Post on 09-Nov-2015
description
1
...
1.1 t
X
= (X1, . . . ,Xn), Y
= (Y1, . . . , Ym)
Xi N (1, 2), i = 1, . . . , n,
Yi N (2, 2), i = 1, . . . ,m, 1, 2 ()
2 .
H0 : 1 = 2 H1 : 1 6= 2.
H0 (..)
|t| := |X Y |
Sp
1
n+
1
m
> tn+m2,/2, (1.1.1)
X =n
i=1 Xi/n, Y =m
i=1 Yi/m ,
S2p =(n 1)S2X + (m 1)S2Y
n+m 2 =n
i=1(Xi X)2 +m
i=1(Yi Y )2n+m 2 -
(pooled) ()
tn+m2,/2 (/2) t n+m 2 .
( 1.1.)
, (1.1.1)
1 = 2
1
2 1.
1 6= 2. , X 1 Y 2, 1 6= 2 .1 :
: -
X, Y ,
X = Y .
-
. .
, ,
1 2
. ,
, |X Y |, 1 6= 2 .
1 6= 2 |X Y | . . |x y| , 1 6= 2; 1 = 2 |x y| = 10 20 100;
X, Y
X N(1,
2
n
), Y N
(2,
2
m
)
X, Y .
X Y N(1 2, 2
(1
n+
1
m
)).
1 = 2 ( )
X Y 0. , C > 0
P(|X Y | > C) = 1 P(|X Y | 6 C)= 1 P(C 6 X Y 6 C)
= 1 P( C1/n + 1/m
6X Y
1/n+ 1/m
6C
1/n + 1/m
)
= 1{
(C
1/n+ 1/m
)
( C1/n+ 1/m
)}
1 , !
;
2
1.1. t 3
= 2
{1
(C
1/n + 1/m
)}(1.1.2)
> 0,
.
(z) = 1 (z), z, ( H0)
X Y1/n+ 1/m
N (0, 1)
( H0)
X Y . : , C , X Y , C
.
, C ,
|X Y | > C C
|X Y | > C . , C
|X Y | > C C , .. 5% 1%.
. (
,
.) -
S2p
2. ( 2
S2p .)
: (X Y )/{1/n + 1/m} ( ) ,
(X Y )/{Sp1/n + 1/m} ( Sp
) !
Sp . ,
! .
1.1.1. t. Z N (0, 1), V 2p . Z/
V/p t p ,
tp.
3
4 1.
Z =X Y
1/n + 1/m
V =(n+m 2)S2p
2.
. (
,
.) ,
. , 2n+m2
( H0) -
(n1)S2X/2 2n1 (m1)S2Y /2 2m1. , H0,
X YSp
1n +
1m
d=
ZV/(n +m 2) ,
d= ,
. 1.1.1 tn+m2
tn+m2,/2
. (1.1.1)
(
I) .
1.2 p
, -
p (p-value). , (.. SPSS, R),
p. p
(significance).
1.2.1. ( p) p
.
, : p -
.
, p :
1.2.2. ( p - )
T = T (X). (
4
1.2. p 5
T C.) T (x) T
p PH0{T > T (x)}.
p. p
. :
(.. ).
.
p
, -
.
1.2.1. ,
H0 : 1 = 2 H1 : 1 6= 2 |t| t = (X Y )/{Sp
1/n + 1/m}. H0,
tn+m2. n = 8, m = 10
t = (x y)/{sp1/n+ 1/m} = 2.02. p
P(|tn+m2| > | 2.02|) = P(t16 < 2.02) + P(t16 > 2.02)= 0.030227 + 0.030227 = 0.060454.
( 1.1.) P(t16 > 2.02)
MS Excel
=tdist(2.02;16;1)
1 2, Excel
P(|t16| > 2.02).2
1.2.1. , p
, ,
, p,
.
p .
-
2 MS Excel .
2,02 2.02.
, Excel
;.
5
6 1.
2.02|t|
2.02|t|
P(t16 > 2.02) 0.0302
1.1: 1.2.1, p
t16 ( t ) t =2.02, |t| = 2.02, .
: 1.2.1, -
p
.
1.2.1 (). p 0.060454. 0.05 < p < 0.10,
0.10
0.05. , > 0.060454
6 0.060454 .
1.2.2. ( .) p , , .
T H0 F0, 1.2.2,
p = PH0{T > T (X)} = 1 F0{T (X
)}.
: X F F (X) 1 F (X) (0, 1),U(0, 1). , F X F1 . , x (0, 1),
P{F (X) 6 x} = P{X 6 F1(x)} = F{F1(x)} = x
P{1 F (X) 6 x} = P{F (X) > 1 x} = P{X > F1(1 x)} = 1 F{F1(1 x)} = x
F (X) 1 F (X) U(0, 1). , H0, p U(0, 1). , p . ( , , p 1/2.)
6
1.3. 7
.. - ,
PH0( H0 ).
, 1.2.1, H0 p . U U(0, 1) P(U < u) = u, u (0, 1), H0 p
PH0( p < ) = ,
, H0 p U(0, 1). p ( 0.060454 1.2.1) p.
1.3 t
t H0 : 1 = 2
H1 : 1 6= 2 .. |tobs| > tn+m2,/2,
tobs =x y
sp
1n +
1m
(observed, obs t)
t =X Y
Sp
1n +
1m
.
:
1.3.1. H0
100(1)% 12.
. 100(1)% 12
[X Y tn+m2,/2Sp
1
n+
1
m, X Y + tn+m2,/2Sp
1
n+
1
m
]
( ). , H0
|t| 6 tn+m2,/2 tn+m2,/2 6 t 6 tn+m2,/2 tn+m2,/2 6
X YSp
1n +
1m
6 tn+m2,/2
7
8 1.
tn+m2,/2Sp
1
n+
1
m6 X Y 6 tn+m2,/2Sp
1
n+
1
m
X Y tn+m2,/2Sp
1
n+
1
m6 0 6 X Y + tn+m2,/2Sp
1
n+
1
m
,
0 [X Y tn+m2,/2Sp
1
n+
1
m, X Y + tn+m2,/2Sp
1
n+
1
m
].
, t
/2- tn+m2,
100(1 )% 1 2 H0
.
.
1.3.1. ( ), -
1 2 . ,
1 2, 1 = 2 ( ) . ,
, .
1.4 t F
|t| > tn+m2,/2 : /2-
(0, 1). /2- t
= 1 t,/2 t,0.5 = 0, . , t,/2 1 < 2
1/2 = P(t > t,1/2) < P(t > t,2/2) = 2/2
t,1/2 > t,2/2 (= 1)
/2-
t . ,
|t| > tn+m2,/2
8
1.4. F 9
t2 > t2n+m2,/2,
(X Y )2S2p(
1n +
1m
) > t2n+m2,/2.
:
1.4.1. () W t W 2 F1, , F .
() (0, 1) t2,/2 = F1,,, - F1, .
. () Z N (0, 1) V 2 . ,
Wd=
ZV/
( t). , Z2 21,
W 2d=Z2/1
V/ F1,
( F1,).() , (0, 1) t,/2 > 0. W t ,
= P(|W | > t,/2) = P(W 2 > t2,/2)
t2,/2 - F1, () W 2.
, 1 =
2,
t2 =(X Y )2S2p(
1n +
1m
) F1,n+m2. |t| > tn+m2,/2 t2 > t2n+m2,/2 = F1,n+m2,, ..
(X Y )2S2p(
1n +
1m
) > F1,n+m2,. (1.4.3)
() (1.4.3): F .
9
10 1.
1.5
X
= (X1, . . . ,Xn) (n )
Y
= (Y1, . . . , Ym) (m ).
, X
Y
.. ; n
m. , .
, Y .
, Y
1 Y
2
. , n1
n2. n1
Y
1 = (Y11, . . . , Y1n1).
n2 Y
2 = (Y21, . . . , Y2n2). , Yij j
i : Y1j Y2j .
,
Yij, i = 1, 2, j = 1, . . . , ni.
i 1 ( ), j
1 n1. , i 2 ( ),
j 1 n2.
X Y . ,
Y1 Y2. ,
, . -
Y11, . . . , Y1n1 . ,
n1
j=1 Y1j/n1.
( j) ( 1),
Y1 (Y 1 ): .
, n2
j=1 Y2j/n2 Y2 (Y 2 ). ,
Yi = 1ninij=1
Yij
10
1.6. 11
. ( !)
1 = 2, ()
, n + m (
n1 + n2) ( 1.1).
nX +mY
n+m
nX Xi mY Yi.
, 2i=1
nij=1 Yij . ;
. i
1 2, : i = 1
i = 2. i = 1 n1
j=1 Y1j
i = 2 n2
j=1 Y2j.
. n1 + n2
. ( i)
( j), Y .
Y = 1n1 + n22
i=1
nij=1
Yij
.
1.6
, -
. . ,
,
:
1.6.1. X .
= X (error) X.
.
1.6.1. X N (, 2) + N (0, 2).
. + X.
1.6.1, X.
, .
11
12 1.
X = + ,
+ . , ( )
.
,
.
1.6.2. . = X (residual) X.
:
, X
X
. , ,
. , X
, . ,
-
. ,
. :
-
.
.
. Y
1 =
(Y11, Y12, . . . , Y1n1) N (1, 2), Y
2 = (Y21, Y22, . . . , Y2n2)
N (2, 2) .
Y11 = 1 + 11 Y21 = 2 + 21
Y12 = 1 + 12 Y22 = 2 + 22...
...
Y1n1 = 1 + 1n1 Y2n2 = 2 + 2n2
11, . . . , 1n1 , 21, . . . , 2n2 n1 + n2 N (0, 2) . ( .
n1 n2 .) 1.6.1,
ij
.
12
1.6. 13
. -
, . -
Yij = i + ij, i = 1, 2, j = 1, . . . , ni,
ij N (0, 2) . , i Yij ,
( )
.
. , Yij
( ), ij
.
1, 2 -
Y1, Y2. , ij- ij = Yij i = Yij Yi, i = 1, 2, j = 1, . . . , ni. (1.6.4)
i
Cov(ij , ik) = Cov(Yij Yi, Yik Yi)= Cov(Yij , Yik) Cov(Yij , Yi) Cov(Yi, Yik) + Cov(Yi, Yi)= 0
2
ni
2
ni+2
ni
= 2
ni6= 0,
Cov(Yij , Yik) = 0 j 6= k ( Yij Yik )
Cov(Yij , Yi) = Cov(Yij ,
1
ni
ni=1
Yi
)=
1
ni
ni=1
Cov(Yij , Yi) =2
ni
Cov(Yij , Yi) = j Cov(Yij , Yij) =
Var(Yij) = 2,
Cov(Yi, Yi) = 2/ni ( ) Cov(Yi, Yi) = Var(Yi) = 2/ni.
, ij ik ,
ij ik . -
.
: 1.2.
13
14 1.
1.7
. , ,
.
