Modul Ajar Statistika Inferensia ke-13: Analisis Variansi, Eksperimentasi Faktorial

Analisa Variansi:
Eksperimentasi Faktorial
ARIF RAHMAN
1

Statistika
Statistika adalah cabang ilmu matematika yang
mempelajari metode ilmiah untuk mengumpulkan,
mengorganisasi, merangkum, menyederhanakan,
menyajikan, menginterpretasikan, menganalisa dan
mensintesa data (numerik atau nonnumerik) untuk
menghasilkan informasi dan/atau kesimpulan, yang
membantu dalam penyelesaian masalah dan/atau
pengambilan keputusan.
2

Statistika
3
Mengorganisasi,
Merangkum,
Menyederhanakan,
Menyajikan,
Menginterpretasikan
Menganalisa
Mensintesa
Mengumpulkan data
Menghasilkan informasi dan/atau kesimpulan
Menggeneralisasi
Mengestimasi,
Menguji hipotesa,
Menilai relasi,
Memprediksi
Menyelesaikan masalah Mengambil keputusan

Statistika Inferensia
Statistika inferensia adalah cabang statistika yang
menganalisa atau mensintesa data untuk
menggeneralisasi sampel terhadap populasi,
mengestimasi parameter, menguji hipotesa, menilai
relasi, dan membuat prediksi untuk menghasilkan
informasi dan/atau kesimpulan.
Terdapat banyak alat bantu statistika (statistical tools)
yang dapat dipergunakan untuk menginferensi
populasi atau sistem yang menjadi sumber asal data
sampel
4

Statistika Inferensia
5
Tujuan studi terhadap populasi Observasi atau eksperimen pada sampel
SAMPLING
INFERENSI
Parameter :
N (banyaknya anggota populasi),
μ (rata-rata populasi),
σ (simpangan baku populasi),
π (proporsi populasi)
Statistik :
n (banyaknya anggota sampel),
ẋ (rata-rata sampel),
s (simpangan baku sampel),
p (proporsi sampel)

Tipe Data
Data Nominal, data yang hanya berupa simbol (meski berupa
angka) untuk membedakan nilainya tanpa menunjukkan tingkatan
Data Ordinal, data yang mempunyai nilai untuk menunjukkan
tingkatan, namun tanpa skala yang baku dan jelas antar tingkatan.
Data Interval, data yang mempunyai nilai untuk menunjukkan
tingkatan dengan skala tertentu sesuai intervalnya. Nilai nol hanya
untuk menunjukkan titik acuan (baseline).
Data Rasio, data yang mempunyai nilai untuk menunjukkan
tingkatan dengan skala indikasi rasio perbandingan. Nilai nol
menunjukkan titik asal (origin) yang bernilai kosong (null).
6

Tipe Data
Data Parametrik, data kuantitatif yang mempunyai
sebaran variabel acak mengikuti pola distribusi
probabilitas dengan parameter tertentu (independent
and identically distributed random variables)
Data Nonparametrik, data yang tidak mempunyai
distribusi probabilitas (distribution-free)
7

Tipe Data
Data Diskrit, data hasil pencacahan atau
penghitungan, sehingga biasanya dalam angka
bilangan bulat.
Data Kontinyu, data hasil pengukuran yang
memungkinkan dalam angka bilangan nyata
(meskipun dapat pula dibulatkan)
8

Statistika Alat Bantu Problem Solving
9
Penting memperhatikan
cara memperoleh
data yang akan diolah
Demikian pula
cara mengolah data
juga penting diperhatikan

Statistika Alat Bantu Problem Solving
10
Metode statistika bukan
ramuan sihir
Alat statistika bukan
tongkat sihir

Ketelitian &
Tipe Kesalahan
11

Akurasi dan Presisi
Akurasi (accuracy), kesesuaian hasil pengukuran
terhadap nilai obyek sesungguhnya (bias kecil)
Presisi (precision), tingkat skala ketelitian
pengukuran dari alat pengukur, atau ketersebaran
yang relatif mengumpul (variansi atau deviasi kecil)
12

