rak week 10/11 - 1 / 47 rancangan acak kelompok (rak) diterapkan pada percobaan yang dilakukan pada...
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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