, 1,
, 2, + 2. , 2 = ( +
2) = 2 1 1 = 2 1 6= 2 2 = 0 2 6= 0 . 1 0, i = + i, i = 1, 2, 1 = 2 1 = 2 = 0. , 1 = 0
,
. ,
1, 2 , 1, 2,
1 0. .
.
. -
1, 2. c [0, 1]
= c1 + (1 c)2,1 = 1 = (1 c)(1 2),2 = 2 = c(2 1).
i = + i, i = 1, 2, 1, 2
c1 + (1 c)2 = 0. , 1, 2 , 1, 2
1, 2. ( c = 1
.)
1 = 2 1 6= 2; ,
1 = 2 = 0.
,
1, 2
1 = 2 = 0.
14
1.8. t 15
, c = 1/2. , -
1, 2 1 + 2 = 0 ( ;).
1.6, Y
1 = (Y11, . . . , Y1n1)
Y
2 = (Y21, . . . , Y2n2) N (1, 2) N (2, 2)
Yij = + i + ij, i = 1, 2, j = 1, . . . , ni,
ij N (0, 2), i = 1, 2, j = 1, . . . , ni, 1 + 2 = 0.
1.8 t
1.8.1
t
.
.
(
:
.)
KolmogorovSmirnov. -
. (
1/n
) -
(
). Fn F0 -
, F0
Dn = supxR
|Fn(x) F0(x)|. (1.8.5)
Dn
F0 ( ),
Dn . ( Glivenko-
Cantelli.) .
KolmogorovSmirnov
,
.
Lilliefors
15
16 1.
ShapiroWilk. SPSS
, Q-Q plot.
1.8.2
t
() . 21 = 22
F = S2X/S2Y (1.8.6)
, , F n 1 m 1 . F < Fn1,m1,1/2 F > Fn1,m1,/2. ( 1.3.) ,
t. , :
( )
.
,
t .
(1.8.6) . SPSS
, Levene
.
1.9 t SPSS
SPSS
. .
,
,
g ml . ( )
:
37.2 41.7 32.1 38.3 40.5 39.4 53.2 40.8 43.7 45.4
32.7 36.4 39.3 42.5 38.4 27.9 30.1 33.7 37.2 41.8
.
SPSS.
16
1.9. t SPSS 17
1.9 SPSS.
, t
. SPSS
Analyze > Compare Means > Independent-Samples T Test
Test Variable Grouping Variable ( ). SPSS
Grouping Variable Define Groups, Group 1 1 Group 2 2 Continue. OK :
Group Statistics
10 41.2300 5.57017 1.76144
10 36.0000 4.83896 1.53021
N Mean Std. Deviation
Std. Error
Mean
17
18 1.
Independent Samples Test
.005 .944 2.241 18 .038 5.23000 2.33329 .32794 10.13206
2.241 17.655 .038 5.23000 2.33329 .32107 10.13893
Equal variances
assumed
Equal variances
not assumed
F Sig.
Levene's Test for
Equality of Variances
t df Sig. (2-tailed)
Mean
Difference
Std. Error
Difference Lower Upper
95% Confidence
Interval of the
Difference
t-test for Equality of Means
, -
: , ,
( )
. -
n = m = 10, x = 41.23, y = 36.00, sx 5.57 sy 4.84 sx/
n 5.57/10 1.76
sy/m 4.84/10 1.53, .
.
Levene
. F ( ) p 0.944
. (
p
.)
t . -
t = (X Y )/{Sp
1n +
1m
} 2.241,
t
n+m 2 = 10 + 10 2 = 18 p 0.038:
p = P(|t18| > |2.241|) = P(t18 < 2.241) + P(t18 > 2.241)= 0.0189 + 0.0189 = 0.0378 0.038
( 1.2).
: t = 2.241.
, SPSS t
. p
, .
18
1.9. t SPSS 19
2.241|t|
2.241|t|
P(t18 > 2.241) 0.0189
1.2: 1.9 p
t18 ( t - ) t = 2.241, |t| = 2.241, .
p 0.05, -
. ,
.
SPSS. KolmogorovSmirnov (K-S)
Analyze > Nonparametric Tests > Sample K-S
Test Variable List -
.
-
. .
Exact. -
: Asymptotic only ( ) , MonteCarlo ( ) Exact ( ).
Exact Monte Carlo -
. SPSS
.
, .
K-S -
. .
: , -
19
20 1.
. ,
. SPSS,
Data > Split File
Compare groups Organizeoutput by groups. ( ,
.) (
) Groups Based on.
OK. , SPSS
.
Analyze all cases, do not create groups
K-S :
N
Mean
Std. Deviation
Absolute
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
Exact Sig. (2-tailed)
Point Probability
Normal Parametersa
Most Extreme Differences
N
Mean
Std. Deviation
Absolute
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
Exact Sig. (2-tailed)
Point Probability
Normal Parametersa
Most Extreme Differences
,000
,984
,994
,420
-,133
,090
,133
4,83896
36,0000
10
,000
,904
,945
,526
-,135
,166
,166
5,57017
41,2300
10
One-Sample Kolmogorov-Smirnov Test
a. Test distribution is Normal.
(
) ( ).
p. p:
(.945) (.904).
. ( ,
.
.)
p
.
20
1.9. t SPSS 21
p () K-S .984.
() t .
SPSS .
.
; ;
, SPSS
F0
( ) N (y1, s21) Dn (1.8.5). ,
.
-
Q-Q plot. Q-Q
quantile-quantile (quantile = ).
SPSS.
Analyze > Descriptive Statistics > Q-Q Plots
Variables , .
.
1.3 Q-Q plots, . -
,
.
, (
.. -
). (
)
.
, (
) .
( ) -
. , -
.
-
21
22 1.
Observed Value
555045403530
Ex
pe
cte
d N
orm
al
Va
lue
50
45
40
35
30
Omada:Astheneis
Normal Q-Q Plot of sygkentrosh
Observed Value
4540353025
Ex
pe
cte
d N
orm
al V
alu
e
45
40
35
30
25
Omada:Oxi astheneis
Normal Q-Q Plot of sygkentrosh
1.3: Q-Q plots 1.9.
.
. -
Yij
i ij. , n1 + n2 -
,
. ( .)
-
.
ij ij.
SPSS
.
Data > Split File
Analyze all cases, do not create groups.
Analyze > General Linear Model > Univariate
Dependent Variable Fixed Factor .( dependent variable, , fixed factor, -
, .) Save
Residuals > Unstandardized. Continue OK. SPSS .
22
1.9. t SPSS 23
RES_1.
.
Q-Q Plots . :
Observed Value
151050-5-10
Exp
ecte
d N
orm
al V
alu
e
10
5
0
-5
-10
Normal Q-Q Plot of Residual for sygkentrosh
. -
.
Q-Q plot
Q-Q plot. x1, x2, . . . , xn
(
) F0.
.
x(1) 6 x(2) 6 . . . 6 x(n) .
.
, n n+1 :
(, x(1)], (x(1), x(2)], . . . , (x(n1), x(n)], (x(n),). F0, x(1) ( n )
n/(n + 1)- F0,
1/(n + 1) n/(n + 1). F0
, F10 (1/(n + 1)). ,
x(2) ( n 1 ) (n 1)/(n + 1)- F0,
23
24 1.
2/(n + 1) (n 1)/(n + 1). F10 (2/(n + 1)). , x(k) (
k n k+1 ) (n k + 1)/(n + 1)- F0, k/(n + 1) (n k + 1)/(n + 1) F10 (k/(n + 1)).
(x(k), F
10 (k/(n + 1))
), k = 1, 2, . . . , n,
. F0,
. Q-Q plot
,
.
, -
( 2),
. , -
.3
1.10
1.1. t :
X
= (X1, . . . ,Xn), Y
= (Y1, . . . , Ym), n,m > 2,
, N (1, 2) N (2, 2). = (1, 2,
2) = R2 (0,).
H0 : 1 = 2
H1 : 1 6= 2.
() -
(1.1.2) C = z/21/n+ 1/m z/2
(/2)- .
() H1 1, 2 X, Y
3 SPSS (n k + 5/8)/(n + 1/4)- (n k + 1)/(n + 1).
Q-Q Plots Van der Waerdens Proportion Estimation Formula. , n .
24
1.10. 25
,
2 =
ni=1(Xi X)2 +
mi=1(Yi Y )2
n+m=n+m 2n+m
S2p.
() H0 ()
= (nX +mY )/(n +m)
20 =
ni=1(Xi )2 +
mi=1(Yi )2
n+m.
:
(n+m)20 = (n+m 2)S2p + {n(X )2 +m(Y )2} (1.10.7)= (n+m 2)S2p +
nm
n+m(X Y )2. (1.10.8)
() x, y
(x, y
) =
maxH1
L(1, 2, 2|x, y
)
maxH0
L(1, 2, 2|x, y
)=
L(x, y, 2|x, y
)
L(, , 20 |x, y
)
.
(1.10.8), C1 C2
(x, y
) > C1 |x y|
sp
1
n+
1
m
> C2.
1.2. ij (1.6.4). E(ij) = 0 Var(ij) = (1 1/ni)2.
1.3. F pi: X
= (X1, . . . ,Xn), Y
= (Y1, . . . , Ym), n,m > 2,
, N (1, 21) N (2, 22). = (1, 2,
21 ,
22) = R2 (0,)2.
() 21 22
21 =
ni=1(XiX)2/n 22 =
mi=1(YiY )2/m
. , 21 = 22 =
2, 2 (0,) , 2 20 = (n
21 +m
22)/(n +m).
() H0 : 21 =
22
H1 : 21 6= 22.
(x, y
) =
maxH1
L(|x, y
)
maxH0
L(|x, y
)> C wn/2(1 w)m/2 < C , (1.10.9)
w = n21/(n21 +m
22) C, C
.
() g(w) = wn/2(1w)m/2 w (0, 1)
25
26 1.
w < n/(n +m) w > n/(n +m).
(1.10.9)
w < C1 w > C2 (1.10.10)
0 < C1 < C2 < 1 .
() w s2X/s2Y , s
2X , s
2Y
() .
(1.10.10)
s2X/s2Y < C3 s
2X/s
2Y > C4 (1.10.11)
C3 < C4 .
() H0 : 21 =
22, S
2X/S
2Y
Fn1,m1
(X, Y) =
1, S2XS2Y
< Fn1,m1,1/2 S2XS2Y
> Fn1,m1,/2,
0,
I .
26
2
: , -
. (
) , , .
2.1 k > 2
k > 2
. .
y11, . . . , y1n1 (n1 )
y21, . . . , y2n2 (n2 )
...
yk1, . . . , yknk (nk k- ).
n = n1 + n2 + + nk. , -
Y11, . . . , Y1n1
Y21, . . . , Y2n2...
Yk1, . . . , Yknk .
, :
i = 1, . . . , k, i , Yi1, . . . , Yini,
( ).
27
28 2.
k .
k .
k 2.
k
k , -
.
1, . . . , k.
n
Yij N (i, 2), i = 1, . . . , k, j = 1, . . . , ni.
