20180119handans 資料置き場 hustat2017 20180119hand ans

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13 http://sites.google.com/site/hustat2017/

[email protected]

January 19th, 2018

2

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. . . 5

. . . 6

. . . 8

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. . . 10

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13 . . . 14

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2 / 18

■ A

□ A I 1×I

■

□ I {p1, p2, p3, . . . , pI}

□ n {np1, np2, np3, . . . , npI}

{X1, X2, X3, . . . , XI}

Table 1:

A1 A2 . . . AI

X1 X2 · · · XI n

n·p1 n·p2 · · · n·pI n

■

H0: {p1, p2, p3, . . . , pI}

■

H1:

■

X2≡ X2(p1, p2, . . . , pI) = I

∑

i=1

(Xi−n·pi)2

npi

(1)

13 – 3 / 18

0.1 ( ).

n

I

{

p

1

, p

2

, . . . , p

_I

}

X

i

n

· p

i

X

2

(p

1

, p

2

, . . . , p

_I

) =

I

∑

i=1

(X

i

−

n

· p

i

)

2

np

i

−→

χ

2

(I

₋

1)

X

2

(p

1

, p

2

, . . . , p

_I

)

n

(3)

■ H0 n

I−1 1

■ H1 n

I−1 1

Figure 1:

H

0

:

p

1

, p

2

, . . . , p

_I

H

1

:

H

0

■ C= (c,∞) α

I−1

13 – 4 / 18

■ ( ) 120 ( )

48 ( ) 36 ( ) 13

Table 2:

(1. ) (2. ) (3. ) (4. )

120 48 36 13 217

217

■

H0: p1= , p2= , p3= , p4=

■ 1 H0

3 (= 4−1) C= ( ,∞)

■

x2= {120−217·(9/16)}

2

217·(9/16) +

{48−217·(3/16)}2

217·(3/16) +

{36−217·(3/16)}2

217·(3/16) +

{13−217·(1/16)}2

217·(1/16) ≈

(4)

■ A B

■ A I B J

■ I×J

Table 3:

I

×

J

B

A B1 B2 · · · BJ

A1 X1,1 X1,2 · · · X1,J X1,•

A2 X2,1 X2,2 · · · X2,J X2,•

. .

. ... ... . .. ... ...

AI XI,1 XI,2 · · · XI,J XI,•

X_•,1 X•,2 · · · X•,J n(=

∑I i=1

∑J

j=1Xi,j)

X_•,j =∑I_i₌₁Xi,j Xi,_•=∑_jJ₌₁Xi,j n=∑I_i₌₁∑J_j₌₁Xi,j

Table 4:

I

_×

J

B A B1 B2 · · · BJ

A1 pˆ1,1 pˆ1,2 · · · pˆ1,J pˆ1,•

A2 pˆ2,1 pˆ2,2 · · · pˆ2,J pˆ2,•

. .

. ... ... . .. ... ... AI pÎ,1 pÎ,2 · · · pÎ,J pÎ,_•

ˆ

p•,1 pˆ•,2 · · · pˆ•,J 1

ˆ

p•,j =∑ I

i=1Xi,j/n pˆi,•=

∑J

j=1Xi,j/n ˆ

pi,j=Xi,j/n

5

6

Table 5:

1625 5 1630

1022 11 1033

2647 16 2663

Table 6:

YS MA NK

40 23 9 72

705 544 149 1398

215 272 96 583

960 840 254 2053

(5)

■ A i Ai

■ B j Bj

■ (i, j) Ai∩ Bj

■ A I B

■ i, j i̸=j

pi,j=pi,_•×p_•,j (2)

pi,j ≡Pr[Ai∩ Bj] pi,•≡Pr[Ai] =∑ J

j=1Pr[Ai∩ Bj] p•,j ≡Pr[Bj] =∑ I

i=1Pr[Ai∩ Bj]

□ 5

□ 6

13 – 7 / 18

■ n (i, j)

E[Xi,j] =n·pi,j (3)

■ 2

E[Xi,j] =n·pi,j =n·pi,_•·p_•,j (4)

■ Xi,j 4 (i, j)

Xi.j

{Xi,j−n·pˆi,•·pˆ•,j}2

n·pˆi,_•·pˆ_•,j

={Xi,j−Xi,•·X•,j/n}2

Xi,_•·X_•,j/n

ˆ

pi,•=

Xi,_•

n =

1

n

J

∑

j=1

Xi,j, pˆ•,j =

X_•,j

n =

1

n

I

∑

i=1 Xi,j

■

X2= I

∑

i=1

J

∑

j=1

{Xi,j−n·pˆi,_•·pˆ_•,j}2

n·pˆi,•·pˆ•,j

= I

∑

i=1

J

∑

j=1

{Xi,j−Xi,_•·X_•,j/n}2

Xi,•·X•,j/n

(5)

(6)

0.2 ( ).

