"f ) Áº[O \"f TOLS

Ä»¸ n1+ n2− 2_ H t ìrí<Ê`¦ s6 x # &ñ ¦ eÜ¼ 9 :r 7HëH\"f¸ s\¦ Õª@/

Ð G×þôÇ.

é ß

íH > >íß÷&H OLS :x>|¾ÓÉr Ãºu&hÜ¼Ð îß&ñ H ©&hs etëß @/ÂÒìr_ â Ä

º\ #Q ìøÍ6£xÃºçß\ 1lx{9 ©'a$í`¦ tm¦ et ·ú§l M:ëH\ sQôÇ ©'a$í`¦ ×æÜ¼

Ð ¿º¦ ½+ËíßôÇ GLS :x>|¾Ós 8 H &ñ§4`¦ t> )a.

2.1.2. GLS

GLS :x>|¾ÓÉr y ³ðïro)a &ñ½©¼#  Zk\ ³ðïro)a Y_ijk^∗ ÐÂÒ' ÆÒ&ñ)a ©'a'§>=_

% i

(inverse)`¦ ×æÜ¼Ð ½+ËíßôÇ :x>|¾Ós 9, "f y ìøÍ6£xÃº_ l# &ñ¸ ØÔ> )a

. 7£¤, GLS :x>|¾ÓÉr 6£§õ °ú .

TGLS= J^TRˆ⁻¹Z q

J^TRˆ⁻¹J

. (2.6)

s

 TGLS %ir O⁰Brien (1984)Ér ) Áº[O \"f Ä»¸ n1+ n2− 2_ H t ìrí<Ê`¦ s 6

x # &ñ ¦ eÜ¼ 9 :r 7HëH\"f¸ s\¦ Õª@/Ð G×þôÇ. ìøÍ6£xÃº_ Ãº éßt ¿º >h

âÄº\H OLS :x>|¾Óõ GLS :x>|¾Ós {9u > ÷&tëß, ìøÍ6£xÃº !Ó s© âÄº\

H GLS :x>|¾Ó_ &ñ§4s 8¹¡¤ ß¼¦ ·ú94R e (Lehmacher 1px, 1991).

OLSü< GLS :x>|¾ÓÉr ³ðïro)a ìøÍ6£xÃº\ H # >íß÷&HX< #Q ìøÍ6£xÃº

ª

ôÇ «Ñ +þAI âÄº, :£¤y síß+þAs íH"f+þA(ordinal), Òqt>r «Ñ\¦ 1lx{9 #30A\¦ t¸ 2

¤ ³ðïrorvH {9s "îu ·ú§. #Q ìøÍ6£xÃº\¦ y ìøÍ6£xÃºZ> íH0AÐ ¨8ôÇ¦ K

¸ ëH]jH K÷&t ·ú§H. "f sQôÇ ªôÇ ìøÍ6£xÃº_ ìr$3\H 6£§ ]X\"f è>h

H P °úכ_ #î½+Ës 8¹¡¤ |ÃÐf .

2.2. P ÇH?ê :êÔ± 2.2.1. #H B!@ ^lv¶Ýä®ÃÅ'H P ÇH?ê 5_lv Ñä=ê

e

©r+«>\"f Ãº|9÷&H ìøÍ6£xÃº_ +þAIÐH 5Åq+þA «ÑÐ"f &ñ½© ¢¸H q&ñ½© ìrí

«Ñ e¦, sü@\¸ síß+þA, íH"f+þA x9 Òqt>r «Ñ e. u«Ñ´òõ >rFôÇH @/wn

[O \"f y +þAI_ ìøÍ6£xÃº\ @/K éß8£¤ &ñ_ P °úכ`¦ ½¨ H õ&ñ`¦ ÷&¸2¤ Âúª> "f Õ

ü tôÇ.

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Ä

º kP: ìøÍ6£xÃº_ 5Åq+þA «Ñ &ñ½© ¢¸H H &ñ½© ìrí H âÄº 6£§_ T :x

>

|¾Ós Ä»¸ n1+ n2− 2 t ìrí<Ê`¦ s6 x # &ñôÇ.

Tk = Y¯1.k− ¯Y2.k

S..k

p(1/n1) + (1/n2). (2.7)

#

l"f S²_..kH ¿º çH_ ½+Ë#îÆÒ&ñ|¾Ós 9, P °úכÉr Pk= Pr(T ≥ Tk)\ _K ½¨ôÇ.

