L.T. Gama Luís Telo da Gama [email protected]Estação Zootécnica Nacional Faculdade de Medicina Veterinária - UTL Luís Telo da Gama [email protected]Estação Zootécnica Nacional Faculdade de Medicina Veterinária - UTL Statistics and Experimental Design Basic Principles L.T. Gama Basic principles Statistics Statistics is is often often used used as a as a drunk drunk man man uses a uses a street street lamp lamp… More for More for support support than than for for illumination illumination! ! L.T. Gama Test Test hypotheses hypotheses based based on on information information from from a a sample sample Obtain Obtain inferences inferences ( prediction prediction and and decision decision- making making) for a ) for a global global population population Information Data Knowledge Statistical analyses Biological integration Statistics – The basic problem… Statistics – The basic problem… L.T. Gama Corn Corn or or barley barley for for pigs pigs?? ?? Comparison Comparison of of effect effect of of: One One factor factor • Source Source of of energy energy ) With With two two treatments treatments • Corn Corn or or barley barley One One response response variable variable ) Growth Growth rate rate in in pigs pigs Example 1 Example 1 L.T. Gama Example 2 Example 2 What What is is the the effect effect of of the the level level of of PMSG PMSG administered administered on on ovulation ovulation rate rate in in sows sows? Comparison Comparison of of effect effect of of : : One One factor factor • Level Level of of PMSG PMSG administered administered ) With With several several treatments treatments • 250, 500, 750 UI 250, 500, 750 UI One One response response variable variable ) Ovulation Ovulation rate rate L.T. Gama Example 3 Example 3 Effects Effects of of bST bST administration administration to to dairy dairy cows cows? What What is is the the effect effect in in Holstein Holstein and and Jersey Jersey cows cows? What What is is the the optimum optimum level level? Is Is the the optimum optimum level level the the same same for for the the two two breeds breeds? Comparison Comparison of of effects effects of of: Two Two factors factors studied studied ) Breed Breed and and level level of of bST bST Treatments Treatments ) Breed Breed: : Holstein Holstein and and Jersey Jersey ) bST bST/day day: 0, 15, 30, 45 mg/ : 0, 15, 30, 45 mg/day day One One response response variable variable Milk Milk yield yield per per lactation lactation
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Estação Zootécnica NacionalFaculdade de Medicina Veterinária - UTL
Statistics andExperimental Design
Basic Principles
Statistics andExperimental Design
Basic Principles
L.T. Gama
Basic principlesBasic principles
StatisticsStatistics isis oftenoften usedusedas a as a drunkdrunk manman uses a uses a streetstreet lamplamp……More for More for supportsupport thanthan for for illuminationillumination! !
L.T. Gama
TestTest hypotheseshypotheses basedbasedonon informationinformation fromfrom a a samplesample
ObtainObtain inferencesinferences((predictionprediction andanddecisiondecision--makingmaking) for a ) for a global global populationpopulation
Information
Data
Knowledge
Statisticalanalyses
Biologicalintegration
Statistics – The basic problem…Statistics – The basic problem…
Characteristics of the normal distributionCharacteristics of the normal distribution
68%
95%
99%
Approximate!!!
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Steps in statistical analysisSteps in statistical analysis
DescriptiveDescriptive statisticsstatisticsPossiblePossible needneed for for transformationtransformation
Examples oftransformations
Original data
Transformed data
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Steps in statistical analysisInferentialInferential statisticsstatistics
InIn a a probabilisticprobabilistic mannermanner, , obtainobtain conclusionsconclusionswhichwhich cancan bebe appliedapplied to to thethe populationpopulation ofof interestinterest::
IsIs corncorn betterbetter thanthan barleybarley for for pigpig feedingfeeding??HowHow muchmuch does does ovulationovulation rate rate changechange perper additionaladditional IU IU ofof PMSG ?PMSG ?WhatWhat isis thethe optimumoptimum dosagedosage ofof bSTbST inin JerseyJersey andandHolsteinHolstein cowscows??
AnalysisAnalysis ofof variancevariance, , regressionregression, etc., etc.
Major objective Major objective inin statisticalstatistical analysisanalysis!!