Akurat dan Presisi
Tidak presisi, akibat pola sebaran sampel
lebih melebar daripada pola sebaran
populasi menyebabkan deviasi yang besar.
Tidak akurat, akibat pergeseran
pemusatan sampel menjauh dari
pemusatan populasi menyebabkan bias
yang besar.
Akurat dan presisi, bias dan deviasi kecil,
membutuhkan sampel sedikit.
13

Kesalahan Pengambilan Kesimpulan
Galat tipe 1 () : kesalahan menyimpulkan karena
menolak hipotesa yang semestinya diterima
Galat tipe 2 () : kesalahan menyimpulkan karena
menerima hipotesa yang semestinya ditolak
14
 

Kesalahan Pengambilan Kesimpulan
15
The true state of nature
Decision H0 is true H0 is false
Reject H0 Type I error Exact decision
Fail to reject H0 Exact decision Type II error
The true state of nature
Decision H0 is true H0 is false
Reject H0  1 – 
Fail to reject H0 1 –  

Ukuran Ketelitian Pendugaan
Tingkat keberartian (significance level, ), probabilitas
penolakan data observasi, karena menyimpang signifikan terhadap
sasaran.
Tingkat kepercayaan (confidence coefficient,1-), persentase
data observasi yang diyakini tidak berbeda signifikan dengan target.
Kuasa statistik (power,1-), persentase data observasi yang
diyakini berbeda signifikan dengan target.
Derajat kebebasan (degree of freedom, df=n-k), besaran
yang menunjukkan bebas terhadap bias dari n data observasi.
16

Sumber Kesalahan atau Error
17

Sumber Kesalahan atau Error
random error is an uncontrollable difference from one trial to
another due to environment, equipment, or other issues that
reduce the repeatability of an observation
systematic error is a reproducible deviation of an
observation that biases the results, arising from procedures,
instruments, or ignorance
illegitimate error is an error introduced when an engineer
does mistakes, blunders, or miscalculations (e.g. measures
at the wrong time, notes the wrong value)
18

Kesalahan Pengukuran
unusual value (outlier) is an observation in a sample that are
so far from the main body of data that they give rise to the
question that they may be from another population.
missing value is any relevant data which are missing, since
there may be transcription or recording errors or may not
have been collected and archived.
bias is an effect that systematically distorts a statistical result
or estimate, preventing it from representing the true quantity
of interest.
19

Observasi dan Eksperimen
In an observational study, the engineer observes the
process or population, disturbing it as little as possible, and
records the quantities of interest.
In a designed experiment the engineer makes deliberate or
purposeful changes in the controllable variables of the
system or process, observes the resulting system output
data, and then makes an inference or decision about which
variables are responsible for the observed changes in output
performance.
21

Eksperimen dirancang namun keluarannya acak
Designed experiment is an experiment in which the tests
are planned in advance and the plans usually incorporate
statistical models
Random experiment is an experiment that can result in
different outcomes, even though it is repeated in the same
manner each time.
Outcome is an element of a sample space.
Event is a subset of a sample space.
Sample space is the set of all possible outcomes of a
random experiment.
25

Unit Eksperimen atau Trial
Experimental unit or trial is a single testing in scientific
investigation through observations or experiments that is
reproducible in the same condition or treatment to observe the
response variable. It is an entity which is the primary unit of
interest in a specific research objective for researcher to make
inferences about (in the population) based on the sample (in the
experiment). Thus it needs adequate replication of experimental
units. The sample size is the number of experimental units per
group.
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Panduan Desain dan Analisa Eksperimen
33

Prinsip Dasar Desain Eksperimen
Replication, to provide an estimate of experimental error;
Randomization, to ensure that this estimate is statistically
valid; and
Local control, to reduce experimental error by making the
experiment more efficient
34