ì ø

Í6£xÃº &ñ½© ìrí t ·ú§H âÄº, q¸Ãº ~½ÓZO ]49qH(Wilcoxon) íH0A½+Ë &ñÜ¼

Ð ¿º çH`¦ q§ôÇ. ]49qH íH0A½+Ë &ñÉr ¿º çH «Ñ\¦ íH0AÐ ¨8ôÇ Êê ôÇ çH_ íH0A

½ +

Ë Wk\¦ :x>|¾ÓÜ¼Ð 6 x 9 ³ð:rÃº n1s n2 20Ð ß¼ 6£§_ :x>|¾Ós &ñ½© H

<Ê`¦ s6 x # &ñôÇ.

Z_k =Wk− E(Wk)

Var(Wk) . (2.8)

#

l"f E(Wk)ü< Var(Wk)H yy Wk_ ¨îçHõ ìríßÜ¼Ð"f 6£§õ °ú .

E(Wk) = n1(n1+ n2+ 1)

2 , Var(Wk) = n1n2

12 (n1+ n2+ 1). (2.9) 1

l

x&h «Ñ e, 0A_ ìríß /BNdÉr Ð&ñ ÷&#Q ôÇ (Hollanderü< Wolfe, 1999). P °úכ

É

r Pk = Pr(Z ≥ Zk)\ _K ½¨ôÇ.

s

íß+þA ìøÍ6£xÃº âÄº\H qÖ¦ &ñÜ¼Ð ³ð:rÃº t ·ú§ qÖ¦ _ H &ñ

½

&

ñ

|¾Ó{9 M:, qÖ¦ &ñ :x>|¾ÓÉr

Z_k = Pˆ1k− ˆP2k

qPˆp(1 − ˆPp)(_n¹

1 +_n¹

2)

(2.10)

s

9, P °úכÉr Pk= Pr(Z ≥ Zk)\ _K ½¨ôÇ.

] X

éß «Ñ(censored data) eH Òqt>r «Ñ_ âÄºH Gehan (1965)_ :x>|¾Ó W_s=

n1

X

j=1

|u_j| (2.11)

`

¦ s6 xôÇ. #l"f ujH Mantel (1966)_ Û¼ïÐ"f 'Í P: |9éß_ jP: 'a8£¤°úכÐ H

¿

ºP: |9éß_ 'a8£¤°úכ\"f 'Í P: |9éß_ jP: 'a8£¤°úכÐ Ér ¿ºP: |9éß_ 'a8£¤°úכ`¦

É

°úכs. Ws_ l@/°úכÉr 0s¦, ìríßÉr Var(Ws) = n1n2

(n1+ n2)(n1+ n2− 1)

nX1+n2

j=1

(u^∗_j)² (2.12)

s

9, u^∗_jH D¥½+Ë ³ð:r_ jP: 'a8£¤°úכÐ H 'a8£¤°úכ\"f jP: 'a8£¤°úכÐ Ér 'a8£¤°úכ`¦

É

°úכÜ¼Ð &ñ_ôÇ. n1, n2 Øæìry ß¼ :x>|¾Ó Zk = Ws/p

Var(Ws)s H &ñ½© ìr

í<Ê`¦ s6 x # &ñ 9, P °úכÉr Pk = Pr(Z ≥ Zk)\ _K ½¨ôÇ.

s

ü< °ú s #Q +þAI ìøÍ6£xÃºÐÂÒ' ½¨ôÇ P °úכÉr ©'a$ís eH P °úכÜ¼Ð"f K × 1 7 '

 P = (P1, . . . , Pk)^TÐ ³ð&³ôÇ. s]j, #Q P °úכ`¦ #î½+Ë # _ 7áx½+Ë)a P °úכ(global P -value)`¦ ]jr H ~½ÓZO`¦ [O"îôÇ.

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2.2.2. æÐüÊPèh²¿ P ÇH?ê :êÔ± 1

l

qwn)a P °úכ_ #î½+ËÜ¼Ð #Q t ~½ÓZO[þts ]jr÷&%3Ü¼ 9 :r 7HëH\"fH © V,o æ¼ s

H Fisher (1950)_ ~½ÓZOõ Good (1958)_ ¸o¨îçH(harmonic mean)`¦ s6 xôÇ ~½ÓZO`¦ ¦

9ôÇ. :r 7HëH_ ÅÒ 3lq&hÉr 1lqwn)a P °úכ_ #î½+Ës m 'a)a P °úכ_ #î½+ËÜ¼Ð"f s\¦ /

B

I [O"î > )a.

