COMPLETELY RANDOMIZED BLOCK DESIGN

• A randomized complete block design is an experimental design for comparing ttreatments in b blocks.

• The blocks consist of t homogeneous experimental units.

• Treatments are randomly assigned to experimental units within a block, with each treatment appearing exactly once in every block.

DEFINITION

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Motivation: to deal with non-homogenous groups (experimental material, area or time) subdivide experimental units into homogenous groupsBLOCKS : controlling the variability of experimental material, area or time that are not homogenous.

Design:number of experimental units in a block = number (or multiple) of studied treatmentsTreatments assigned at random to experimental units within block

Objective: by blocking isolate, remove from the error term variation attributable to blocks

Examples of blocks: breed of animal, age, laboratory, day, etc.

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ADVANTAGES OF THE RANDOMIZED COMPLETE BLOCK DESIGN

1. The design is useful for comparing t treatment means in the presence of a single extraneous source of variability.

2. This design to be more accurate than CRD because the elimination of SS block from SS error usually results in a decrease in the MS error.

3. The statistical analysis is simple.

4. The design is easy to construct.

5. It can be used to accommodate any number of treatments in any number of blocks.

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DISADVANTAGES OF THE RANDOMIZED COMPLETE BLOCK DESIGN

1. Because the experimental units within a block must be homogeneous, the design is best suited for a relatively small number of treatments.

2. This design controls for only one extraneous source of variability (due to blocks). Additional extraneous sources of variability tend to increase the error term, making it more difficult to detect treatment differences.

3. The effect of each treatment on the response must be approximately the same from block to block.

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RANDOMISASI RCBD

Blok I

Blok II

Blok III

Blok IV

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RANDOMIZED COMPLETE BLOCK DESIGN, LAY OUT

Location

1 2 3 4

P2 P1P3P4

P2P4P1P3

P1P3P4P2

P1P2P4P3

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BAK UKURAN 1 X 1 M2

20 ikan

30 ikan30 ikan40 ikan

40 ikan

20 ikan40 ikan30 ikan

30 ikan

40 ikan20 ikan

20 ikan

Blok IIBlok III

Blok I

Blok IV

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PEMBUATAN ROTI DENGAN 4 MACAM ADONAN A, B, C DAN D

Oven lamaModel A

Oven baruModel B

Oven baruModel A

Oven lamaModel B

Awas panasnya tidak sama!

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Blok I

Blok 2

Blok 3

POPULATION PARAMETERS 2T BY 2B –NO INTERACTION

B =1 B=2 A = 1

11µ 12µ A = 2

21µ 22µ

ijjiji eX += µ

22211211 µµµµ −=−Subject to:

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FIGURE : TREATMENT MEANS IN A RANDOMIZED BLOCK DESIGN

100

90

80

70

60

50

40

1 2 3 Treatment

ijµ

34

1

234

1

2

34

1

2

Plot of Treatment Mean by Treatment

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THE HYPOTHESES FOR TESTING TREATMENT MEAN DIFFERENCES

1 2: ...: At least one differs from the rest

o t

a i

HH

µ µ µµ

= = =

The null hypothesis : is no difference among treatment means

versus

The research hypothesis: treatment means differ.

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BlocksTreatment 1 2 3 … r Total Mean1 x11 x12 x13 … x1r x1. x1.2 x21 x22 x23 … x2r x2. x2.3 x31 x32 x33 … x3r x3. x3.. . . . . . . .. . . . . . . .. . . . . . . .t xt1 xt2 xt3 … xtr xt. xt.

Total x.1 x.2 x.3 x.r x..

Mean x.1 x.2 x.3 x.r x..

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µ CONSTANTβj BLOCK EFFECT τi TREATMENT EFFECTeij ERROR TERM: sources of variation other than treatment and

block

Constant

Treatment effect

Error term

Block effect

ijjiji eX +++= βτµ

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Two-way ANOVA: observation analyzed under two criteria, the block and the treatment group to which it belongs

Assumptions:1. Xij is a random independent sample from one of txr populations2. Each of txr populations ~ N(µij,σ2) => eij are independent, N(0, σ2)3. Block and treatment effects are additive, i.e. there is no interaction

between treatments and block

Block-treatment effect = block effect + treatment effect

Its violation => misleading results. Concern when: largest mean>50% smallest

When assumptions hold-> τj and βi are fixed constants -> fixed-effects

16

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Hypotheses: Ho: τj=0, j=1,2,…,k vs. Ha: not all τj=0

No block effect because: (1) primary interest is on treatment effect and (2) blocks are obtained non-randomly

SST=SSBl+SSTr+SSE

∑∑=

−=t

i

r

jji xxSStotal

1

2..)( ∑=

−=t

iiitreat xxrSS

1

2..)(.