I

_×

J

n

■ 4

X

2

(I

₋

1)(J

₋

1)

■ 4

X

2

(I

−

1)(J

−

1)

(I

₋

1)(J

₋

1)

■

□ H0 4 □ H1 4

■ X2

□ H0 X2 (I−1)(J−1) □ H1 X2 H0

■

□ α H0 X2 (I−1)(J−1)

□ χ2

α

Pr[X2≥χ2α] =α

■

(7)

Table 7:

0.608 0.004 0.612

0.386 0.388

0.994 0.006 1.000

Table 8:

1620.2 9.8 1026.8 6.2

1. 5

□

0.612·0.994≈0.608, . . . , 0.388·0.006≈0.002

□

2663·0.608≈1620.2, . . . , 2663·0.00233≈6.2

□ = (2−1)·(2−1) 3.84

(3.84,∞)

□

x2= (1625−1620.2)

2

1620.2 +· · ·+

(11−6.2)2

6.2 ≈6.72

6.084865

Table 9:

YS MA NK

0.016 0.014 0.004 0.035

0.318 0.278 0.084 0.681

0.133 0.116 0.035 0.284

0.467 0.409 0.124 1.000

Table 10:

YS MA NK

33.6 29.4 8.9

653.2 571.5 172.9

272.7 238.6 72.2

2. 6

□

□ = (3−1)·(3−1) 9.49

(9.49,∞)

□

x2= (40−33.6)

2

33.6 +

(23−29.4)2

29.4 +· · ·+

(96−72.2)2

72.2 ≈37.41

(8)

■

■ p1

■ p2

■ n,m

■

{X1, X2, . . . , Xn}, {Y1, Y2, . . . , Ym}

Xi p1 1−p1 Yi p2 1−p2

■

ˆ

p1=

1

n

∑

i=1

Xi, pˆ2=

1

m

∑

i=1 Yi

■ H0: p1=p2

Z= √ pˆ1−pˆ2

¯

p(1−p¯)/n+ ¯p(1−p¯)/m, p¯=

n·pˆ1+m·pˆ2

n+m (6)

■

Table 11:

135 465 600

116 284 400

251 749 1000

Table 12: :

150.6 449.4 0.6

100.4 299.6 0.4

0.251 0.749 1.000

■ Z2 a

■

■ 29.0 22.5

600 400

■

x2=(135−150.6)

2

150.6 +

(465−449.4)2 449.4 +

(116−100.4)2 100.4 +

(284−299.6)2 299.6 ≈5.39

■ = (2−1)·(2−1) 3.84

(3.84,∞)

a

(9)

■ : Cochran rule

□

■

□

■ ( )

□

□ 0.5

I

∑

i=1

{|Xi−npi| −0.5}2

npi

n·max{0,|X11X22−X21X12| −n/2}2 X1,•X2,•X•,1X•,2

2 ∑

i=1 2 ∑

j=1

(Xij−Xi,_•X_•,j/n)2

Xi,•X•,j

=n·(X11X22−X21X12)

2 X1,•X2,•X•,1X•,2

(7)

□

■ Exact Test

□

1. 1630 1625 1630C1625

2. 1033 1022 1033C1022

3. 2663 (= 1630 + 1033)

2647 (= 1625 + 1022) 2663C2647

1630C1625×1033C1022 2663C2647

= 0.01106942

4.

0.01106942

■

5. 16 0.011

0.01884755

(10)

13 / 18

a

■

□

□ 13 14 (1) (2)

Table 13: S ppm

10 20 50 100 200 200

457 480 310 94 39 23 1403

38 11 4 1 1 0 55

495 491 314 95 40 23 1458

Table 14: S ppm

10 20 50 100 200 200

38 11 4 1 1 0 55

32 7 3 1 1 0 44

70 18 7 2 2 0 99

■ 5 b

□ 13

x2=

2 ∑

i=1 6 ∑

j=1

{xi,j−1458·pi,_•·p_•,j}2 1458·pi,•·p•,j

= 32.378

5

5 (= (2−1)·(6−1) ) (11.0705,∞)

□ 200ppm

14

x2=

2 ∑

i=1 5 ∑

j=1

{xi,j−99·pi,•·p•,j}2 99·pi,•·p•,j

= 0.328

5

4 (= (2−1)·(5−1) ) (9.48773,∞)

a

(1971) ฀ Vol.41, No.10

b

(11)

cont.

