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Chap. 2 ÄÄÄuuu������©©©ÙÙÙ���uuu���
2.1 ������©©©ÙÙÙ���uuu���
��©Ù
�X1, ..., Xn iid ∼ B(1, p), Kp�¿©��ÚOþ�
T =n∑i=1
Xi
´�
I T ∼ B(n, p)I T�CDFP�F (t;n, p) = P (T ≤ t),K
F (t;n, p) = 1− Γ(n+ 1)Γ(t+ 1)Γ(n− t)
∫ p
0ut(1− u)n−t−1du,
=F (t;n, p)´p�ëYüN~¼ê"I F (t;n, p) = 1− Fν1,ν2(ν2
ν1p
1−p),Ù¥ ν1 = 2(t+ 1),ν2 = 2(n− t), Fν1,ν2´gdÝ�ν1, ν2�F©Ù�CDF"
��©Ù
�X1, ..., Xn iid ∼ B(1, p), Kp�¿©��ÚOþ�
T =n∑i=1
Xi
´�
I T ∼ B(n, p)I T�CDFP�F (t;n, p) = P (T ≤ t),K
F (t;n, p) = 1− Γ(n+ 1)Γ(t+ 1)Γ(n− t)
∫ p
0ut(1− u)n−t−1du,
=F (t;n, p)´p�ëYüN~¼ê"I F (t;n, p) = 1− Fν1,ν2(ν2
ν1p
1−p),Ù¥ ν1 = 2(t+ 1),ν2 = 2(n− t), Fν1,ν2´gdÝ�ν1, ν2�F©Ù�CDF"
'up�n«b�/ª
(1) H0 : p ≤ p0 ↔ H1 : p > p0I u�ÚOþµ T
I áý�µ{T > c0}I �.�µ
1− F (c0;n, p0) ≤ α
´�, T �lÑ�§¤±����êc0¦�þª¤á"
I p�µ
P value = P (T ≥ Tobs|p = p0) = 1− F (Tobs − 1;n, p0).
�n > 20 ½ö25�§þª�d��©ÙCqO�§
P value = 1− Φ(Tobs − np0 − 0.5√
np0(1− p0)
).
'up�n«b�/ª
(1) H0 : p ≤ p0 ↔ H1 : p > p0I u�ÚOþµ T
I áý�µ{T > c0}I �.�µ
1− F (c0;n, p0) ≤ α
´�, T �lÑ�§¤±����êc0¦�þª¤á"
I p�µ
P value = P (T ≥ Tobs|p = p0) = 1− F (Tobs − 1;n, p0).
�n > 20 ½ö25�§þª�d��©ÙCqO�§
P value = 1− Φ(Tobs − np0 − 0.5√
np0(1− p0)
).
'up�n«b�/ª
(2) H0 : p ≥ p0 ↔ H1 : p < p0I u�ÚOþµ T
I áý�µ{T < c0}I �.�µ
c0−1∑k=0
Cknpk0(1− p0)n−k ≤ α <
c0∑k=0
Cknpk0(1− p0)n−k.
I p�µ
P value = P (T ≤ Tobs|p = p0) = F (Tobs;n, p0).
�n > 20 ½ö25�§þª�d��©ÙCqO�§
P value = Φ(Tobs − np0 + 0.5√
np0(1− p0)
).
'up�n«b�/ª
(2) H0 : p ≥ p0 ↔ H1 : p < p0I u�ÚOþµ T
I áý�µ{T < c0}I �.�µ
c0−1∑k=0
Cknpk0(1− p0)n−k ≤ α <
c0∑k=0
Cknpk0(1− p0)n−k.
I p�µ
P value = P (T ≤ Tobs|p = p0) = F (Tobs;n, p0).
�n > 20 ½ö25�§þª�d��©ÙCqO�§
P value = Φ(Tobs − np0 + 0.5√
np0(1− p0)
).