Yij = i + ij,
Yij = i + ij , i = 1, . . . , k, j = 1, . . . , ni,
ij N (0, 2) ( ) . , k -
k .
Yi = 1ninij=1
Yij
i , i = 1, . . . , k.
( )
Y = 1nki=1
nij=1
Yij .
niYi = nij=1 Yij , k
Y =ki=1
ninYi , (2.1.1)
.
2.1.1. y1, . . . , ym
m
i=1 ciyi ci 0
1 :m
i=1 ci = 1. y1, . . . , ym
yi. :
yi ymin ci ciyi >
ciymin i, m
i=1 ciyi >m
i=1 ciymin =
yminm
i=1 ci = ymin. ( yi .)
2.2. 29
2.2
.
-
yij, i = 1, . . . , k, j = 1, . . . , ni.
2.2.1.
ki=1
nij=1
(yij y)2 =ki=1
ni(yi y)2 +ki=1
nij=1
(yij yi)2. (2.2.2). yi
ki=1
nij=1
(yij y)2 =ki=1
nij=1
(yij yi+ yi y)2
=
ki=1
nij=1
{(yij yi)2 + (yi y)2 + 2(yij yi)(yi y)}
=
ki=1
{ nij=1
(yij yi)2 + ni(yi y)2 + 2(yi y)nij=1
(yij yi)}
=
ki=1
nij=1
(yij yi)2 +ki=1
ni(yi y)2
ni
j=1(yij yi) = 0 i ( ;). (2.2.2)
yij y, ki=1nij=1(yij y)2, :
ki=1 ni(yi y)2
k k
i=1
nij=1(yijyi)2
yij .
. , k
, ,
.
(2.2.2) Yij :
ki=1
nij=1
(Yij Y)2 =ki=1
ni(Yi Y)2 +ki=1
nij=1
(Yij Yi)2. , SST
(Total Sum of Squares, ), SSB (Sum of Squares
30 2.
Between groups, ) SSW (Sum of
Squares Within groups, ), .
, SSTotal, SSBetween, SSWithin.
.
2.2.1. () SSB SSW .
() SSW/2 2nk() E(SSB) = (k 1)2 +ki=1 ni(i )2, :=kj=1 njj/n.() 1 = = k SSB/2 2k1.
. () (2.1.1) -
k . S2i =ni
j=1(Yij Yi)2/(ni 1),i = 1, . . . , k, k .
SSW =
ki=1
(ni 1)S2i . (2.2.3)
SSB SSW
.
k , SSB SSW
.
() m
2, S2 (m 1)S2/2 2m1. Vi = (ni 1)S2i /2 2ni1, i = 1, . . . , k. SSW/2 =
ki=1 Vi. k
( k ),
k
i=1 Vi -
k
i=1(ni 1) = n k.
()
E(SSB) = E
{ ki=1
ni(Yi Y)2}
=ki=1
niE{(Yi Y)2}
=
ki=1
ni{Var(Yi Y) + [E(Yi Y)]2}. (2.2.4)
2.2. 31
(2.1.1) j
j,
E(Yi Y) = E(Yi) E(Y)= i E
{kj=1njYj/n}
= i k
j=1njE(Yj)/n= i
kj=1njj/n
= i . (2.2.5)
, i
Cov(Yi, Y) = Cov(Yi,
k=1
nnY)=
k=1
nn
Cov(Yi, Y)=nin
Cov(Yi, Yi) = nin Var(Yi) = nin 2
ni=2
n,
Var(Yi Y) = Var(Yi) + Var(Y) 2Cov(Yi, Y)=
2
ni+2
n 2
2
n
=
(1
ni 1n
)2. (2.2.6)
, (2.2.4)
E(SSB) =
ki=1
ni
{(1
ni 1n
)2 + (i )2
}
=
ki=1
(1 ni
n
)2 +
ki=1
ni(i )2
= (k 1)2 +ki=1
ni(i )2.
() k
. n
S2 = SST/(n 1). W := (n 1)S2/2 = SST/2 2n1. 2m (1 2t)m/2, t < 1/2. W1 := SSB/
2, W2 := SSW/2. ()
() W2 2nk. ,
W = W1 + W2
32 2.
, W ,
W1, W2
MW (t) = MW1(t)MW2(t)(1 2t)(n1)/2 = MW1(t)(1 2t)(nk)/2, t < 1/2,
MW1(t) = (1 2t)(k1)/2, t < 1/2, 2k1.
2.2.1. ( )
() (
) () (
). () ()
.
2.2.1 :
2.2.1. 1 = = k F := SSB/(k 1)SSW/(n k) Fk1,nk.
. 2.2.1 W1 = SSB/2, W2 = SSW/
2 -
. 2nk 1 = = k 2k1. , 1 = = k,
F =SSB/(k 1)SSW/(n k) =
W1/(k 1)W2/(n k) Fk1,nk
F .
2.2.2.
H0 : 1 = = k
H1 : H0.
() H0 ..
F =SSB/(k 1)SSW/(n k) > Fk1,nk,.
. ,
= (1, . . . , k, 2) = Rk (0,).
2.2. 33
i ()
, H0 0 = R (0,). ()
(y
) =max
L(|y
)
max0
L(|y
)=
L(|y
)
L(0|y
),
0, H0, H1
L(|y
) =1
n(2)n/2exp
{ 1
22
[ n1j=1
(y1j 1)2 + +nkj=1
(ykj k)2]}
=1
n(2)n/2exp
{ 1
22
ki=1
nij=1
(yij i)2}
.
0 = (0, 20) = (Y,SST/n).
, H0
n N (, 2). Y 2 Y n ( ). , H1 k
, 1, . . . , k k
Y1, . . . , Yk 2 2 =
1
n
{ n1j=1
(Y1j Y1)2 + +nkj=1
(Ykj Yk)2}=
SSW
n(2.2.7)
( 2.2). ,
(y
) =
1
n(2)n/2exp
{ 1
22
ki=1
nij=1
(yij yi)2}
1
n0 (2)n/2
exp
{ 1
220
ki=1
nij=1
(yij y)2}
=
1
(SSW/n)n/2exp
{ 1
2SSW/nSSW
}
1
(SST/n)n/2exp
{ 1
2SST/nSST
}
=
(SST
SSW
)n/2
C > 0,
(y
) > C (SST
SSW
)n/2> C SST
SSW> C2/n
34 2.
SSB+ SSWSSW
> C2/n SSBSSW
> C2/n 1
SSB/(k 1)SSW/(n k) > C
:=n kk 1 (C
2/n 1).
, .. (0, 1),
= PH0{(Y) > C} = PH0(F > C)
C = Fk1,nk, .
2.2.2. , F
E(F ) = E
{SSB/(k 1)SSW/(n k)
}
=n kk 1E
{SSB
SSW
}
=n kk 1E(SSB)E
{1
SSW
}
( SSB, SSW)
=n kk 1
{(k 1)2 +
ki=1
ni(i )2}
1
(n k 2)2( 2.2.1 E(1/2) = 1/( 2) > 2)
=n k
n k 2 +n k
(k 1)(n k 2)ki=1
ni(i )22
.
(nk)/(nk2) ( Fk1,nk)
ni(i)2 =
0, i ,
1 = = k. , ni(i )2, i
1, . . . , k,
F .
F
, .
F .
2.3 (ANOVA Table)
k > 2
, -
.
2.3. 35
ANOVA Table ANOVA ANalysis
Of VAriance.
k = 3
(n1, n2, n3) = (4, 7, 5). n = 4 + 7 + 5 = 16.
SSB = 65.5, SSW = 135.2 () SST = 200.7 = 65.5 + 135.2.
SSB, SSW SST k 1 = 3 1 = 2,n k = 16 3 = 13 n 1 = 16 1 = 15, . ( 15 = 2 + 13 n 1 = (k 1) + (n k).) F SSB/(k 1) =65.5/2 = 32.75 SSW/(n k) = 135.2/13 = 10.40. , F = 32.75/10.40 3.149. :
F
65.5 2 32.75 3.149
135.2 13 10.40
200.7 15
,
F2,13 F . F
.
, p. ,
p P(F2,13 > 3.149) ( F
F ).
0.0767 Excel. ,
Excel
=fdist(3.149;2;13)
Enter. p -
0.05.
: 0.1 ; (-
0.1 F .)
36 2.
2.4 H0:
H0 : 1 = = k k . k :
,
. (
) .
,
(k2
)= k(k 1)/2, .
( )
1 6= 2 1 6= 3 1 6= k2 6= 3 2 6= k...
k2 6= k1 k2 6= kk1 6= k
i j ,
H0,ij : i = j H1,ij : i 6= j .
( ij i j.)
i j, (
)
t .
tij =Yi Yj
Sij1/ni + 1/nj
,
S2ij
( S2p),
H0,ij |tij| > tni+nj2,/2. : -
k ,
2 = SSW/(n k). 2 S2ij k .
() : 2
24/(n k) S2ij 24/(ni + nj 2) ( 2.3). tij S
2ij
2;
SSW k , 2
2.4. 37
Yi Yj . , (n k)2/2 2nk Yi Yj N (i j, 2(1/ni + 1/nj)). , H0,ij :i j = 0,
tij =Yi Yj
1/ni + 1/nj
=(Yi Yj)/{1/ni + 1/nj}
2/2d=
N (0, 1)2nk/(n k)
d= tnk.
(
!)
, H0,ij H1,ij
|tij| > tnk,/2. 1.3 100(1)% - i j. ,
[Yi Yj tnk,/2
1/ni + 1/nj , Yi Yj+ tnk,/2
1/ni + 1/nj
](2.4.8)
.
, (k2
)
: (k2
)
. -
-
.
.
. ( (2.4.8)
.) .
2.4.1 LSD Fisher
LSD least significant difference ( ).
(2.4.8).
YiYj ( i j , i = j)
tnk,/2 1/ni + 1/nj . (2.4.9)
38 2.
, (2.4.9)
() :
.
2.4.2 Bonferroni
(..) . , -
I12 I13 1 2 1 3 1 ,
P(1 2 I12, 1 3 I13) < P(1 2 I12) = 1 . P(AB) 6 P(A) P(AB) >0.
.
P(AB) 6 P(A) +P(B).1 m > 2 :
P(mi=1Ai) 6m
i=1 P(Ai). (2.4.10)
Bonferroni
.
,
(k2
) .
m -
1, . . . , m. , J1, . . . , Jm. ,
P( mi=1 {i Ji}) = 1 P([ mi=1 {i Ji}])
= 1 P( mi=1 {i Ji})( de Morgan)
= 1 P( mi=1 {i / Ji})> 1mi=1P(i / Ji)
( Bonferroni ).
J1, . . . , Jm 11, . . . , 1m , P(i / Ji) = i, i = 1, . . . ,m.
P( mi=1 {i Ji}) > 1mi=1 i.
1 A B . ,
P(A B).
2.4. 39
m
1, i . i = /m, i = 1, . . . ,m.
, -
m =(k2
) m =
(k2
) .