Replication is an independent repeat run of each factor combination. It
is the repetition of experiment under identical conditions. It refers to the
number of distinct experimental units under the same treatment.
 Replication is useful for obtaining homogeneous data.
 Replication improves the accuracy of response estimates by mapping confidence
intervals at a specific significance level.
 Replication helps detect outliers due to errors in experimentation, measurement errors
or other confounding factors.
35

Randomization is the cornerstone underlying the use of statistical
methods in experimental design to randomly determine the order in which
the individual runs of the experiment are to be performed. Through
randomization, every experimental unit will have the same chance of
receiving any treatment.
 Randomization is useful to ensure that each experiment is independent, and that the
effect of nuisance factors is reduced.
 Randomization reduces the risk of experimental bias due to nuisance factors having
clustered effects on the same experimental treatment.
 The randomization helps increase the confidence of the statistical analysis of the
experimental results.
36

37
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!
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(
ts
Arrangemen
Possible
2
1
2
1
a
a
n
n
n
n
n
n
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38

Local control is the control of all factors except the design factors
which are investigated. It refines the relatively heterogeneous experimental
subset into homogeneous subset by removing extraneous sources of
variability. It refers to the amount of balancing, blocking and grouping of
the experimental units.
 Grouping is arranging a homogeneous set of experiments in groups that receive the
same treatment.
 Blocking is allocating experiments in blocks, so that each block contains
homogeneous experiments.
 Balancing is controlling the grouping and blocking processes so that the experiment is
in a balanced configuration or formation.
39

40

 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’s of no interest to the experimenter…however, 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?)
41

 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 remove the effect of the nuisance
factor from the analysis
 If the nuisance factor is unknown and uncontrollable (a “lurking”
variable), 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
42

43

44

45

46

47

48

49
Randomization between blocks
Replicate Block Random
1 1 2047
1 2 5589
2 1 3255
2 2 7641
3 1 6859
3 2 3455
4 1 4328
4 2 1391

50
Randomization within blocks
Repl-Block Test Random
4.2 abc 2997
4.2 a 8220
4.2 b 3507
4.2 c 1522
: : :
2.2 abc 9450
2.2 a 5147
2.2 b 2085
2.2 c 4735

Faktor dan Perlakuan
Factors are the potential sources of variability that influence
the performance of a process or system.
Treatments are specific levels of the design factors (factors
of interest). They are deliberate changes of a set of design
factors at various level to observe the changes in the system
performance.
Factor level is the settings (or conditions) used for a factor in
an experiment.
52

Effects are the impact of treatment to response variables.
They are the mean change to the response due to the
presence of the treatment.
Interaction is interdependence of several factors. Two
factors are said to interact if the effect of one variable is
different at different levels of the other variables. In general,
when variables operate independently of each other, they do
not exhibit interaction. An interaction is the failure of one
factor to produce the same effect on the response at different
levels of another factor.
53

The potential design factors are those factors that the
experimenter may wish to vary in the experiment.
Design factors are the factors actually selected for study in the
experiment.
Held-constant factors are variables that may exert some effect on
the response, but for purposes of the present experiment these
factors are not of interest, so they will be held at a specific level.
Allowed-to-vary factors are variables that are usually
nonhomogeneous, but for ignoring this unit-to-unit variability, it
relies on randomization to balance out any effect.
54

Nuisance factors may have large effects that must be
accounted for, yet the experimenter may not be interested in
them in the context of the present experiment.
A controllable nuisance factor is one whose levels may be set by
the experimenter
An uncontrollable nuisance factor is a nuisance factor that is
uncontrollable in the experiment, but it can be measured. An
analysis procedure called the analysis of covariance can be used to
compensate for its effect.
A noise factor is a factor that varies naturally and uncontrollably in
the process.
55

Fixed effect factor is a design factor of experiment with
specific treatment at certain levels. All the levels of interest
for the factor are included in the experiment.
Random effect factor is a design factor of experiment with
treatment by random sample from some population of factor
levels. There may be unknown levels between treatment
(level numbers are only nominal).
56