2

1..).( xxtSS

r

jjblocks −= ∑

=

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df= (tr-1)

df= (r-1)

df= (t-1)

df= (t-1)(r-1)

Xij - µ= i + βj + eij, I , βj & eij independenΣ Σ (Xij - µ)2 = Σ Σ ( i + βj + eij)2

Σ Σ (Xij - µ)2 = Σ Σ i2 + Σ Σ βj

2 + Σ Σ eij2

JK total JK perlakuan JK Blok JK sesatan

Σ Σ (Xij - µ)2= Σ Σ Xij2 + trµ2 – 2trµ2

= Σ Σ Xij2 - trµ2

= Σ Σ Xij2 – tr(Σ Σ Xij)2/(tr)2

= Σ Σ Xij2 – (Σ Σ Xij)2/tr

∑∑∑∑

= =

= =

−=t

i

r

j

t

i

r

j

tr

XijXijJKtotal

1 1

2

1 1218

ijjiji eX +++= βτµ

τ τττ

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JK PERLAKUANJK plk =r Σ i

2

∑ −= 2..)( XXr i

22

..

.)(

2

+−= ∑∑∑∑∑

rtX

trrtX

rX jijii

2

1 11

2

.

.

rt

X

r

XJKplk

t

i

r

jij

t

ii

−=∑∑∑= ==

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τ

JK BLOKJK blok =t Σ βj

2

∑ −= 2. ..)( XXt j

22

..

.)(

2

+−= ∑∑∑∑∑

rtX

trrtX

tX jijij

2

1 11

2.

.rt

X

t

XJKblok

t

i

r

jij

r

jj

−=∑∑∑= ==

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21

∑∑∑∑= =

∧

= =

−−−==t

i

r

jjiji

t

i

r

jijSesa xeJK

1 1

2

1 1

2tan )ˆˆ( βτµ

tr

XCF

t

i

r

jij

2

1 1)(∑∑

= ==

JKbloknJKperlakuaJKtotalJKError −−=

2

1 11

2.

1

2.

1 1 .tan

rt

X

t

X

r

XXJKsesa

t

i

r

jij

r

jj

t

iit

i

r

jij

+−−=∑∑∑∑

∑∑ = ===

= =

TOTAL SUM OF SQUARES

2

2

.

..

rt

X

r

XSStreat

t

i

r

j

t

ii ji

−=∑∑∑

∑∑∑∑

= =

= =

−=t

i

r

j

t

i

r

j

tr

XijijXSStotal

1 1

2

1 12

2

2

.

..

rt

X

t

XSSblocks

t

i

r

j

t

ij ji

−=∑∑∑

Partition of TSS:

∑∑=

−=t

i

r

jji xxSStotal

1

2..)(

∑∑==

=−=t

iii

t

iiitreat rxxrSS

1

2

1

2..)(. τ

∑∑==

=−=r

jj

r

jjblocks txxtSS

1

22

1..).( β

∑∑∑∑= =

∧

= =

−−−==t

i

r

jjiji

t

i

r

jijError xeSS

1 1

2

1 1

2 )ˆˆ( βτµ

SSblockSStreatSStotalSSerror −−= .tr

XCF

t

i

r

jij

2

1 1)(∑∑

= ==

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ANALYSIS OF VARIANCE TABLE FOR A RANDOMIZED COMPLETE BLOCK DESIGN

Statistical Decision: when Ho is true,

MSTr/MSE ~ Fα;(t-1);,(t-1)(r-1)

if F stat ≥ F reject

Source df SS MS F

Blocks r-1 SSB MSB=SSB/(r-1) MSB/MSE

Treatment t-1 SST MST=SST/(t-1) MST/MSE

Error (t-1)(r-1) SSE MSE=SSE/(t-1)(r-1)

Total tr-1 SSTot

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EXPECTED MEAN SQUARES

When is true, both MST and MSE are unbiased estimates of , the variance of the experimental error.