,
1/m, 1 . , k = 5 - 95% (
1 = 0.95, = 0.05), 1/m = 0.995, m = (52) = 10.
, Bonferroni
m =(k2
)
[Yi Yj tnk,/(2m)
1/ni + 1/nj , Yi Yj+ tnk,/(2m)
1/ni + 1/nj .
](2.4.11)
,
.
2.4.3 Scheffe
, -
. ,
k = 4 ,
.
1, 2, 3, 4,
-
( ). ,
()
1 + 22
3 + 42
,
21 + 2
3 3 + 4
2.
4
i=1 cii ci
4
i=1 ci = 0: ci 1/2, 1/2, 1/2 1/2
40 2.
2/3, 1/3, 1/2 1/2. , .
, 1 2 4
i=1 cii ci 1,
1, 0 0.
2.4.1. k
i=1 cii ci ki=1 ci = 0 (contrast).
( ).
2.4.1. C0 c= (c1, . . . , ck)
k
i=1 ci = 0.
P
( cC0
{ ki=1
ciYi (k 1)Fk1,nk,
ki=1c
2i /ni 6
ki=1
cii 6
ki=1
ciYi+ (k 1)Fk1,nk,
ki=1c
2i /ni
})= 1 . (2.4.12)
: (2.4.12) -
c C0. (2.4.12)
k
i=1 cii
1 c C0.
. (2.4.12)
P
( c
C0
{(k 1)Fk1,nk, 6
k
i=1ciYi ki=1 cii
k
i=1c2i/ni
6
(k 1)Fk1,nk,
})= 1
P
( c
C0
{ [ki=1
ci(Yi i)]2(k 1)2k
i=1c2i/ni
6 Fk1,nk,})
= 1 .
( c
C0
c
) Fk1,nk, Fk1,nk,.2
P
(maxc
C0
[k
i=1ci(Yi i)]2
(k 1)2ki=1
c2i/ni
6 Fk1,nk,)= 1 .
2
.
2.4. 41
=
k
i=1nii/n. ( 2.2.1.)
Y i, ki=1 ci(Y ) = 0., c
C0,
[k
i=1ci(Yi i)]2
(k 1)2ki=1
c2i/ni
=
[k
i=1ci{(Yi i) (Y )}]2(k 1)2k
i=1c2i/ni
( 0 =
k
i=1ci(Y ))
=
[k
i=1(ci/
ni)ni{(Yi i) (Y )}]2
(k 1)2ki=1
c2i/ni
(2.4.13)
( i ni).
CauchySchwarz.
2.4.1. ( CauchySchwarz) a1, . . . , ak, b1, . . . , bk
( ki=1
aibi
)26
( ki=1
a2i
)( ki=1
b2i
)
ai bi ai = bi i = 1, . . . , k.
(2.4.13) ai = ci/ni bi =
ni{(Yii) (Y )}
(k
i=1c2i/ni)[
k
i=1ni{(Yi i) (Y )}2]
(k 1)2ki=1
c2i/ni
=
k
i=1ni{(Yi i) (Y )}2
(k 1)2 . (2.4.14)
(2.4.14) . , CauchySchwarz
ci = ni{(Yi i) (Y )}, i = 1, . . . , k.( c
C0.) ,
maxc
C0
[k
i=1ci(Yi i)]2
(k 1)2ki=1
c2i/ni
=
k
i=1ni{(Yi i) (Y )}2
(k 1)2 .
(2.4.14) Fk1,nk ( Fk1,nk, 1 ). .
Xij = Yij i, i = 1, . . . , k, j = 1, . . . , ni.
Xij N (0, 2). ( ),
k
i=1ni(Xi X)2/(k 1)
k
i=1
ni
j=1(Xij Xi)2/(n k)
42 2.
Fk1,nk. ,
Xi = 1ninij=1
Xij =1
ni
nij=1
(Yij i) = 1ni
{ nij=1
Yij nii}= Yi i,
Xij Xi = (Yij i) (Yi i) = Yij Yi
X = 1nk
i=1
niXi = 1nk
i=1
ni(Yi i) = 1n{ k
i=1
niYi k
i=1
nii
}= Y .
,
k
i=1ni(Xi X)2/(k 1)
k
i=1
ni
j=1(Xij Xi)2/(n k) =
k
i=1ni{(Yi i) (Y )}2/(k 1)k
i=1
ni
j=1(Yij Yi)2/(n k)
.
2.4.1. ,
:
P
( c
Rk
{ ki=1
ciYi kFk,nk,
k
i=1c2i/ni 6
ki=1
cii 6
ki=1
ciYi + kFk,nk,
k
i=1c2i/ni
})= 1 . (2.4.15)
c
ci = 0.
(2.4.15) k
i=1ni(Yii)2/2 2k. (2.4.15) (2.4.12)
c ( ci) :
F k nk (2.4.12) k1 nk.3
Scheffe,
[Yi Yj
(k 1)Fk1,nk,
(1ni
+ 1nj
), Yi Yj+
(k 1)Fk1,nk,
(1ni
+ 1nj
)]
i, j
.
Scheffe -
.
.
.
3 :
ci = 0 c
(2.4.12) . , k n k
(2.4.15), k 1 n k (2.4.12).
2.4. 43
2.4.4 Tukey
Scheffe
( Scheffe )
k
. (
.)
n1 = = nk . n = k n k = k( 1). , q (
k, n
1 )
P
( 16i
44 2.
! , (2.4.16)
, .
max(Yi i)min(Yi i)
k n ( F Scheffe). qk,n, , -
Tukey
[Yi Yj qk,n,/ , Yi Yj+ qk,n,/].
qk,n,
.
2.5
, k
-
( SPSS).
, ,
.
nj=1
(xj x)2 =nj=1
x2j nx2 =nj=1
x2j 1
n
( nj=1
xj
)2. (2.5.17)
( -
.)
F
SSB =
ki=1
ni(Yi Y)2 SSW =ki=1
nij=1
(Yij Yi)2. SSW k (2.5.17). ,
i, ni
j=1(Yij Yi)2, - i , (2.5.17) ni n, Yij
xj Yij ( Yi) xj ( x). nij=1
(Yij Yi)2 =nij=1
Y 2ij niY 2i =nij=1
Y 2ij 1
ni
( nij=1
Yij
)2
2.5. 45
SSW =
ki=1
nij=1
(Yij Yi)2 =ki=1
nij=1
Y 2ij ki=1
niY2i =
ki=1
nij=1
Y 2ij ki=1
1
ni
( nij=1
Yij
)2.
SSW k+1
: ,k
i=1
nij=1 Y
2ij , k
,n1
j=1 Y1j, . . . ,nk
j=1 Ykj.
SSB
SST =ki=1
nij=1
(Yij Y)2 SSB
SSB = SST SSW.
SST (2.5.17) -
(
).
SST =ki=1
nij=1
Y 2ij nY 2 =ki=1
nij=1
Y 2ij 1
n
( ki=1
nij=1
Yij
)2.
-
. k
k
i=1
nij=1 Yij.
F =SSB/(k 1)SSW/(n k)
Fk1,nk.
2.5.1. :
1: 8, 10, 6
2: 7, 9
3: 11, 15, 9, 9
4: 14, 16, 14, 12
k = 4, n1 = 3, n2 = 2, n3 = n4 = 4 n = 3 + 2 + 4 + 4 = 13.
ki=1
nij=1
y2ij = 82 + 102 + 62 + 72 + 92 + 112 + 152 + 92 + 92 + 142 + 162 + 142 + 122
46 2.
= 64 + 100 + 36 + 49 + 81 + 121 + 225 + 81 + 81 + 196 + 256 + 196 + 144
= 1630,n1j=1
y1j = 8 + 10 + 6 = 24,
n2j=1
y2j = 7 + 9 = 16,
n3j=1
y3j = 11 + 15 + 9 + 9 = 44,
n4j=1
y4j = 14 + 16 + 14 + 12 = 56,
ki=1
nij=1
yij = 24 + 16 + 44 + 56 = 140.
,
SSW =
ki=1
nij=1
y2ij ki=1
1
ni
( nij=1
yij
)2= 1630
(242
3+
162
2+
442
4+
562
4
)
= 1630 (576
3+
256
2+
1936
4+
3136
4
)= 1630 (192 + 128 + 484 + 784)
= 1630 1588 = 42,
SST =
ki=1
nij=1
y2ij 1
n
( ki=1
nij=1
yij
)2= 1630 140
2
13
= 1630 1960013
1630 1507.69 = 122.31
SSB = SST SSW 122.31 42 = 80.31.
F =SSB/(k 1)SSW/(n k)
80.31/3
42/9 26.77
4.67 5.73.
:
F
80.31 3 26.77 5.73
42 9 4.67
122.31 12
2.5. 47
3.86F3,9,0.05
5.73F
6.99F3,9,0.01
p = P(F3,9 > 5.73) 0.0179
2.1: 2.5.1, p -
F3,9 F = 5.73. , p 0.01 0.05 - F 0.05- 0.01- F .
F F3,9,0.05 = 3.86 F3,9,0.01 = 6.99.
= 0.05
( F = 5.73 > 3.86)
. = 0.01
( F = 5.73 6 6.99).
0.05
0.01, p
0.01 0.05. , p
P(F3,9 > 5.73) 0.0179 ( 2.1).
2.5.1. F p Excel. 5%-
=finv(0.05;3;9)
p
=fdist(5.73;3;9).
: SSB, SSB F -
48 2.
( e-class).
SPSS; ( , . !)
2.5.1. ()
= 0.05. , -
. 100(1 )% = 95% .
1 2, 1 3, 1 4, 2 3, 2 4 3 4. y1 = 24/3 = 8, y2 = 16/2 = 8, y3 = 44/4 = 11,y4 = 56/4 = 14, 2 = SSW/(n k) = 4.67 . LSD Fisher.
tnk,/2 = t9,0.025 = 2.262. 95% 1 2
[y1 y2 tnk,0.025
1n1
+ 1n2 , y1 y2+ tnk,0.025
1n1
+ 1n2
]
=[8 84.67 2.262
13 +
12 , 8 8 +
4.67 2.262
13 +
12
]
[4.462, 4.462],
1 3 [y1 y3 tnk,0.025
1n1
+ 1n3 , y1 y3+ tnk,0.025
1n1
+ 1n3
]
=[8 114.67 2.262
13 +
14 , 8 11 +
4.67 2.262
13 +
14
]
[6.733, 0.733],
1 4 [y1 y4 tnk,0.025
1n1
+ 1n4 , y1 y4+ tnk,0.025
1n1
+ 1n4
]
=[8 144.67 2.262
13 +
14 , 8 14 +
4.67 2.262
13 +
14
]
[9.733, 2.267],
2 3[y2 y3 tnk,0.025
1n2
+ 1n3 , y2 y3+ tnk,0.025
1n2
+ 1n3
]
=[8 114.67 2.262
12 +
14 , 8 11 +
4.67 2.262
12 +
14
]
[7.233, 1.233],
2.5. 49
2 4[y2 y4 tnk,0.025
1n2
+ 1n4 , y2 y4+ tnk,0.025
1n2
+ 1n4
]
=[8 144.67 2.262
12 +
14 , 8 14 +
4.67 2.262
12 +
14
]
[10.233, 1.767]
3 4[y3 y4 tnk,0.025
1n3
+ 1n4 , y3 y4+ tnk,0.025
1n3
+ 1n4
]
=[11 144.67 2.262
14 +
14 , 11 14 +
4.67 2.262
14 +
14
]
[6.456, 0.456].
