Robust Brown-Forsythe and Robust Modified Brown-Forsythe ANOVA Tests Under

16 Pages·2016·0.63 MB·English

Revista Colombiana de Estadística
January 2016, Volume 39, Issue 1, pp. 17 to 32
DOI: http://dx.doi.org/10.15446/rce.v39n1.55135
Robust Brown-Forsythe and Robust Modified
Brown-Forsythe ANOVA Tests Under
Heteroscedasticity for Contaminated Weibull
Distribution
ANOVAS robustos heterocedásticos Brown-Forsythe y
Brown-Forsythe modificado para la distribución Weibull contaminada
Derya Karagöz1,a, Tülay Saraçbaşı1,b
1Department of Statistics, Faculty of Science, Hacettepe University, Ankara,
Turkey
Abstract
Inthisstudy,robustBrown-ForsytheandrobustModifiedBrown-Forsythe
ANOVA tests are proposed to take into consideration heteroscedastic and
non-normalitydatasetswithoutliers. Thenon-normaldataisassumedtobe
a two parameters Weibull distribution. Robust proposed tests are obtained
by using robust mean and variance estimators based on median/MAD and
median/Q methods instead of maximum likelihood. The behaviors of the
n
robust proposed and classical ANOVA tests are examined by simulation
study. The results shows that the proposed robust tests have good per-
formanceespeciallyinthepresenceofheteroscedasticityandcontamination.
Key words:Brown-Forsythe, Modified Brown-Forsythe, ANOVA, Weibull
Distribution.
Resumen
En este estudio se proponen tests Brown-Forsythe y robustos Brown-
Forsythe ANOVA para tener en cuenta la no-normalidad en datos debida a
lapresenciadedatosatípicos. Seasumequelosdatosno-normalestienenuna
distribución Weibull de dos parámetros. Estos tests se construyen en base
aestimadoresrobustosdemediayvarianzaobtenidosconmétodosbasados
en la mediana en vez de métodos de máxima verosimilitud. Se examina en
comportamientodeestostestscondatossimulados. Losresultadosmuestran
queéstostienenunbuendesempeño,especialmenteenpresenciadeatípicos
y datos contaminados.
Palabrasclave:ANOVA,Brown-Forsythe,Brown-Forsythemodificado,dis-
tribución Weibull.
aPhD.E-mail: [email protected]
bProfessor.E-mail: [email protected]
17
18 Derya Karagöz & Tülay Saraçbaşı
1. Introduction
The procedures to test equal means in the conventional analysis of variance
(ANOVA) are based on assumptions of normality, independence, and equality of
the error variances.
If the assumptions of normality and homogeneity of variances are invalid and
alsooutliersarepresent,classicalANOVAdoesnotgiveaccurateresults. Theone-
way ANOVA under the violation of assumptions has been studied extensively. To
deal with non-normal data and/or heteroscedastic variances across groups, many
alternativessuchasQ,Welch,Brown-ForsytheandModifiedBrown-Forsythetests
have been developed instead of classical ANOVA. The statistic Q has been exten-
sively studied by many authors under a variety of assumptions. James (1951) and
Welch (1951) derived improved approximations for the distribution of Q under
thenullhypothesis;theseapproximationsaremoreaccurateforsmallsamplesizes
of groups. A number of authors have discussed extensions of the Welch methods
based on the use of robust estimators for the population location and scale pa-
rameters. Notable among these are the efforts of Wilcox (1995), Wilcox (1997),
and the references contained in these papers. Kulinskaya & Dollinger (2007) con-
sider three common robust estimators: Huber’s proposed two estimators of lo-
cation and scale, Hampel’s M-estimator of location with scale estimated by the
median absolute deviation (MAD), and the trimmed mean with scale estimated
by the Winsorized standard deviation. In the presence of heteroscedasticity, the
actual size of the ANOVA (which uses the F-test) can exceed the nominal level.