Fisher (1950)_ P °úכ #î½+Ë~½ÓZOÉr Littellõ Folk (1971, 1973) t&h %i1pws 1lqwn)a

&

ñ

Ü¼ÐÂÒ' ½¨K P °úכ_ #î½+Ë~½ÓZO îrX< ´òÖ¦$ís Z}¦ ·ú94R e. P °úכÉr {9ª ìr

í Ù¼Ð, Pr(−2 log_e P > t) = Pr(P < e^−t/2) = e^−t/2ÐÂÒ' Ä»¸ 2_ s]jYL ìrí\¦

2£§`¦ ·ú Ãº e. "f 1lqwn)a #Q P °úכ_ ½+ËÉr Ä»¸ 2K s]jYL ìrí\¦ Ér.

X²= XK k=1

−2 log_ePk. (2.13)

Õ

ªQÙ¼Ð 7áx½+Ë)a P °úכÉr PFisher= Pr(χ²_(2K)≥ X²)s )a. #l"f χ²_(2K)H Ä»¸ 2K_ s

]jYL ìrís.

Good (1958)Ér 6£§õ °ú s ìøÍ6£xÃº_ Ãº\¦ 1lqwn)a P °úכ_ %iÃº_ ½+ËÜ¼Ð èH °úכ`¦ 7

á

x½+Ë)a P °úכÜ¼Ð &ñôÇ.

PGood= K XK k=1

P_k⁻¹

. (2.14)

2.2.3. î|fy;êc ÎPòKûý P ÇH?ê Ã=êh²¿ כ¬Þ2 cëàf£ :êÔ±

Brown (1975)Ér ©'a$ís eH P °úכ_ #î½+Ë~½ÓZOÜ¼Ð ·ú¡\"f /åLôÇ Fisher (1950)_ ~½ÓZO

`

¦ Ãº&ñ # &h6 xôÇ. ©'a$ís eH P °úכ`¦ #î½+Ë H âÄº 0A_ d (2.13)\ ]jr)a X²_

¨ î

çHõ ìríß`¦ ½¨KÐ 6£§õ °ú .

E(X²) = 2K, (2.15)

Var(X²) = XK i=1

Var(−2 log_ePi) + 2 XK X^K

i<j

Cov(−2 log_ePi, −2 log_ePj)

= 4K + 2 XK X^K

i<j

Cov(−2 log_ePi, −2 log_ePj). (2.16)

Brown (1975)Ér ìríß_ ¸ÉrAá¤ ½Ó\ eH Cov(−2 log_ePi, −2 log_ePj)\¦ iP:ü< jP: ìøÍ 6

£

xÃº_ ©'a>Ãº ρij_ <ÊÃºÐ ³ð&³ #, ρij ªÃº{9 M: ρij(3.25 + 0.75 × ρij)Ð, 6£§Ãº {

9

M: ρij(3.27 + 0.71 × ρij)Ð ]jr %i. ÕªQÙ¼Ð õ ½¨ÐÂÒ' ρij_ °úכ`¦ ·ú> ÷& Var(X²)\¦ ½¨½+É Ãº e. s]j X²_ ¨îçHõ ìríßs ½¨K&¦, s]jYL ìríü< 'a f± H {

9

s z e.

(7)

χ²_{(f )} Ä»¸ f _ s]jYL ìrí{9 M: d (2.13)_ X²\¦ cχ²_{(f )}Ð Hr~´ Ãº e¦

&

ñ

, X²_ ¨îçHõ ìríßÉr 6£§õ °ú .

E(X²) = cf, Var(X²) = 2c²f. (2.17) Õ

ªQÙ¼Ð d (2.15), (2.16)õ (2.17)`¦ s6 x cü< f H 6£§õ °ú s &ño)a.

c = Var(X²)

2E(X²), f =2[E(X²)]²

Var(X²) . (2.18) s

]j E(X²)ü< Var(X²)\¦ s6 x # cü< f _ °úכ`¦ ½¨ ¦, sÐÂÒ' Ä»¸ f _ s]jYL ì

rí\¦ ØÔH 6£§_ :x>|¾Ó`¦ %3H.

X²

c ≈ χ²_{(f )} (2.19)

"f 7áx½+Ë)a P °úכÉr PBrown = Pr(χ²_{(f )} ≥ X²/c)s. sü< °ú s ©'a$ís eH #Q P °úכ`¦ #î½+Ë # PBrown`¦ Ä»¸ H ~½ÓZO`¦ ÄºoH ¼#_© ‘~½ÓZO B’ ÂÒØÔx.