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Important conceptsPopulationPopulation -- groupgroup ofof interestinterest for for thetheresearcherresearcher
UsuallyUsually notnot knownknown inin detaildetailParametersParameters ofof thethe populationpopulation are are notnot knownknown, , butbutare are estimableestimable
SampleSample –– partpart ofof thethe populationpopulation selectedselected for for thethe experimentexperiment
ConclusionsConclusions are are onlyonly validvalid ifif thethe samplesample isisselectedselected atat randomrandom ((i.ei.e., ., representativerepresentative ofof thethepopulationpopulation))
Similar to Similar to surveysurvey//pollpoll vsvs. . electionelectionL.T. Gama
Important conceptsWeWe studystudy a a smallsmall partpart ofof a a populationpopulation to to makemake judgmentsjudgments aboutabout thatthat populationpopulation
SampleSampleResultsResults are are statisticsstatistics
MeanMean, standard , standard deviationdeviation, , relationshipsrelationships amongamongvariablesvariables, , etcetc. . ObservableObservable inin thethe samplesample
statisticsstatistics are are estimatorsestimators ofof parametersparameters ininthethe populationpopulation
e.u.e.u. = = unitunit ofof material to material to whichwhich a a treatmenttreatment isis appliedapplied..animalanimaltreattreat. . appliedapplied to a to a groupgroup ofof animalsanimals
Nominal (no Nominal (no scalescale))BrucelosisBrucelosis, IBR, etc., IBR, etc.
Não paridas
Paridas
Fertility in a group of 200 cows
InfertilidadeMamitesPésOutros
Causes of culling in a dairy herd
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Most frequent case…Most frequent case…TheThe majoritymajority ofof response response variablesvariables inin biologybiologyhashas a a continuouscontinuous andand normal normal distributiondistribution
0 1 2 3 4 5 6 7 8 9 10 11 12
Level of metabolite X
Freq
uenc
y
Characterized by:- Mean- Variance
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Mean and varianceMean and variance
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Level of metabolite X
Freq
uenc
y
1
2 3
Level of metabolite X in 3 breeds
μ1 = μ2 < μ3 σ21 > σ2
2 = σ23
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Importance of residual variationImportance of residual variation
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Residual Residual variabilityvariability
Experimental Experimental errorerror
VariabilityVariability amongamongobservationsobservations subjectsubject to to thethe samesame treatmenttreatment
IfIf thethe ratio ratio isis highhighWeWe concludeconclude thatthat treattreat. are . are indeedindeed differentdifferent
ReductionReduction ofof thethe denominatordenominator ((expexp. . errorerror) ) betterbetter capacitycapacity to to detectdetect real real differencesdifferences amongamongtreatmentstreatments ((betterbetter precisionprecision))
i.e. i.e. attemptattempt to to reducereduce ““background background noisenoise””HowHow??
more more homogeneoushomogeneous e.ue.u. . maymay limitlimit spacespace ofof inferenceinference
stratificationstratification intointo homogeneoushomogeneous groupsgroupsblocksblocks –– buildingbuilding, , litterlitter, etc., etc.
statisticalstatistical adjustmentadjustment for for variablesvariables whichwhich cancan bebeidentifiedidentified//controledcontroled–– covariablescovariables
initialinitial weightweight, age , age ofof cowcow, , etcetc. .
Every experiment may be said to exist only in order to give the facts a chance of disproving the null hypothesis.
Sir R. Fisher - The Design of Experiments, 1935
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Steps in experimentation1. 1. PlanningPlanning
a) a) DefinitionDefinition ofof workwork hypothesishypothesisImportanceImportance; ; simplesimple, precise, preciseCanCan bebe verifiedverified inin thethe experienceexperienceResultsResults shouldshould allowallow thethe researcherresearcher to determine to determine thethe probabilityprobability ofof beingbeing wrongwrong inin hishis conclusionsconclusions
b) b) DefinitionDefinition ofof populationpopulation ofof inferenceinferenceAnimalsAnimals, , facilitiesfacilities, , managementmanagement, etc., , etc., usedused inin thetheexperienceexperience are are representativerepresentative ofof thethe populationpopulation to to whichwhich wewe wantwant to to applyapply thethe conclusionsconclusions??