The challenge of how to accurately compare independent heteroscedastic normal
means of three or more groups is referred to as the generalized Behrens-Fisher
problem. Various solutions to this problem have been proposed. In practice,
some approximate procedures such as the weighted F, Welch, Brown Forsythe
(BF) and modified Brown Forsythe (MBF) tests are widely used (cf. Gamage
& Weerahandi 1998, Welch 1951, Brown & Forsythe 1974). Weerahandi (1995)
introduced a test using the notion of generalized p-value for comparing the means
ofk populationswithunequalerrorvariances. Senoglu(2005)andSenoglu(2007)
obtainedrobustandefficientestimatorsoftheparametersinthedifferentfactorial
designsbyusingthemethodologyknownasmodifiedmaximumlikelihood(MML)
andproposednewteststatisticsbasedonMMLestimatorstotestthemaineffects
andtheinteractionswhenthedistributionoferrortermsisgeneralizedlogistic. In
this paper we consider the problem of comparing the means of k populations not
only with heteroscedastic variances and non-normality, but also with the outliers.
Normality is the one of the important assumptions of ANOVA. However, in
theapplicationnormalityassumptiondoesnotworkforthereallifedatamodeled
by the exponential, Weibull or lognormal distributions; especially in the field of
reliability, engineering and life sciences. The characteristics of these distributions
can be explained by the Weibull distribution, which is also known as Extreme
Value Type III minimum distribution. This has made it extremely popular in
reliability engineering, biology and medicine. This distribution is the most com-
monly used distribution for modeling reliability data, because it represents a wide
range of asymmetric distributions. Moreover, ANOVA cannot handle censored or
RevistaColombianadeEstadística39(2016)17–32
Robust Brown-Forsythe and Robust Modified Brown-Forsythe ANOVA 19
intervaldatabecauseofnon-normality. Thesimplestpossiblelifetimedistribution
is exponential distribution. However, its constant hazard rate is improper and
unrealistic in many cases. Gamma distribution is another candidate distribution
for lifetimes. Nevertheless, distribution function or survival function of gamma
distribution cannot be expressed in a closed form if the shape parameter is not
an integer. It is in terms of an incomplete gamma function, that one needs to
obtainthedistributionfunction,survivalfunctionorthehazardratebynumerical
integration. This makes gamma distribution a little unpopular compared to the
Weibull distribution, which has a nice distribution function, survival function and
hazard function Gupta & Kundu (2001). It has been used in many different fields
such as material science, engineering, physics, chemistry, meteorology,medicine,
pharmacy, quality control, biology, geology, geography, economics and business.
The purpose of this study is to develop test statistics by using robust methods
for the non-normal distribution with an outlier for the one-way ANOVA. For this
reasontheWeibulldistributionwiththeparametersλandβ isdiscussedthemost
frequently encountered in the literature as a non-normal distribution. Robust
estimators of Weibull distribution parameters are based on the median/median
absolute deviation (median/MAD) and median/Q methods. To test the hy-
n
pothesis of equal means under the assumption of heteroscedastic variances the
standard BF and MBF tests were proposed. To obtain these standard tests, the
maximumlikelihood(ML)meanandvarianceestimatesareused. However,incase
of the outliers these tests can exceed the nominal level. This is why we recom-
mend replacing the ML mean and variance estimates with the robust estimates
in the formulation of the tests. The equations from the robust tests are obtained
by using the robust estimators of mean and variance based on median/MAD and
median/Q methods and tests are called Robust Brown-Forsythe (RBF) and ro-
n
bust Modified Brown-Forsythe (RMBF). The behavior of the developed robust
tests is examined by simulation study.
TherobustmeanandvarianceestimationsofWeibulldistributionisoneofthe
important subjects of this paper, since proposed robust tests are obtained based
on these estimators. The mean and variance of a Weibull random variable are the
functions of the parameters. Therefore, we consider robust estimators of Weibull
distributionparametersthatarethemedian/MAD proposedbyOlive(2006)and
the median/Q proposed by Boudt, Caliskan & Croux (2011). These are two
n
robust estimator alternatives to the maximum likelihood estimator. In this paper
they use the log-Weibull density as a location-scale family with location µ=logλ
andscaleσ =1/β (seeBoudtetal.2011). Aninitialorauxiliaryestimateofscale
is frequently required in robust estimation. This paper usually uses the median
absolute deviation MAD = 1.4826med |x −med x |, because it has a simple
n i i j j
explicit formula, needs little computation time, and is very robust as witnessed
by its bounded influence function and its 50% breakdown point. However MAD
n
is aimed at symmetric distributions and it has a low (37%) Gaussian efficiency.
