Chap 3 ???????????

1
Chap 3 ???????????

• Single-Factor Design
• 1??????? Completely Randomized Design (Chap3)
• 2????? (Chap4)
• Complete Randomized Block Design
• Latin Squares Design
• Balanced Imcomplete Block Design
• 3?????? Randomized Design with Covariate (15-3)
• 4??????? Randomized Design with Random Factor
• (Chap 14)

2
Completely Randomized Design, CRD

• Single-factor experiment with a treatments, A1
  , , Aj , , Aa
• The number of the observations under Aj is
  nj.
• The experiment is performed in a random
  order.
• (???????????)

3
3-1 EXAMPLE (p61) ?? RF power setting ? etch
rate ??? Factor RF power Response
etch rate ? 4 levels (treatments) 160w,
180w, 200w, 220w ??level ?? 5 ?,? ?5
replicates these 20 runs are in a random
order (see p62)
Data and average
4
???
???? ?????????????????????
(1) RF power setting affects the etch rate, (2)
higher power settings result in increased etch
rate. (3) The variability in etch rate does not
depends on the power setting.
5
(a )
(b )
(c )
???????,????????????????????????????????????,?????
????????????
6
3-2 Aanlysis of Variance (ANOVA) ?????
Data
?????????
Model
fixed effects model for balanced complete
randomized design Yij µ ti eij ,
i 1,, a, j 1,, n Yij
the (ij)th observation µ the overall
mean ti the ith treatment
effect eij a random error
7
?? ???? a ? treatments , ?? treatment ? n ?
replicates ?????,?? fixed effect
design ???????,?? balanced design
?????? linear statistical models
? 1?model Yij µ ti eij , ? model Yij
µi eij ?? 2?balanced design ? unbalanced
design ???????? 3?? a ? treatments ??????,??ti
?? random variable, ?? random effect model, ?
chap 12 ??????
8
Subject
???????????
Yij µ ti eij
??? ?????????????
Yij µi eij
9
? ?
?? Total SS
The fundamental ANOVA identity
SST SSTr SSE
Thus, where SST total corrected sum of squares
( SSY) SSTr sum of squares due to
treatments (?????) SSE sum of squares due
to error (?????)
10
? 1? ?Yij????,?? ??Y??? 2?????????????????
??(?????)??? ? ??(????)???
? a ???? pooled sample varianced, ?????????
???
Mean Square
11
???
???
12
??
2. SSTr ? SSE ????? SST ???
?Thm 3-1 Cochrans theorem Q chi-square
( v ), and Q Q1 Qs , Qs has vi degrees
of freedom. Q1, , Qs are independent
chi-square random variables with v1, , vs d.
f. if and only if v v1 v2 vs
13
???
???
When H0 is not true, E(MSTr) gt E(MSE)F0
????? ?????? F0 ??,???? H0,? F0
??F??,???????a?,Fa(a-1,N-a) ?????
?? When F0 gt Fa (a-1 , N-a ) , reject H0
?????????,?,????????????
( p-? P(Fgt F0),? p-? lt a?,?? H0 )
14
ANOVA Table
The ANOVA Table for 1-way Fixed-effects Model
Source SS df MS F p-value
Treatment SSTr a-1 MSTr F0 P(FgtF0)
Error SSE N-a MSE
Total SST N-1
?
?
??
N na, SST SSY
15
3-1 EXAMPLE (p61) ?? RF power setting ? etch
rate ??? Factor RF power Response
etch rate ? 4 levels (treatments) 160w,
180w, 200w, 220w ??level ?? 5 ?,? ?5
replicates these 20 runs are in random
order (see p62)
Data and average
16
???? RF power ? etch rate ??????
Syij2 , Syij , Syi.2
, SST
SSTr SSE
??SAS Output etch_rate
17
( p-? 0.0001 )
?? ? 0.01????, RF power ? etch rate??????? (
????? etch rate ?????? )
18
???
??
?????µi , µi µi ?? S ci µi ????????
(one-at-a-time C. I.)
????
? µi , µi µi ???,? ti ????
19
?????
?????????,F-???? ??????(coding) ,
?????? ?3-1????? 600,?????? F0-?
Why?
20
Unbalanced Data
?????????
Model Yij µ ti eij , i 1,, a, j
1,, ni where Sti0
??
? unbalanced data ? balanced data ????????
?? balanced data ??? 1????????,???????????????
2????????,????????
21
3-4 Model Adequacy Checking

• ???? eij NID( 0, s2),
• ??????
• Normality ???
• Independence ???
• Homogeneity ??? ( Constant variance )

?? residual ???????????? ANOVA
( t-???)
22
Checking Guide
1. ???? normal probability plot (Fig
3-4)????????????? 2. ?????????,????? (Fig
3-5) 3. ???????????????? (Fig 3-6) 4. ?
Bartletts ? Levenes ?????????
5. ??????????,?????????
6. ???????????
23
Figure 3.6 (p. 79)???,??? Y ???
Figure 3.5 (p. 78)???,???????
24
???????
Bartletts test (???????)
?????????????????????, Bartletts test
?????????
Modified Levene test ??????? (absolute
deviation), dij ,??????,??ANOVA? dij,? F-test ?
p-????????????? ?????????????????
25
??????????????
??
????? YY1-a ,???? constant var.
26
?? 1. ? yij ???????? ? Si ??????,?? ?
??,???a?? (Fig.3.7, fig.3.8)

1. Box-Cox ? ????????,?? SSE(?),????????SSE(?)??

27
Figure 3.8 (p. 84)Plot of log Si versus log
y-barfor the peak discharge data from Example
3.5.
Figure 3.7 (p. 83)???,??? Y ???
28
EXAMPLE 3.5 (p 81) ????????????? Factor ??
Response ???? 5 treatments, 6
replicates
Analysis
method N Mean
Std Dev 1 6
0.7100000 0.66109001 2
6 2.6266667 1.19220244
3 6 7.9300000
1.64707013 4 6
14.7150000 2.80089093
?????????????,?log (S) ? log(mean) ??,??vy
???????????? (fig3-8 ???0.5)
29
Y ? vY ????
???
30
vY ? ANOVA ?,???????????
????,???????????????,?????????,??????????????
31
??????????

• ???? ??????????????
• ?????????????????

???????
?????