Parameter Diagram (P Diagram)
59

Signal
Factors
(m)
Noise
Factors
(x)
Control
Factors
(z)
Scaling
Factors
(r)
Response
Variables
(y)
F(x,m,z,r)
60
Inputs
Controllable Factors
(x)
Uncontrollable Factors
(z)
Output
(y)
F(x,z)

 Response variables (y) are the dependent variables (that are affected some
factors) as observed output characteristics (that are designed to meet the target).
 Signal factors (M) are the parameter values set by the user at specified point or
within an acceptable range to attain the desired output.
 Control factors (Z) are the parameter values set by the engineer at least at two-
levels to select the best level for the desired output.
 Noise factors (X) are not controllable by the engineer or the user. However, for
the purpose of optimization, these factors may be set at one or more levels.
 Scaling factors (R) are special cases of control factors that are adjusted to
achieve the desired functional relationship as a ratio between the signal factor and the
response.
 Leveling factors (D) are special cases of control factors that are adjusted to
achieve the desired functional relationship as a constant between the signal factor and
the response.
61

 External noise factors are sources of variation that are external to the product
or process. They include environmental noise factors and load-related noise factors.
The environmental noise factors are temperature, humidity, dust, electromagnetic
interference, etc. The load-related noise factors are the period of time the product
works continuously, the pressures to which it is subjected simultaneously..
 Internal noise factors are sources of variation that are internal to the product or
process. They include time-dependent deterioration factors such as wear of
components, spoilage of materials, fatigue of parts, and operational errors, such as
improper settings on product or equipment.
 Unit-to-unit noise factors are inherent random variations in the process or
product caused by variability in raw materials, machinery and human participation.
62

Strategi Desain Eksperimen
Best-guess experiments
One-factor-at-a-time (OFAT) experiments
Statistically-designed experiments
Factorial experiments
Fractional factorial experiments
64

Best-guess Experiments
Advantages
 The experimenter reasonably selects
an arbitrary combination of the design
factors, test them, and see what
happens
 The experimenter switches the levels
of one or two (or perhaps several)
factors for the next test, based on the
outcome of the current test.
 There is a great deal of technical or
theoretical knowledge of the system, as
well as considerable practical
experience.
Disadvantages
 The approach could be continued
almost indefinitely.
 The initial best-guess does not produce
the desired results. So the
experimenter has to take another
guess at the correct combination of
factor levels. This could continue for a
long time, without any guarantee of
success.
 The initial best-guess produces an
acceptable result. And the
experimenter is tempted to stop testing,
although there is no guarantee that the
best solution has been found.
65

One-factor-at-a-time (OFAT) Experiments
Advantages
 The experimenter selects a starting
point, or baseline set of levels, for each
factor, and then successively varying
each factor over its range with the
other factors held constant at the
baseline level.
 The experimenter analyzes how the
response variable is affected by
varying each factor with all other
factors held constant.
 The interpretation is straightforward,
conclude the interaction.
Disadvantages
 It assumes factors were independent. If
the experimenter varies a factor, he
assumes that the other factors have
virtually no effect.
 It fails to consider any possible
interaction between the factors. A
factor may produce the different effect
on the response at different levels of
another factor.
 If the interactions between factors
occur, it will usually produce poor
results
66

Statistically-designed (Factorial) Experiments
Advantages
 All possible combinations of the design
factors across their levels are used in
the design
 A reasonable plan would be at each
combination of factor levels
 The experimental design would enable
the experimenter to investigate the
individual effects of each factor (or the
main effects) and to determine whether
the factors interact.
Disadvantages
 The number of factors of interest
increases, the number of runs required
increases rapidly.
67