'up�n«b�/ª
(2) H0 : p ≥ p0 ↔ H1 : p < p0I u�ÚOþµ T
I áý�µ{T < c0}I �.�µ
c0−1∑k=0
Cknpk0(1− p0)n−k ≤ α <
c0∑k=0
Cknpk0(1− p0)n−k.
I p�µ
P value = P (T ≤ Tobs|p = p0) = F (Tobs;n, p0).
�n > 20 ½ö25�§þª�d��©ÙCqO�§
P value = Φ(Tobs − np0 + 0.5√
np0(1− p0)
).
'up�n«b�/ª
(3) H0 : p = p0 ↔ H1 : p 6= p0I u�ÚOþµ T
I áý�µ{T < c1 or T > c2}, c1 < c2I �.�µ
P (T < c1|H0) + P (T > c2|H0) ≤ α.
Ï~�/e§
P (T < c1|H0) ≤ α/2, P (T > c2|H0) ≤ α/2.
I p� (p0 = 0.5)µ
P value = 2 min{P (T ≤ Tobs|p = p0), P (T ≥ Tobs|p = p0)}.
'up�n«b�/ª
(3) H0 : p = p0 ↔ H1 : p 6= p0I u�ÚOþµ T
I áý�µ{T < c1 or T > c2}, c1 < c2I �.�µ
P (T < c1|H0) + P (T > c2|H0) ≤ α.
Ï~�/e§
P (T < c1|H0) ≤ α/2, P (T > c2|H0) ≤ α/2.
I p� (p0 = 0.5)µ
P value = 2 min{P (T ≤ Tobs|p = p0), P (T ≥ Tobs|p = p0)}.
p�
I éué¡©Ù§~X��©Ù§V>u��°(p��
P value =P (|Z| ≥ |Zobs||H0)=2 min{P (Z ≤ Zobs|H0), P (Z ≥ Zobs|H0)}.
I �p0 6= 0.5�§
P value =P (ÏéX*ÿ(J�U5@o�½���(J|p = p0)=
∑k:P (T=k|H0)≤P (T=tobs|H0)
P (T = k|H0).
I ���/e§þãu��Y²�uα,e���α, �±^�Åzu�"
p�
I éué¡©Ù§~X��©Ù§V>u��°(p��
P value =P (|Z| ≥ |Zobs||H0)=2 min{P (Z ≤ Zobs|H0), P (Z ≥ Zobs|H0)}.
I �p0 6= 0.5�§
P value =P (ÏéX*ÿ(J�U5@o�½���(J|p = p0)=
∑k:P (T=k|H0)≤P (T=tobs|H0)
P (T = k|H0).
I ���/e§þãu��Y²�uα,e���α, �±^�Åzu�"
p�
I éué¡©Ù§~X��©Ù§V>u��°(p��
P value =P (|Z| ≥ |Zobs||H0)=2 min{P (Z ≤ Zobs|H0), P (Z ≥ Zobs|H0)}.
I �p0 6= 0.5�§
P value =P (ÏéX*ÿ(J�U5@o�½���(J|p = p0)=
∑k:P (T=k|H0)≤P (T=tobs|H0)
P (T = k|H0).
I ���/e§þãu��Y²�uα,e���α, �±^�Åzu�"
~f
~µâ�O§3�c£�c�|J�Ãâ¥��k��¾<k,«A½�B�^§�~�B�^�u)§FDAïÄ�«#�Ãâ�Y§uy 19¶¾<¥k3�<Ñyù«B�^§K#�Y'�5��Y�Ðíº)µPp�TB�^u)�VǧKu�¯K�
H0 : p ≥ 0.5↔ H1 : p < 0.5.
K
P value = P (T ≤ 3|p = 0.5) = F (3; 19, 0.5) = 0.0022.
�3α = 0.05eAáýH0,=@�#�YÐ"áý��T ≤ 5, ý¢u�Y²�
α′ = P (T ≤ 5|p = 0.5) = 0.0318 < P (T ≤ 6|p = 0.5) = 0.0835.
