Pengujian AsumsiKuliah 8 | Perancangan Percobaan
Asumsi-asumsi Analisis Ragam
Pengaruh perlakuan & lingkungan bersifat aditif
Galat percobaan memiliki ragam yg homogen
Galat percobaan saling bebas
Galat percobaaan menyebar normal
Jika asumsi dilanggar….
Dapat mempengaruhi kepekaan uji F atau t
#1Asumsi keaditifan model
Asumsi keaditifan model
Ilustrasi:
𝑌𝑖𝑗 = 𝜇 + 𝜏𝑖 + 𝛽𝑗 + 𝜀𝑖𝑗
Aditif, artinya 𝑌𝑖𝑗 adalah hasil PENJUMLAHANkomponen 𝜇 , 𝜏𝑖 , 𝛽𝑗 , 𝜀𝑖𝑗.
Asumsi keaditifan model
Ketidakaditifan model keheterogenan galat
Akibatnya:
• Ragam galat gabungan tidak efisien
• Dapat memberi tingkat nyata yg palsu
Pengujian Asumsi
UJI TUKEY
Hipotesis:
H0: model aditif Vs H1: model non-aditif
Statistik uji
𝐹ℎ𝑖𝑡𝑢𝑛𝑔 =𝐽𝐾(𝑛𝑜𝑛 𝑎𝑑𝑖𝑡𝑖𝑓)
𝐽𝐾𝐺 𝑑𝑏𝑔~𝐹𝛼(1,𝑑𝑏𝑔)
dengan:
𝐽𝐾(𝑛𝑜𝑛 𝑎𝑑𝑖𝑡𝑖𝑓) =𝑄2
𝑟 𝑌𝑖∙− 𝑌∙∙ 2 𝑌∙𝑗− 𝑌∙∙2
𝑟 =banyaknya ulangan
𝑄 = 𝑌𝑖∙ − 𝑌∙∙ 𝑌∙𝑗 − 𝑌∙∙ 𝑌𝑖𝑗
Jika 𝐹ℎ𝑖𝑡𝑢𝑛𝑔 ≤ 𝐹𝛼(1,𝑑𝑏𝑔) maka keaditifan model dapat diterima.
#2Asumsi Kehomogenan
Asumsi Kehomogenan
Pemeriksaan AsumsiUJI BARTLETT
Hipotesis:
H0: ragam homogen
H1: ragam tidak homogen
Statistik Uji:
𝜒2 = 2.3026 𝑖 𝑟𝑖 − 1 𝑙𝑜𝑔 𝑠2 − 𝑖 𝑟𝑖 − 1 𝑙𝑜𝑔 𝑠𝑖2
Kriteria :
𝜒2 < 𝜒𝛼 (𝐾−1)2 maka terima H0 ragam homogen
dengan 𝐾 = 1 +1
3 𝑡−1 𝑖
1
𝑟𝑖−1−
1
𝑟𝑖−1
Pemeriksaan Asumsi
Pemeriksaan Asumsi
Way to solve the problem of Heterogeneous variances
The data can be separated into groups such that the variances within each group are homogenous
An advance statistic tests can be used rather than analysis of variance
Transform the data in such a way that data will be homogenous
Remedial Measures for Heterogeneous Variances
• Studies that do not involve repeated measures
• If normality is violated, the data transformation necessary to normalize data will usually stabilize variances as well
• If variances are still not homogeneous, non-ANOVA tests might be an option
#3Asumsi Kebebasan
Asumsi Kebebasan
Possible Causes of Serial Correlated Error
1) omitted variables
2) ignoring nonlinearities
3) measurement errors
Consequences of Serial Correlated Error
1. The OLS estimators are still unbiased and consistent
2. In large samples, the error may be still normally distributed
3. The estimators are no longer efficient no longer BLUE.
4. The estimated standard error may be underestimated,
5. the tests using the t and F distribution, may no longer be appropriate
Asumsi Kebebasan
Asumsi Kebebasan
Pemeriksaan Asumsi
Residual Plot
Durbin Watson test
Runs Test
Etc.
Remedial Measures for Dependent Data
• First defense against dependent data is proper study design and randomization• Designs could be implemented that takes correlation into account,
e.g., crossover design
• Look for environmental factors unaccounted for • Add covariates to the model if they are causing correlation, e.g.,
quantified learning curves
• If no underlying factors can be found attributed to the autocorrelation• Use a different model, e.g., random effects model
• Transform the independent variables using the correlation coefficient
#4Asumsi Kenormalan
Asumsi Kenormalan Galat
Asumsi Kenormalan Galat
Berlaku terutama utk pengujian hipotesis
Jika sebaran galat menjulur, komponen galat dariperlakuan cenderung merupakan fungsi dariperlakuan, akibatnya ragamnya menjadi tidakhomogen.
