Title: Incomplete Block Design ?????????????????????????????
1Incomplete Block Design??????????????????????????
???
- ?????????
- ?????????????? ???????
- ????????????????? ?????????????? ??.
2Incomplete Block Design
- ????????????????????????????????
- ????????????????????????????? ???? ??
???????????????????? ?????????????????????????????
- ?????????????????????????????
3Objective
- ??.???????????????????????????????????????????????
?????????????????????????????? - ??.??????????????? ????? ?????????????????
??????????????????????????????????????????????????
?
4Incomplete 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)
-
6Balanced 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))
7Balanced 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
8Balanced Incomplete Block Design BIB
- ??????????????? lay out
- ?????????? t ??? k
- ??????????????? ??????? b rt/k ???? tCk
- ????????????????????????????????????????????
??????? ? r(k-1)/(t-1) (?????????????????????????
) - ???????????????? (????????????????)
??????????????????????????????
9Balanced 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
?????????????
10Balanced 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
11Balanced 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
12Balanced Incomplete Block Design BIB
13Balanced 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
-
14Balanced Incomplete Block Design BIB
SOV Df
Treatment t-1
Block b-1
Error rt-t-b1 (Residual)
Total rt-1
15Balanced Incomplete Block Design BIB with
Replicated Groups
- Definition
- ??????????????????????????????????????????????????
???????????? - ????????????????????????????????????????? 1 ???
- ?????????????????????????????????? ??????????????
????????????????????
16Balanced 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
17Balanced Incomplete Block Design BIB with
Replicated Groups
18Balanced 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
- ?????????????????????????
- ??????????
20Incomplete 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
21Incomplete Latin Square Design
- Proc GLM data.
- class trt blk rep
- model Y trt blk rep
- LSMEANS trt
- RUN
22Incomplete Latin Square Design
SOV df
Row r-1
Column c-1
Treatment t-1
Error residuals
Total n-1
23- Confounding
- Experiments
- ??????????????????????????
24Confounding Experiments
- Definition
- ?????????????????
- ??????????????????? ???????????????????????????
- ???????????????????????????????????????????????
- ???????????????????????????? factorial (t.c.)
- ?????????????? block size lt trt
- ??????????????????????????????????????????????????
???????????????????????????????????????????
25Confounding 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)
26Confounding Experiments
- Objective
- ??. ??????????????????????? confounding
experiments ??? - ??.???????????????????????????????????????????????
?? layout - ??.??????????????? ????? ?????????????????????????
? confounding experiments ???
27Confounding Experiments
28Confounding Experiments
- Ex. ??????????????????? 23 factorial ??? ?? block
size 4 - ?????????????????????????????????????????????????
23 factorial - 2x2x2 8 t.c.
- a1b1c1
- a1b1c2
- a1b2c1
- a1b2c2
- a2b1c1
- a2b1c2
- a2b2c1
- a2b2c2
Treatment combination
29Confounding Experiments
- 2n Confounding experiments ???????????????????????
????? - ???????? n ?????????????????????? 2 ?????
- ????????????????????? treatment combination
????????????? - confounding experiments ?????????????? 2n level
system
30Confounding Experiments
Treatment A B AB C AC BC ABC
(1)
a
b
ab
c
ac
bc
abc
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
???????????????? ???????????????? code
???????????? ??? -
????????????????????????????????
31Confounding 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)
32Confounding Experiments
Confound ??????? 2 ???????????????????????????????
? confound ????????? ???????????????? generalized
interaction (GI) ??????? ????????
confound ??????? AC ??? BC generalized
interaction (GI) ???????????????????? ACBC
ABC2 AB (modulus 2) ??? ????????????????????????
??????????????????? 1 ???????? ABC2 AB1 (AB)
33Confounding 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
34Confounding 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
35Confounding 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.
36Confounding Experiments
???????????????????????????? confound
????????????????????????????????????????? complet
e confounding ????????????????????????????
confound ????????????????????????????????????????
partial confounding Note complete
confounding ???????????????????? partial
confounding partial confounding
???????????????????????????????????
???????????????????
37Confounding Experiments
38Confounding 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
39Confounding 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)
40Single replication Experiments?????????????????
?????????
41Single replication Experiments
- Definition
- ??????????????????????????????????????????????????
???????? - ????????????????? ??????????????????????????? 1
??? - ???? ??????????????? / ???????????????????????????
? - ??????????????????????????????????????????????????
? error - ???????????????????????????????????????????
42Single 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
43Single 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
44Single 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
45Single replication Experiments
46Single replication Experiments
????????? 1. error ???????????????????
(non-homogeneity) ???????????????
??????????????????? error ??? 2. ??? interaction
???????????????????????????????? error
??????????????? ???????
47Fractional replication Experiments?????????????
??????????????????
48Fractional replication Experiments
- Definition
- ???????????????? fractional factorial experiments
- ??????????????????????????????????????????????????
???????? - ????????????????????? ????????????????????????????
??? - ???? ??????????????? / ???????????????????????????
? - ??????????????????????????????????????????????????
? error - ???????????????????????? ½ , ¼ ???? 1/8 ??????
- ??????????????????????????????????????????????
49Fractional 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
50Fractional replication Experiments
- ??????????????????
- Defining contrast (I) ??? ??????????
confound ?????????????????????????? - Alias effects ???????????????????????????????
????????????????????
51Fractional 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
52Fractional 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
53Fractional replication Experiments
54- Lattice design
- ????????????????????
55Lattice Design
- ???????????????????????????????????????????????
- ????????????? t k2 (t treatment, k block
size) - ????????????? BIB ?????????????? layout
???????????
56??????????????????????? Lattice
- Balanced Lattice Design
- ?????????????????????????
- Balanced Lattice Square Design
- ????????????????????????????????
- 3. Rectangular Lattice Design
- ????????????????????????????????
571. 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)
?????????????
581. 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
591. 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 ?
60Layout
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
611. 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
621. Balanced Lattice Design
ANOVA
SOV Df
Replication r-1
Treatment k2-1
Block(rep) r(b-1)
Error residuals
Total n-1
Statistical model
632. Balanced Lattice Square Design
- ???????????????? balanced lattice design (tk2)
- ??????????????????????????????????????????????????
???????????????????????????? (?1) - ??????????????????????????????????????????????????
?????????????????????????????? 1 ??? - ???????????????????????????????
- ????????????????????????????
642. 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
652. 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
662. 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
673. Rectangular Lattice Design
- ?????????????????? BIB ?????????? trt k(k1)
- ??????????????????????????????????????????????????
???????????????????????????? (?1) - ???????????????????????????????
- ?????????????????????????????????? (k1)x(k1)
683. 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
693. 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
713. 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