2.2 ©©© êêêuuu���
ëY�/
(1) éuëY.�ÅCþX,b�
H0 : X�p∗© ê�x∗ ⇐⇒ H0 : p = P (X ≤ x∗) = p∗.
(2) �X1, ..., Xn iid��, ½Â
Yi = I(Xi ≤ x∗),
KY1, ..., Yn iid ∼ B(1, p),@o�±¦^��u��{u�d© êu�"
l�/
I (1) éulÑ.�ÅCþX,b�
H0 : X�p∗© ê�x∗
⇐⇒ H0 : P (X ≤ x∗) ≥ p∗, P (X ≥ x∗) ≥ 1− p∗
⇐⇒ H0 : P (X ≤ x∗) ≥ p∗, P (X < x∗) ≤ p∗
I (2) �X1, ..., Xn iid��, ½Â
T1 =n∑i=1
I(Xi ≤ x∗), T2 =n∑i=1
I(Xi < x∗),
KT1ÚT2ÑÑl��©Ù"
l�/
I (1) éulÑ.�ÅCþX,b�
H0 : X�p∗© ê�x∗
⇐⇒ H0 : P (X ≤ x∗) ≥ p∗, P (X ≥ x∗) ≥ 1− p∗
⇐⇒ H0 : P (X ≤ x∗) ≥ p∗, P (X < x∗) ≤ p∗
I (2) �X1, ..., Xn iid��, ½Â
T1 =n∑i=1
I(Xi ≤ x∗), T2 =n∑i=1
I(Xi < x∗),
KT1ÚT2ÑÑl��©Ù"
l�/
I (1) éulÑ.�ÅCþX,b�
H0 : X�p∗© ê�x∗
⇐⇒ H0 : P (X ≤ x∗) ≥ p∗, P (X ≥ x∗) ≥ 1− p∗
⇐⇒ H0 : P (X ≤ x∗) ≥ p∗, P (X < x∗) ≤ p∗
I (2) �X1, ..., Xn iid��, ½Â
T1 =n∑i=1
I(Xi ≤ x∗), T2 =n∑i=1
I(Xi < x∗),
KT1ÚT2ÑÑl��©Ù"
© êu�(1) Výu�
H0 : X�p∗© ê�x∗ ←→ H1 : X�p∗© êØ´x∗.
I Tu�¯K�du
H0 : P (X ≤ x∗) ≥ p∗, P (X < x∗) ≤ p∗ ↔ H1 : P (X ≤ x∗) < p∗ or P (X < x∗) > p∗
I áý� {T1 ≤ t1} ∪ {T2 ≥ t2}I �½wÍY²α, t1, t2÷v
P ({T1 ≤ t1} ∪ {T2 ≥ t2}|H0) ≤ α
I -P ({T1 ≤ t1}|H0) + P ({T2 ≥ t2}|H0) ≤ α, KP ({T1 ≤ t1}|H0) + P ({T2 ≥ t2}|H0) ≤ F (t1;n, p∗) + 1− F (t2 − 1;n, p∗)
I �α1, α2 ÷vα1 + α2 = α,¦�F (t1;n, p∗) ≤ α1, F (t2 − 1;n, p∗) ≥ 1− α2.
© êu�(1) Výu�
H0 : X�p∗© ê�x∗ ←→ H1 : X�p∗© êØ´x∗.
I Tu�¯K�du
H0 : P (X ≤ x∗) ≥ p∗, P (X < x∗) ≤ p∗ ↔ H1 : P (X ≤ x∗) < p∗ or P (X < x∗) > p∗
I áý� {T1 ≤ t1} ∪ {T2 ≥ t2}I �½wÍY²α, t1, t2÷v
P ({T1 ≤ t1} ∪ {T2 ≥ t2}|H0) ≤ α
I -P ({T1 ≤ t1}|H0) + P ({T2 ≥ t2}|H0) ≤ α, KP ({T1 ≤ t1}|H0) + P ({T2 ≥ t2}|H0) ≤ F (t1;n, p∗) + 1− F (t2 − 1;n, p∗)
I �α1, α2 ÷vα1 + α2 = α,¦�F (t1;n, p∗) ≤ α1, F (t2 − 1;n, p∗) ≥ 1− α2.