Pemeriksaan Asumsi
1.Histogram and/or box-plot of all residuals (eij).
2.Normal probability (Q-Q) plot.
3.Formal test for normality.
Pemeriksaan Asumsi
Pengujian Asumsi
• Shapiro-Wilk’s W
• Lilliefors-Kolmogorov-Smirnov Test
• Kolmogorov-Smirnov D
• Ryan-Joiner test
• Anderson-Darling A2
• Etc.
Pengujian Asumsi
Asumsi Kenormalan
Asumsi Kenormalan
The Consequences of Non-Normality
• F-test is very robust against non-normal data, especially in a fixed-effects model
• Large sample size will approximate normality by Central Limit Theorem (recommended sample size > 50)
• Simulations have shown unequal sample sizes between treatment groups magnify any departure from normality
• A large deviation from normality leads to hypothesis test conclusions that are too liberal and a decrease in power and efficiency
Remedial Measures for Non-Normality
• Data transformation
• Be aware - transformations may lead to a fundamental change in the relationship between the dependent and the independent variable and is not always recommended.
• Don’t use the standard F-test. • Modified F-tests
• Adjust the degrees of freedom• Rank F-test (capitalizes the F-tests robustness)
• Randomization test on the F-ratio • Other non-parametric test if distribution is unknown• Make up our own test using a likelihood ratio if distribution is
known
Penanganan Data terhadapPelanggaran Asumsi
Data Transformation
There are two ways in which the anova assumptions can be violated:
1. Data may consist of measurement on an ordinal or a nominal scale
2. Data may not satisfy at least one of the four requirements
Two options are available to analyze data:
1. It is recommended to use non-parametric data analysis
2. It is recommended to transform the data before analysis
Square Root Transformation
It is used when we are dealing with counts of rare events
The data tend to follow a Poisson distribution
If there is account less than 10. It is better to add 0.5 to the value
ii yz
i i
k2 This transformation works when we notice the variance changes as a linear function of the mean.
• Useful for count data (Poisson Distributed).
• For small values of Y, use Y+.5.
Typical use: Counts of items when countsare between 0 and 10.
Square Root Transformation
k>0
Response is positive and continuous.
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
0 10 20 30 40
Sample Mean
Sam
ple
Vari
an
ce
Logaritmic Transformation
It is used when the standard deviation of samples are roughly proportional to the means
There is an evidence of multiplicative rather than additive
Data with negative values or zero can not be transformed. It is suggested to add 1 before transformation
This transformation tends to work when the variance is a linear function of the square of the mean
• Replace Y by Y+1 if zero occurs.• Useful if effects are multiplicative (later).• Useful If there is considerable heterogeneity
in the data.
Z Y ln( )
2 2ki i
Typical use: 1. Growth over time.2. Concentrations.3. Counts of times when counts
are greater than 10.
Logarithmic Transformation
k>0
Response is positive and continuous.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40
Sample Mean
Sam
ple
Vari
an
ce
Arcus sinus or angular Transformation
It is used when we are dealing with counts expressed as percentages or proportion of the total sample
Such data generally have a binomial distribution
Such data normally show typical characteristics in which the variances are related to the means
With proportions, the variance is a linear function of the mean times (1-mean) where the sample mean is the expected proportion.
• Y is a proportion (decimal between 0 and 1).• Zero counts should be replaced by 1/4, and
N by N-1/4 before converting to percentages
YarcsinYsinZ 1
i i i
k2 1
Response is a proportion.
Typical use: 1. Proportion of seeds germinating.2. Proportion responding.
ARCSINE SQUARE ROOT
Response is positive and continuous.
This transformation works when the variance is a linear function of the fourth power of the mean.
• Use Y+1 if zero occurs• Useful if the reciprocal of the original
scale has meaning.
ZY
1
i i
k2 4
Typical use: Survival time.
Reciprocal Transformation
n
i
i
i
i
i
yn
y
y
z
1
1
ln1
exp
0ln
01
suggestedtransformation
geometric mean of the original data.
Exponent, 𝒍, is unknown. Hence the model can be viewed as having an additional parameter which must be estimated (choose the value of l that minimizes the residual sum of squares).
Box/Cox Transformations (advanced)
Metode Non-parametrik
• Uji Kruskal Walis RAL
• Uji Friedman RAK
Daftar Pustaka
1) Mattjik, A.A dan I M Sumertajaya. 2002. Perancangan Percobaan dengan Aplikasi SAS danMinitab, Jilid I. IPB Press. Bogor.
2) Pustaka lain yg relevan.