1 4 2 4, 1 2
4. 1 2
. (
y1 y2.) 3
1 3 2 3 3 4 .
Bonferroni. () , -
95%. , Bonferroni -
() .
95% Bonferroni
tnk,0.025 tnk,0.025/m m
.
( ), t9,0.025
t9,0.025/6 t9,0.00417 3.364. MS Excel
=tinv(0.05/6;9)
Bonferroni
50 2.
:
1 2 : [6.636, 6.636], 1 3 : [8.552, 2.552], 1 4 : [11.552, 0.448], 2 3 : [9.296, 3.296], 2 4 : [12.296, 0.296], 3 4 : [8.140, 2.140].
( !) , -
1 4.
2.5.2.
. ,
-
. .. k 1 > 1
placebo.
k
k1 . m = k 1 (k2) . , m
.
: 95% -
Bonferroni
. MS Excel.
2.5.1. () Scheffe.
Scheffe. -
Scheffe ,
k
i=1 cii k
i=1 ci = 0. -
i j i, j, 1 22 33 + 44, 121 + 2 + 3 524.
-
, . , -
2.6. 51
tnk = t9 -
0.05- ( = 0.05)
Fk1,nk = F3,9 k 1 = 3, 3F3,9,0.05
3 3.863 3.404. :
1 2 : [6.715, 6.715], 1 3 : [8.618, 2.618], 1 4 : [11.618, 0.382], 2 3 : [9.371, 3.371], 2 4 : [12.371, 0.371], 3 4 : [8.202, 2.202].
Bonferroni. ( .)
:
() , -
. , 2.9, SPSS
.
;
() internet.
One-way ANOVA example
google, .
-
. : ,
, .
2.6 4
-
y x. ,
(y) 1.20
(x) , (y)
(x) .
4
.
52 2.
y = h(x). (2.6.18)
, y ,
x. x . , x y
.
(2.6.18)
.
. ,
(2.6.18). x
, y
Y x
y. , x y
(2.6.18),
Y . , Y x:
FY () = h( ;x).
, , Y
(2.6.18) .
E(Y ) = h(x).
,
, , x
, Y
E(Y ) = + x.
. , x
k ,
Y () x.
, ,
k k .
, x i, i = 1, . . . , k,
Y
E(Y ) = i.
2.7. 53
:
(classification) (treatment).
, -
( ) ,
. , IQ, ,
, , . ,
. ,
, , -
, ,
.
, ,
. ,
. ,
.
2.7
( ) .
-
. , ()
.
2.7.1.
, , ,
() .
- .
k , .
k () : -
.
k ;
;
54 2.
() , F ,
k . ,
( ).
-
.
. -
.
(
one-way ANOVA)
. , . -
, . , . ( two-way ANOVA,
three-way ANOVA .)
,
k . ,
, , ,
, -
.
( ). ,
,
. , (
), -
:
. ,
()
. .
2.8
, ()
1, . . . , k k , k , -
:
1 = + 1
2 = + 2...
...
k = + k
(2.8.19)
2.8. 55
, 1, . . . , k ( 1, . . . , k).
: -
, 1, . . . , k
k :
i i .
, : k -
k + 1 .
, k = 2
1 = + 1 2 = + 2.
1 2 . , 1
2;
! .
, 1, . . . , k .
( )
k ,
. -
.
, 1, . . . , k
. 1 = 0 k = 0 ki=1 i = 0. ( -
, ) ( )
. ,
.
2.8.1. k = 3
.
.
.
1 = 0.
2.8.2.
. -
( ),
56 2.
-
. , 1, 2
1 + 2 = 0.
i , -
. ,
i .
,
ki=1
cii = 0.
( .) ci
k
i=1 ci 6= 0 1, . . . , k! k = 2:
12 = 0 ( c1 = 1 c2 = 1, c1+c2 = 0), 1 = 2, .
2.8.1
, k 1, . . . , k
1 = = k, i = j + i = + j i = j .
ki=1 cii = 0, i
0 =ki=1
cii =ki=1
ci1 = 1
ki=1
ci
1 = 0, ci k
i=1 ci 6= 0.
H0 : 1 = = k = 0( ci).
i
H1 : i 6= 0 i.
2.8.2
.
2.8.1. k
i=1 cii = 0 k
i=1 ci 6= 0, 1, . . . , k
=
ki=1 ciYiki=1 ci
i = Yi , i = 1, . . . , k. (2.8.20)
2.8. 57
. i (2.8.19) ci, i =
1, . . . , k,
c11 = c1+ c11
c22 = c2+ c22...
...
ckk = ck+ ckk
ki=1
cii =
ki=1
ci +
ki=1
cii =
ki=1
ci
k
i=1 cii = 0, ,
=
ki=1 ciiki=1 ci
k
i=1 ci 6= 0. 1, . . . , k
Y1, . . . , Yk, . , ,
=
ki=1 ciiki=1 ci
=
ki=1 ciYiki=1 ci
.
, i = i, i i = i Yi .
, c1, . . . , ck.
c1 6= 0 c2 = = ck = 0, 1 = 0,:
= Y1, 1 = 0 () i = Yi Y1 i = 2, . . . , k., ,
. -
,
.
: SPSS k = 0.
58 2.
,
,
Y, .
ki=1 nii = 0
=
ki=1 niYiki=1 ni
= Y i = Yi Y i = 1, . . . , k.: , :
.
k
i=1nii; ,
, ni,
! i.
1, . . . , k
k
, 1, . . . , k.
k
i=1 i = 0 c1 = = ck = 1. ,
=1
k
kj=1
Yj i = Yi 1kk
j=1
Yj i = 1, . . . , k.
2.9 SPSS
SPSS. 2.5.1. ( 2.2.)
SPSS
Analyze > Compare Means > One-Way ANOVA
Dependent List ( y) Factor (
group). OK :
ANOVA
y
80.308 3 26.769 5.736 .018
42.000 9 4.667
122.308 12
Between Groups
Within Groups
Total
Sum of
Squares df Mean Square F Sig.
( 46.)
2.9. SPSS 59
2.2: 2.5.1 SPSS.
F
.
. SPSS
Analyze > Compare Means > One-Way ANOVA
Options Homogeneity of variance test. Continue OK :
Test of Homogeneity of Variances
y
.493 3 9 .696
Levene
Statistic df1 df2 Sig.
60 2.
SPSS Levene
k = 4 .
H0 : 21 =
22 =
23 =
24 H1 : H0.
p .696
.
Levene
K ,
Yij , i = 1, . . . ,K, j = 1, . . . , ni, Yij N (i, 2i ). ( K k k
.) Levene
H0 : 21 = = 2K H1 : H0
1, . . . , K . (.. i =
+ i
.) 1, . . . , K
ij = Yij i, i = 1, . . . ,K, j = 1, . . . , ni, . Levene
FLev =
Ki=1 ni(Ui U)2/(K 1)K
i=1
nij=1(Uij Ui)2/(n K) ,
Uij = |ij |, i = 1, . . . ,K, j = 1, . . . , ni. FLev () FK1,nK H0 . ( Levene
Uij.)
.
Analyze > General Linear Model > Univariate
y Dependent Variable group Fixed Factor - Save Residuals > Unstandardized. Continue OK. SPSS (
)
2.9. SPSS 61
RES_1. Q-Q plot: (
.)
, .
! SPSS ,
Levene;
SPSS.
, p .018
H0 : 1 = 2 = 3 = 4 .. 5%. -
SPSS 95%
.
Analyze > Compare Means > One-Way ANOVA
Post Hoc. (
.) .
LSD, Bonferroni Scheffe .
(
.)
62 2.
.
5%. Continue OK. SPSS .
Multiple Comparisons
Dependent Variable: y
.000 1.972 1.000 -6.71 6.71
-3.000 1.650 .398 -8.62 2.62
-6.000* 1.650 .036 -11.62 -.38
.000 1.972 1.000 -6.71 6.71
-3.000 1.871 .497 -9.37 3.37
-6.000 1.871 .066 -12.37 .37
3.000 1.650 .398 -2.62 8.62
3.000 1.871 .497 -3.37 9.37
-3.000 1.528 .337 -8.20 2.20
6.000* 1.650 .036 .38 11.62
6.000 1.871 .066 -.37 12.37
3.000 1.528 .337 -2.20 8.20
.000 1.972 1.000 -4.46 4.46
-3.000 1.650 .102 -6.73 .73
-6.000* 1.650 .005 -9.73 -2.27
.000 1.972 1.000 -4.46 4.46
-3.000 1.871 .143 -7.23 1.23
-6.000* 1.871 .011 -10.23 -1.77
3.000 1.650 .102 -.73 6.73
3.000 1.871 .143 -1.23 7.23
-3.000 1.528 .081 -6.46 .46
6.000* 1.650 .005 2.27 9.73
6.000* 1.871 .011 1.77 10.23
3.000 1.528 .081 -.46 6.46
.000 1.972 1.000 -6.63 6.63
-3.000 1.650 .614 -8.55 2.55
-6.000* 1.650 .033 -11.55 -.45
.000 1.972 1.000 -6.63 6.63
-3.000 1.871 .860 -9.29 3.29
-6.000 1.871 .064 -12.29 .29
3.000 1.650 .614 -2.55 8.55
3.000 1.871 .860 -3.29 9.29
-3.000 1.528 .487 -8.14 2.14
6.000* 1.650 .033 .45 11.55
6.000 1.871 .064 -.29 12.29
3.000 1.528 .487 -2.14 8.14
(J) group
2
3
4
1
3
4
1
2
4
1
2
3
2
3
4
1
3
4
1
2
4
1
2
3
2
3
4
1
3
4
1
2
4
1
2
3
(I) group
1
2
3
4
1
2
3
4
1
2
3
4
Scheffe
LSD
Bonferroni
Mean
Difference
(I-J) Std. Error Sig. Lower Bound Upper Bound
95% Confidence Interval
The mean difference is significant at the .05 level.*.
2.5.
2.10
2.1. (2.2.2) .
2.2.
(2.2.7).
2.3. () X1, . . . ,Xn, n > 2, N (, 2) S2 . Var(S2) = 24/(n 1).() 2 S2ij 36.
k > 3
;
() E(S2ij |2) = 2. ;
2.10. 63
2.4. 51.
2.5. k , H0 : 1 = = k .. ,
. Bonferroni, -
100(1 )%() i j i < j() i j j = i 1.
= 0.05, k = 5, n = 25,
MS Excel.
3
Blocks
3.1 , blocking
(nuisance factor) -
.
() () .
