RAK

Week 10/11 - 1 / 47

Rancangan Acak Kelompok (RAK)

Diterapkan pada percobaan yang dilakukan pada lingkungan tidak homogen (heterogen)

Struktur Data RAK

Week 10/11 - 2 / 47

  Perlakuan

Kelompok 1 2 … t1 x11 x21 xt12 x12 x22 xt2… … … … …… … … … …b x1b x2b   xtb

Week 10/11 - 3 / 47

�̂�𝑅𝐴𝐾2 =

(𝑟 −1 ) 𝐾𝑇𝑆𝑅𝐴𝐾+𝑟 (𝑡−1 )𝐾𝑇𝑆𝑅𝐴𝐿

𝑡𝑟−1

Week 10/11 - 4 / 47

ER untuk memperoleh sensitivitas RAL yang sama dengan RAK maka ulangan yang digunakan dalam menerapkan RAL harus ER kali dari ulangan yang digunakan dalam RAK.

FAKTORIAL - RAL

Week 10/11 - 5 / 47

Dr. Ir. Rahmat Kurnia, M.Si

Week 10/11 - 6 / 47

Two-Way ANOVA

Examines the effect of Two factors of interest on the dependent

variable e.g., Percent carbonation and line speed on soft drink

bottling process Interaction between the different levels of these

two factors e.g., Does the effect of one particular carbonation

level depend on which level the line speed is set?

Week 10/11 - 7 / 47

Two-Way ANOVA

Assumptions

Independent random samples are drawn

Populations have equal variances Populations are normally distributed

(continued)

Week 10/11 - 8 / 47

Two-Way ANOVA Sources of Variation

Two Factors of interest: A and B

a = number of levels of factor A

b = number of levels of factor B

r = number of replications for each cell

n = total number of observations in all cells(n = abr)

Xijk = value of the kth observation of level i of factor A and level j of factor B

Week 10/11 - 9 / 47

Two-Way ANOVA Sources of Variation

SSTTotal Variation

SSAFactor A Variation

SSBFactor B Variation

SSABVariation due to interaction

between A and B

SSERandom variation (Error)

Degrees of Freedom:

a – 1

b – 1

(a – 1)(b – 1)

ab(r – 1)

n - 1

SST = SSA + SSB + SSAB + SSE

(continued)

Week 10/11 - 10 / 47

Two Factor ANOVA Equations

r

1i

c

1j

n

1k

2ijk )XX(SST

2r

1i

..i )XX(ncSSA

2c

1j

.j. )XX(nrSSB

Total Variation:

Factor A Variation:

Factor B Variation:

Week 10/11 - 11 / 47

Two Factor ANOVA Equations

2r

1i

c

1j

.j...i.ij )XXXX(nSSAB

r

1i

c

1j

n

1k

2.ijijk )XX(SSE

Interaction Variation:

Sum of Squares Error:

(continued)

Week 10/11 - 12 / 47

Two Factor ANOVA Equations

where:Mean Grand

nrc

X

X

r

1i

c

1j

n

1kijk

r) ..., 2, 1, (i A factor of level i of Meannc

X

X th

c

1j

n

1kijk

..i

c) ..., 2, 1, (j B factor of level j of Meannr

XX th

r

1i

n

1kijk

.j.

ij cell of Meann

XX

n

1k

ijk.ij

r = number of levels of factor A

c = number of levels of factor B

n’ = number of replications in each cell

(continued)

Week 10/11 - 13 / 47

Mean Square Calculations

1r

SSA Afactor square MeanMSA

1c

SSBB factor square MeanMSB

)1c)(1r(

SSABninteractio square MeanMSAB

)1'n(rc

SSEerror square MeanMSE

Week 10/11 - 14 / 47

Two-Way ANOVA:The F Test Statistic

F Test for Factor B Effect

F Test for Interaction Effect

H0: μ1.. = μ2.. = μ3.. = • • •

H1: Not all μi.. are equal

H0: the interaction of A and B is equal to zero

H1: interaction of A and B is not zero

F Test for Factor A Effect

H0: μ.1. = μ.2. = μ.3. = • • •

H1: Not all μ.j. are equal

Reject H0

if F > FUMSE

MSAF

MSE

MSBF

MSE

MSABF

Reject H0

if F > FU

Reject H0

if F > FU

Week 10/11 - 15 / 47

Two-Way ANOVASummary Table

Source ofVariation

Sum ofSquares

Degrees of Freedom

Mean Squares

FStatistic

Factor A SSA r – 1MSA

= SSA /(r – 1)MSAMSE

Factor B SSB c – 1MSB

= SSB /(c – 1)MSBMSE

AB(Interaction) SSAB (r – 1)(c – 1)

MSAB= SSAB / (r – 1)(c – 1)

MSABMSE

Error SSE rc(n’ – 1)MSE =

SSE/rc(n’ – 1)

Total SST n – 1

Week 10/11 - 16 / 47

Features of Two-Way ANOVA F Test

Degrees of freedom always add up n-1 = rc(n’-1) + (r-1) + (c-1) + (r-1)(c-1)

Total = error + factor A + factor B + interaction

The denominator of the F Test is always the same but the numerator is different

The sums of squares always add up SST = SSE + SSA + SSB + SSAB

Total = error + factor A + factor B + interaction

Week 10/11 - 17 / 47

Examples:Interaction vs. No Interaction

No interaction:

Factor B Level 1

Factor B Level 3

Factor B Level 2

Factor A Levels

Factor B Level 1

Factor B Level 3

Factor B Level 2

Factor A Levels

Mea

n R

espo

nse

Mea

n R

espo

nse

Interaction is present:

Week 10/11 - 18 / 47

Chapter Summary

Described one-way analysis of variance The logic of ANOVA ANOVA assumptions F test for difference in c means The Tukey-Kramer procedure for multiple comparisons

Described two-way analysis of variance Examined effects of multiple factors Examined interaction between factors