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2. Design2. DesignCharacteristicsCharacteristics to to measuremeasureFactorsFactors to to studystudy; ; treatmentstreatments to to bebe usedusedMinimizationMinimization ofof uncontrolleduncontrolled influencesinfluences andandsubjectivitysubjectivityDetermine Determine thethe desireddesired precisionprecision andand differencesdifferenceswhichwhich are are expectedexpected andand justifiablejustifiable
ConsiderationsConsiderations ofof statisticalstatistical, , economiceconomic, etc. , etc. naturenature
To To callcall inin thethe statisticianstatistician afterafter thetheexperimentexperiment isis donedone maymay bebe no more no more thanthanaskingasking himhim to to performperform a a postmortempostmortemexaminationexamination: : hehe maymay bebe ableable to to saysay whatwhatthethe experimentexperiment dieddied ofof..
Sir R. Fisher , 1938
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3. 3. ExecutionExecutionAllocationAllocation ofof differentdifferent treatmentstreatments to experimental to experimental unitsunitsRigorousRigorous carryingcarrying outout ofof experimentationexperimentation
4. 4. AnalysesAnalysesStatisticalStatistical analysisanalysis ofof thethe resultsresults shouldshould leadlead to to thetheconfirmationconfirmation, , rejectionrejection oror changechange ofof thethe original original hypothesishypothesisStatementStatement, , inin a a probabilisticprobabilistic mannermanner, , aboutabout thethepossibilitypossibility ofof thethe researcherresearcher beingbeing mistakenmistaken inin hishisconclusionsconclusions..
AdvicesAdvices onon publicationpublicationHaveHave somethingsomething newnew to to saysaySaySay ititShutShut upup afterafter youyou’’veve saidsaid ititGiveGive thethe texttext anan appropriateappropriatetitletitle andand orderorder RamonRamon y y CajalCajal,,
18991899
No research is finished until it has been published!
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PublicationPublicationThe massacre of P<0.05
The fact that a difference is statistically significantdoes not imply that there is a “true” difference
In every 20 results where P<0.05, on average 1 result is nottruly different
NS results usually are not publishedSame experience repeated by different scientists
Sentences such as “the differences were not statisticallysignificant, but they are of biological importance” willcause any referee to jump on his/her chair
Lack of statistical significance only means that, given theexisting variability among observations, the difference thatwas found could very well be due to chance alone
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Data available for statistical analysisData available for statistical analysis
1.1. DesignedDesigned experimentsexperimentsTreatmentsTreatments appliedapplied to to e.u.e.u. andand responses responses observedobservedExampleExample::
2.2. SurveySurvey studiesstudiesData Data collectedcollected inin a a samplesample, , accordingaccording to to prepre--defineddefined criteriacriteriaThereThere are no are no treatstreats. . appliedapplied as as suchsuch to to thethe e.ue.u. . ExampleExample::
surveyssurveys aimedaimed atat characterizingcharacterizing managementmanagement practicespractices andand milkmilkproductionproduction inin farmersfarmers ofof a a givengiven breedbreedfarmersfarmers chosenchosen randomlyrandomly, , usingusing some some stratificationstratification criteriacriteria ((herdherdsizesize, , farmerfarmer’’s age, s age, levellevel ofof educationeducation, , etcetc.).)studystudy ofof thesethese andand otherother identifyableidentifyable factorsfactors ((seasonseason, , levellevel ofofsupplementationsupplementation, use , use ofof silagesilage, etc.) , etc.) onon dairydairy performances.performances.
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Data available for statistical analysisData available for statistical analysis
3.3. ObservationObservation studiesstudiesLikeLike thethe previousprevious case, case, butbut withwith no no obviousobvious criterioncriterion ininchoosingchoosing e.ue.u. . ExamplesExamples::
use use ofof hospital hospital recordsrecords to to studystudy thethe influenceinfluence ofof age, age, sexsex, , seasonseason, , yearyear, , regionregion, , etcetc., ., onon thethe incidenceincidence ofof influenza influenza ininhumanshumans. . Use Use ofof dairydairy recordsrecords to to evaluateevaluate effectseffects ofof age, age, monthmonth ofof calvingcalving, , herdherd, , etcetc. . onon milkmilk yieldyield
Cases 2 Cases 2 andand 33ThereThere isis no no clearclear attributionattribution ofof treatstreats. to . to e.ue.u..HoweverHowever, , therethere are are factorsfactors whichwhich cancan bebe identifiedidentified as as havinghaving a a potentialpotential influenceinfluence onon thethe response response variablesvariablesTheseThese factorsfactors are are consideredconsidered inin statisticalstatistical analysesanalyses inin a a waywaysimilar to similar to treatmentstreatments inin a a designeddesigned experimentexperiment. .
