DESAIN EKSPERIMEN SIMPLE COMPARATIVE EXPERIMENTS · Desain Eksperimen - Semester Genap 2017/2018. Introduction (1/3) • Formulation of a cement mortar • Original formulation and

DESAINEKSPERIMENSIMPLECOMPARATIVE

EXPERIMENTS

SemesterGenap2017/2018JurusanTeknikIndustriUniversitasBrawijaya

Outline•  Introduction•  BasicStatisticalConcepts•  TheHypothesisTestingFramework

–  Two-Samplet-Test–  TheTwo-Sample(Pooled)t-Test–  Two-samplet-test:Choiceofsamplesize

•  AnIntroductiontoExperimentalDesign-Howtosample?–  CompletelyRandomizedDesign–Howtosample

•  Importanceofthet-Test•  Two-samplet-test—ConfidenceIntervals•  Othertopics

2/33 12/02/18 Desain Eksperimen - Semester Genap 2017/2018

Introduction(1/3)

•  Formulationofacementmortar•  Originalformulationandmodifiedformulation•  10samplesforeachformulation•  Onefactoràformulation•  Twoformulations:àtwotreatmentsàtwolevelsofthefactorformulation

3 12/02/18 Desain Eksperimen - Semester Genap 2017/2018

Introduction(2/3)

4

Results:

12/02/18 Desain Eksperimen - Semester Genap 2017/2018

Introduction(3/3)

5

Dot diagram


BasicStatisticalConcepts(1/27)

•  Experiencesfromaboveexample–  Run–eachofaboveobservations–  Noise,experimentalerror,error–theindividualrunsdifference

–  Statisticalerror–arisesfromvariationthatisuncontrolledandgenerallyunavoidable

–  Thepresenceoferrormeansthattheresponsevariableisarandomvariable

–  Randomvariablecouldbediscreteorcontinuous



•  Describingsampledata–  Graphicaldescriptions

•  Dotdiagram—centraltendency,spread•  Boxplot–•  Histogram


8





10

• Discrete vs continuous



•  Probabilitydistribution– Discrete

– Continuous

11

∑ =

==

≤≤

jy

j

jjj

jj

yp

yypyyPyyp

of valuesall

1)(

of valuesall )()( of valuesall 1)(0

1)(

)()(

)(0

=

=≤≤

≤

∫

∫∞

∞−dyyf

dyyfbyap

yfb

a



•  Probabilitydistribution– Mean—measureofitscentraltendency

– Expectedvalue–long-runaveragevalue

12

⎪⎩

⎪⎨

⎧= ∑∫∞

∞−

yallyyypydyyyf

discrete )(continuous )(

µ

⎪⎩

⎪⎨

⎧== ∑∫∞

∞−

yallyyypydyyyf

yE

discrete )(continuous )(

)(µ



•  Probabilitydistribution– Variance—variabilityordispersionofadistribution

13

2

22

(

2

2

)(

])[(

discrete )()(continuous )()(

σ

µσ

µ

µσ

=

−=

⎪⎩

⎪⎨

⎧

−

−= ∑∫∞

∞−

yVor

yEor

yypyydyyfy

yall



•  Probabilitydistribution– Properties:cisaconstant

•  E(c)=c•  E(y)=μ•  E(cy)=cE(y)=cμ•  V(c)=0•  V(y)=σ2•  V(cy)=c2σ2•  E(y1+y2)=μ1+μ2



•  Probabilitydistribution– Properties:cisaconstant

•  V(y1+y2)=V(y1)+V(y2)+2Cov(y1,y2)•  V(y1-y2)=V(y1)+V(y2)-2Cov(y1,y2)•  Ify1andy2areindependent,Cov(y1,y2)=0•  E(y1*y2)=E(y1)*V(y2)=μ1*μ2•  E(y1/y2)isnotnecessaryequaltoE(y1)/V(y2)