2.2.4. î|fy;êc ÎPòKûý P ÇH?ê æÐüÊPè Ýäsìë ¦Í :êÔ±

©

'a$ís eH P °úכ_ #î½+ËÐH 1lqwn)a P °úכ_ #î½+Ë\ @/ôÇ s:r[þts 8¹¡¤ ´ú§s ]jr

÷

&%36£§s z´s. ÕªQÙ¼Ð ©'a$ís eH P °úכ`¦ 1lqwn)a P °úכÜ¼Ð ¨8r& s\¦ #î½+Ë

H ~½ÓZO¸ ¢¸ôÇ 0px . Zaykin 1px (2002)Ér Ä»<Æ_ ©S!, 7£¤ Ãºëß>h_ Ä» &ñÐ\¦ ì

r$3K H âÄº\ BÄº &h]XôÇ Wilkinson (1951)_ #QÖ¼ °úכ s _ P °úכëß`¦ YL # 7áx

½ +

Ë H Truncation Product ~½ÓZO`¦ ]jîß H õ&ñ\"f ¢¸ôÇ 1lqwn)a P °úכÜ¼Ð_ ¨8`¦ [O

"

î

%i. :r 7HëH\"fH e©r+«>\_ &h6 xÜ¼Ð ©'a$ís eH P °úכ`¦ 1lqwn)a P °úכÜ¼Ð

¨ 8

r Êê\ ´òÖ¦$ís Z}¦ µ1ß) Fisher (1950)_ ~½ÓZOõ Good (1958)_ ~½ÓZO`¦ ]X3lq r

.

#

Q ìøÍ6£xÃº_ ©'a'§>= Rs ª&ñu '§>=(positive definite matrix){9 M:, R = MM^T`¦ ë

ß

7á¤ H c+tYUÛ¼v כ¹è(Cholesky factor) K × K'§>= Ms >rFôÇ. 1lqwn)a K>h_ P °úכ Ü

¼Ð sÀÒ#Q K × 1 7'\¦ PIÐ &ñ_ , s\ ©6£x H ³ðïr &ñ½© ìrí_ Z°úכ 7'H ZI = Φ⁻¹(J − PI)s )a. #l"f Φ(·)H ³ðïr &ñ½© ìrí_ SXÒ¦ìrí <ÊÃºs 9, JH 1]X\

"

f &ñ_ %i1pws K ×1 7' J = (1, . . . , 1)^Ts. s]j ©'a$ís eH P °úכ 7'\¦ PÐ &ñ_

sH Äº MZI ¨îçHs 0s¦, ìríßs Var(MZI) = MVar(ZI)M^T = MM^T = R

½Ó &ñ½© ìrí\¦ ÉrH כ `¦ s6 x # ½¨ôÇ.

P = J − Φ{MZI}

= J − Φ{MΦ⁻¹(J − PI)}. (2.20)

#

l"f P °úכ\ ©6£x H Z°úכ 7'_ ©'a'§>= Var(ZI)H #Q ìøÍ6£xÃº_ ©'a'§>=õ 1lx {

9

¦ &ñ %iHX<, sH ©'a>ÃºH éß¸ ¨8(monotone transformation)\ _K H

&hÜ¼Ð Ô¦ H z´, 7£¤, Corr{g(Pi), g(Pj)} ≈ Corr{Pi, Pj}`¦ s6 xôÇ כ s (Zaykin 1

p x, 2002).

(8)

s

ÐÂÒ' 1lqwn)a P °úכ 7' PIH 6£§õ °ú s ½¨K.

PI = J − Φ{M⁻¹Φ⁻¹(J − P)}. (2.21) s

ü< °ú s ©'a$ís eH P °úכ_ 7' P\¦ ¨8 # 1lqwn)a P °úכ_ 7' PI\¦ ½¨ %i¦ s ]

j PI_ y כ¹è 1lqwn)a P °úכ_ #î½+ËÜ¼Ð"f ·ú¡\"f è>hôÇ Fisher (1950)_ P °úכ #î½+Ëõ Good (1958)s ]jrôÇ ¸o¨îçH ~½ÓZO`¦ &h6 xôÇ. yy_ ~½ÓZO\ _ôÇ 7áx½+Ë)a P °úכ_ #î½+Ë

`

¦ ¼#_© ‘~½ÓZO F’ü< ‘~½ÓZO G’ ÂÒØÔx.