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Data available for statistical analysisData available for statistical analysis
UnbalancedUnbalanced datadataNNºº ofof obsevobsev. . notnot equalequal for for differentdifferenttreatstreats.; .; sometimessometimes ““emptyempty cellscells””
frequentfrequent withwith fieldfield datadataStatisticalStatistical methodsmethods are are approximateapproximateSometimesSometimes problemsproblems inin interpretinginterpretingadjustedadjusted meansmeans
seesee laterlater
AgeAge
SexSex
1010101022
1010101011
FFMM
AgeAge
SexSex
00101022
1515202011
FFMM
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The foundation of all statistical analyses
Central limit theorem
The foundation of all statistical analyses
Central limit theorem
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ExampleExampleSMART SMART hashas receivedreceived complaintscomplaintsthatthat itsits carscars are are tootoo smallsmall for for thethePortuguesePortuguese youngyoung populationpopulation!!
TheyThey givegive youyou a a grantgrant to to estimateestimate thethe averageaverage heightheightofof PortuguesePortuguese peoplepeople
YouYou hirehire 20 20 studentsstudentsAskAsk themthem to to gogo to to theirtheir favouritefavourite bar, bar, andand measuremeasure 10 10 peoplepeople chosenchosen randomlyrandomly ((e.ge.g., ., thethe firstfirst onesones arrivingarriving atat thethebar bar afterafter 3 3 a.ma.m. . onon a a FridayFriday))EachEach studentstudent calculatescalculates thethe meanmean ofof itsits samplesample ofof 10 10 individualsindividuals
WhatWhat cancan bebe expectedexpected fromfrom thethe samplingsampling thatthat youyouaskedasked youryour studentsstudents to to carrycarry outout??
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Example resultsExample results
1.4 1.5 1.6 1.7 1.8 1.9
Height (m)
Freq
uenc
y
1.4 1.5 1.6 1.7 1.8 1.9
Height (m)
Freq
uenc
y
1.4 1.5 1.6 1.7 1.8 1.9
Height (m)
Freq
uenc
y
Student 1Mean = 1.59
Student 2Mean = 1.72
Student 3Mean = 1.66
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Example resultsExample results
PossiblePossible resultsresults
0
0.05
0.1
0.15
0.2
0.25
1.4 1.5 1.6 1.7 1.8 1.9
Height (m)
Freq
uenc
y
0
2
4
6
8
10
12
Nº m
eans
Distribution of heights in theoriginal population
Distribution ofthe mean of 10
people, collectedby 20 students
Notice that sample means are closer to the mean of the original population.Does it make sense? L.T. Gama
Basic concepts – CLTBasic concepts – CLT
LevelLevel ofof metabolitemetabolite X X inin thousandsthousandsofof micemice ofof strainstrain AA
0 1 2 3 4 5 6 7 8 9 10 11 12
Level of metabolite X
Freq
uenc
y
Normal distribution
6=μ
2=σ
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Basic conceptsBasic conceptsIfIf wewe taketake severalseveral samplessamples ofof nn micemice fromfrom thisthispopulationpopulation, , whatwhat do do wewe expectexpect to to getget? ?
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Level of metabolite X
Freq
uenc
y
Distribution of the meansof samples of n individuals
n=4
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Basic conceptsBasic conceptsIfIf wewe taketake severalseveral samplessamples ofof nn micemice fromfrom thisthispopulationpopulation, , whatwhat do do wewe expectexpect to to getget? ?
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Level of metabolite X
Freq
uenc
y
Distribution of the meansof samples of n individuals
n=4
n=9
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Basic conceptsBasic conceptsIfIf wewe taketake severalseveral samplessamples ofof nn micemice fromfrom thisthispopulationpopulation, , whatwhat do do wewe expectexpect to to getget? ?
