Analysis Diakimansis
109
Κεφάλαιο 1 Σύνδεση με τα προηγούμενα... 1.1 Ο έλεγχος t για δύο κανονικούς πληθυσμούς ΄Εστω X ˜ =(X 1 ,...,X n ), Y ˜ =(Y 1 ,...,Y m ) δύο ανεξάρτητα τυχαία δείγματα από κανονικές κατανομές· X i ∼N (μ 1 ,σ 2 ), i =1,...,n, και Y i ∼N (μ 2 ,σ 2 ), i =1,...,m, με τις μέσες τιμές μ 1 ,μ 2 και την (κοινή) διασπορά σ 2 να είναι άγνωστες παράμετροι. Ας υποθέσουμε ότι έχουμε το πρόβλημα ελέγχου τής H 0 : μ 1 = μ 2 κατά τής H 1 : μ 1 = μ 2 . ΄Οποιος έχει παρακολουθήσει το μάθημα τών Ελέγχων Υποθέσεων γνωρίζει ότι ο έλεγχος λόγου πιθανοφανειών απορρίπτει την H 0 σε επίπεδο σημαντικότητας (ε.σ.) α αν |t| := | ¯ X − ¯ Y | S p 1 n + 1 m >t n+m−2,α/2 , (1.1.1) όπου • ¯ X = ∑ n i=1 X i /n, ¯ Y = ∑ m i=1 Y i /m είναι οι δύο δειγματικοί μέσοι, • S 2 p = (n − 1)S 2 X +(m − 1)S 2 Y n + m − 2 = ∑ n i=1 (X i − ¯ X) 2 + ∑ m i=1 (Y i − ¯ Y ) 2 n + m − 2 είναι ο συνδυα- σμένος (pooled) αμερόληπτος εκτιμητής τής (κοινής) διασποράς και • t n+m−2,α/2 είναι το (α/2)−ποσοστιαίο σημείο τής κατανομής t με n + m − 2 ϐαθμούς ελευθερίας. (Για την κατασκευή του ελέγχου δείτε ΄Ασκηση 1.1.) Πριν ϑυμηθούμε γιατί αυτός ο έλεγχος έχει μέγεθος α, ας παρατηρήσουμε ότι ο (1.1.1) είναι ένας πολύ λογικός κανόνας απόρριψης τής μηδενικής υπόθεσης μ 1 = μ 2 υπέρ τής 1
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Transcript of Analysis Diakimansis
-
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).
