Incomplete Block Design ?????????????????????????????

1
Incomplete Block Design??????????????????????????
???

• ?????????
• ?????????????? ???????
• ????????????????? ?????????????? ??.

2
Incomplete Block Design

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

3
Objective

• ??.???????????????????????????????????????????????
  ??????????????????????????????
• ??.??????????????? ????? ?????????????????
  ??????????????????????????????????????????????????
  ?

4
Incomplete Block Design

• ???????????????????????????????? (block size, k)
  ????????????
• k lt t ???? ?????????? 4 trt ?????????????????????
  ???? 2 ???/???

block1
t1
t3
1 replication
block2
t2
t4
5
????????????????????????????????????

• ????????????????? 2 ??? ???
• 1. ?????????????????????????????????????
• (Balanced Incomplete Block Design BIB)
• 2. ?????????????????????????????????????????????
• ????????????????????
• (Balanced Incomplete Block Design with
  Replicate Groups
• BIB with Replicate Groups)

6
Balanced Incomplete Block Design BIB

• Definition
• 1.????????????????? block size ???????????????
• 2.????????????????????????????? (1 ???, 2 ???
  ???? 3 ???)
• 3.??????????? trt ??????????????? (??? 1, 2, 3 ?
  rt bk)
• 4.??? trt ???????????????????????????????????????
  1 ?????
• 5.??????????????????????????????????????? (?)
  ??????????? (??? 4 ? ?(t-1)r(k-1))

7
Balanced Incomplete Block Design BIB
???????? layout
blk 1 1 2 4

blk 2 1 3 6

blk 3 2 3 5

blk 4 4 5 6

????????????? trt 6, k 3
8
Balanced Incomplete Block Design BIB

• ??????????????? lay out
• ?????????? t ??? k
• ??????????????? ??????? b rt/k ???? tCk
• ????????????????????????????????????????????
  ??????? ? r(k-1)/(t-1) (?????????????????????????
  )
• ???????????????? (????????????????)
  ??????????????????????????????

9
Balanced Incomplete Block Design BIB

• ???????????????? layout ??? BIB (? 1)
• ???????? t 4, k 2
• ??????????????? tCk 4C2 4!/(2!2!) 6
  blk
• ????? ? r(k-1)/(t-1) r(2-1)/(4-1)
• r(1)/(3)
• 3(1)/(3)
  1

?????????????
10
Balanced Incomplete Block Design BIB

• ???????????????? layout ??? BIB (? 2)
• ???????? t 6, k 3
• ??????????????? tCk 6C3 6!/(3!3!) 20
  blk
• ????? ? r(k-1)/(t-1) r(3-1)/(6-1)
• r(2)/(5)
• 5(2)/(5)
  2

11
Balanced Incomplete Block Design BIB
blk 1 4 1 2

blk 2 2 4 3

blk 3 1 4 3

blk 4 3 1 2

????????????? trt 4, k 3
??????? block 4 rep 3
? 2
12
Balanced Incomplete Block Design BIB

• Statistical model

13
Balanced Incomplete Block Design BIB

• SAS data set / case t 6, k 3/
• Data
• input blk trt observ
• Cards
• x x x
• x x x
• Proc glm data .
• class trt blk
• model observ trt blk
• lsmeans trt
• Run

14
Balanced Incomplete Block Design BIB
SOV Df
Treatment t-1
Block b-1
Error rt-t-b1 (Residual)
Total rt-1
15
Balanced Incomplete Block Design BIB with
Replicated Groups

• Definition
• ??????????????????????????????????????????????????
  ????????????
• ????????????????????????????????????????? 1 ???
• ?????????????????????????????????? ??????????????
  ????????????????????