Normal distribution
6=X
ns σ=
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Level of metabolite X
Freq
uenc
y
Distribution of the meansof samples of n individuals
n=4
n=9
n=16
Central LimitTheorem
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Effect of sample sizeEffect of sample size
Sampling: n=2, 5, 10, 25; 10000 replications
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Effect of the original distributionEffect of the original distributionSampling: n=10, 10000 replications
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ConclusionConclusion
TheThe meanmean ofof a a smallsmall samplesample tendstends to to havehave a a widerwiderdistributiondistribution aroundaround thethe meanmean ofof thethe populationpopulation fromfromwherewhere itit waswas collectedcollectedTheThe distributiondistribution getsgets narrowernarrower as as nn increasesincreases
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Level of metabolite X
Freq
uenc
y
n=4
n=9
n=16
66709
2 .s ==⇒
14
2==⇒ s
50162 .s ==⇒
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Hypothesis testingBasic principles
Hypothesis testingBasic principles
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FundamentalsFundamentals
HypothesisHypothesis testingtesting isis basedbased onon some some typetypeofof ““testtest ofof significancesignificance””
TestsTests t, F, t, F, etcetc..
InIn essenceessence, , wewe wantwant to compare:to compare:
Ratio Ratio isis highhigh SignificantSignificant!!!!Ratio Ratio isis smallsmall N.S.!!!N.S.!!!
eatments within trobserv. among eatmentsbetween tr
i.e., i.e., thethe factfact thatthat wewe are are workingworking withwith a a samplesample((andand notnot withwith thethe populationpopulation) ) maymay leadlead to to wrongwrongconclusionsconclusions. .
CorrectCorrectErrorError !!((TypeType 2)2)
TreatTreat. do . do notnotdifferdiffer
ErrorError !!((TypeType 1)1)
CorrectCorrectTreatTreat. . differdiffer
TreatTreat. do . do notnotdifferdiffer
TreatTreat. . differdiffer
“REALITY”
SAM
PLE
Prob. α
Prob. βL.T. Gama
Sequence of the analysis1. 1. WeWe startstart byby assumingassuming thatthat thethe twotwo treatstreats. do . do
notnot differdifferi.e. i.e. thethe twotwo groupsgroups ((oneone correspondingcorresponding to to eacheachtreattreat.) .) actuallyactually come come fromfrom thethe samesame conceptual conceptual populationpopulation, , andand theythey onlyonly differdiffer duedue to to thethesamplingsampling processprocess
2. 2. TestTest ifif itit isis plausibleplausible thatthat thethe observedobserveddifferencesdifferences are are onlyonly duedue to to thethe samplingsamplingprocessprocess
ProbabilityProbability ofof errorerror inin a a givengiven experimentexperiment::typetype I I isis controlablecontrolable ((αα))typetype II II isis notnot ((ββ) ) –– resultsresults fromfrom thethe expectedexpecteddistributiondistribution inin thethe samplingsampling processprocess
SameSame differencedifference betweenbetween μμAA andand μμBB easiereasier to to detectdetect whenwhen n n
For a For a givengiven situationsituation αα => => ββ
PowerPower ofof a a testtest = 1= 1--ββL.T. Gama
Power of a test as a function of nPower of a test as a function of n
Difference of 10% almost undetectablewhen n is small
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Risks of ignoring the power of a testRisks of ignoring the power of a test
ExampleExamplePacemakerPacemaker A A isis standard standard inin thethe marketmarketBrandBrand B B wantswants to prove to prove thatthat itit hashas a a productproduct as as goodgoodas as brandbrand A A HH00 : : μμAA = = μμBB
Assume Assume thatthat::inin realityreality B B isis worseworse thanthan AADueDue to to lacklack ofof resourcesresources, , experienceexperience isis smallsmall, , withwith lowlowpowerpowerwewe makemake a a typetype II II errorerror, , andand do do notnot rejectreject HH00
brandbrand B B maymay (?) (?) claimclaim thatthat itit isis as as goodgood as A (?) as A (?)
ConsequencesConsequences??????L.T. Gama
““TheThe resultsresults ofof thethe experienceexperience werewere::oneone thirdthird ofof thethe animalsanimals showedshowed clearclearimprovementimprovement withwith thethe treatmenttreatment;;oneone thirdthird didndidn’’t show t show anyany improvementimprovement; ; thethe thirdthird mouse mouse ranran awayaway!!””