(
) (randomi-
zation). -
( ) .
,
( ). -
( .. IQ
), (.. ) -
. ,
.
-
, ( )
.
. ..
IQ -
(Analysis of Covariance,
ANCOVA). .
-
blocking. blocks -
block.
65
66 3. Blocks
Blocks
1 2 3 4
1 X X X X
2 X X X X
3 X X X X
Blocks
1 2 3 4
1 X X X2 X X 3 X X X
3.1: blocks . -
X
. .
block,
( ) ,
. (design)
.
block, (complete).
(incomplete). , -
(
blocks)
( 3.1).
3.2 Blocks
-
Blocks (Randomized Complete Block Designs, RCBD).
(
) r c blocks. -
. ,
:
Blocks
1 2 . . . j . . . c
1 Y11 Y12 . . . Y1j . . . Y1c
2 Y21 Y22 . . . Y2j . . . Y2c...
......
......
i Yi1 Yi2 . . . Yij . . . Yic...
......
......
r Yr1 Yr2 . . . Yrj . . . Yrc
3.2. Blocks 67
N = r c. (row) r (
) (column) c blocks. (
r c row column.)
:
N = rc .
Yij, i = 1, . . . , r, j = 1, . . . , c, -
.
N .
ij Yij ,
ij = + i + j . (3.2.1)
, i
j block (
), i i j j block. ,
, :
c blocks, ()
(
i).
r -
block ( j).
,
1, . . . , r 1, . . . , c.
ri=1
i = 0
cj=1
j = 0
. ,
( ).
, SPSS r = 0 c = 0.
Yij = + i + j + ij, i = 1, . . . , r, j = 1, . . . , c,
r
i=1 i =c
j=1 j = 0 ij, i = 1, . . . , r, j = 1, . . . , c,
N (0, 2) . ij
68 3. Blocks
Yij .
, (errors).
, , 1, . . . , r
() ( ).
, , i -
.
i = 0 . ,
H0 : 1 = = r = 0 H1 : i 6= 0 i.
Yi Y.
Yi = 1cc
j=1
Yij , i = 1, . . . , r, Y = 1Nr
i=1
cj=1
Yij .
Yj = 1rr
i=1
Yij , j = 1, . . . , c.
.
3.2.1. yij , i = 1, . . . , r, j = 1, . . . , c,
() :
ri=1
cj=1
(yij y)2 =
c
ri=1
(yi y)2 + rc
j=1
(yj y)2 +r
i=1
cj=1
(yij yi yj + y)2.. yi, yj y
ri=1
cj=1
(yij y)2 =r
i=1
cj=1
(yij yi+ yi yj + yj y+ y y)2
=
ri=1
cj=1
{(yi y) + (yj y) + (yij yi yj + y)}2. (3.2.2)
(+ + )2 = 2 + 2 + 2 + 2 + 2 + 2
3.2. Blocks 69
, , (3.2.2),
ri=1
cj=1
{(yi y)2 + (yj y)2 + (yij yi yj + y)2+
2(yi y)(yj y) + 2(yi y)(yij yi yj + y)+
2(yj y)(yij yi yj + y)}
= c
ri=1
(yi y)2 + rc
j=1
(yj y)2 +r
i=1
cj=1
(yij yi yj + y)2 . ,
i j
.
cj=1
(yij yi) =c
j=1
yij cyi =c
j=1
yij c 1c
cj=1
yij = 0 i, (3.2.3)
ri=1
(yij yj) =r
i=1
yij r yj =r
i=1
yij r 1r
ri=1
yij = 0 j, (3.2.4)
ri=1
(yi y) =r
i=1
yi r y =r
i=1
1
c
cj=1
yij r 1rc
ri=1
cj=1
yij = 0 (3.2.5)
cj=1
(yj y) =c
j=1
yj c y =c
j=1
1
r
ri=1
yij c 1rc
ri=1
cj=1
yij = 0. (3.2.6)
(3.2.3) (3.2.6)
cj=1
(yij yi yj + y) =c
j=1
(yij yi)c
j=1
(yj y) = 0 (3.2.7) (3.2.4) (3.2.5)
ri=1
(yij yi yj + y) =r
i=1
(yij yj)r
i=1
(yi y) = 0. (3.2.8) :
ri=1
cj=1
2(yi y)(yj y) = 2r
i=1
(yi y)c
j=1
(yj y) = 0 (3.2.6),
ri=1
cj=1
2(yi y)(yij yi yj + y) = 2r
i=1
(yi y)c
j=1
(yij yi yj + y) = 0
70 3. Blocks
(3.2.7)
ri=1
cj=1
2(yj y)(yij yi yj + y) = 2c
j=1
(yj y)r
i=1
(yij yi yj + y) = 0 (3.2.8).
, Yij, i = 1, . . . , r, j = 1, . . . , c,
:
ri=1
cj=1
(Yij Y)2 =
cr
i=1
(Yi Y)2 + rc
j=1
(Yj Y)2 +r
i=1
cj=1
(Yij Yi Yj + Y)2.,
SSTotal =
ri=1
cj=1
(Yij Y)2
:
SSTreatment = c
ri=1
(Yi Y)2
SSBlock = rc
j=1
(Yj Y)2
SSError =
ri=1
cj=1
(Yij Yi Yj + Y)2.
, blocks , ,
Yij + i + j + ij .
Yi = 1cc
j=1
Yij =1
c
cj=1
(+ i + j + ij)
=1
c
(c+ ci + 0 +
cj=1
ij
)= + i + i,
Yj = 1rr
i=1
Yij =1
r
ri=1
(+ i + j + ij)
=1
r
(r+ 0 + rj +
ri=1
ij
)= + j + j,
3.2. Blocks 71
Y = 1rcr
i=1
cj=1
Yij =1
rc
ri=1
cj=1
(+ i + j + ij)
=1
rc
(rc+ 0 + 0 +
ri=1
cj=1
ij
)= +
Yi Y = (+ i + i) (+ ) = i + i,Yj Y = (+ j + j) (+ ) = j + j ,Yij Yi Yj + Y = (+ i + j + ij) (+ i + i) (+ j + j) + (+ )
= ij i j + .
c
ri=1
(Yi Y)2 = cr
i=1
(i + i)2,r
ci=1
(Yj Y)2 = rc
j=1
(j + j)2
ri=1
cj=1
(Yij Yi Yj + Y)2 =r
i=1
cj=1
(ij i j + )2. i
, j
blocks .
E(SSTreatment) = E
{c
ri=1
(i + i)2}
= c
ri=1
E{(i + i)2}
= cr
i=1
{Var(i + i) + [E(i + i)]2}
= (r 1)2 + cr
i=1
2i
Var(i + i) = Var(i) + Var() 2Cov(i, )= Var(i) + Var() 2Cov
(i, 1rc
rk=1
cj=1
kj
)
72 3. Blocks
= Var(i) + Var() 2Cov(i, 1r
rk=1
k)
= Var(i) + Var() 2r Cov(i, i)= Var(i) + Var() 2r Var(i)=
2
c+2
rc 2
2
rc
=1
c
(1 1
r
)2
E(i + i) = E(i) E() + i = 0 + 0 + i = i. , ( !)
E(SSBlock) = (c 1)2 + rc
j=1
2j
E(SSError) = (r 1)(c 1)2.
.
3.2.1. , -
:
()SSError2
2(r1)(c1).
() 1 = = r = 0, SSTreatment2
2r1.
() 1 = = c = 0, SSBlock2
2c1.() .
.
. .
3.2.1. H0 : 1 = = r = 0,
F =SSTreatment/(r 1)
SSError/[(r 1)(c 1)] Fr1,(r1)(c1).
. -
F .
3.2. Blocks 73
3.2.1.
H0 : 1 = = r = 0 H1 : i 6= 0 i
F > Fr1,(r1)(c1),.
. .
, i, j 2. -
.
.
3.2.2.
:
= Yi = Yi Y, i = 1, . . . , r,j = Yj Y, j = 1, . . . , c,
2 =1
rc
ri=1
cj=1
2ij
ij = Yij (+ i + j) = Yij Yi Yj + Y ij-.. ( , i, j
2).
logL(|y
) = rc2
log 2 rc2
log(2) 122
ri=1
cj=1
(yij i j)2, ,
Rr+c+1 (0,) ri=1 i = 0
cj=1 j = 0.
logL 2 -
. r =
r1k=1 k c = c1k=1 k
logL =
1
2
ri=1
cj=1
(yij i j)
ilogL =
1
2
{ cj=1
(yij i j)c
j=1
(yrj +
r1k=1
ak j)}
74 3. Blocks
=1
2
{ cj=1
(yij i j)c
j=1
(yrj r j)}, i = 1, . . . , r 1,
jlogL =
1
2
{ ri=1
(yij i j)r
i=1
(yic i +
c1k=1
k
)}
=1
2
{ ri=1
(yij i j)r
i=1
(yic i c)}, j = 1, . . . , c.
0 =
ri=1
cj=1
(yij i j) =r
i=1
cj=1
yij rc cr
i=1
i rc
j=1
j
i j
=1
rc
ri=1
cj=1
yij = y. i
0 =c
j=1
(yij i j)c
j=1
(yrj r j)
=
( cj=1
yij c ci c
j=1
j
)( c
j=1
yrj c cr c
j=1
j
)
= c(yi i yr+ r), i = 1, . . . , r 1,
r i = yr yi, i = 1, . . . , r 1. (3.2.9) i 1 r 1
(r 1)r r1i=1
i = (r 1)yrr1i=1
yi rr = ryr
ri=1
yi r = yr y (3.2.10)
y = ri=1 yi/r. (3.2.10) (3.2.9)
i = yi y, i = 1, . . . , r 1, . j = yj y, j = 1, . . . , c. logL
2 2 . ( !)
3.3. 75
3.2.1. 3.2.1 -
logL. 3.4.
3.2.2.
blocks.
H0,Block : 1 = = c = 0 H1,Block : j 6= 0 j.
H0,Block ..
FBlock =SSBlock/(c 1)
SSError/[(r 1)(c 1)] > Fc1,(r1)(c1),.
3.2.3.
SSTreatment = cr
i=1
(Yi Y)2 = cr
i=1
2i ,
SSBlock = r
cj=1
(Yj Y)2 = rc
j=1
2j
SSError =
ri=1
cj=1
(Yij Yi Yj + Y)2 =r
i=1
cj=1
2ij .
, H0 : 1 =
= r = 0
F = c(c 1)r
i=1 2ir
i=1
cj=1
2ij
,
i i
.
F : i
. i
H0
r
i=1 2i
. ,
2:
. F
,
. FBlock ;
76 3. Blocks
3.3
SSTotal =
ri=1
cj=1
(Yij Y)2 =r
i=1
cj=1
Y 2ij (r
i=1
cj=1 Yij
)2rc
rc
.
r
i=1
cj=1
(Yij Y)2 =r
i=1
c(Yi Y)2 +r
i=1
cj=1
(Yij Yi)2(
). ,
SSTreatment = c
ri=1
(Yi Y)2
=r
i=1
cj=1
(Yij Y)2 r
i=1
cj=1
(Yij Yi)2
=
{ ri=1
cj=1
Y 2ij (r
i=1
cj=1 Yij
)2rc
}{ r
i=1
cj=1
Y 2ij r
i=1
(cj=1 Yij
)2c
}
=1
c
ri=1
( cj=1
Yij
)2 1rc
( ri=1
cj=1
Yij
)2.