16
Balanced Incomplete Block Design BIB with
Replicated Groups
? 1, t 6, k 2
Blk Trt Trt
1 1 2
2 1 3
3 1 4
4 1 5
5 1 6
6 2 3
7 2 4
8 2 5
9 2 6
10 3 4
11 3 5
12 3 6
13 4 5
14 4 6
15 5 6
Block with Replicated Groups
Rep Block Block Block
1 1 10 15
2 2 8 14
3 3 9 11
4 4 7 12
5 5 6 13
17
Balanced Incomplete Block Design BIB with
Replicated Groups

• Statistical model

18
Balanced Incomplete Block Design BIB with
Replicated Groups
SOV Df
Replication r-1
Treatment t-1
Block(rep) r(b-1)
Error rt-t-b1 (Residual)
Total rt-1
19

• Incomplete
• Latin Square Design
• ?????????????????????????
• ??????????

20
Incomplete Latin Square Design
Blk 1 1 2 3 5
Blk 2 2 3 4 1
Blk 3 3 4 5 2
Blk 4 4 5 1 3
Blk 5 5 1 2 4
Youden square Youden square Youden square Youden square Youden square

Blk 1 1 2 3 4 5
Blk 2 2 3 4 5 1
Blk 3 3 4 5 1 2
Blk 4 4 5 1 2 3
Blk 5 5 1 2 3 4
Regular latin square Regular latin square Regular latin square Regular latin square Regular latin square

• Youden Square Design
• (???????????????????????????)
• Row treatment column
• (k lt t)

????????????????
Cyclic latin square
orthogonal latin square
21
Incomplete Latin Square Design

• Proc GLM data.
• class trt blk rep
• model Y trt blk rep
• LSMEANS trt
• RUN

22
Incomplete Latin Square Design
SOV df
Row r-1
Column c-1
Treatment t-1
Error residuals
Total n-1
23

• Confounding
• Experiments
• ??????????????????????????

24
Confounding Experiments

• Definition
• ?????????????????
• ??????????????????? ???????????????????????????
• ???????????????????????????????????????????????
• ???????????????????????????? factorial (t.c.)
• ?????????????? block size lt trt
• ??????????????????????????????????????????????????
  ???????????????????????????????????????????

25
Confounding Experiments
SUM
Blk 1 a1b1 a2b1

Blk 2 a1b2 a2b2
b1..
b2..
SUM
a1..
a2..
Effect Blk ½ (blk2..- blk1..)
½ (a1b2a2b2-a1b1a2b1) A ½
(a2..- a1..) ½
(a2b1a2b2-a1b1a1b2) B ½ (b2..- b1..)
½
(a1b2a2b2-a1b1a2b1) AxB ½ (a1b1a2b2 -
a1b2a2b1)
26
Confounding Experiments

• Objective
• ??. ??????????????????????? confounding
  experiments ???
• ??.???????????????????????????????????????????????
  ?? layout
• ??.??????????????? ????? ?????????????????????????
  ? confounding experiments ???

27
Confounding Experiments

• ?????????

28
Confounding Experiments

• Ex. ??????????????????? 23 factorial ??? ?? block
  size 4
• ?????????????????????????????????????????????????
  23 factorial
• 2x2x2 8 t.c.
• a1b1c1
• a1b1c2
• a1b2c1
• a1b2c2
• a2b1c1
• a2b1c2
• a2b2c1
• a2b2c2

Treatment combination
29
Confounding Experiments

• 2n Confounding experiments ???????????????????????
  ?????
• ???????? n ?????????????????????? 2 ?????
• ????????????????????? treatment combination
  ?????????????
• confounding experiments ?????????????? 2n level
  system

30
Confounding Experiments
Treatment A B AB C AC BC ABC
(1)
a
b
ab
c
ac
bc
abc
-
-
-



-

-
-
-
-


-
-
-



-

-

-
-
-


-
-


-
-
-

-

-
-

-

-

-
-








???????????????? ???????????????? code
???????????? ??? -
????????????????????????????????
31
Confounding Experiments

• ?????????????????????????????? confound
  ???????????????????????? ?????????????????????????
  confound ?????????????? block size
• ???? ????? 8 ????????? ????? block size ????? 4
  ??????????????????????????????? 1 ?????
• 8 ? 4 ????????????????? confound 1 ???????
• 8 ? 4 ? 2 ????????????????? confound 2 ???????