HoweverHowever::ttcrcríítt.. alreadyalready dependsdepends onon nn ((i.ei.e. . d.f.ed.f.e.).)WeWe are are ignoringignoring TypeType II II errorerror
crít.BA
XX
BAobs t
ns
XXs
XXt >−
=−
=−
2221
2
2
2
⎟⎟⎠
⎞⎜⎜⎝
⎛ −≥
sXX
t nBA
.crít
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Correct way to determine nCorrect way to determine n
Define Define previouslypreviously::DifferenceDifference whichwhich wewe wantwant//expectexpect to to detectdetect ((μμ11-- μμ22))ExpectedExpected variabilityvariability amongamong e.u.e.u. submittedsubmitted to to thethesamesame treatmenttreatment ((σσ oror ss))TolerableTolerable probabiltiesprobabilties for for typetype I (I (αα) ) andand typetype II (II (ββ) ) errorserrors
SeveralSeveral methodsmethods are are possiblepossibleUse Use ofof thethe Z Z distributiondistribution
SimplestSimplest methodmethodSimilar to t Similar to t distributiondistribution, , butbut independentindependent ofof d.f.ed.f.e. .
Nº observ. with differences expressed as %Nº observ. with differences expressed as %
XsCV =
n n mustmust bebe highhigh ifif::SmallSmall differencesdifferences are are expectedexpectedVariabilityVariability amongamong e.ue.u. . isis highhigh
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NNºº observobserv././treattreat. as a . as a functionfunction ofof::DifDif. . amongamong μμ expressedexpressed inin s s unitsunitsProbProb. . typetype II II errorerror ((ββ))AssumingAssuming αα=0.05=0.05
β
Bilateral test!!
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Internet resourcesInternet resources
www.stat.uiowa.edu/~rlenth/
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Design of the experimentDesign of the experiment
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Design of experimentDesign of experiment
TreatmentTreatment structurestructureSimpleSimple
eacheach e.u.e.u. subjectsubject to to oneone treattreat. . FactorialFactorial
eacheach e.u.e.u. subjectsubject to a to a combinationcombination ofof treatstreats. . allowsallows studystudy ofof interactionsinteractions
Experimental design Experimental design AttributionAttribution ofof treatstreats. . takingtaking intointo accountaccount identifiableidentifiablefactorsfactors whichwhich cause cause additionaladditional variationvariation(background (background noisenoise))
TreatTreat. A . A vsvs. . treattreat. B . B –– TraditionalTraditional ANOVAANOVAEffectoEffecto ofof antibioticantibiotic X X comparedcompared withwith Y Y
IncreasedIncreased levelslevels ofof a a givengiven factorfactorp.ep.e. . concentrationconcentration ofof a a givengiven drugdrug, , levellevel ofof metabolizablemetabolizableenergyenergy, , etcetc..
AnalysisAnalysis ofof regressionregressionLinear Linear regressionregression
QuadraticQuadratic regressionregressionOneOne ofof thethe mostmost frequentfrequent inin biologybiologyRelationshipRelationship betweenbetween Y Y andand X X isis curvilinearcurvilinearExamplesExamples
Age Age ofof cowcow andand fertilityfertilityLevelLevel ofof energyenergy andand growthgrowthLevelLevel ofof bSTbST andand milkmilk yieldyield
One factorOne factor
Y=b0+b1X+b2X2
5000
6000
7000
8000
9000
10000
11000
0 5 10 15 20 25 30 35 40bST (mg)
PL (k
g)
Maximum at28 mg
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Example – simple factorialExample – simple factorialResponse Response variablevariable
Factorial experimentFactorial experimentWant to study joint effect of two factors, each one Want to study joint effect of two factors, each one with several levels with several levels Interaction among factors!Interaction among factors!ExampleExample
ExampleExampleTest of two protein sources in the dietTest of two protein sources in the diet
Soybean meal (SBM) and fishmeal (FM)Soybean meal (SBM) and fishmeal (FM)Each diet has 4 levels of crude protein (12, 14, 16 and 18%)Each diet has 4 levels of crude protein (12, 14, 16 and 18%)Trial conducted with 16 pigs (2 per combination source of Trial conducted with 16 pigs (2 per combination source of protprot. x level of CP). x level of CP)The effect of level of CP (continuous factor) is the same for The effect of level of CP (continuous factor) is the same for the two protein sources (discontinuous factor)?the two protein sources (discontinuous factor)?
