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.