Confound ??????? 2 ???????????????????????????????
? confound ????????? ???????????????? generalized
interaction (GI)
32
Confounding Experiments
Confound ??????? 2 ???????????????????????????????
? confound ????????? ???????????????? generalized
interaction (GI) ??????? ????????
confound ??????? AC ??? BC generalized
interaction (GI) ???????????????????? ACBC
ABC2 AB (modulus 2) ??? ????????????????????????
??????????????????? 1 ???????? ABC2 AB1 (AB)
33
Confounding Experiments

• ????????????????????????????? confound
• ????? confound ????????????????????????
  (interaction ??????????, ABC)
• ???????? confound ????????????????????????????????
  ????????

Treatment ABC effect
(1) -
a
b
ab -
c
ac -
bc -
abc
blk1 (-) blk2 ()
(1) a
ab b
ac c
bc abc
??????????? 8 ?????????????? block size 4
34
Confounding Experiments
Treatment AB AC BC blk
(1) 1
a - - 2
b - - 3
ab - - 4
c - - 4
ac - - 3
bc - - 2
abc 1
blk1 () blk2 (--) blk3 (-) blk4 (-)
(1) a b c
abc bc ac ab
??????????? 8 ?????????????? block size 2
35
Confounding Experiments

• ????????????? confounding ???????????? 1 ???
• ???????????????????????????????????????

blk1 (-) blk2 ()
(1) a
ab b
ac c
bc abc
blk1 (-) blk2 ()
(1) a
ab b
ac c
bc abc
1st rep.
1st rep.
blk3 (-) blk4 ()
(1) a
ab b
ac c
bc abc
blk3 (-) blk4 ()
(1) c
a ac
b bc
ab abc
2st rep.
2st rep.
36
Confounding Experiments
???????????????????????????? confound
????????????????????????????????????????? complet
e confounding ????????????????????????????
confound ????????????????????????????????????????
partial confounding Note complete
confounding ???????????????????? partial
confounding partial confounding
???????????????????????????????????
???????????????????
37
Confounding Experiments
38
Confounding Experiments
SOV df
rep r-1
block(rep) r(b-1)
a a-1
b b-1
ab (a-1)(b-1)
c c-1
ac (a-1)(c-1)
bc (b-1)(c-1)
abc (a-1)(b-1)(c-1)
Error Residuals
total n-1
39
Confounding Experiments
Proc GLM data . class rep blk model
y rep blk(rep) a b ab c ac bc abc /ss3
solution y rep blk(rep) abc
/ss3 Run ????????????????? 1. ????????????
Interaction ???? (???????????????????? factorial)
40
Single replication Experiments?????????????????
?????????
41
Single replication Experiments

• Definition
• ??????????????????????????????????????????????????
  ????????
• ????????????????? ??????????????????????????? 1
  ???
• ???? ??????????????? / ???????????????????????????
  ?
• ??????????????????????????????????????????????????
  ? error
• ???????????????????????????????????????????

42
Single replication Experiments

• Statistical model ??? ANOVA ??????????????
• ???????? factorial ???????????????????????????????
  ?????
• ??????????????????? error ????

SOV Df
Trt 10
A 1
B 1
C 1
D 1
AB 1
AC 1
AD 1
BC 1
BD 1
CD 1
Error 5
Total 15
????????????????????????? ABC, ABD, ACD, BCD, ABCD
43
Single replication Experiments
Rep 1
22
a a (1) (1) ab ab b b

abc a b ab ac c bc (1)
23
Rep 1
Lay Out
44
Single replication Experiments

• Data
• Input A B C D observe
• Cards
• 0 0 0 0 135
• 1 0 0 0 139
• 0 1 0 0 135
• 1 1 0 0 157
• 0 0 1 0 133
• 1 0 1 0 141
• 0 1 1 0 103
• 1 1 1 0 163
• 0 0 0 1 206
• 1 0 0 1 216
• 0 1 0 1 207
• 1 1 0 1 235
• 0 0 1 1 207
• 1 0 1 1 200
• 0 1 1 1 209
• 1 1 1 1 231