OneOne factor (factor (withwith ii levelslevels); ); possiblypossibly a factorial a factorial NotNot possiblepossible to to groupgroup e.ue.u. . EachEach animal animal submittedsubmitted to to oneone treattreat. (. (oror combinationcombinationofof treatstreats.), .), withwith oneone observationobservation..ExampleExample::
GroupGroup ofof 20 20 lambslambs, , ofof thethe samesame sexsex andand breedbreed, , withwith similar similar ages (50 d).ages (50 d).TestTest ofof 4 4 differentdifferent dietsdiets; ; effecteffect onon weightweight atat 100 d 100 d
OtherOther examplesexampleseffect of two drugs used in dermatology;effect of two drugs used in dermatology;
both used simultaneously in the same animal both used simultaneously in the same animal animal considered as a blockanimal considered as a block
comparison of two ways to consolidate fractures comparison of two ways to consolidate fractures surgicallysurgically
forced fracture of the radius in the two members of dogs forced fracture of the radius in the two members of dogs dog considered as a blockdog considered as a block
effect of supplementation with calcium in laying effect of supplementation with calcium in laying henshens
each building split in the middle ; each building split in the middle ; 2 treat. used in the same building; 2 treat. used in the same building; experience repeated in 3 buildings .experience repeated in 3 buildings .Building is the blockBuilding is the block
Randomized Complete Blocks Design (RCBD)
Randomized Complete Blocks Design (RCBD)
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Applies the principles of RCBD; Applies the principles of RCBD; in this case there are two factors which can be used to group in this case there are two factors which can be used to group the experimental units.the experimental units.
However, number of levels of the two factors must be However, number of levels of the two factors must be the same (or multiple), and equal to the number of the same (or multiple), and equal to the number of treats. treats.
Example:Example:3 litters in mice (9 animals)3 litters in mice (9 animals)3 weeks of trial3 weeks of trial3 treats. (A, B, C).3 treats. (A, B, C).
each treat. appears only once in each row and each treat. appears only once in each row and columncolumnassumes that interactions do not exist!!assumes that interactions do not exist!!
Latin squareLatin square
BBAACC33
CCBBAA22
AACCBB11
ZZYYXX
Litter
Week
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ExampleExample
Test of efficacy of 4 antidotes of Test of efficacy of 4 antidotes of substsubst. X. XA=controlA=control B=B=phenobarbitalphenobarbital, , C=ammonium chloride C=ammonium chloride D=lactoseD=lactose
rabbits (n=16) treated previously with the antidote. rabbits (n=16) treated previously with the antidote.
trial carried out in 4 consecutive days trial carried out in 4 consecutive days
on the day of the trial injected with substance X with on the day of the trial injected with substance X with intervals of .5, 1, 1.5 or 2 minutes intervals of .5, 1, 1.5 or 2 minutes
response variable is the lethal dose of Xresponse variable is the lethal dose of X
Latin squareLatin square
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Latin square usedLatin square usedLetalLetal dose expressed in (log mg X/ log Kg PV), presented in dose expressed in (log mg X/ log Kg PV), presented in ( ) ( ) –– average of 4 rabbits/treat.average of 4 rabbits/treat.
Latin squareLatin square
A(1.168)
D(1.139)
B(1.240)
C(0.665)
4
C(0.934)
B(1.394)
A(0.925)
D(1.266)
3
D(0.935)
A(1.031)
C(1.432)
B(1.220)
2
B(1.231)
C(1.161)
D(1.231)
A(1.576)
1
4321Day of trial
Interval
15Total
6Error
3Antidote
3Interval
3Days
d.f.Source of variation
ANOVA
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SameSame individual individual submmitedsubmmited sucessivelysucessively to to severalseveral treatmentstreatmentsTreatTreat. are . are comparedcompared withinwithin thethe individual, individual, thusthus removingremoving thethe noisenoiseintroducedintroduced byby variabilityvariability amongamong individualsindividuals..e.g. e.g. differentdifferent drugsdrugs to to controlcontrol hipertensionhipertension
Similar to Similar to latinlatin squaresquareIndividual = Individual = columncolumnPeriodPeriod = = linelineTreatmentsTreatments assignedassigned sequentiallysequentially
AllAll indivindiv. . subjectsubject to to everyevery treatmenttreatment
assumptionassumption inin ANOVAANOVAItIt isis ofof interestinterest to to studystudy thethe evolutionevolution ofof a a phenomenonphenomenon overover timetime
Objective Objective isis to to studystudy evolutionevolution ofof FSH FSH overover timetime, , andandhowhow itit isis affectdaffectd byby treattreat. .
interactioninteraction treat.*timetreat.*timetesttest linear linear andand quadraticquadratic evolutionevolution ofof FSH (FSH (timetime consideredconsidered as as a a continuouscontinuous variablevariable))