1 2: ...o tH α α α= = =2εσ

( ) ( )2 2MST MSEE Eε εσ σ= =

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BERBAGAI NILAI DUGA DENGAN MODEL

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∑∑∑∑= =

∧

= =

−−−=t

i

r

jjiji

t

i

r

jij Xe

1 1

2

1 1

2 )ˆˆ( βτµ

..ˆ XtrXij

=ΣΣ=µ

.... XX

rXi

ii −=−Σ=∧∧

µτ

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....

1XX

tjX

j

r

jj −=−=

∧

=

∧

∑ µβ

.... XXXXe

Xe

jiijij

jiijij

+−−=

−−−=

∧

∧∧∧∧

βτµ

EXAMPLE: PROTEIN CONTENT (%) OF THE SOYBEAN TREATED BY FOLIAGE FERTILIZER A, B, C AND D IN THE FIELD WITH DIFFERENT FERTILITY

A=18 C=10 B=8 D=12

B=8 D=8 A=13 C=7

C=8 A=11 B=5 D=12

D=8 B=7 C=11 A=14

What is the experimental design ?

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SS => Linear model

i j Xij µ ζi βJ eij1 1 18 10 4 2 21 2 13 10 4 -1 01 3 11 10 4 -1 -21 4 14 10 4 0 02 1 8 10 -3 2 -12 2 8 10 -3 -1 22 3 5 10 -3 -1 -12 4 7 10 -3 0 03 1 10 10 -1 2 -13 2 7 10 -1 -1 -13 3 8 10 -1 -1 03 4 11 10 -1 0 24 1 12 10 0 2 04 2 8 10 0 -1 -14 3 12 10 0 -1 34 4 8 10 0 0 -2Σ ( ….) 160 160 0 0 0Σ( …..)2 1762 1600 104 24 34

TSS CF SST SSB SSE10/22/2012 28

ALTERNATE METHOD

160044

1602

==x

CF

1044

40362856 2222

=−+++

= CFSST

244

40363648 2222

=−+++

= CFSSB

16216008....1318 222 =−+++=TSS

3424104162 =−−=SSE

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Analysis of Variance ProcedureClass Level Information

Class Levels Values

TREAT 4 1 2 3 4

BLOCK 4 1 2 3 4

Number of observations in data set = 16

Analysis of Variance Procedure

Dependent Variable: PROTEIN

Source DF Sum of Squares Mean Square F Value Pr > FModel 6 128.00000000 21.33333333 5.65 0.0109Error 9 34.00000000 3.77777778Corrected Total 15 162.00000000

R-Square C.V. Root MSE PROTEIN Mean0.790123 19.43651 1.94365063 10.00000000

Source DF Anova SS Mean Square F Value Pr > F

BLOCK 3 24.00000000 8.00000000 2.12 0.1681TREAT 3 104.00000000 34.66666667 9.18 0.0042

SAS Program for RCBD

data agus; Iinput treat block protein; cards;1 1 181 2 13……4 3 124 4 8;proc print uniform;proc anova;class treat block;model protein=block treat;run; quit;

Tell me the conclusion!30

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Model linear:

Xijk= Data perlakuan ke-i, blokn ke-j, sampel ke-kµ= rerata umumµ = pengaruh perlakuan ke-iβj=pengaruh blok ke-jεij = pengaruh blok ke-j pada perlakuan ke-iδijk = pengaruh sesatan ke-k pada perlakuan ke-i dan

blok ke-ji = perlakuan ke-ij = blok ke-jk = sampel ke-ki = 1, 2, 3, ……., tj = 1, 2, 3, ……, rk = 1, 2, 3, ……. , s

ijkijjijkiX δεβτµ ++++=

iτ

RCBD DG BEBERAPA PENGAMATAN TIAPEXPERIMENTAL UNIT

Perlakuan 1 Perlakuan 2

Blok 1 Blok 2 Blok 3 Blok 1 Blok 2 Blok 3

Sample 1 X111 X121 X131 X211 X221 X231Sample 2 X112 X122 X132 X212 X222 X232Sample 3 X113 X123 X133 X213 X223 X233Sample 4 X114 X124 X134 X214 X224 X234

X11. X12. X13. X21. X22. X23.