Rousseeuw&Croux(1993)setouttoconstructexplicitand50%breakdownscale
estimators that are more efficient. Rousseeuw & Croux (1993) considered the
estimator Q given by the 0.25 quantile of the distances |x −x |;i<j. The
n i j
advantages of the Q are that it does not need any location estimate and the
n
Gaussian efficiency of Q is 82%, Rousseeuw & Croux (1993).
n
RevistaColombianadeEstadística39(2016)17–32
20 Derya Karagöz & Tülay Saraçbaşı
Theremainderofourpaperisorganizedasfollows. Section2introducesrobust
and explicit estimators of the mean and variance of Weibull distribution. Section
3 explains the robust proposed test statistics. A numerical example is given in
Section4. Toshowtheperformanceofproposedteststatistics,asimulationstudy
and the results are presented in Section 5. Finally, the last section summarizes
the study’s.
2. Robust Estimators
The density and cumulative distribution functions of Weibull distribution are
given by
β
f (x) = (x/λ)β−1exp[−(x/λ)β], (1)
λ,β λ
F (x) = 1−exp[−(x/λ)β] (2)
λ,β
x,λ,β > 0. Since f (x) = 1f (x), the parameter λ is called the scale pa-
λ,β λ 1/λ,β
rameter. The parameter β is the shape parameter. The mean and variance of a
Weibull random variable can be expressed as
E(X) = λΓ(1+1/β) (3)
Var(X) = λ2[Γ(1+2/β)−Γ2(1+1/β)]. (4)
As shown in equation (3) and equation (4), the mean and variance of a Weibull
random variable are the functions of shape β and scale λ parameters of Weibull
distribution. After obtaining robust parameters estimators, the robust mean and
variance estimators of this distribution can be obtained. We consider robust es-
timators of Weibull distribution parameters that are median/MAD, proposed by
Olive(2006)andmedian/Q proposedbyBoudtetal.(2011). InthenextSection
n
2.1. these robust location-scale estimators with be explained.
2.1. Location-Scale Estimators
The proposed robust estimators are based on the log-Weibull density which is
a location-scale family with location µ = logλ and scale σ = 1/β. Estimation
of Weibull parameters can thus be seen as an estimation problem of the location
andscaleoflogx ,...,logx . Olive(2006)presentsthecorrectionfactorsmaking
1 n
theseestimatorsconsistentforthelog-Weibulldistribution. Olive(2006)proposed
the standard location and scale estimators with a 50% breakdown point, which
were the median and median absolute deviation
σˆ =1.3037med |logx −med logx | (5)
MAD j j i i
µˆ =med logx −σˆ loglog2, (6)
med/MAD i i MAD
where x ∼ W(λ,β) and y = logx have log-Weibull distribution. By using
i i i
the location and scale estimators based on med/MAD given by (equation 5) and
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Robust Brown-Forsythe and Robust Modified Brown-Forsythe ANOVA 21
(equation6),respectively,Boudtetal.(2011)proposedthemed/MADestimators
of scale and shape parameters of Weibull distribution:
λˆ =exp(µˆ ) (7)
med/MAD med/MAD
βˆ =1/σˆ . (8)
med/MAD MAD
By considering robust med/MAD estimators given by equation (7) and equation
(8),themed/MADestimatorsofthemeanandthevarianceofWeibulldistribution
are obtained as per the following
µˆ = λˆ Γ(1+1/βˆ ) (9)
Wmed/MAD med/MAD med/MAD
σˆ = λˆ2 [Γ(1+2/βˆ )−Γ2(1+1/βˆ )]. (10)
Wmed/MAD med/MAD med/MAD med/MAD
Q scale-estimator is as a more efficient alternative and the the same breakdown
n
point of 50%. Boudt et al. (2011) recommended estimating σ using the Q scale-
n
estimator proposed by Rousseeuw & Croux (1993). It was given by Boudt et al.
