Pengujian AsumsiKuliah 8 | Perancangan Percobaan

rahmaanisa@apps.ipb.ac.id

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.