X1.. X2.. X…

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BERBAGAI NILAI DUGA DENGAN MODEL

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∑∑∑∑∑∑= =

∧∧

== = =

−−−−=t

i

r

jijjijki

s

k

t

i

r

jijk

s

kX

1 1

2

11 1

2

1)ˆˆ( εβτµδ

...ˆ XtrsXijk

=ΣΣΣ=µ

....... XX

rsXi

ii −=−Σ=∧∧

µτ

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....... XX

tsjX

jj −=−Σ=∧∧

µβ

.ijijkijk

ijjiijkijk

XX

X

−=

−−−−=

∧

∧∧∧∧∧

δ

εβτµδ

........

.

XXXX

X

jiijij

jiijij

+−−=

−−−=

∧

∧∧∧∧

ε

βτµε

TOTAL SUM OF SQUARES

2

2

..

...

srt

X

rs

XSStreat

t

i

r

j

s

k

t

ii ji

−=∑∑∑∑

∑∑∑∑∑

∑= =

= = =

=

−=t

i

r

j

t

i

r

j

s

kijk

s

k trs

XijkXSStotal

1 1

2

1 1 12

1

2

2.

...

..

srt

X

st

XSSblocks

t

i

r

j

s

k

t

ij jki

−=∑∑∑∑

∑∑∑= = =

−=t

i

r

jjki

s

kXXSStotal

1 1

2

1...)(

∑∑==

=−=t

iii

t

iijitreat sjrXXsrSS

1

2

1

2...)..(. τ

∑∑==

=−=r

jj

r

jjblocks tsXXtsSS

1

22

1. ...).( β

∑∑∑∑∑∑= =

∧∧

== = =

−−−−==t

i

r

jijjiji

s

k

t

i

r

jijk

s

kError kXSS

1 1

2

11 1

2

1)ˆˆ( εβτµδ

.. atSSblokxtreSSblockSStreatSStotalSSerror −−−=trs

XCF

t

i

r

jijk

s

k

2

1 1 1)(∑∑∑

= = ==

35

∑∑∑∑====

=−=r

jij

t

i

r

jji

t

ikstreatxbloc sXXsSS

1

2

1

2

1.

1...)( ε

2

2..

22

..

..

srt

X

ts

X

rs

X

s

XokSStreatxbl

t

i

r

j

s

k

r

jj

t

ii

r

jij

t

iji

+−−=∑∑∑∑∑∑∑

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ANALYSIS OF VARIANCE TABLE FOR A RCBD SUBSAMPLE

Statistical Decision: when Ho is true,

MSTr/MSE ~ Fα;(t-1);,(s-1)tr

if VR≥F reject

Source df SS MS F

Blocks r-1 SSB MSB=SSB/(r-1) MSB/MSE

Treatment t-1 SST MST=SST/(t-1) MST/MSE

BlocksXTreat (r-1)(t-1) SSTXB MSBxT=SSBxT/(r-1)(t-1) MSBxT/MSE

Error (s-1)tr SSE MSE=SSE/(s-1)tr

Total trs-1 SSTot

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SOAL LATIHAN

BLOK SAMPLETreatment

1 2 3 4 5

1 5 6 9 10 12Blok I 2 4 7 9 8 10

3 2 6 8 10 111 2 4 9 9 9

Blok II 2 3 6 8 8 113 2 6 10 10 111 3 5 9 10 9

Blok III 2 3 6 8 9 103 4 6 9 9 11

Buatlah analisis varian dengan α = 5%Apabila terdapat perbedaan pengaruh yang nyata, lakukan uji lanjut dg DMRT α = 5%

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BLOK SAMPLETREATMENT

1 2 3 4 5 Total

1 5 6 9 10 12 42

Blok I 2 4 7 9 8 10 38

3 2 6 8 10 11 37

1 2 4 9 9 9 33

Blok II 2 3 6 8 8 11 36

3 2 6 10 10 11 39

1 3 5 9 10 9 36

Blok III 2 3 6 8 9 10 36

3 4 6 9 9 11 39

Total 28 52 79 83 94

Buatlah analisis varian dengan α = 5%Apabila terdapat perbedaan pengaruh yang nyata, lakukan uji lanjut dg DMRT α = 5%

SOAL LATIHAN

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