Practical Interpretation of Results
A Regression Model
Comparisons Among Treatment Means
Graphical Comparisons of Means
Contrasts
Orthogonal Contrasts
Scheffé’s Method for Comparing All Contrasts
Comparing Pairs of Treatment Means
Tukey’s Test
Tukey–Kramer procedure
The Fisher Least Significant Difference (LSD) Method
87

an Example: Etching Process Experiment
88

89

90

91
Power Replication Random
160 1 57102
160 2 29337
160 3 24621
160 4 63548
160 5 40062
180 1 32318
180 2 71834
180 3 84675
180 4 77216
180 5 43289
200 1 49271
200 2 36481
200 3 94037
200 4 89323
200 5 12417
220 1 49813
220 2 21238
220 3 52286
220 4 67710
220 5 18369
Test Sequence Random (Sorted) Power
1 12417 200
2 18369 220
3 21238 220
4 24621 160
5 29337 160
6 32318 180
7 36481 200
8 40062 160
9 43289 180
10 49271 200
11 49813 220
12 52286 220
13 57102 160
14 63548 160
15 67710 220
16 71834 180
17 77216 180
18 84675 180
19 89323 200
20 94037 200

92

93

94

95

96

97

98

99

100

101

Pengantar Desain Faktorial 2k
103

104

Pengantar Desain Faktorial 22
105

Desain Faktorial 22
111























high
treatment
low
high
high
low
high
low
treatment
low
high
high
low
low
low
high
high
low
treatment
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
if
1
)
2
/
)
((
)
2
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)
((
if
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)
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)
((
)
2
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)
((
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)
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119

120

121

122

123

124

125

126

138

139

140

an Example: Syrup Frothing Experiment
147

148

149

150

151

152

153

154

155

156

157
78
.
126
54
165
3
.
2
16
75
74
)
(
2
2
2
2
2
2





 AB
AB
I
I(AB) J(AB)
-198 -106 -155 -222 -79 -158
331 255 377 238 440 285
-59 -74 -206 -144 -40 -155
74 75 16 -128 321 -28
11
.
174
,
6
54
165
3
.
2
)
28
(
321
)
128
(
)
(
2
2
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 AB
AB
J
89
.
300
,
6
11
.
174
,
6
78
.
126 


AB
SS

158

159

160

161

162
SSAB = 6,300.89 = 126.78 + 6,174.11
SSABC = 4,628.76 = 3,804.11 + 221.77 + 18.77 + 584.11

Nonparametric
Kruskal-Wallis Test
165

Kruskal-Wallis H-Test
Tests that more than two (k) independent samples
are from identical distributions
Corresponds to ANOVA for more than two means
Used to analyze completely randomized experimental
designs
Uses 2 distribution with k – 1 df
— if sample size nj ≥ 5
169

Kruskal-Wallis H-Test for Comparing k
Probability Distributions
H0: μ1= μ2= . . . = μk
H1: All means are not all equal. at least one mean is
different.
170
Test statistic:
)
1
(
3
)
1
(
12
)
(
)
1
(
12
2
2

















n
n
R
n
n
R
R
n
n
n
H
j
j
j
j

where
nj =Number of measurements in sample j
Rj = Rank sum for sample j, where the rank of each
measurement is computed according to its relative
magnitude in the totality of data for the k samples
͞Rj=Rj/nj = Mean rank sum for j-th sample
͞R =Mean of all ranks = (n + 1)/2
n = Total sample size = n1 + n2 + . . . + nk
171

Rejection region:
H > χ2 with (k – 1) degrees of freedom
Ties: Assign tied measurements the average of the ranks
they would receive if they were unequal but occurred in
successive order. For example, if the third-ranked and fourth-
ranked measurements are tied, assign each a rank of (3 +
4)/2 = 3.5. The number should be small relative to the total
number of observations.
172

Conditions Required for the Validity of
the Kruskal-Wallis H-Test
1. The k samples are random and independent.
2. There are five or more measurements in each
sample.
3. The k probability distributions from which the
samples are drawn are continuous
173

Kruskal-Wallis H-Test Procedure
1. Assign ranks, Ri , to the n combined observations
 Smallest value = 1; largest value = n
 Average ties
2. Sum ranks for each group
3. Compute test statistic
174
 
 
2
12
3 1
1
j
j
R
H n
n n n
 
  
 
 

 