•  Samplingandsamplingdistribution– Randomsamples--ifthepopulationcontainsNelementsandasampleofnofthemistobeselected,andifeachofN!/[(N-n)!n!]possiblesampleshasequalprobabilitybeingchosen

– Randomsampling–aboveprocedure– Statistic–anyfunctionoftheobservationsinasamplethatdoesnotcontainunknownparameters



•  Samplingandsamplingdistribution– Samplemean

– Samplevariance

17

n

yy

n

ii∑

== 1

1

)(1

2

2

−

−=∑=

n

yys

n

ii



•  Samplingandsamplingdistribution– Estimator–astatisticthatcorrespondtoanunknownparameter

– Estimate–aparticularnumericalvalueofanestimator

– Pointestimator:toμands2toσ2– Propertiesonsamplemeanandvariance:

•  Thepointestimatorshouldbeunbiased•  Anunbiasedestimatorshouldhaveminimumvariance

18

y



•  Samplingandsamplingdistribution– Sumofsquares,SSin

19

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

∑=

1

)()( 1

2

2

n

yyESE

n

ii

Sumofsquares,SS,canbedefinedas

∑=

−=n

ii yySS

1

2)(



•  Samplingandsamplingdistribution– Degreeoffreedom,v,numberofindependentelementsinasumofsquarein

20

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

∑=

1

)()( 1

2

2

n

yyESE

n

ii

Degreeoffreedom,v,canbedefinedas1−= nv



•  Samplingandsamplingdistribution– Normaldistribution,N

21

∞<<∞= −− y- 2

1)(2]/))[(2/1( σµ

πσyeyf



•  Samplingandsamplingdistribution– StandardNormaldistribution,z,anormaldistributionwithμ=0andσ2=1

22

),z~N(ei

yz

10.,.

σµ−

=



•  Samplingandsamplingdistribution– CentralLimitTheorem–Ify1,y2,…,ynisasequenceofnindependentandidenticallydistributedrandomvariableswithE(yi)=μandV(yi)=σ2andx=y1+y2+…+yn,thenthelimitingformofthedistributionof

asnà∞,isthestandardnormaldistribution

23

2σ

µ

nnxzn

−=



•  Samplingandsamplingdistribution– Chi-square,χ2,distribution–Ifz1,z2,…,zkarenormallyandindependentlydistributedrandomvariableswithmean0andvariance1,NID(0,1),therandomvariable

followsthechi-squaredistributionwithkdegreeoffreedom.

24

222

21 ... kzzzx +++=

2/1)2/(2/ )2/(21)( xk

k exk

xf −−

Γ=



•  Samplingandsamplingdistribution– Chi-squaredistribution–exampleIfy1,y2,…,ynarerandomsamplesfromN(μ,σ2),distribution,

– SamplevariancefromNID(μ,σ2),

25

212

1

2

2 ~)(

−=∑ −

= n

n

ii yy

SSχ

σσ

21

222 )]1/([~ .,. 1 −−

−= nnSeinSSS χσ



•  Samplingandsamplingdistribution–  tdistribution–Ifzandareindependentstandardnormalandchi-squarerandomvariables,respectively,therandomvariable

followstdistributionwithkdegreesoffreedom

26

2kχ

/2 kztk

kχ

=



•  Samplingandsamplingdistribution– pdfoftdistribution–

μ=0,σ2=k/(k-2)fork>2

27

∞<<∞−+Γ

+Γ= + t

ktkkktf k

]1)/[(1

)2/(]2/)1[()( 2/)1(2π





•  Samplingandsamplingdistribution–  Ify1,y2,…,ynarerandomsamplesfromN(μ,σ2),thequantity

isdistributedastwithn-1degreesoffreedom

29

nSyt/µ−

=



•  Samplingandsamplingdistribution– Fdistribution—Ifandaretwoindependentchi-squarerandomvariableswithuandvdegreesoffreedom,respectively

followsFdistributionwithunumeratordegreesoffreedomandvdenominatordegreesoffreedom