(2011)
σˆ = 1.9577{|logx −logx |;i<j} (11)
Qn i j (l)
µˆ = med logx −σˆ loglog2, (12)
med/Qn i i Qn
where the last part is the lth ordered value among this set of (cid:0)n(cid:1) values, where
2
l = (cid:0)h(cid:1) ≈ (cid:0)n(cid:1)/4 with h = (cid:98)n/2(cid:99)+1. The correction factor 1.9577 ensures Fisher
2 2
consistency. It equals the inverse of the 1/4 quantile of the distribution of the ab-
solute difference between two log-Weibull random variables. These estimators are
called med/Q estimators for scale and shape parameters. By using the med/Q
n n
location-scale estimators in equation (11) and equation (12), Boudt et al. (2011)
proposed the robust scale and shape estimators of Weibull distribution based on
med/Q
n
λˆ = exp(µˆ ) (13)
med/Qn med
βˆ = 1/σˆ . (14)
med/Qn Qn
Byconsideringrobustestimatorsinequation(13)andequation(14), themed/Q
n
mean and variance estimators of Weibull distribution are obtained as per the
following
µˆ = λˆ Γ(1+1/βˆ ) (15)
Wmed/Qn med/Qn med/Qn
σˆ = λˆ2 [Γ(1+2/βˆ )−Γ2(1+1/βˆ )]. (16)
Wmed/Qn med/Qn med/Qn med/Qn
In this study, to define the breakdown point of estimations, previous studies are
taken into account, especially Boudt et al. (2011). The breakdown point of the
mean estimator and the variance estimator for Weibull distribution based on the
proposed robust estimator are obtained a 50% as the breakdown of mean and
variance depends on the the breakdown point of shape and scale estimators of the
Weibull distribution.
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22 Derya Karagöz & Tülay Saraçbaşı
3. Robust BF and MBF ANOVA Tests
TheANOVAfornon-normaldatawithheteroscedasticvariancehasbeenstud-
ied extensively. In the case of disruption of assumptions, instead of classical
ANOVA, many tests have been developed such as BF and MBF. Brown &
Forsythe (1974) proposed the BF test that can be shown by
(cid:80)k n (µˆ −µˆ )2
BF = i=1 i i. .. (17)
(cid:80)k (1−n /N)σˆ2
i=1 i i
where µˆ , σˆ and n denoted the ML estimator of the sample mean and variance
i. i i
and size for the ith k group, respectively, µˆ is the ML estimator of overall mean,
..
and N = (cid:80)k n . The BF test has a F distribution with k −1 and v
i=1 i k−1,v1 1
degrees of freedom. v is defined as
1
[(cid:80)k (1−n /N)σˆ2]2
v = i=1 i i . (18)
1 (cid:80)k (1−n /N)2σˆ4/(n −1)
i=1 i i i
Mehrotra (1997) proposed the MBF test given by
(cid:80)k n (µˆ −µˆ )2
MBF = i=1 i i. .. , (19)
(cid:80)k (1−n /N)σˆ2
i=1 i i
whichisthesameastheequation(17). ButtheMBF testhasF distribution
v2,v1
with v and v degrees of freedom. By using Mehrotra’s (1997) approach the
2 1
numerator degrees of freedom v was defined as
2
[(cid:80)k (1−n /N)σˆ2]
v = i=1 i i . (20)
2 (cid:80)k σˆ4+[(cid:80)ki=1niσˆi2]2−2(cid:80)ki=1niσˆi4
i=1 i N N
In this study, two robust tests are proposed to test the equality of population
meansundertheWeibulldistribution. Theproposedtestscorrespondtothestan-
dard BF and MBF test statistics in which the maximum likelihood mean and
variance estimators are replaced with robust estimators. We propose using the
RBF andRMBF testsbasedonthemed/MAD andmed/Q meanandvariance
n
estimators of Weibull distribution. The RBF test is given by
(cid:80)k n (µˆ −µˆ )2
RBF = i=1 i ri. r.. , (21)
(cid:80)k (1−n /N)σˆ2
i=1 i ri
whereµˆ ,σˆ andn denotedtherobustestimatorsofmeanandvarianceandsize
ri. ri i
fortheithkgroup,µˆ istherobustestimatorofoverallmean,andN =(cid:80)k n .
