Regression Tutorial 201 With NumXL

7/30/2019 Regression Tutorial 201 With NumXL

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Regression201Tutorial 1 SpiderFinancialCorp,2013

Tutorial:Regression201

Thisisthethirdentryinourregressionanalysisandmodelingseries.Inthistutorial,wecontinuethe

analysisdiscussionwestartedearlierbyleveragingamoreadvancedtechniqueinfluentialdata

analysis tohelpusimprovethemodel,and,asaresult,thereliabilityoftheforecast.

Again,wewilluseasampledatasetgatheredfrom20differentsalespersons.Theregressionmodel

attemptstoexplainandpredicttheweeklysalesforeachperson(dependentvariable)usingtwo

explanatoryvariables:intelligence(IQ)andextroversion.

DataPreparationSimilartowhatwedidinourearliertutorial,weorganizeoursampledatabyplacingthevalueofeach

variableinaseparatecolumnandeachobservationinaseparaterow.

Next,weintroducethemask.ThemaskisaBooleanarray(0,1)thatchooseswhichvariableis

included(orexcluded)intheanalysis.

Initially,atthetopofthetable,letsinsertthemaskcellsarray;eachwithavalueof1(i.e.included).

Thearrayisshownbelowhighlightedbelow:

Inthisexample,wehave20observationsandtwoindependent(explanatory)variables.Theresponseor

dependentvariableistheweeklysales.

ProcessNowwearereadytoconductourregressionanalysis. First,selectanemptycellinyourworksheet

whereyouwishtheoutputtobegenerated,thenlocateandclickontheregressioniconintheNumXL


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tab(ortoolbar).

NowtheRegressionWizardwillappear.

Selectthecellsrangefortheresponse/dependentvariablevalues(i.e.weeklysales).Selectthecells

rangefortheexplanatory(independent)variablesvalues.ForVariables(X)Mask,selectthecellsatthe

topofthedatatable(Booleanarray).

Notes:1. Thecellsrangeincludes(optional)theheading(Label)cell,whichwouldbeusedintheoutput

tableswhereitreferencesthosevariables.

2. Theexplanatoryvariables(i.e.X)arealreadygroupedbycolumns(eachcolumnrepresentsavariable),sowedontneedtochangethat.

3. Bydefault,theoutputcellsrangeissettothecurrentselectedcellinyourworksheet.Pleasenotethat,onceweselecttheXandYcellsrange,theoptions,ForecastandMissingValues

tabsbecomeavailable(enabled).

Next,selecttheOptionstab.


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Initially,thetabissettothefollowingvalues:

Theregressionintercept/constantisleftblank.Thisindicatesthattheregressioninterceptwillbeestimatedbytheregression.Tosettheregressiontoafixedvalue(e.g.zero(0)),enterit

there.

Thesignificancelevel(aka. )issetto5% Inoutputsection,themostcommonregressionanalysisisselected. Forautomodeling,checkthisoption.

Now,clicktheMissingValuestab.


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Inthistab,youcanselectanapproachtohandlemissingvaluesinthedataset(XandY).Bydefault,any

missingvaluefoundinXorinYinanyobservationwouldexcludetheobservationfromtheanalysis.

Thistreatmentisagoodapproachforouranalysis,soletsleaveitunchanged.

Now,clickOKtogeneratetheoutputtables.

Toassesstheinfluencethateachobservationexertsonourmodel,wecalculateacoupleofstatistical

measures:leverageandCooksdistance.


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Selectthecellnexttotheresponsevariable. Intheformulabar,typeintheMLR_FITTEDfunction,thenclickthefxbutton.

TheFunctionWizardpopsup.Selecttheinputcellsrange,mask,andaReturntypeof4fortheleveragestatistics.ClickOK.

MLR_FITTEDreturnsanarrayofvalues,butyouwillinitiallyonlyseethe1stvalue.

To

display

the

full

array,

select

all

the

cells

below

(to

the

end

of

the

sample).

Press

F2,

then

press

CTRL+SHIFT+ENTERtocopythearrayformula.