Proc GLM class A B C D model
observe A B C D AB AC BC
BD CD Run
45
Single replication Experiments
46
Single replication Experiments
????????? 1. error ???????????????????
(non-homogeneity) ???????????????
??????????????????? error ??? 2. ??? interaction
???????????????????????????????? error
??????????????? ???????
47
Fractional replication Experiments?????????????
??????????????????
48
Fractional replication Experiments

• Definition
• ???????????????? fractional factorial experiments
• ??????????????????????????????????????????????????
  ????????
• ????????????????????? ????????????????????????????
  ???
• ???? ??????????????? / ???????????????????????????
  ?
• ??????????????????????????????????????????????????
  ? error
• ???????????????????????? ½ , ¼ ???? 1/8 ??????
• ??????????????????????????????????????????????

49
Fractional replication Experiments

• Example.
• ??????????? ?? 16 trt ????? blocksize ????? 8
• 16 ? 8 ?????????????????? ½
  ???
• 16 ? 8 ? 4 ?????????????????? ¼ ???
• 16 ? 8 ? 4 ?2 ?????????????????? 1/8 ???

?????? defining contrast (I) 1 ???
?????? defining contrast (I) 2 ?????????????? 1 GI
?????? defining contrast (I) 3 ?????????????? 3
GI
50
Fractional replication Experiments

• ??????????????????
• Defining contrast (I) ??? ??????????
  confound ??????????????????????????
• Alias effects ???????????????????????????????
  ????????????????????

51
Fractional replication Experiments

• Example
• ????????????????? ABCD ???? defining contrast
  (I) ??????? alias effects ?????????????? ?????
  defining contrast (I) ????????????????????????????
  ????????? modulus 2
• ????
• I A ABCDA A2BCD BCD
• I B ABCDB AB2CD ACD
• I C ABCDC ABC2D ABD
• I AB ABCDAB A2B2CD CD

52
Fractional replication Experiments
SOV Df
A (BCD) 1
B (ACD) 1
C (ABD) 1
D (ABC) 1
Error residuals
Total n-1
????????? ????????????????
alias effect
??????????????? 241 ?? SAS book
?????????????????????????????????????????????
error
53
Fractional replication Experiments
54

• Lattice design
• ????????????????????

55
Lattice Design

• ???????????????????????????????????????????????
• ????????????? t k2 (t treatment, k block
  size)
• ????????????? BIB ?????????????? layout
  ???????????

56
??????????????????????? Lattice

• Balanced Lattice Design
• ?????????????????????????
• Balanced Lattice Square Design
• ????????????????????????????????
• 3. Rectangular Lattice Design
• ????????????????????????????????

57
1. Balanced Lattice Design

• ? ???????????????? Balanced Simple Lattice Design
• ? ?????????????????? BIB ?????????? t k2
• ? ??????????????????????????????????? (Replicated
  Groups)
• ?????????????????????????????????????????????? (?
  1)
• ? ???????????????????
• 1. ??????????????????????????????? ????????? k
  x k (????????? k 3)
• ????????????????????????????????????????
  1,2,,t ?????????
• ????? (???????? t 9 ????????? t k2)
  ?????????????

58
1. Balanced Lattice Design
Col1 Col2 Col3
Row1 A 1 a B 2 ß C 3 ?
Row2 B 4 ? C 5 a A 6 ß
Row3 C 7 ß A 8 ? B 9 a

• Row Column Trt Greek

59
1. Balanced Lattice Design

• 2. ????????????????????????????????????????
  ?????????, ???????, ???????????????

Rep Blk Trt Trt Trt Note
1 1 1 2 3 Row1
2 4 5 6 Row2
3 7 8 9 Row3
2 4 1 4 7 Col1
5 2 5 6 Col2
6 3 6 9 Col3
3 7 1 6 8 Letter A
8 2 4 9 Letter B
9 3 5 7 Letter C
4 10 1 5 9 Letter a
11 2 6 7 Letter ß
12 3 4 8 Letter ?
60
Layout
1 2 3