r.. i=1 i
TheRBF testhasaF distributionwithk−1andv degreesoffreedom. v
k−1,vr r r
is defined as
[(cid:80)k (1−n /N)σˆ2]2
v = i=1 i ri . (22)
r (cid:80)k (1−n /N)2σˆ4/(n −1)
i=1 i ri i
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Robust Brown-Forsythe and Robust Modified Brown-Forsythe ANOVA 23
The RMBF tests is given by
(cid:80)k n (µˆ −µˆ )2
RMBF = i=1 i ri. r.. , (23)
(cid:80)k (1−n /N)σˆ2
i=1 i ri
which is the same as the equation (21). But the RMBF test has a F distri-
vr2,vr
bution with v and v degrees of freedom. By using Mehrotra’s (1997) approach
r2 r
the numerator degrees of freedom v is defined as
r2
[(cid:80)k (1−n /N)σˆ2]
v = i=1 i ri . (24)
r2 (cid:80)k σˆ4 +[(cid:80)ki=1niσˆr2i]2−2(cid:80)ki=1niσˆr4i
i=1 ri N N
In the above equations equation (21), equation (22), equation (23) and equation
(24), µˆ , µˆ , σˆ robust mean and variance estimators are based on med/MAD
ri. r.. ri
and med/Q methods.
n
4. A Numerical Example
Asanexample,threeWeibullpopulations,eachwithasizeof5,000,aregener-
ated with shape the following and scale parameters: The shape parameter of the
first population is 1.5 and the scale parameter is 1. With the second population,
the shape parameter is 2.5 and the scale parameter is 1. The shape parameter
of the third population is 3.5 and the scale parameter is 1. Supposing that three
random samples with a size of 20 are taken from each Weibull population, the
sample data are shown in Table 1.
Table 1: Three random samples with a size of 20 taken from each of three Weibull
populations.
Sample1 Sample2 Sample3
0.7110 0.6601 0.9159
0.4247 0.9464 0.5436
0.8905 0.7786 0.9487
1.4064 0.7703 0.9932
0.8012 1.2813 0.9775
2.0574 0.5002 1.0214
0.6533 1.6926 0.9030
1.7747 1.0385 1.3542
0.3197 0.6782 0.9577
1.8344 0.8490 0.9496
0.6231 1.2822 0.6689
0.6943 1.1174 1.2202
1.2458 0.5139 1.0077
1.0185 0.8083 0.4797
0.2401 0.8469 1.0241
2.6065 1.0146 0.8206
0.8898 0.5889 0.5370
0.2850 2.0002 0.7586
1.2093 1.5597 1.0501
2.3521 0.4992 0.4278
RevistaColombianadeEstadística39(2016)17–32
24 Derya Karagöz & Tülay Saraçbaşı
The Weibull P-P plot for each sample of this data is presented in Figure 1.
The results show that each sample of data is Weibull.
Figure 1: Weibull Probability Plot of Three Samples
The Anderson-Darling test Anderson & Darling (1952) is used to test if a
sample of data comes from a specific distribution. It is a modification of the
Kolmogorov-Smirnov (K-S) test and gives more weight to the tails than the K-S
test. The K-S test is distribution free in the sense that the critical values do not
depend on the specific distribution being tested. The Anderson-Darling (A-D)
test makes use of the specific distribution in calculating critical values. This has
the advantage of allowing a more sensitive test and the disadvantage that critical
values must be calculated for each distribution Trujillo-Ortiz, Hernandez-Walls,
Barba-Rojo, Castro-Perez & Lavaniegos-Espejo (2007). The sampled populations
have a Weibull distribution. The results of the Anderson-Darling tests for three
samplesareshowninTable2. Thus,thesesampleshavebeendrawnfromaWeibull
population with estimated parameters that are given in Table 2.
Weibull data are used to test the equality of the population means using
ANOVA. We consider not only clean samples but also contaminated samples. To
generate contaminated samples, 4 observations from each sample are randomly
replaced with 100. The classical ANOVA, RBF and RGBF tests are obtained for
RevistaColombianadeEstadística39(2016)17–32
Robust Brown-Forsythe and Robust Modified Brown-Forsythe ANOVA 25
Table 2: Results of Anderson-Darling statistic.
