Matakuliah
Tahun
: I0134 – Metode Statistika
: 2007
Analisis Varians Klasifikasi Satu Arah
Pertemuan 23
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Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa akan dapat menyusun simpulan tentang
sumber variasi, jumlah kuadrat, derajat bebas dan
kuadrat tengah dan uji F.
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Outline Materi
•
•
•
•
•
Konsep dasar analisis varians
Klasifikasi satu arah ulangan sama
Klasifikasi satu arah ulangan tidak sama
Prosedur uji F
Pembandingan perlakuan
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Analysis of Variance and Experimental Design
• An Introduction to Analysis of Variance
• Analysis of Variance: Testing for the Equality of
k Population Means
• Multiple Comparison Procedures
• An Introduction to Experimental Design
• Completely Randomized Designs
• Randomized Block Design
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An Introduction to Analysis of Variance
•
•
•
•
Analysis of Variance (ANOVA) can be used to test for the equality of
three or more population means using data obtained from observational
or experimental studies.
We want to use the sample results to test the following hypotheses.
H0: 1 = 2 = 3 = . . . = k
Ha: Not all population means are equal
If H0 is rejected, we cannot conclude that all population means are
different.
Rejecting H0 means that at least two population means have different
values.
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Assumptions for Analysis of Variance
• For each population, the response variable is
normally distributed.
• The variance of the response variable, denoted 2, is
the same for all of the populations.
• The observations must be independent.
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Analysis of Variance:
Testing for the Equality of K Population Means
•
•
•
•
Between-Samples Estimate of Population Variance
Within-Samples Estimate of Population Variance
Comparing the Variance Estimates: The F Test
The ANOVA Table
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Between-Samples Estimate
of Population Variance
• A between-samples estimate of 2 is called the mean
square between (MSB).
_
=
k
MSB 
2
 n j ( x j  x) 2
j1
k1
• The numerator of MSB is called the sum of squares
between (SSB).
• The denominator of MSB represents the degrees of
freedom associated with SSB.
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Within-Samples Estimate
of Population Variance
• The estimate of 2 based on the variation of the sample
observations within each sample is called the mean square within
(MSW).
k
MSW 
2
 (n j  1) s 2j
j1
nT  k
• The numerator of MSW is called the sum of squares within (SSW).
• The denominator of MSW represents the degrees of freedom
associated with SSW.
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Comparing the Variance Estimates: The F Test
• If the null hypothesis is true and the ANOVA assumptions are
valid, the sampling distribution of MSB/MSW is an F distribution
with MSB d.f. equal to k - 1 and MSW d.f. equal to nT - k.
• If the means of the k populations are not equal, the value of
MSB/MSW will be inflated because MSB overestimates 2.
• Hence, we will reject H0 if the resulting value of MSB/MSW
appears to be too large to have been selected at random from
the appropriate F distribution.
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Test for the Equality of k Population Means
•
Hypotheses
H0: 1 = 2 = 3 = . . . = k
Ha: Not all population means are equal
•
Test Statistic
F = MSB/MSW
•
Rejection Rule
Reject H0 if F > F
where the value of F is based on an F distribution with k - 1
numerator degrees of freedom and nT - 1 denominator degrees of
freedom.
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Sampling Distribution of MSTR/MSE
• The figure below shows the
rejection region associated with a
level of significance equal to 
where F denotes the critical value.
Do Not Reject H0
Reject H0
F
Critical Value
MSTR/MSE
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The ANOVA Table
Source of
Sum of
Variation Squares
Treatment SSTR
Error
SSE
Total
SST
Degrees of
Freedom
k-1
nT - k
nT - 1
Mean
Squares
F
MSTR MSTR/MSE
MSE
SST divided by its degrees of freedom nT - 1 is simply the
overall sample variance that would be obtained if we
treated the entire nT observations as one data set.
k
nj
SST   ( xij  x) 2  SSTR  SSE
j 1 i 1
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Fisher’s LSD Procedure
•
Hypotheses
H0: i = j
Ha: i  j
•
•
Test Statistic
Rejection Rule
xi  x j
t
MSW( 1 n  1 n )
i
j
Reject H0 if t < -ta/2 or t > ta/2
where the value of ta/2 is based on a t distribution
with nT - k degrees of freedom.
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_
_
Fisher’s LSD Procedure
Based on the Test Statistic xi - xj
• Hypotheses
• Test Statistic
• Rejection Rule
H0: i = j
Ha: i j 
_
_
xi - xj
Reject H0 if |xi - xj| > LSD
_
_
where
LSD  t  /2 MSW ( 1 n  1 n )
i
j
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ANOVA Table for a
Completely Randomized Design
Source of
Variation
Treatments
Sum of
Squares
Degrees of
Freedom
SSTR
Mean
Squares
F
k-1
MSTR 
Error
Total
SSE
SST
nT - k
nT - 1
SSTR
k-1
MSTR
MSE
SSE
MSE 
nT - k
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• Selamat Belajar Semoga Sukses.
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