3. = . ãë à N M

#

Q +þAI_ ìøÍ6£xÃºÐÂÒ' 7áx½+Ë)a :r`¦ ?/9 H âÄº ¥y µ1ÏÒqt 9 6£§_

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H tØ¦½Ó3lqÜ¼Ð #î"é¶ {9"é¶u«Ñ t&h÷&"f @"é¶_ Ô¦9כ¹ôÇ t, q´òÖ¦&h u«Ñ

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rÛ¼%7Ü¼Ð ôÇ {9"é¶{9Ãº_ 7£x 1px, ÂÒ&h&ñ {9"é¶{9Ãº\¦ ×¦eÜ¼Ð+ q6 x`¦ ×¦s¦

H ¸§4s ÂÒy÷&¦ e.

:

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§½+É 3lq&hÜ¼Ð ÂÒ&h&ñ {9"é¶{9Ãº\¦ Ãº|9 %i (Kim, 2004). ÂÒ&h&ñ {9"é¶{9Ér Gertmanõ Restuccia (1981)\ _K >hµ1Ï)a Appropriate Evaluation Protocol(AEP)`¦ s6 x # &ñ_

9, 11>h_ _«Ñ "fqÛ¼ x9 '\ 'aº)a ½Ó3lq, 7>h_ çß ñ 1px_ Ð¸ "fqÛ¼\ 'aº)a

½

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9

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[þt\ _K Y> YV Ãº&ñ`¦ 2; ¸½¨Ð {9"é¶{9s 0A_ ½Ó3lq ×æ #QÖ¼ כ \¸ ÂÒ½+Ë÷&t ·ú§

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í ß

Ø¦÷& 9, © ¥y æ¼sH '¸H y ¨8_ 8úx {9"é¶{9Ãº ×æ ÂÒ&h&ñ {9"é¶ {9Ãº_ qÖ¦s

. ÕªQ 5ÅqÃº s ÂÒ&h&ñÖ¦Ér @/|ÄÌ&h '¸Ð"f Øæìry [jÂÒ&hst 3lw . ÂÒ&h

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Ér síß+þA Ãºs. %6£§ ÂÒ&h&ñ{9s r)a sÊêÐ 6£§ ÂÒ&h&ñ{9s l t

\

¦ _ :x>|¾Ó(run statistics)Ü¼Ð &ñ_ # Ãº(run number)\¦ ?/H íH"f+þA Ã

º ¢¸ôÇ _p e. ôÇ \VÐ"f, {9"é¶lçßs 8úx 21{9s ÷&H ¨8_ «Ñ\"f 0`¦ &h&ñ {

9

Ð, 1`¦ ÂÒ&h&ñ{9Ð ³ð&³ # 0-0-0-1-1-1-1-1-1-0-0-0-1-1-1-1-0-1-0-0-0ü< °ú ¦ . 8úx 11{9_ ÂÒ&h&ñ {9ÃºÐ {9"é¶{9\ @/ôÇ ÂÒ&h&ñÖ¦Ér 11/21=0.524s ÷& 9, ÃºH 4s. ¢¸ôÇ {

9

"é¶ 'Í ±úõ @"é¶ ±ú ¸¿º &h&ñôÇ u«Ñ\¦ ~ÃÎ¤. #Q Ãº <Êa ÂÒ&h&ñôÇ &ñ¸\¦

·p.

³

ð 3.1\ Iõü< Sõ ¨8_ y Ãº\ @/ôÇ כ¹ Ãºu ]jr÷&%3Ü¼ 9, #Q '¸\¦ 7áx

½ +

Ë #"f ¿º «Ñõ\¦ q§ ¦ ôÇ. {9"é¶ 'Í ±ú, @"é¶ ±úÉr ^ ¨8 ×æ\"f ÂÒ&h&ñ {

9

`¦ ¨8Ãº ]jr÷&%3¦, Ãº_ 4H Õª s©`¦ í<Ê H #3ÅÒÐ"f Ãº\ ]jr)a

(9)

³

ð 3.1: y Ãº\ @/ôÇ כ¹ Ãºu

¨ 8

Ãº {9"é¶ @"é¶ Ãº

Â Ò&h&ñÖ¦ '