Sample1 Sample2 Sample3
A-Dstatistic 0.2440 0.4757 0.6327
A-Dadjustedstatistic 0.2549 0.4970 0.6610
Prob. A-Dstatistic 0.6577 0.2210 0.0872
Significance 0.050 0.050 0.050
WeibullpopulationW(λˆ,βˆ) W(1.2409,1.7039) W(1.0974,2.5538) W(0.9661,4.2887)
clean and contaminated samples. The results are shown in Table 3. F , RBF
obt obt
and RMBF indicate the calculated tests. F , RBF and RMBF indicate the
obt t t t
Ftablevalueatthesignificantof5%(α=0.05). RBF andRMBF arebased
obt obt
on med/MAD and med/Q robust methods. In the contamination case classical
n
ANOVA is badly deteriorated. For clean and contaminated data, since calculated
tests<Ftablevalue,therearenotsignificantdifferencesbetweenthethreepopu-
lation means. The results shows that proposed robust tests give accurate results.
Table 3: Three random samples with a size of 20 taken from each of three Weibull
populations.
CleanSample ContaminatedSample
Tests
med/MAD med/Qn med/MAD med/Qn
Fobt 0.6905 1.0558 0.0003 0.0009
Ft 3.1588 3.1588 3.1588 3.1588
RBFobt 0.6219 0.4548 1.7068 0.4410
RBFt 3.2287 3.2715 3.3308 3.2954
RMBFobt 0.6219 0.4528 1.7068 0.4410
RMBFt 5.1432 3.3778 4.7062 3.5294
5. Simulation Study
In the simulation study the reference distribution is W(1,β). The value of the
shape parameters are selected, as in Table 4, with respect to the different exper-
imental designs we want to create. In this table for equal means, homogeneous
variances EA, EB are used for a balanced sample size, EC, ED are used for an
unbalanced sample size. For unequal means, heterogenous variances UA, UB are
usedforbalancedsamplesizeandUC,UDareusedforanunbalancedsamplesize.
As we mentioned, previously to generate data with equal means + homogeneous
variances and unequal means + heterogenous variances, we just change the shape
parameterofWeibulldistribution. Sincethemeanandvarianceofthedistribution
have been changed with respect to the shape and scale of the parameters of the
Weibull distribution, the creation of different experiment designs depends on the
parameters of the distribution.
In this table
• DesignEA:balancedsamplesize(n=5)withequalmeansandhomogenous
variances,
• Design UA: balanced sample size (n=5) with unequal means and heteroge-
nous variances,
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26 Derya Karagöz & Tülay Saraçbaşı
• DesignEB:balancedsamplesize(n=10)withequalmeansandhomogenous
variances,
• DesignUB:balancedsamplesize(n=10)withunequalmeansandheteroge-
nous variances,
• Design EC: unbalanced sample size (n = 5,10,15) with equal means and
homogenous variances,
• Design UC: unbalanced sample size (n=5,10,15) with unequal means and
heterogenous variances,
• Design ED: unbalanced sample size (n = 10,20,30) with equal means and
homogenous variances,
• DesignUD:unbalancedsamplesize(n=10,20,30)withunequalmeansand
heterogenous variances.
Table 4: Experimental designs for k=3, k=6 and k=9.
K=3 K=6 K=9
EA ni 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
β 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
UA ni 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
β 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5
EB ni 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
β 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
UB ni 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
β 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5
EC ni 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15
β 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
UC ni 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15
β 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5
ED ni 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30
β 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
UD ni 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30
β 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5
The simulation experiment stork with k =3 groups. The balanced and unbal-
anced sample sizes with homogenous and heterogeneous variances are considered.
Toseetheeffectofthenumberofgroups,k,onthelevelofthetests,wereplicated
the experiment for k = 3 groups two and three times to provide the simulation
experiment for k =6 and k =9 groups, respectively. By using the proposed esti-
matorsmed/MAD andmed/Q theTypeIerrorsoftherobustteststatisticsand
n
classical ANOVA are obtained with 10,000 repetitions. Four different models are
discussed below to test the behaviors of the tests, when the model is deteriorated
andinthepresenceofoutliers. Inmodelstheresponsevariableisassumedtohave
a Weibull distribution. The models for simulation study are:
RevistaColombianadeEstadística39(2016)17–32