© êu�
I @o
t1 = sup{k : F (k;n, p∗) ≤ α1},t2 = inf{k : F (k − 1;n, p∗) ≥ 1− α2}
= inf{k : F (k;n, p∗) ≥ 1− α2}+ 1
I P �
P value = 2 min{P (T1 ≤ T1obs|p = p∗), P (T2 ≥ T2obs|p = p∗)}.
© êu�
I @o
t1 = sup{k : F (k;n, p∗) ≤ α1},t2 = inf{k : F (k − 1;n, p∗) ≥ 1− α2}
= inf{k : F (k;n, p∗) ≥ 1− α2}+ 1
I P �
P value = 2 min{P (T1 ≤ T1obs|p = p∗), P (T2 ≥ T2obs|p = p∗)}.
© êu�(2) �ýu�
H0 : X�p∗© êØ�ux∗ ←→ H1 : X�p∗© ê�ux∗.
I Tu�¯K�du
H0 : P (X ≤ x∗) ≥ p∗ ↔ H1 : P (X ≤ x∗) < p∗
I u�ÚOþµT1"
I �Óu��u���ýu�"
I P �
P value = P (T1 ≤ T1obs|p = p∗)= F (T1obs;n, p∗) (n ≤ 20)
≈ Φ(T1obs − np∗ + 0.5√
np∗(1− p∗)
)(n > 20)
© êu�(2) �ýu�
H0 : X�p∗© êØ�ux∗ ←→ H1 : X�p∗© ê�ux∗.
I Tu�¯K�du
H0 : P (X ≤ x∗) ≥ p∗ ↔ H1 : P (X ≤ x∗) < p∗
I u�ÚOþµT1"
I �Óu��u���ýu�"
I P �
P value = P (T1 ≤ T1obs|p = p∗)= F (T1obs;n, p∗) (n ≤ 20)
≈ Φ(T1obs − np∗ + 0.5√
np∗(1− p∗)
)(n > 20)
© êu�(3) mýu�
H0 : X�p∗© êØ�ux∗ ←→ H1 : X�p∗© ê�ux∗.
I Tu�¯K�du
H0 : P (X < x∗) ≤ p∗ ↔ H1 : P (X < x∗) > p∗
I u�ÚOþ T2I �Óu��u��mýu�"
I P �
P value = P (T2 ≥ T2obs|p = p∗)= 1− F (T2obs − 1;n, p∗) (n ≤ 20)
≈ 1− Φ(T2obs − np∗ − 0.5√
np∗(1− p∗)
)(n > 20)
© êu�(3) mýu�
H0 : X�p∗© êØ�ux∗ ←→ H1 : X�p∗© ê�ux∗.
I Tu�¯K�du
H0 : P (X < x∗) ≤ p∗ ↔ H1 : P (X < x∗) > p∗
I u�ÚOþ T2I �Óu��u��mýu�"
I P �
P value = P (T2 ≥ T2obs|p = p∗)= 1− F (T2obs − 1;n, p∗) (n ≤ 20)
≈ 1− Φ(T2obs − np∗ − 0.5√
np∗(1− p∗)
)(n > 20)
~f
~µ²�L²§,�Æ\Æ�Á©ê�75% © ê´193§y�,¥Æ.�Æ)¤1§�ÅÄ�15¶Æ)§¤1Xe
189, 233, 195, 160, 212, 176, 231, 185, 199, 213, 202, 193, 174, 166, 248
Á¯T¥Æ.�)¤175%© ê´193íº)µ(1) u�u�¯K�
H0 :T¥Æ.�)¤175%© ê´193↔ H1 :T¥Æ.�)¤175%© êØ´193.