Í

±ú ±ú 1 2 3 4

Sõ 47 25 27 11 13 16 7

(53%) (57%) (23%) (28%) (34%) (15%) 0.22

Iõ 63 15 25 30 19 7 7

(24%) (40%) (48%) (30%) (11%) (11%) 0.20

110 40 52 41 32 23 14

(36%) (47%) (37%) (29%) (21%) (13%) 0.21

Ã

ºH y #3ÅÒ\ 5Åq H ¨8Ãº\¦ ·p. síß+þAõ íH"f+þA Ãº_ âÄº\ Fc ñ îß\ q Ö

¦`¦ ]jr %i. Sõ\"f {9"é¶ 'Í ±ú ÂÒ&h&ñ{9_ qÖ¦s Z}Ü¼ 9 ¢¸ôÇ Ãº H âÄº_ q Ö

¦¸ Z}. ÕªQ ÂÒ&h&ñÖ¦Ér Sõ IõÐ ÂÒ&h&ñôÇ &ñ¸ Z}H z´`¦ ìøÍ%ò t 3lw

¦ e.

y

ÃºZ>Ð Sõ IõÐ ÂÒ&h&ñ '¸ Z}H כ `¦ &ñ½+É 3lq&hÜ¼Ð z´rôÇ éß

|

¾

Ó ìr$3\"f {9"é¶ 'Í ±úõ @"é¶ ±ú ÃºH qÖ¦ &ñÜ¼Ð Z°úכs yy 3.169õ 1.846s

¦, ÃºH ]49qH íH0A½+Ë &ñÜ¼Ð Z°úכs 2.797s. 5Åq+þA ÂÒ&h&ñÖ¦Ér ¸¨îçH q§Ð t &ñ`¦ z´r ¦ @/³ð:r H Z:x>|¾ÓÜ¼Ð 0.39s ½¨K&. Ä»_Ãºïr 5%\"f ÂÒ&h&ñÖ¦

É

r Sõ_ ÂÒ&h&ñôÇ u«Ñ\ @/ôÇ Ä»_ôÇ H ÷&t 3lw H ìøÍ, Qt [j ÃºH Ä»_ôÇ

H\¦ ]jrôÇ.

:

r 7HëH\"f [O"îôÇ ×æ ìøÍ6£xÃº\¦ 7áx½+ËôÇ &ñ~½ÓZO`¦ &h6 xK :r. y Ãºçß ©'a '

§>=Ér õ ½¨ÐÂÒ' ½¨ H כ s þj&hstëß s «Ñõ ìr$3\"fH ³ð:r\"f ÆÒ&ñôÇ

כ

`¦ 6 xôÇ.

R =







1 0.2306 0.4970 0.5612 1 0.5298 0.5387 1 0.4111 1





.

©

'a$í eH P °úכ #î½+Ë~½ÓZO ~½ÓZO B\"f ½¨ôÇ X²x9 ¨îçH E(X²)õ ìríß Var(X²)H 6£§ õ

°ú .

X² = XK k=1

−2 log_ePk = 14.352 + 6.857 + 11.92 + 2.106 = 35.235, E(X²) = 2K = 8,

Var(X²) = 4K + 2 XK X^K

i<j

Cov(−2 log_ePi, −2 log_ePj)

= 16 + 2 × (0.791 + 1.801 + 2.06 + 1.933 + 1.97 + 1.462)

= 36.026.

(10)

³

ð 3.2: 7áx½+Ë)a ìr$3 õ

OLS GLS ~½ÓZO B ~½ÓZO G ~½ÓZO F

&ñ :x>|¾Ó 2.656 2.957 15.648 - 26.185 7

á

x½+Ë)a P °úכ 0.004 0.0016 0.0023 0.0004 0.001

"f c = Var(X²)/2E(X²) = 36.026/(2 × 8) = 2.252, f = 2E(X²)²/Var(X²) = (2 × 8²)/36.026 = 3.553`¦ s6 x # %3Ér X²/c = 15.648H Ä»¸ f  s]jYL ìrí\¦ Ér.

s

M: PBrown°úכÉr 0.0023Ü¼Ð, Ä»_Ãºïr 5%\"f Iõü< Sõ {9"é¶¨8_ ÂÒ&h&ñôÇ &ñ¸\ s

\OH ) Áº[O`¦ lyôÇ.

6£§Ér ~½ÓZO Fü< G_ >íß\"f כ¹½¨÷&H Mõ M⁻¹s.

M =







1 0.2306 0.4970 0.5612 0 0.9731 0.4267 0.4206 0 0 0.7556 −0.0625 0 0 0 0.7101





,

M⁻¹=







1 −0.2369 −0.5240 −0.6960 0 1.0277 −0.5803 −0.6599 0 0 1.3234 0.1165 0 0 0 1.4082





.

s

H Z°úכ 7' Φ⁻¹(J−P) = (3.1690, 1.8461, 2.7968, 0.3885)^T\ _K Φ{M⁻¹Φ⁻¹(J−P)} = (0.8403, 0.5071, 0.9999, 0.7079)^Ts >íß÷&#Q, 1lqwn ¨8)a P °úכ 7' PI = (0.1597, 0.4929, 0.0001, 0.2921)^T\¦ Ä»¸ôÇ.