4 5 6

7 8 9

1 4 7

2 5 6

3 6 9

1 6 8

2 4 9

3 5 7

1 5 9

2 6 7

3 4 8
blk1
blk2
Rep1
blk3
blk4
Rep2
blk5
blk6
blk7
Rep3
blk8
blk9
blk10
Rep4
blk11
blk12
61
1. Balanced Lattice Design

• ????????????????????????? k1
• ??????????????? (letter) ??????????? k-1

TRT 9 16 25 49 64
Block Size 3 4 5 6 7
Rep 4 5 6 7 8
Letter 2 3 4 5 6
62
1. Balanced Lattice Design
ANOVA
SOV Df
Replication r-1
Treatment k2-1
Block(rep) r(b-1)
Error residuals
Total n-1
Statistical model
63
2. Balanced Lattice Square Design

• ???????????????? balanced lattice design (tk2)
• ??????????????????????????????????????????????????
  ???????????????????????????? (?1)
• ??????????????????????????????????????????????????
  ?????????????????????????????? 1 ???
• ???????????????????????????????
• ????????????????????????????

64
2. Balanced Lattice Square Design
Col1 Col2 Col3
Row1 A 1 a B 2 ß C 3 ?
Row2 B 4 ? C 5 a A 6 ß
Row3 C 7 ß A 8 ? B 9 a

• Row Column Trt Greek

65
2. Balanced Lattice Square Design

• 2. ????????????????????????????????????????
  ?????????, ???????, ???????????????

Rep Blk Trt Trt Trt Note
1 1 1 2 3 Row1
2 4 5 6 Row2
3 7 8 9 Row3
2 4 1 4 7 Col1
5 2 5 6 Col2
6 3 6 9 Col3
3 7 1 (Aa) 6 (Aß) 8 (A?) Letter A
8 9 (Ba) 2 (Bß) 4 (B?) Letter B
9 5 (Ca) 7 (Cß) 3 (C?) Letter C
4 10 1 (aA) 9 (aB) 5 (aC) Letter a
11 6 (ßA) 2 (ßB) 7 (ßC) Letter ß
12 8 (?A) 4 (?B) 3 (?C) Letter ?
Order by greek letter
Order by eng letter
66
2. Balanced Lattice Square Design
ANOVA
SOV Df
Replication r-1
Treatment k2-1
row(rep) Column(rep) r(r-1) r(c-1)
Error residuals
Total n-1
Statistical model
67
3. Rectangular Lattice Design

• ?????????????????? BIB ?????????? trt k(k1)
• ??????????????????????????????????????????????????
  ???????????????????????????? (?1)
• ???????????????????????????????
• ?????????????????????????????????? (k1)x(k1)

68
3. Rectangular Lattice Design
Col1 Col2 Col3 Col4
Row1 A B 1 C 2 D 3
Row2 D 4 C B 5 A 6
Row3 B 7 A 8 D C 9
Row4 C 10 D 11 A 12 B
A
C
D
B
69
3. Rectangular Lattice Design
Rep Blk Trt Trt Trt Note Group
1 1 2 3 Row1 X
2 4 5 6 Row2
3 7 8 9 Row3
4 10 11 12 Row4
5 4 7 10 Col1 Y
6 1 8 11 Col2
7 2 5 12 Col3
8 3 6 9 Col4
9 6 8 12 Letter A Z
10 1 5 7 Letter B
11 2 9 10 Letter C
12 3 4 11 Letter D
70
?????????

• ???????????????????????????????????????????
• ????????????????????????????????????????????
• ??????????????????????????????????????? 2 ???
  ???? 3 ????????
• ??????????????? 2 ????????????? simple
  rectangular lattice
• ??????????????? 3 ????????????? tripple
  rectangular lattice

71
3. Rectangular Lattice Design
ANOVA
SOV Df
Replication r-1
Treatment k2-1
block(rep) r(b-1)
Error residuals
Total n-1
Statistical model
72
????

• ??????????? BIB, Latiice, Confounding, Single
  replication ??? Fractional ???????????????????????
  ????
• ???????????????????????????????????????????
• ??????????????????????????????????????????????????
  ????????????
• ??????????????????????????????????????????????????
  ???????????????????????????

73

• ??????