30

2uχ

2vχ

vuF

v

uvu /

/2

2

, χχ

=



•  Samplingandsamplingdistribution– pdfofFdistribution–

31

∞<<+ΓΓ

+Γ= +

−

xxvuvxuxvuvuxh vu

uu

0 ]1)/)[(2/()/(

)/](2/)[()( 2/)(

1)2/(2/



•  Samplingandsamplingdistribution– Fdistribution–exampleSupposewehavetwoindependentnormaldistributionswithcommonvarianceσ2,ify11,y12,…,y1n1isarandomsampleofn1observationsfromthefirstpopulationandy21,y22,…,y2n2isarandomsampleofn2observationsfromthesecondpopulation

32

1 ,122

21

21~ −− nnFS

S


TheHypothesisTestingFramework

•  Statisticalhypothesistestingisausefulframeworkformanyexperimentalsituations

•  Originsofthemethodologydatefromtheearly1900s

•  Wewilluseaprocedureknownasthetwo-samplet-test


Two-Samplet-Test(1/8)

–  Supposewehavetwoindependentnormal,ify11,y12,…,y1n1isarandomsampleofn1observationsfromthefirstpopulationandy21,y22,…,y2n2isarandomsampleofn2observationsfromthesecondpopulation



–  Amodelfordata

εisarandomerror

35

),0(~,,...,2,12,1

{ 2iij

jijiij NID

nji

y σεεµ=

=++



•  Samplingfromanormaldistribution•  Statisticalhypotheses:

36

0 1 2

1 1 2

::

HH

µ µ

µ µ

=

≠12/02/18 Desain Eksperimen - Semester Genap

2017/2018


•  H0iscalledthenullhypothesisandH1iscallalternativehypothesis.

•  One-sidedvstwo-sidedhypothesis•  TypeIerror,α:thenullhypothesisisrejectedwhenitistrue

•  TypeIIerror,β:thenullhypothesisisnotrejectedwhenitisfalse

37

false) is |reject tofail()error II type() trueis |reject ()error I type(

00

00

HHPPHHPP

==

==

β

α



•  Powerofthetest:

•  TypeIerroràsignificancelevel•  1-α=confidencelevel

38

false) is |reject (1 00 HHPPower =−= β



•  Two-sample-t-test–  Hypothesis:

–  Teststatistic:

where

39

11

210 11

nnS

yyt

p +

−=

0 1 2

1 1 2

::

HH

µ µ

µ µ

=

≠

)2()1()1(

21

222

2112

−+−+−

=nn

SnSnSp



40

1

2 2 2

1

1 estimates the population mean

1 ( ) estimates the variance 1

n

ii

n

ii

y yn

S y yn

µ

σ

=

=

=

= −−

∑

∑




Example–Two-Samplet-Test(1/6)

121

1

1

16.76

0.1000.31610

ySSn

=

=

=

=

222

2

2

17.04

0.0610.24810

ySSn

=

=

=

=

42

Formulation 1

“New recipe”

Formulation 2

“Original recipe”



43

1 2

22y

Use the sample means to draw inferences about the population means16.76 17.04 0.28

Difference in sample meansStandard deviation of the difference in sample means

This suggests a statistic:

y y

nσ

σ

− = − = −

=

1 20 2 2

1 2

1 2

Z y y

n nσ σ

−=

+



44

2 2 2 21 2 1 2

1 22 2

1 2

1 2

2 2 21 2

2 22 1 1 2 2

1 2

Use and to estimate and

The previous ratio becomes

However, we have the case where Pool the individual sample variances:

( 1) ( 1)2p

S Sy yS Sn n

n S n SSn n

σ σ

σ σ σ

−

+

= =

− + −=

+ −



•  Valuesoft0thatarenearzeroareconsistentwiththenullhypothesis•  Valuesoft0thatareverydifferentfromzeroareconsistentwiththe

alternativehypothesis•  t0isa“distance”measure-howfaraparttheaveragesareexpressedin

standarddeviationunits•  Noticetheinterpretationoft0asasignal-to-noiseratio

45

1 20

1 2

The test statistic is

1 1

p

y ytS

n n

−=

+



46

2 22 1 1 2 2

1 2

1 20

1 2

( 1) ( 1) 9(0.100) 9(0.061) 0.0812 10 10 2

0.284

16.76 17.04 2.201 1 1 10.284

10 10

The two sample means are a little over two standard deviations apartIs t

p

p

p

n S n SSn n

S

y ytS

n n

− + − += = =

+ − + −

=

− −= = = −

+ +

his a "large" difference?