■ a

■

15 16

Table 15: ppm

10 20 50 100 200 200

451 475 308 91 38 23 1386

44 16 6 4 2 0 72

495 491 314 95 40 23 1458

Table 16: ppm

10 20 50 100 200 200

44 16 6 4 2 0 72

23 3 2 1 0 0 29

67 19 8 5 2 0 101

15 16

a

(12)

■ 15

19

□ 15

x2₌ 2 ∑

i=1 6 ∑

j=1

{xi,j−1458·pi,•·p•,j}2 1458·pi,•·p•,j

=

5

(11.0705,∞)

■ 16

19

□ 200ppm

16

x2=

2 ∑

i=1 5 ∑

j=1

{xi,j−101·pi,•·p•,j}2 101·pi,_•·p_•,j

=

5

(9.48773,∞)

Table 17:

1458

_·

p

i,•

· p

•,j

10 20 50 100 200 200

470.556 466.753 298.494 90.309 38.025 21.864 0.951

24.444 24.247 15.506 4.691 1.975 1.136 0.049

0.340 0.337 0.215 0.065 0.027 0.016

Table 18: ppm

101

· p

i,•

· p

•,j

10 20 50 100 200 200

47.762 13.545 5.703 3.564 1.426 0.713

19.238 5.455 2.297 1.436 0.574 0.287

0.663 0.188 0.079 0.050 0.020

■ (7)

(13)

■ 15

19

□ 15

x2₌ 2 ∑

i=1 6 ∑

j=1

{xi,j−1458·pi,•·p•,j}2 1458·pi,•·p•,j

= 26.841

5

5 (= (2−1)·(6−1) )

(11.0705,∞)

■ 16

19

□ 200ppm

16

x2₌ 2 ∑

i=1 5 ∑

j=1

{xi,j−101·pi,•·p•,j}2 101·pi,•·p•,j

= 3.627

5

4 (= (2−1)·(5−1) )

(9.48773,∞)

Table 19:

1458

· p

i,•

· p

•,j

10 20 50 100 200 200

470.556 466.753 298.494 90.309 38.025 21.864 0.951

24.444 24.247 15.506 4.691 1.975 1.136 0.049

0.340 0.337 0.215 0.065 0.027 0.016

Table 20: ppm

101 _·

p

i,•

· p

•,j

10 20 50 100 200 200

47.762 13.545 5.703 3.564 1.426 0.713

19.238 5.455 2.297 1.436 0.574 0.287

0.663 0.188 0.079 0.050 0.020

(14)

Table 21:

B1 B2 total

A1 X11 X12 n1=X1,•

A2 X21 X22 n2=X2,_•

total X•,1 X•,2 n

■ Xi,•=Xi1+Xi2,X•,j =X1j+X2j,n=∑2i=1Xi,•=

∑2

j=1X•,j =∑2i=1

∑2

j=1Xij ■

(Xij−Xi,•X•,j/n)2= 1

n2 (

Xij· n ∑ s=1 2 ∑ t=1

Xst−Xi,•X•,j

)2

= 1

n2(X11X22−X12X21) 2

(Xi1−Xi,•X•,1/n)2 Xi,•X•,1/n +

(Xi2−Xi,•X•,2/n)2 Xi,•X•,2/n =

1

n

(X11X22−X12X21)2 Xi,•

(

1

X•,1 +

1

X•,2 )

= (X11X22−X12X21)

2 Xi,•X•,1X•,2

X = (X11−X1,•X•,1/n)2 X1,•X•,1 +

(X12−X1,•X•,2/n)2 X1,•X•,2 +

(X21−X2,•X•,1/n)2 X2,•X•,1 +

(X22−X2,•X•,2/n)2 X2,•X•,2

= n·(X11X22−X12X21)

2 X1,•X•,2X•,1X•,2

■ B1 A1 B2 A2

□

□ pˆ1=X11/(X11+X12) =X11/X1,•,pˆ2=X21/(X21+X22) =X21/X2,• □ (pˆ= (X11+X21)/n)

ˆ

p1−pˆ2 √

ˆ

p(1−pˆ)/n1+ ˆp(1−pˆ)/n2

ˆ

p = X11+X21

n , 1−pˆ=

X12+X22 n

ˆ

p(1−pˆ)

n1 =

X11+X21 n

X12+X22 n

1

n1 =

1

n2

X•,1X•,2 X1,•

, pˆ(1−pˆ) n2 =

1

n2

X•,1X•,2 X2,•

ˆ

p(1−pˆ)

n1

+pˆ(1−pˆ)

n2

= 1

n2

X_•,1X•,2

X1,• + 1

n2

X_•,1X•,2

X2,• = 1

n

X_•,1X•,2

X1,•X2,•

ˆ

p1−pˆ2 = X11 X1,_• −

X21 X2,_•

=X11X22−X12X21

X1,_•X2,_•

ˆ

p1−pˆ2 √

p(1−p)

n + p(1−p)

n =

X11X22−X12X21 X₁,•X2,•

√ 1

n

X•,1X•,2 X X

= √

n(X11X22−X12X21) √