(Randomized Complete Block Design) Randomized Block Design Rancangan Acak Kelompok (RAK)

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RCBD

(Randomized Complete Block Design)

Randomized Block DesignRancangan Acak Kelompok

(RAK)

EPI809/Spring 2008 2

Types of Experimental Designs

ExperimentalDesigns

One-Way Anova

Completely Randomized

Randomized Block

Two-Way Anova

Factorial

Kondisi Percobaan Yang sesungguhnya:-Ada nuisance factor (pengganggu), homogenitas materi terganggu :

-(Data HETEROGEN)

Misalnya: Pengaruh ransum terhadap ADG (kg)

Umur juga berpengaruh terhadap ADG

sehingga : umur mrpk faktor pengganggu

Pilihan:1. Umur juga diteliti : RAL Pola Faktorial,

umur sebagai faktor perlakuan juga

2. menggunakan umur untuk pengelompokan (sebagai BLOK):

Mengeluarkan variasi yang bersumber pada umur dari variasi error

percob.

Asumsi TIDAK ADA interaksi antar perlakuan

Catatan: jika ragu-ragu dengan Asumsi . Sebaiknya

faktor pengganggu dijadikan perlakuan , gunakan

RAL Faktorial.

EPI809/Spring 2008

4

Graphs of Interaction

Effects of Gender (Jantan-Betina) & dietary group (Rendah,Sedang,Tinggi) energi terhadap pertumbuhan

Interaction No Interaction

AverageResponse

RDH SDG TINGGI

male

female

AverageResponse

RDH SDG TINGGI

male

female

Occurs When Effects of One Factor Vary According

to Levels of Other Factor

Detected : In Graph , Lines Cross

Persyaratan RAK :

Keuntungan;

Kerugian;

EPI809/Spring 2008 7

Randomized Block Design

1.Experimental Units (Subjects) Are Assigned Randomly within Blocks

Blocks are Assumed Homogeneous

2.One Factor or Independent Variable of Interest

2 or More Treatment Levels or Classifications

3. One Blocking Factor

The Blocking Principle

Blocking is a technique for dealing with nuisance factors

A nuisance factor is a factor that probably has some effect on the response, but it is of no interest to the experimenterhowever, the variability it transmits to the response needs to be minimized

Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units

Many industrial experiments involve blocking (or should)

Failure to block is a common flaw in designing an experiment (consequences?)

The Blocking Principle

If the nuisance variable is known and controllable, we use blocking

If the nuisance factor is known and uncontrollable, sometimes we can use the analysis of covariance to statistically remove the effect of the nuisance factor from the analysis

If the nuisance factor is unknown and uncontrollable , we hope that randomization balances out its impact across the experiment

Sometimes several sources of variability are combined in a block, so the block becomes an aggregate variable

Randomized Complete Block Design

An experimental design in which there is one independent

variable, and a second variable known as a blocking

variable, that is used to control for confounding or

concomitant variables.

It is used when the experimental unit or material are

heterogeneous

There is a way to block the experimental units or materials

to keep the variability among within a block as small as

possible and to maximize differences among block

The block (group) should consists units or materials which

are as uniform as possible

A Randomized Block Design

Individual

observations

.

.

.

.

.

.

.

.

.

.

.

.

Single Independent Variable

Blocking

Variable

.

.

.

.

.

MSE

MST

EPI809/Spring 2008 12

Randomized Block Design

Factor Levels:

(Treatments) A, B, C, D

Experimental Units

Treatments are randomly

assigned within blocks

Block 1 A C D B

Block 2 C D B A

Block 3 B A D C.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Block ... D C A B

EPI809/Spring 2008 13

Randomized Block F-Test Hypotheses

H0: 1 = 2 = ... = p All Population Means

are Equal

No Treatment Effect

Ha: Not All j Are Equal At Least 1 Pop. Mean is

Different

Treatment Effect

1 2 ... p Is wrong

X

f(X)

1 = 2 = 3

X

f(X)

1 = 2 3

Randomized Block Design

100 Subjects

New Medication- 25 subjects

Old Medication-25 subjects

Compare level of pain relief as reported by subjects

Ran

do

m A

ssig

nm

ent

50 Women

50 Men

New Medication- 25 subjects

Old Medication-25 subjects

Compare level of pain relief as reported by subjects

Ran

do

m A

ssig

nm

entB

lock

by

Gen

der

EPI809/Spring 2008 15

Randomized Block F-Test Test Statistic

1. Test Statistic

F = MST / MSE

MST Is Mean Square for Treatment

MSE Is Mean Square for Error

2. Degrees of Freedom

1 = p -1

2 = n b p +1

p = # Treatments, b = # Blocks, n = Total Sample Size

Partitioning the Total Sum of Squares

in the Randomized Block Design

SStotal

(total sum of squares)

SST

(treatment

sum of squares)

SSE

(error sum of squares)

SSB

(sum of squares

blocks)

SSE

(sum of squares

error)

ANOVA Table for a

Randomized Block Design

Source of Sum of Degrees of Mean

Variation Squares Freedom Squares F

Treatments SST t 1 SST/t-1 MST/MSE

Blocks SSB r - 1

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

Total SSTot tr - 1

Contoh:Percobaan mengetahui efek Level lemak (L1.L2.L3) terhadap pertambahan BB

Bloking dilakukan terhadap BB sbb

Perlakuan

Blok

1 2 3 4 5 6

L1 89 89 87 92 92 85

L2 96 94 96 98 94 100

L3 96 97 99 101 102 103

281 280 282 291 288 298

SSY = 356,44

SSP =248,44

SSB = 82.444

SSE = 25.556

Sumber variasi

df SS MS F-stat F Tabel

Perlakuan 2 248.444 122.222 48.60 0.001,2,10=7.56

Blok 5 82.444 16.489

Error 10 25,556 2.556

Total 17 356,444

Tabel ANOVA:F- stat lebih

besar dari F-Tab.

Kesimpulan:

terdapat

perbedaan efek

Lemak (P

Extension of the ANOVA to the RCBD

ANOVA partitioning of total variability:

t

1i

r

1j

2

...ji.ij

r

1j

2

...j

t

1i

2

..i.

t

1i

r

1j

2

...ji.ij...j..i.

t

1i

r

1j

2

..ij

)yyy(y)yy(t)yy(r

)yyy(y)yy()yy()y(y

EBlocksTreatmentsT SSSSSSSS

Extension of the ANOVA to the RCBD

The degrees of freedom for the sums of squares in

are as follows:

Ratios of sums of squares to their degrees of freedom result in mean squares, and

The ratio of the mean square for treatments to the error mean square is an F statistic used to test the hypothesis of equal treatment means

EBlocksTreatmentsT SSSSSSSS

)]1)(1[( )1( )1( 1 rtrttr

ANOVA Procedure

The ANOVA procedure for the randomized block design requires us to partition the sum of squares total (SST) into three groups: sum of squares due to treatments, sum of squares due to blocks, and sum of squares due to error.

The formula for this partitioning is

SSTot = SST + SSB + SSE

The total degrees of freedom, nT - 1, are partitioned such that k - 1 degrees of freedom go to treatments,

b - 1 go to blocks, and (k - 1)(b - 1) go to the error term.

Example: Eastern Oil Co.

Automobile Type of Gasoline (Treatment) Blocks

(Block) Blend X Blend Y Blend Z Means

1 31 30 30 30.333

2 30 29 29 29.333

3 29 29 28 28.667

4 33 31 29 31.000

5 26 25 26 25.667

Treatment

Means 29.8 28.8 28.4