•  P-value–Thesmallestlevelofsignificancethatwouldleadtorejectionofthenullhypothesis.

•  Computerapplication

Two-SampleT-TestandCISample NMeanStDevSEMean1 1016.7600.3160.102 1017.0400.2480.078Difference=mu(1)-mu(2)Estimatefordifference:-0.28095%CIfordifference:(-0.547,-0.013)T-Testofdifference=0(vsnot=):T-Value=-2.20P-Value=0.041DF=18BothusePooledStDev=0.2840


48

WilliamSealyGosset(1876,1937)

Gosset'sinterestinbarleycultivationledhimtospeculatethatdesignofexperimentsshouldaim,notonlyatimprovingtheaverageyield,butalsoatbreedingvarietieswhoseyieldwasinsensitive(robust)tovariationinsoilandclimate.Developedthet-test(1908)GossetwasafriendofbothKarlPearsonandR.A.Fisher,anachievement,foreachhadamonumentalegoandaloathingfortheother.Gossetwasamodestmanwhocutshortanadmirerwiththecommentthat“Fisherwouldhavediscovereditallanyway.”


TheTwo-Sample(Pooled)t-Test(1/3)

•  Sofar,wehaven’treallydoneany“statistics”

•  Weneedanobjectivebasisfordecidinghowlargetheteststatistict0reallyis

•  In1908,W.S.Gossetderivedthereferencedistributionfort0…calledthetdistribution

•  Tablesofthetdistribution–seetextbookappendix

49

t0 = -2.20


•  Avalueoft0between–2.101and2.101isconsistentwithequalityofmeans

•  Itispossibleforthemeanstobeequalandt0toexceedeither2.101or–2.101,butitwouldbea“rareevent”…leadstotheconclusionthatthemeansaredifferent

•  CouldalsousetheP-valueapproach

50

t0 = -2.20


•  TheP-valueisthearea(probability)inthetailsofthet-distributionbeyond-2.20+theprobabilitybeyond+2.20(it’satwo-sidedtest)

•  TheP-valueisameasureofhowunusualthevalueoftheteststatisticisgiventhatthenullhypothesisistrue

•  TheP-valuetheriskofwronglyrejectingthenullhypothesisofequalmeans(itmeasuresrarenessoftheevent)

•  TheP-valueinourproblemisP=0.042

51

t0 = -2.20


CheckingAssumptions–TheNormalProbabilityPlot


Two-samplet-testChoiceofsamplesize(1/4)

•  ThechoiceofsamplesizeandtheprobabilityoftypeIIerrorβarecloselyrelatedconnected

•  Supposethatwearetestingthehypothesis

•  AndThemeanarenotequalsothatδ=μ1-μ2

•  BecauseH0isnottrueàwecareabouttheprobabilityofwronglyfailingtorejectH0àtypeIIerror

53/72

0 1 2

1 1 2

::

HH

µ µ

µ µ

=

≠



•  Define

•  Onecanfindthesamplesizebyvaryingpower(1-β)andδ

54

σδ

σµµ

2221 =

−=d



Testingmean1=mean2(versusnot=)Calculatingpowerformean1=mean2+differenceAlpha=0.05Assumedstandarddeviation=0.25 SampleTargetDifferenceSize PowerActualPower0.25 27 0.950.9500770.25 23 0.900.9124980.25 10 0.550.5620070.50 8 0.950.9602210.50 7 0.900.9290700.50 4 0.550.656876Thesamplesizeisforeachgroup.