K¦^Vý© êu��{§T1obs = 7, T2obs = 6.
P value =2 min{P (T1 ≤ 7|p = 0.75, n = 15), P (T2 ≥ 6|p = 0.75, n = 15)}=2 min{0.0173, 0.999} = 0.0346
�3α = 0.05e§áýH0.(2) e±áý��8�§duP (T1 ≤ 7|p = 0.75, n = 15) = 0.0173, P (T1 ≤ 8|p = 0.75, n = 15) = 0.0566,Kα1 = 0.0173, t1 = 7, α2 = α− α1 = 0.0327. duP (T2 ≤ 14|p = 0.75, n = 15) = 0.987, P (T2 ≤ 13|p = 0.75, n = 15) = 0.920,¤±�t2 = 15,K P (T2 ≥ 15|p = 0.75, n = 15) = 0.0134.
~f
ý¢u�Y²�
α′ = 0.0173 + 0.0134 = 0.0307.
áý��{T1 ≤ 7} ½ö {T2 ≥ 15}.
2.3 ÎÎÎÒÒÒuuu���
ÎÒu�
I ��{(Xi, Yi), i = 1, ..., n′},b���ÕáÓ©Ù§êâ´kSºÝe�ëYÿþ�"
(a) ¤éêâ(b) X Ú Y Õá (Mann-Whitney u�)
I u�¯K
(1)H0 : P (X > Y ) = P (X < Y )←→ H1 : P (X > Y ) 6= P (X < Y ).
(2)H0 : P (X > Y ) ≥ P (X < Y )←→ H1 : P (X > Y ) < P (X < Y ).
(3)H0 : P (X > Y ) ≤ P (X < Y )←→ H1 : P (X > Y ) > P (X < Y ).
I u�ÚOþ
S+ = #{Xi > Yi}, S− = #{Xi < Yi}, S0 = #{Xi = Yi},
Ù¥S0´ê⥓("��ê,ÎÒu�¥“("Øå�^"
I Pn = S+ + S−, p+ = P (X > Y ), p− = P (X < Y ).
ÎÒu�
I ��{(Xi, Yi), i = 1, ..., n′},b���ÕáÓ©Ù§êâ´kSºÝe�ëYÿþ�"
(a) ¤éêâ(b) X Ú Y Õá (Mann-Whitney u�)
I u�¯K
(1)H0 : P (X > Y ) = P (X < Y )←→ H1 : P (X > Y ) 6= P (X < Y ).
(2)H0 : P (X > Y ) ≥ P (X < Y )←→ H1 : P (X > Y ) < P (X < Y ).
(3)H0 : P (X > Y ) ≤ P (X < Y )←→ H1 : P (X > Y ) > P (X < Y ).
I u�ÚOþ
S+ = #{Xi > Yi}, S− = #{Xi < Yi}, S0 = #{Xi = Yi},
Ù¥S0´ê⥓("��ê,ÎÒu�¥“("Øå�^"
I Pn = S+ + S−, p+ = P (X > Y ), p− = P (X < Y ).
ÎÒu�
(1) u�¯K
H0 : p+ = p− ↔ H1 : p+ 6= p−.
I 3H0e, S+ ∼ B(n, 0.5), u´�Óuu�
H0 : p+ = 0.5↔ H1 : p+ 6= 0.5.
I d��u��{�§ÙP �
P value = 2 min{P (S+ ≤ S+obs|p = 0.5), P (S+ ≥ S+
obs|p = 0.5)}.
(2) u�¯K
H0 : p+ ≤ p− ↔ H1 : p+ > p−.
I �Óuu�
H0 : p+ ≤ 0.5↔ H1 : p+ > 0.5.
d��u��{�§ÙP �
P value = {P (S+ ≥ S+obs|p = 0.5)}.
(3) u�¯K
H0 : p+ ≥ p− ↔ H1 : p+ < p−.