PI\¦ s6 x # P

P_k⁻¹ = 6.2617 + 2.0288 + 10000 + 3.4235 = 10011.71`¦ ìr¸Ð

H PGood = 4/10011.71 = 0.0004s >íß)a. Õªo¦ Ä»¸ 8_ s]jYL ìrí\¦ ØÔH X_Fisher² = 3.6692 + 1.4148 + 18.6401 + 2.4610 = 26.1851_ P_Fisher°úכÉr 0.0010s. ~½ÓZO G ü

< F ¢¸ôÇ Ä»_Ãºïr 5%\"f Iõü< Sõ_ ÂÒ&h&ñôÇ &ñ¸\ s \OH ) Áº[O`¦ ly ô

Ç.

OLSü< GLS :x>|¾ÓÉr éß|¾Ó ìr$3 õ Ä»¸ôÇ &ñ :x>|¾Ó Z 7'ÐÂÒ' TOLS= J^TZ/

p

J^TRJ = 8.2013/3.0881 = 2.656õˆ  TGLS = J^TRˆ⁻¹Z/

q

J^TRˆ⁻¹J = 3.8574/1.3045 = 2.957`¦ yy >íßôÇ. ¸H ~½ÓZO\"f 7áx½+Ë)a P °úכs Ä»_Ãºïr 5% \"f Ä»_ . ³ð 3.2\ OLS, GLS x9 ~½ÓZO B, G, F_ õ\¦ כ¹ %i.

4. ¬? êÍ P æB

4.1. ¬?êÍPæB 5¨ÉÌ

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(11)

&

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a$ís eH 4>h_ |¾Ó ìøÍ6£xÃº\¦ Òqt$í 9 y çH\"f 1lx{9 ³ð:rÃº n(= n1= n2)=20, 50 âÄº\¦ ¦9ôÇ. ¿º |9éß ×æ u«ÑçH(treatment group)Ér T Ð, @/¸çH(control group)Ér CÐ ³ðr ¦, T çH_ ìøÍ6£x°úכs CçHÐ ß¼H @/wn[O \"f éß8£¤ &ñ`¦ 1000 ìøÍ 4

¤ # ]j 17áx ¸ÀÓü< &ñ§4`¦ q§ôÇ.

:

0AK Äº ¸H ~½ÓZO`¦ 5Åq+þA ìøÍ6£xÃº\ &h6 x # q§ôÇ. 5Åq+þAÜ¼ÐH |¾Ó &ñ

½

1 − c²Xijk\ lí # ½ ü

R =







1 c² c² c² 1 c² c² 1 c² 1





.

\

¨ 8

â

Äº\¦ LN s tgAôÇ.

5Åq+þA «Ñ\ ¸H &ñZO, 7£¤, OLSü< GLS :x>|¾Óõ ~½ÓZO B, F, G\¦ q§ l 0AK Äº

&hÜ¼Ð ÃºZ>Ð ³ðïro\¦ z´rôÇ. ³ðïro)a «Ñ\ OLSü< GLS :x>|¾ÓÉr t :x>|¾Ó`¦

½

¨ # #î½+Ë ¦, ~½ÓZO B, F, GH y Ãº ¸¿º t &ñ`¦ z´r # ½¨ôÇ P °úכ`¦ #î½+ËôÇ.

³

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Q +þAI ìøÍ6£xÃº_ «Ñ Òqt$íÉr 6£§õ °ú . 4>h_ Ãº\¦ íH"f@/Ð 0A\"f /åLôÇ

5Åq+þA ìrí N, CN, LN Ü¼Ð Òqt$íôÇ Êê, [j P: ÃºH síß+þAÜ¼Ð, W1 P: ÃºH íH"f +

þ

H ìr0AÃº\¦ ]Xéß&h(cutoff point)Ü¼Ð #, T çHõ CçH_ «Ñ ]Xéß&hõ °ú `¦

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Äº\¦ 0Ü¼Ð, H âÄº\¦ 1Ð ¨8ôÇ. síß qÖ¦`¦ 0.3, 0.5, 0.8Ð 7£xr& ¸_z´+«>`¦ z´ r

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.