SSBlock =1
r
cj=1
( ri=1
Yij
)2 1rc
( ri=1
cj=1
Yij
)2.
,
SSError = SSTotal SSTreatment SSBlock.
,
r
c
.
3.3. 77
3.3.1. r = 3 c = 4 blocks.
1 2 3 4
1 5 8 4 4
2 7 10 6 5
3 7 11 7 5
-
:
1 2 3 4
1 5 8 4 4 21
2 7 10 6 5 28
3 7 11 7 5 30
19 29 17 14 79
,
ri=1
cj=1
= 52 + 82 + + 52 = 575.
, ,
SSTotal = 575 792
12 575 520.08 = 54.92
SSTreatment =1
4(212 + 282 + 302) 79
2
12 441 + 784 + 900
4 520.08
=2125
4 520.08 = 531.25 520.08 = 11.17
SSBlock =1
3(192 + 292 + 172 + 142) 79
2
12 361 + 841 + 289 + 196
3 520.08
=1687
3 520.08 562.33 520.08 = 42.25
SSError 54.92 11.17 42.25 = 1.5.
F =SSTreatment/(r 1)
SSError/[(r 1)(c 1)] 11.17/2
1.5/6 5.585
0.25 22.34.
F blocks
FBlock =SSBlock/(c 1)
SSError/[(r 1)(c 1)] 42.25/3
1.5/6 14.083
0.25 56.33.
:
78 3. Blocks
. F
11.17 2 5.585 22.34
Blocks 42.25 3 14.083 56.33
1.50 6 0.25
54.92 11
F F2,6,0.01 = 10.92. F = 22.34 > 10.92, .. 1% ( H0 : 1 = =r = 0 .. = 0.01). blocks
, F3,6,0.01 = 9.78 FBlock = 56.33 > 9.78.
3.4 SPSS
:
( ) 1 r
1, blocks 1 c
. ,
( block)
.
Analyze > General Linear Model > Univariate
Dependent Variable Fixed Factor(s) . OK, Model -
Custom ( Full Factorial). Factors & Covariates - Model.
( SPSS),
Build Terms (Type) Main Effects ( Interaction) Model. Continue OK.
.
1 -
( ) ValueLabels.
3.5. 79
3.5
3.1. H0 : 1 = = r = 0 .. .
() H0.
3.2. : Var(ij), Cov(ij, ij), Cov(ij , ij), i 6= i j 6= j. Cov(ij , ij).
3.3. , i j .
. Yij, i = 1, . . . , r, j = 1, . . . , c,
.
3.4.
, :
h = h(x1, . . . , xm) , m m ij- h2/(xixj). m m aij (y1, . . . , ym) 6= (0, . . . , 0)
mi=1
mj=1 aijyiyj < 0.
logL , 1, . . . , r1, 1, . . . , c1 ( r c -
)
.
3.2.2.
4
4.1
-
. ,
, . ,
:
;
;
;
, -
;
:
4.1.1.
.
4.1.1.
:
. ,
.
4.1.2. -
. -
, :
,
( , -
, , )
.
81
82 4.
,
, , ,
, .
, .
, -
. , (
1) . -
,
.
.
r > 2 c > 2 .
i j
nij .
Yij1, Yij2, . . . , Yijnij .
4.1 .
i j ij-.
ij- nij > 1 Y :
, .
, Y241 24- (-
).2
N =
ri=1
cj=1
nij
. ,
nij n i, j, (balanced data). ,
(balanced design). -
(unbalanced) .
( n = 1) -
(without replications). (
1 ,
.2 -
,
: Y3,10,4 Y3104.
4.1. 83
1 2 . . . j . . . c
1
Y111
Y112...
Y11n11
Y121
Y122...
Y12n12
. . .
Y1j1
Y1j2...
Y1jn1j
. . .
Y1c1
Y1c2...
Y1cn1c
2
Y211
Y212...
Y21n21
Y221
Y222...
Y22n22
. . .
Y2j1
Y2j2...
Y2jn2j
. . .
Y2c1
Y2c2...
Y2cn2c...
......
......
i
Yi11
Yi12...
Yi1ni1
Yi21
Yi22...
Yi2ni2
. . .
Yij1
Yij2...
Yijnij
. . .
Yic1
Yic2...
Yicnic...
......
......
r
Yr11
Yr12...
Yr1nr1
Yr21
Yr22...
Yr2nr2
. . .
Yrj1
Yrj2...
Yrjnrj
. . .
Yrc1
Yrc2...
Yrcnrc
4.1: :
r B c . ij-, i j nij -.
blocks
.) , nij > 0
i, j, (complete)
(incomplete). ,
.3 ,
.
:
3 ..
, . , ,
,
( ) .
84 4.
i, j, ij- .
rc .
ij-
.
rc .
Yijk N (ij, 2), i = 1, . . . , r, j = 1, . . . , c, k = 1, . . . , nij,
ij, i = 1, . . . , r, j = 1, . . . , c, 2 . ,
( )
ij (
).
.
.
4.2
,
ij = + i + j + ()ij , i = 1, . . . , r, j = 1, . . . , c. (4.2.1)
, ij- ,
, i i
, j j ,
()ij
i j . ()
.. . i
(main effects) , j
()ij (interaction).
, , ()
: 1 + r + c + rc
rc . ,
.
ri=1
i = 0,
cj=1
j = 0,
ri=1
()ij = 0, j,c
j=1
()ij = 0, i. (4.2.2)
.
(4.2.2) rc. ,
4.2. 85
i j . (
c j)
()ij c ( j)
r 1 ( r ). , (1 + r + c+ rc) (1 + 1 + c+ r 1) = rc.
(4.2.1) (4.2.2)
.
. (4.2.1) j
cj=1
ij =
cj=1
{+ i + j + ()ij} = c+ ci + 0 + 0 (4.2.3)
(4.2.2). i
ri=1
cj=1
ij =
ri=1
(c+ ci) = rc+ 0
=1
rc
ri=1
cj=1
ij = . (4.2.4) (4.2.3)
i =1
c
cj=1
ij = i .
j = j .,
()ij = ij i j
= ij (i ) (j )= ij i j + .
Yijk = + i + j + ()ij + ijk, i = 1, . . . , r, j = 1, . . . , c, k = 1, . . . , nij ,
r
i=1 i =c
j=1 j =r
i=1()ij =c
j=1()ij = 0 ijk, i = 1, . . . , r,
j = 1, . . . , c, k = 1, . . . , nij , N (0, 2) . , ijk .
86 4.
,
N :=r
i=1
cj=1
nij > rc+ 1
( ).
.
:
. i
1cj=1 nij
cj=1
nijk=1
Yijk = Yi. nij = n, i, j ( ),
Yi = 1cnc
j=1
nk=1
Yijk.
. j
1ri=1 nij
ri=1
nijk=1
Yijk = Yj. nij = n, i, j,
Yj = 1rnr
i=1
nk=1
Yijk.
.
ij
1
nij
nijk=1
Yijk = Yij. nij = n, i, j,
Yij = 1nn
k=1
Yijk.
Y = 1Nr
i=1
cj=1
nijk=1
Yijk.
4.2. 87
(
) ij
ij , ij = Yij, ij
2,
2 =1
N
ri=1
cj=1
nijk=1
(Yijk Yij)2. , i, j ()ij ij -
=1
rc
ri=1
cj=1
ij =1
rc
ri=1
cj=1
Yij,
i =1
c
cj=1
ij = 1c
cj=1
Yij 1rcr
s=1
cj=1
Ysj, i = 1, . . . , r,
j =1
r
ri=1
ij = 1r
ri=1
Yij 1rcr
i=1
ct=1
Yit, j = 1, . . . , c,
()ij = ij i j = Yij 1cc
t=1
Yit 1rr
s=1
Ysj+ 1rcr
s=1
ct=1
Yst,i = 1, . . . , r, j = 1, . . . , c.
:
= Y,i = Yi Y,j = Yj Y
()ij = Yij Yi Yj+ Y( !). ijk-
ijk = Yijk {+ i + j + ()ij} = Yijk Yij. Yijk,
i,j,k aijkYijk. ,
E
( ri=1
cj=1
nk=1
aijkYijk
)=
ri=1
cj=1
nk=1
aijkE(Yijk) =
ri=1
cj=1
nk=1
aijkij
88 4.
, Yijk,
Var
( ri=1
cj=1
nk=1
aijkYijk
)=
ri=1
cj=1
nk=1
a2ijkVar(Yijk) = 2
ri=1
cj=1
nk=1
a2ijk.
4.2. -
.
4.3
nij = n, i, j, N = rcn.
.
4.3.1. yijk, i = 1, . . . , r, j = 1, . . . , c,
k = 1, . . . , n, ()
ri=1
cj=1
nk=1
(yijk y)2 = cnr
i=1
(yi y)2 + rnc
j=1
(yj y)2+
nr
i=1
cj=1
(yij yi yj+ y)2 +r
i=1
cj=1
nk=1
(yijk yij)2. (4.3.5). yi, yj, yij y -
ri=1
cj=1
nk=1
(yijk y)2 =
=
ri=1
cj=1
nk=1
(yijk yi+ yi yj+ yj yij+ yij y+ y y)2
=r
i=1
cj=1
nk=1
{(yi y) + (yj y) + (yij yi yj+ y) + (yijk yij)}2
(4.3.6)
(+ + + )2 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2
, , , (4.3.6),
ri=1
cj=1
nk=1
{(yi y)2 + (yj y)2 + (yij yi yj+ y)2 + (yijk yij)2
4.3. 89
+ 2(yi y)(yj y) + 2(yi y)(yij yi yj+ y)
+ 2(yi y)(yijk yij) + 2(yj y)(yij yi yj+ y)
+ 2(yj y)(yijk yij) + 2(yij yi yj+ y)(yijk yij)}. i, j, k (4.3.5)
. ,
ri=1
cj=1
nk=1
(yi y)(yij yi yj+ y)
=
ri=1
(yi y)n
k=1
cj=1
(yij yi yj+ y)
=
ri=1
(yi y) n( c
j=1
1
n
nk=1
yijk c 1cn
cj=1
nk=1
yijk
c
j=1
1
rn
rs=1
nk=1
ysjk + c1
rcn
rs=1
cj=1
nk=1
ysjk
)
= 0
i . (
.)
, Yijk, i = 1, . . . , r, j = 1, . . . , c, k = 1, . . . , n,
:
ri=1
cj=1
nk=1
(Yijk Y)2 = cnr
i=1
(Yi Y)2 + rnc
j=1
(Yj Y)2+
n
ri=1
cj=1
(Yij Yi Yj+ Y)2 +r
i=1
cj=1
nk=1
(Yijk Yij)2.