I �Óuu�
H0 : p+ ≥ 0.5↔ H1 : p+ < 0.5.
d��u��{�§ÙP �
P value = {P (S+ ≤ S+obs|p = 0.5)}.
~f
~µ,úi#ïuAÚBü« ¬§Ý�½|��)��éü« ¬�UÐ�¹"3,½|�ÅÄ�20 ��Á¯§(J15¶���U�A§3¶�U�B§ 2¶éA§BÑU�" Á¯��é ¬A§B�UдÄk�ɺ
)µPp+ = P (�U�A), p− = P (�U�B). u�u�¯K�
H0 : p+ = p− ↔ H1 : p+ 6= p−.
�K¿�, S+ = 15, S− = 3, n = 18,u´
P value = 2 min{P (S+ ≤ 15|p = 0.5), P (S+ ≥ 15|p = 0.5)} = 0.0075.
�3α = 0.05e§áýH0§@���éA,B� Ðk�É.?�Ú§éub�
H0 : p+ ≤ p− ↔ H1 : p+ > p−.
P �P value = {P (S+ ≥ 15|p = 0.5)} = 0.0037,
�3α = 0.05e§áýH0,K��U�A��U5��.
ü��ÎÒu�üoN �ëê�'�
I �oNXÚY©OäkëY©Ù¼êF (x)ÚG(y), ���{Xi, i = 1, ..., n}Ú{Yi, i = 1, ...,m}"
I b�FÚG�3ù��'X: �3δ, ¦�?�x ∈ R,G(x) = F (x− δ), �Ò´X + δÚYÓ©Ù"�Ä
H0 : δ = 0←→ H1 : δ > 0 ½ö δ < 0.
I d¯K�du
H0 : P (X > Y ) = P (X < Y )←→ H1 : P (X > Y ) 6= P (X < Y ).
d�ü�� ��'���u¥ ê�m�'�§XJδ > 0, KX�©Ù²þ'Y�©Ù�§=
P (X > Y ) =∫ ∞−∞
F (x− δ)dF (x) ≤∫ ∞−∞
F (x)dF (x) = 12 .
ü��ÎÒu�üoN �ëê�'�
I �oNXÚY©OäkëY©Ù¼êF (x)ÚG(y), ���{Xi, i = 1, ..., n}Ú{Yi, i = 1, ...,m}"
I b�FÚG�3ù��'X: �3δ, ¦�?�x ∈ R,G(x) = F (x− δ), �Ò´X + δÚYÓ©Ù"�Ä
H0 : δ = 0←→ H1 : δ > 0 ½ö δ < 0.
I d¯K�du
H0 : P (X > Y ) = P (X < Y )←→ H1 : P (X > Y ) 6= P (X < Y ).
d�ü�� ��'���u¥ ê�m�'�§XJδ > 0, KX�©Ù²þ'Y�©Ù�§=
P (X > Y ) =∫ ∞−∞
F (x− δ)dF (x) ≤∫ ∞−∞
F (x)dF (x) = 12 .
Brown-Mood ¥ êu�I �Äu�¯K
H0 : med(X) = med(Y )←→ H1 : med(X) > med(Y ).
I PmedXY´ü|êâ·Ü��¥ ê§@o3�b�e§medXY , med(X)Úmed(Y )��" u´�E2× 2L�
X Y oÚ
> medXY A B t
< medXY C D m+n-t
oÚ m n m+n
I �m, n Út�½�§A�©Ù3H0e´�AÛ©Ùµ
P (A = k) = CkmCt−kn
Ctm+n, k ≤ min{m, t}.
Brown-Mood ¥ êu�I �Äu�¯K
H0 : med(X) = med(Y )←→ H1 : med(X) > med(Y ).
I PmedXY´ü|êâ·Ü��¥ ê§@o3�b�e§medXY , med(X)Úmed(Y )��" u´�E2× 2L�
X Y oÚ
> medXY A B t
< medXY C D m+n-t
oÚ m n m+n
I �m, n Út�½�§A�©Ù3H0e´�AÛ©Ùµ
P (A = k) = CkmCt−kn
Ctm+n, k ≤ min{m, t}.