(12)

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ð 4.1: 4 "é¶ 5Åq+þA ìrí\"f_ ]j 17áx ¸ÀÓ

ì

rí ³ð:rÃº OLS GLS ~½ÓZO B ~½ÓZO F ~½ÓZO G

α = 0.05

α = 0.01

4.2. ¬?êÍPæB l5æÑä

³

ð 4.1Ér 5Åq+þA «Ñ N, CN, LN yy\"f Ä»_Ãºïrõ ³ð:r_ ß¼l\¦ ²úo #, u«Ñ

´

òõ \OH ) Áº[O \"f_ ]j 17áx ¸ÀÓ\¦ ·p כ s. OLSü< GLS_ ]j 17áx ¸ À

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Ç õ\¦ Ðstëß, ~½ÓZO Fü< GH Ä»_ÃºïrÐ H Ãºu\¦ Ðs¦ e#Q |ÃÐf t ·ú§.

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ð 4.2H 5Åq+þA «Ñ N , CN , LN yy\"f éß8£¤ @/wn[O \"f_ &ñ§4s. ¸H

â

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s

]j N, CN, LN «Ñ\"f [jP: Ãº\¦ síß+þAÜ¼Ð, W1P: Ãº\¦ íH"f+þAÜ¼Ð ¨8 r

âÄº_ ]j 17áx ¸ÀÓ ³ð 4.3\ ]jr÷&%3. @/gA ìrí N õ CN ìrí\"f ³ð:rÃº

50 âÄº\ ~½ÓZO B Ä»_Ãºïr\ îr °úכ`¦ Ðs¦, q@/gA ìrí Ãº [O LN _

â

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ð 4.2: 4 "é¶ 5Åq+þA ìrí\"f_ &ñ§4

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rí ³ð:rÃº OLS GLS ~½ÓZO B ~½ÓZO F ~½ÓZO G

α = 0.05

α = 0.01

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20 3

0.3 0.042 0.064 0.103 0.041 0.064 0.108 0.046 0.074 0.113 0.5 0.043 0.066 0.107 0.041 0.063 0.106 0.048 0.077 0.116 0.8 0.043 0.065 0.103 0.038 0.063 0.105 0.042 0.078 0.116

5

0.3 0.044 0.064 0.106 0.039 0.061 0.105 0.050 0.077 0.113 0.5 0.045 0.065 0.104 0.042 0.062 0.103 0.052 0.080 0.112 0.8 0.043 0.062 0.106 0.039 0.059 0.100 0.047 0.079 0.110

7

0.3 0.044 0.067 0.105 0.042 0.066 0.105 0.048 0.077 0.111 0.5 0.046 0.068 0.108 0.042 0.066 0.101 0.050 0.079 0.112 0.8 0.042 0.066 0.105 0.040 0.063 0.099 0.049 0.076 0.110

50 3

0.3 0.046 0.066 0.106 0.047 0.065 0.098 0.044 0.061 0.088 0.5 0.048 0.068 0.105 0.048 0.064 0.100 0.045 0.060 0.092 0.8 0.044 0.067 0.110 0.044 0.065 0.100 0.042 0.062 0.096

5

0.3 0.049 0.068 0.110 0.046 0.063 0.096 0.049 0.064 0.085 0.5 0.051 0.069 0.109 0.050 0.064 0.099 0.051 0.065 0.087 0.8 0.048 0.066 0.113 0.047 0.062 0.095 0.048 0.066 0.090

7

0.3 0.047 0.066 0.105 0.049 0.063 0.097 0.051 0.063 0.086 0.5 0.050 0.068 0.104 0.051 0.065 0.098 0.051 0.065 0.083 0.8 0.048 0.066 0.108 0.047 0.063 0.093 0.049 0.066 0.089

ì

rí N õ CN ìrí_ âÄº ³ð 4.2_ &ñ§4\H 3lw putëß #3ÅÒÃº &t"f BÄº

îr °úכ`¦ ?/ Ð. sü< ìøÍ@/Ð q@/gA ìrí LN ìrí\"fH ¸y9 P °úכ #î½+Ë_

&ñ§4s ³ð 4.2_ 5Åq+þA ìrí_ OLSü< GLS_ &ñ§4Ð Z}. sH síß qÖ¦õ y #3 Å

Ò_ qÖ¦\ yy_ :x>|¾Ó\ _ôÇ õ 8¹¡¤ Ä»_ ¦ "f P °úכ #î½+Ë~½ÓZOs 8

Methods of Combining P-values for Multiple Endpoints of Various Data Types

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