Squared total of
each group

an Example: Filling Machine
As production manager, you
want to see if three filling
machines have different filling
times. You assign 15
similarly trained and
experienced workers, 5 per
machine, to the machines. At
the .05 level of significance,
is there a difference in the
distribution of filling times?
175
Mach1Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40

H0: μ1= μ2= μ3
H1: The three means are not all equal
α = 0.05
df = p – 1 = 3 – 1 = 2
Critical Value(s): χ2 = 5.991
176
2
0 5.991
 = .05

Raw Data
Mach1 Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40
Ranks
Mach1 Mach2 Mach3
177

Raw Data
Mach1 Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40
Ranks
Mach1 Mach2 Mach3
1
178

Raw Data
Mach1 Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40
Ranks
Mach1 Mach2 Mach3
2
1
179

Raw Data
Mach1 Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40
Ranks
Mach1 Mach2 Mach3
2
1
3
180

Raw Data
Mach1 Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40
Ranks
Mach1 Mach2 Mach3
2
1
4
3
181

Raw Data
Mach1 Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40
Ranks
Mach1 Mach2 Mach3
2
1
4
5 3
182

Raw Data
Mach1 Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40
Ranks
Mach1 Mach2 Mach3
14 9 2
15 6 7
12 10 1
11 8 4
13 5 3
183

Raw Data
Mach1 Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40
Ranks
Mach1 Mach2 Mach3
14 9 2
15 6 7
12 10 1
11 8 4
13 5 3
65 38 17
184

185
58
.
11
48
5
5958
240
12
)
1
15
(
3
5
)
17
(
5
)
38
(
5
)
65
(
)
1
15
(
15
12
)
1
(
3
)
1
(
12
2
2
2
2











































  n
n
R
n
n
H
j
j

H0: μ1= μ2= μ3
H1: The three means are not all equal
α = 0.05
df = p – 1 = 3 – 1 = 2
Critical Value(s): χ2 = 5.991
Test statistic: H = 11.58, P-value = 0.00306
Decision: Reject H0. At least 2 groups have different
distribution
186
2
0 5.991
 = .05

an Example: Propellant Burning Rate
187

188

189

Nonparametric
Friedman Test
190

Friedman Test
provides another method for testing to detect a shift in
location of a set of k populations that have the same
spread (or, scale)
is based on the rank sums of the treatments,
measures the extent to which the k samples differ
with respect to their relative ranks within the blocks
191

Friedman Test
Friedman Fr-Test for a Randomized Block Design
H0: μ1= μ2= . . . = μk
H1: All means are not all equal. at least one mean is
different.
Test statistic:
192
 

 2
)
(
)
1
(
12
R
R
k
k
b
Fr j

Friedman Test
Friedman Fr-Test for a Randomized Block Design
where
b = Number of blocks
k = Number of treatments
Rj = Rank sum of the j-th treatment, where the rank
of each measurement is computed relative to its
position within its own block
Test statistic:
Fr > χ2 with (k – 1) degrees of freedom
193

Friedman Test
Ties: Assign tied measurements the average of the
ranks they would receive if they were unequal but
occurred in successive order. For example, if the
third-ranked and fourth-ranked measurements are
tied, assign each a rank of (3 + 4)/2 = 3.5. The
number should be small relative to the total number of
observations.
194

an Example: Drug Reaction
Consider the data in the table. A pharmaceutical firm
wants to compare the reaction times of subjects under
the influence of three different drugs that it produces.
Apply the Friedman Fr-test to the data. What conclusion
can you draw? Test using  = .05.
195

H0: The average reaction times are identical for the
three drugs
H1: At least one of the three drugs have different
average reaction time
196

For  = .05,  2
.05 = 5.99147, therefore
Rejection region: Fr > 5.99147
197

Conclusion: Because Fr = 8.33 exceeds the critical
value of 5.99, we reject the null hypothesis and
conclude that at least two of the three drugs have
distributions of reaction times that differ in location. That
is, at least one of the drugs tends to yield reaction times
that are faster than the others.
198

199
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Modul Ajar Statistika Inferensia ke-13: Analisis Variansi, Eksperimentasi Faktorial

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