Brown-Mood ¥ êu�I �Äu�¯K
H0 : med(X) = med(Y )←→ H1 : med(X) > med(Y ).
I PmedXY´ü|êâ·Ü��¥ ê§@o3�b�e§medXY , med(X)Úmed(Y )��" u´�E2× 2L�
X Y oÚ
> medXY A B t
< medXY C D m+n-t
oÚ m n m+n
I �m, n Út�½�§A�©Ù3H0e´�AÛ©Ùµ
P (A = k) = CkmCt−kn
Ctm+n, k ≤ min{m, t}.
Brown-Mood ¥ êu�
@o§P �
P value = {P (A ≥ a|H0)} = Phyper(A ≥ a),
Ù¥a´A�*ÿ�"
Brown-Mood ¥ êu�
@o§P �
P value = {P (A ≥ a|H0)} = Phyper(A ≥ a),
Ù¥a´A�*ÿ�"
~f~µ�ïÄØÓ¬ýw«ì3ØÓû|�"Èd�´Ä�3�É,�ÅÂ8ÑÈA¬ý9[û|ÚB¬ý7[û|�"Èd�êâXe:A (X): 698, 688, 675, 656, 655, 648, 640, 639, 620B (Y): 780, 754, 740, 712, 693, 680, 621Á¯ùü�¬ýd�k�Éíº
)µu�¯K�
H0 : med(X) = med(Y )←→ H1 : med(X) < med(Y ).
dêâ��medXY = 676.5ÚXeL�
X Y oÚ
> medXY 2 6 8
< medXY 7 1 8
oÚ 9 7 16
Ka = 2, �H0¤á�§
P value = P (A ≤ 2|H0) = 0.0203.
�3α = 0.05e§áýH0§@�¬ýBd�puA�d�.
ÎÒu��C«�*ÿêâ�á5êⵓ0" Ú“1". ykü�a X Ú Y §¯K´u�(0,1)Ú(1,0)�mVÇ��ɺ
I êâµ{(Xi, Yi), i = 1, ..., n′}, XÚYþ��0½1, Kk
Y = 0 Y = 1
X = 0 a b
X = 1 c d
I b�(Xi, Yi), i = 1, ..., n′�pÕá"�Ä
H0 : P (X = 0, Y = 1) = P (X = 1, Y = 0)←→H1 : P (X = 0, Y = 1) 6= P (X = 1, Y = 0).
I d¯K�du
H0 : P (X = 0) = P (Y = 0)←→ H1 : P (X = 0) 6= P (Y = 0).
Pn = b+ c, K3H0e, b ∼ B(n, 0.5), u´�A^ÎÒu�"
ÎÒu��C«�*ÿêâ�á5êⵓ0" Ú“1". ykü�a X Ú Y §¯K´u�(0,1)Ú(1,0)�mVÇ��ɺ
I êâµ{(Xi, Yi), i = 1, ..., n′}, XÚYþ��0½1, Kk
Y = 0 Y = 1
X = 0 a b
X = 1 c d
I b�(Xi, Yi), i = 1, ..., n′�pÕá"�Ä
H0 : P (X = 0, Y = 1) = P (X = 1, Y = 0)←→H1 : P (X = 0, Y = 1) 6= P (X = 1, Y = 0).
I d¯K�du
H0 : P (X = 0) = P (Y = 0)←→ H1 : P (X = 0) 6= P (Y = 0).
Pn = b+ c, K3H0e, b ∼ B(n, 0.5), u´�A^ÎÒu�"
ÎÒu��C«�*ÿêâ�á5êⵓ0" Ú“1". ykü�a X Ú Y §¯K´u�(0,1)Ú(1,0)�mVÇ��ɺ
I êâµ{(Xi, Yi), i = 1, ..., n′}, XÚYþ��0½1, Kk
Y = 0 Y = 1
X = 0 a b
X = 1 c d
I b�(Xi, Yi), i = 1, ..., n′�pÕá"�Ä
H0 : P (X = 0, Y = 1) = P (X = 1, Y = 0)←→H1 : P (X = 0, Y = 1) 6= P (X = 1, Y = 0).