57/54

AnIntroductiontoExperimentalDesign-Howtosample?

n  Acompletelyrandomizeddesignisanexperimentaldesigninwhichthetreatmentsarerandomlyassignedtotheexperimentalunits.

n  Iftheexperimentalunitsareheterogeneous,blockingcanbeusedtoformhomogeneousgroups,resultinginarandomizedblockdesign.


CompletelyRandomizedDesign–Howtosample?(1/8)

58/54

n  Recall Simple Random Sampling n  Finite populations are often defined by lists

such as: n  Organization membership roster n  Credit card account numbers n  Inventory product numbers


59/54


n  A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected.

n  Replacing each sampled element before selecting subsequent elements is called sampling with replacement.

n  Sampling without replacement is the procedure used most often.


60/54


n  Inlargesamplingprojects,computer-generatedrandomnumbersareoftenusedtoautomatethesampleselectionprocess.

n  Excelprovidesafunctionforgeneratingrandomnumbersinitsworksheets.

n  Infinitepopulationsareoftendefinedbyanongoingprocesswherebytheelementsofthepopulationconsistofitemsgeneratedasthoughtheprocesswouldoperateindefinitely.


61/54


n  A simple random sample from an infinite population is a sample selected such that the following conditions are satisfied. n  Each element selected comes from the same population. n  Each element is selected independently.


62/54


n  Random Numbers: the numbers in the table are random, these four-digit numbers are equally likely.


63/54


n  Most experiments have critical error on random sampling.

n  Ex: sampling 8 samples from a production line in one day n  Wrong method:

n  Get one sample every 3 hoursà not random!


64/54


n  Ex: sampling 8 samples from a production line n  Correct method:

n  You can get one sample at each 3 hours interval but not every 3 hours à correct but not a simple random sampling

n  Get 8 samples in 24 hours n  Maximum population is 24, getting 8 samplesà two digits n  63, 27, 15, 99, 86, 71, 74, 45, 10, 21, 51, … n  Larger than 24 is discarded n  So eight samples are collected at:

15, 10, 21, … hour


65/54


n  In Completely Randomized Design, samples are randomly collected by simple random sampling method.

n  Only one factor is concerned in Completely Randomized Design, and k levels in this factor.


Importanceofthet-Test

•  Providesanobjectiveframeworkforsimplecomparativeexperiments

•  Couldbeusedtotestallrelevanthypothesesinatwo-levelfactorialdesign,becauseallofthesehypothesesinvolvethemeanresponseatone“side”ofthecubeversusthemeanresponseattheopposite“side”ofthecube


Two-samplet-test—ConfidenceIntervals(1/2)

•  Hypothesistestinggivesanobjectivestatementconcerningthedifferenceinmeans,butitdoesn’tspecify“howdifferent”theyare

•  Generalformofaconfidenceinterval

•  The100(1-α)%confidenceintervalonthedifferenceintwomeans:

67

where ( ) 1 L U P L Uθ θ α≤ ≤ ≤ ≤ = −

1 2

1 2

1 2 / 2, 2 1 2 1 2

1 2 / 2, 2 1 2

(1/ ) (1/ )

(1/ ) (1/ )n n p

n n p

y y t S n n

y y t S n nα

α

µ µ+ −

+ −

− − + ≤ − ≤

− + +


68

Two-samplet-test—ConfidenceIntervals(1/2)


OtherTopics

•  Hypothesistestingwhenthevariancesareknown—two-sample-z-test

•  Onesampleinference—one-sample-zorone-sample-ttests

•  Hypothesistestsonvariances–chi-squaretest•  Pairedexperiments


OtherTopics


OtherTopics


OtherTopics


Documents

DESAIN EKSPERIMEN SIMPLE COMPARATIVE EXPERIMENTS · Desain Eksperimen - Semester Genap 2017/2018. Introduction (1/3) • Formulation of a cement mortar • Original formulation and