SST =
ri=1
cj=1
nk=1
(Yijk Y)2
:
SSA = cn
ri=1
(Yi Y)2,
90 4.
SSB = rnc
j=1
(Yj Y)2,
SSAB = n
ri=1
cj=1
(Yij Yi Yj+ Y)2
SSE =
ri=1
cj=1
nk=1
(Yijk Yij)2.
, ,
.
. ,
ri=1
cj=1
nk=1
(Yijk Y)2 = cnr
i=1
2i + rnc
j=1
2j + nr
i=1
cj=1
()2
ij +r
i=1
cj=1
nk=1
2ijk.
4.2 .
.
:
, ,
. ,
:
,
H0,A : 1 = = r = 0 H1,A : i 6= 0 i.
,
H0,B : 1 = = c = 0 H1,B : j 6= 0 j.
,
H0,AB : ()ij = 0 i, j H1,AB : ()ij 6= 0 i, j.
4.3.
91
nij = n, i, j
2
(rc)2
ri=1
cj=1
1
nij
2
rcn
i i2
(rc)2
{(r 1)2
cj=1
1
nij+
rs=1s 6=i
cj=1
1
nsj
}(r 1)2
rcn
j j2
(rc)2
{(c 1)2
ri=1
1
nij+
ri=1
ct=1t6=j
1
nit
}(c 1)2
rcn
()ij ()ij2
(rc)2
{(r 1)2(c 1)2 1
nij+ (r 1)2
ct=1t6=j
1
nit+ (c 1)2
rs=1s 6=i
1
nsj+
rs=1s 6=i
ct=1t6=j
1
nst
}(r 1)(c 1)2
rcn
nij = n, i, j
SSA = cn
ri=1
2i (r 1)2 + cnr
i=1
2i SSB = rn
cj=1
2j (c 1)2 + rnc
j=1
2j
SSAB = n
ri=1
cj=1
()2
ij (r 1)(c 1)2 + nr
i=1
cj=1
()2ij SSE =
ri=1
cj=1
nk=1
2ijk (N rc)2
4.2: .
,
.
92 4.
( )
.
4.3.1. nij = n, i, j n > 2. , :
()SSE
2 2Nrc.
() 1 = = r = 0, SSA2
2r1.
() 1 = = c = 0, SSB2
2c1.
() ()ij = 0, i, j, SSAB2
2(r1)(c1).() .
4.3.1. ()
. : SSE
rc
2.
.
4.3.1. () H0,A : 1 = = r = 0,
FA =SSA/(r 1)SSE/(N rc) Fr1,Nrc .
() H0,B : 1 = = c = 0,
FB =SSB/(c 1)SSE/(N rc) Fc1,Nrc .
() H0,AB : ()ij = 0, i, j,
FAB =SSAB/[(r 1)(c 1)]
SSE/(N rc) F(r1)(c1),Nrc .
. -
F .
4.3.1. ()
H0,A : 1 = = r = 0 H1,A : i 6= 0 i
FA > Fr1,Nrc,.()
H0,B : 1 = = c = 0 H1,B : j 6= 0 j
4.4. SPSS 93
FB > Fc1,Nrc,.()
H0,AB : ()ij = 0 i, j H1,AB : i 6= 0 i, j
FAB > F(r1)(c1),Nrc,.
( ),
. H0,AB,
, .
H0,A H0,B
. , ,
H0,A .
, (
) .
. ,
.
H(A)0,j : i + ()ij = 0, i = 1, . . . , r, H(A)0,j : i + ()ij 6= 0 i
j = 1, . . . , c. , j
..
F(A)j =
nr
i=1(Yij Yj)2/(r 1)SSE/(N rc) > Fr1,Nrc,. (4.3.7)
F(A)j
(
r 1 ) - j
. SSE/(N rc) (
4.3).
4.4 SPSS
SPSS
: ,
94 4.
.
Analyze > General Linear Model > Univariate
Dependent Variable Fixed Factor(s) . OK .
- . Options Homogeneity Tests: SPSS . Continue - . Save, Unstandardized Residuals Continue. , SPSS ijk.
. , Plots.
. (
) Horizontal Axis ( ) SeparateLines. Add. A B. , Add. -
B A. ( , )
.
Continue. OK .
. -
Battery Design Experiment ( e-class). Design and Analysis of Experiments
(Montgomery, 2005). :
.
( )
. ,
. .
, (15oF),
(70oF) (125oF) oF Fahrenheit.
.
. .
. SPSS
4.4. SPSS 95
hours. k i
j , yijk. i 1 r = 3
: (
SPSS material). j 1 c =3 :
( SPSS temperature). ,
k 1 n = 4 ,
.
, SPSS -
:
Sig.FMean SquaredfType III Sum of Squares
Corrected Model
Intercept
material
temperature
material * temperature
Error
Total
Corrected Total 3577646,972
36478547,000
675,2132718230,750
,0193,5602403,44449613,778
,00028,96819559,361239118,722
,0027,9115341,861210683,722
,000593,739400900,0281400900,028
,00011,0007427,028859416,222a
SourceSource
Tests of Between-Subjects Effects
Dependent Variable:Battery Life in Hours
a. R Squared = .765 (Adjusted R Squared = .696)
. material -
, temperature
,
materialtemperature , Error Corrected Total
. : 2 = 3 1 ( ), 4 = (3 1)(3 1) 36 3 3 = 27 .
, p 0.019.
F(r1)(c1),Nrc F4,27 ( FAB H0,AB)
3.560 ( FAB):
PH0,AB(FAB > 3.560) = P(F4,27 > 3.560) = 0.019.
, -
0.019 ( p).
5%,
.
96 4.
. .
:
Temperature (F)
125oF70oF15oF
Esti
mate
d M
arg
inal
Mean
s
150
125
100
75
50
3
2
1
Material Type
Estimated Marginal Means of Battery Life in Hours
Temperature (F)
125oF70oF15oFE
sti
mate
d M
arg
inal M
ean
s
150
125
100
75
50
3
2
1
Material Type
Estimated Marginal Means of Battery Life in Hours
( SPSS -
.)
. , (
) -
.
.
. -
. ,
.
,
.
.
, -
:
. , -
( ),
.
, -
.
, H0,A H0,B
4.4. SPSS 97
p 0.0005.4
-
.
Sig.df2df1F
,529278,902
Levene's Test of Equality of Error Variancesa
Tests the null hypothesis that the error variance of the dependent variable is equal across groups.
a. Design: Intercept + material + temperature + material * temperature
Dependent Variable:Battery Life in Hours
p 0.529.
.
Analyze > Descriptive Statistics > Explore
Dependent List ( RES_1). Display Plots ( ) Plots. None (
boxplot), Stem-and-Leaf ( )
Normality plots with tests. Continue OK. SPSS
Sig.dfStatistic Sig.dfStatistic
Shapiro-WilkKolmogorov-Smirnova
Residual for hours ,61236,976,200*
36,106
Tests of Normality
a. Lilliefors Significance Correction
*. This is a lower bound of the true significance.
: Kolmogorov-
Smirnov ( -
Shapiro-Wilk.
( p 0.2 0.612 ).5 -
qq plot:
4 p
.000. .000 ! .
0.0005 .001. 0.0005.
5 . -
.
.
98 4.
. , :
.
Observed Value
50250-25-50-75
Exp
ecte
d N
orm
al
3
2
1
0
-1
-2
-3
Normal Q-Q Plot of Residual for hours
H(A)0,j
H(A)
0,j,
j = 1, 2, 3.
. , j = 1, :
15oF
;
SPSS , , -
H(A)
0,j. () ,
. SPSS,
OK ( , )
Paste. OK
. ( )
,
OK. ( Paste SPSS ,
.)
File > New > Syntax
:
UNIANOVA hours BY material temperature/METHOD=SSTYPE(3)
4.4. SPSS 99
/INTERCEPT=INCLUDE/CRITERIA=ALPHA(0.05)/LMATRIX "Material difference at 15oF"material 1 -1 0 material*temperature 1 0 0 -1 0 0 0 0 0;material 1 0 -1 material*temperature 1 0 0 0 0 0 -1 0 0;material 0 1 -1 material*temperature 0 0 0 1 0 0 -1 0 0
/LMATRIX "Material difference at 70oF"material 1 -1 0 material*temperature 0 1 0 0 -1 0 0 0 0;material 1 0 -1 material*temperature 0 1 0 0 0 0 0 -1 0;material 0 1 -1 material*temperature 0 0 0 0 1 0 0 -1 0
/LMATRIX "Material difference at 125oF"material 1 -1 0 material*temperature 0 0 1 0 0 -1 0 0 0;material 1 0 -1 material*temperature 0 0 1 0 0 0 0 0 -1;material 0 1 -1 material*temperature 0 0 0 0 0 1 0 0 -1
/DESIGN=material temperature material*temperature.
. ( ) SPSS
DESIGN
,
material temperature (material*temperature). -
hours BY material temperature, material temperature . LMATRIX
. SPSS
15oF (j = 1), 70oF (j = 2) 125oF (j = 3).
. H(A)
0,1
1 + ()11 = 2 + ()21 = 3 + ()31 = 0. (4.4.8)
3 + ()31 = 0 SPSS
i, j ()ij
SPSS
r = 0, c = 0, ()rj = 0 j, ()ic = 0
i. , (4.4.8)
{1 2}+ {()11 ()21} = 0, (4.4.9){1 3}+ {()11 ()31} = 0, (4.4.10){2 3}+ {()21 ()31} = 0. (4.4.11)
,
SPSS, (4.4.8)
100 4.
. SPSS material
(1, 2, 3)
material*temperature
(()11, ()12, ()13, ()21, ()22, ()23, ()31, ()32, ()33
).
(4.4.9)
1,1, 0, -, 1, 0, 0,1, 0, 0, 0,0, 0, , . (4.4.10)
1, 0,1 1, 0, 0, 0, 0, 0,1, 0, 0 . , (4.4.11) 0, 1,1 0, 0, 0, 1, 0, 0,1, 0, 0. LMATRIX:
. LMATRIX 70oF 125oF.
SPSS ,
Run Current.
:
Test Results
Dependent Variable: Battery Life in Hours
886.167 2 443.083 .656 .527
18230.750 27 675.213
Source
Contrast
Error
Sum of
Squares df Mean Square F Sig.
Test Results
Dependent Variable: Battery Life in Hours
16552.667 2 8276.333 12.257 .000
18230.750 27 675.213
Source
Contrast
Error
Sum of
Squares df Mean Square F Sig.
Test Results
Dependent Variable: Battery Life in Hours
2858.667 2 1429.333 2.117 .140
18230.750 27 675.213
Source
Contrast
Error
Sum of
Squares df Mean Square F Sig.
LMATRIX (
H(A)
0,1, H
(A)
0,2 H
(A)
0,3). 15oF 125oF
() -
. , 70oF .
,
Contrast Results (K Matrix).