I d¯K�du
H0 : P (X = 0) = P (Y = 0)←→ H1 : P (X = 0) 6= P (Y = 0).
Pn = b+ c, K3H0e, b ∼ B(n, 0.5), u´�A^ÎÒu�"
ÎÒu��C«�*ÿêâ�á5êⵓ0" Ú“1". ykü�a X Ú Y §¯K´u�(0,1)Ú(1,0)�mVÇ��ɺ
I êâµ{(Xi, Yi), i = 1, ..., n′}, XÚYþ��0½1, Kk
Y = 0 Y = 1
X = 0 a b
X = 1 c d
I b�(Xi, Yi), i = 1, ..., n′�pÕá"�Ä
H0 : P (X = 0, Y = 1) = P (X = 1, Y = 0)←→H1 : P (X = 0, Y = 1) 6= P (X = 1, Y = 0).
I d¯K�du
H0 : P (X = 0) = P (Y = 0)←→ H1 : P (X = 0) 6= P (Y = 0).
Pn = b+ c, K3H0e, b ∼ B(n, 0.5), u´�A^ÎÒu�"
ÎÒu��C«
I �n ≤ 20�§
P value =2 min{P (b ≤ bobs|p = 0.5), P (b ≥ bobs|p = 0.5)}=2P (b ≥ max{bobs, n− bobs})
=2n∑
k=l
Ckn0.5n, l = max{bobs, n− bobs}
I �n > 20�§�^��%C�{
z =b− n/20.5√n
=b− c√b+ c
→ N(0, 1).
Ïd,
z2 =(b− c)2
b+ c→ χ2
1.
(McNemar test)
ÎÒu��C«
I �n ≤ 20�§
P value =2 min{P (b ≤ bobs|p = 0.5), P (b ≥ bobs|p = 0.5)}=2P (b ≥ max{bobs, n− bobs})
=2n∑
k=l
Ckn0.5n, l = max{bobs, n− bobs}
I �n > 20�§�^��%C�{
z =b− n/20.5√n
=b− c√b+ c
→ N(0, 1).
Ïd,
z2 =(b− c)2
b+ c→ χ2
1.
(McNemar test)
~f~µ3oÚÿÀ<?1>ÀúmFØm�c,�ÅN�100<,ί¦�|±=�,uyk84<|±¬ÌÿÀ<, 16< ��u�Ú.3FØm��,2gN�ù100<,uy1/4�<UC|±�Ý,êâXe:
��
¬Ì �Ú
�c¬Ì 63 21
�Ú 4 12
Á¯¬¯�c�|±�Ýk�Éíº
)µu�¯K�
H0 :¬¯�c�|±�ÝvkUC←→H1 :¬¯�c�|±�ÝkUC.
(1) ¦^��u�
P value = 225∑
k=21
Ck250.525 = 0.00091
3α = 0.05e§áýH0§@�¬¯�c�|±�ÝkUC.(2) ¦^McNemar u�
z2 =(b− c)2
b+ c=
28925
= 11.56.
3α = 0.05e§χ21(0.05) = 3.841 < z2, áýH0.
Ka = 2, �H0¤á�§
P value = P (A ≤ 2|H0) = 0.0203.
�3α = 0.05e§áýH0§@�¬ýBd�puA�d�.
~f
(2) ¦^McNemar u�
z2 =(b− c)2
b+ c=
28925
= 11.56.
3α = 0.05e§χ21(0.05) = 3.841 < z2, áýH0.
(3)McNemar u��ëY.?�Yates(1934)
z2 =(|b− c| − 0.5)2
b+ c.
Fleiss (1981)
z2 =(|b− c| − 1)2
b+ c.