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Now,tocalculatetheCooksdistance,selectthecellnexttoLeverageandrepeatthesamesteps,butwiththereturntype=5.

AnalysisNowthatwehavetheleverageandCooksdistancestatistics,letsinterprettheirfindings.


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1. LeverageStatistics(H)

Leveragestatisticsmeasure

thedistance

of

an

observationfromthecenter

ofthedata.Inourexample,

theintelligenceand

extroversionvaluesfor

Salesman11arefurthest

fromtheaverage.Doesthis

meanSalesman11isan

outlier?Doesthismeanhe

exertsinfluenceonthe

calculationof

the

regression

coefficient?

Toexaminethisassumption,letsremoveSalesman11fromourinputdataandexaminetheresulting

regression.Todoso,justinsertan#N/Avalueinanyinputvariableofthisobservation.

(Fulldataset) Omittingsalesman#11

Droppingobservation11madethingsatbestthesameasearlier. Weoptedtorecoverthisobservation

backintothesample.

Insum,theleveragestatisticsdo

notnecessarilyimplyanoutlier,

butmerely

adistant

observation

withfewneighbors.

2. CooksDistance(D)TheCooksdistancecorrectsfor

weaknessintheleverage

statistics,andisthusmore

0%

10%

20%

30%

40%

50%

60%

70%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Cook'sDistance

(D)

0%

5%

10%

15%

20%

25%

30%

35%

40%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Leverage(H)


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indicativeofinfluentialdata.Furthermore,therearefewheuristicsforthethresholdvaluesofCooks

distancetodetectaninfluentialdatum.Forouranalysis,weoftenuse4

Nasathreshold(which

translatesto20%forthe20observationsinourdataset).

Usingthethresholdorjustlookingattheearlierplot,wedetectthatSalesman16exertsthehighest

influenceonourregression,soletsvoidthisobservation(bysetting#N/Ainoneoftheinputvariables).

NotethattheleveragestatisticsandCooksdistancereturn#N/Aforthismissingvalue.

Letsnowexaminetheregressionstatisticsbeforeandafterwedroppedthesixteenthobservation.

(FullDataset) (WithoutSalesman#16)

Asyoumayalreadyhavenoticed,theregressionimprovedsignificantlyoneverydimension(e.g.R

square,stderror,etc.). Salesman#16seemstobeaninfluentialoutlier,sowelldrophim.


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Tohelpexplainwhatmakesanobservationinfluential,letsexaminetheextroversionvs.weeklysales

graphbelow:

Wedrawthelineartrendasaproxyforourregressionmodel.Theblack(circle)datapointrepresents

Salesman16.Itslocation(extroversionandweeklysalesvalue)ispullingtheregression(dashed)line

towardit,affectingthevalueoftheregressionslopeandintercept.

Droppingthisobservationreleasestheregressionline,adjustingittobetterfittheremainingpoints.

LetstakeanotherlookattheCooksdistanceplot(withoutSalesman16,andwithathresholdof

421%

19 )


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TheCooksdistancevaluesforthedifferentplotsaredistributedsomewhatuniformly,andwemaystop

there.

Note:Bearinmindthatourthresholdruleismerelyaheuristic(ruleofthumb),andshouldnotbetaken

rigidly,butratherasaguideline.

ConclusionInthistutorial,wehaveshownthatexcludingobservation#16isbeneficialtoourmodelingeffortsasit

exertsignificant

influence

on

our

coefficient

calculation.

Next,usingtheremaining19observations,letsrecalculate(SHIFT+F9)theregressionstatistics,ANOVA,

residualsdiagnosis,stepwiseregression,etc.


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Theoptimalsetoftheinputvariablesisthesameasearlier.Letsdroptheintelligencevariable(by

settingitsvalueto0inthemask),andrecalculate

Theregressionerroris$307(vs.$332beforeweremovedsalesman#16).


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TheFinalquestionwemayaskourselves;Istheregressionstableoverthesampledataset? Nextissue.

$1,500

$2,000

$2,500

$3,000

$3,500

$4,000

$4,500

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Regression Tutorial 201 With NumXL