Unit12:RoadMap(VERBAL) - - PowerPoint PPT Presentation

• Unit12:RoadMap(VERBAL)

!"#$ %&'( )*'(#+ READING&(&,-, .&'( )/'(#+ 0.&1 RACE& (,2,"3,4&)-,5 6.&1 HOMEWORK7)&(,8$ ,-$ FREELUNCH9!&(,!:(;&"&, ESL!:&::&(,!" ,)!: +57&< +5&7& $:$outliers#< 3+=777linearity normality <

• +=777homoskedasticity <

>+5&::&!" 7)&:!< 8+/(7:&&:!!"7)< +*(7:&(!!"7)< +5(:(:&9&&&< +=777independence :2%'2#<

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• Unit12:RoadMap(Schematic)

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SinglePredictor

ChiSquares ChiSquares Regression ANOVA Polychotomous ChiSquares Dichotomous ChiSquares Logistic Regression Polychotomous Regression ANOVA T(tests Regression Continuous Dichotomous Continuous

Outcome MultiplePredictors

ChiSquares ChiSquares Regression ANOVA Polychotomous ChiSquares Dichotomous ChiSquares Logistic Regression Polychotomous Regression ANOVA Multiple Regression Continuous Dichotomous Continuous

Outcome

Units11(14,19,B: Dealingwith Assumption Violations

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Unit12:Roadmap(SPSSOutput)

Unit Unit 9 9 Unit Unit 16 16 Unit Unit 17 17 Unit Unit 18 18 Unit Unit 8 8 Unit Unit 11 11 Unit Unit 12 12 Unit Unit 13 13 Unit Unit 14 14 Unit Unit 19 19

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Unit12:CheckingGLMAssumptionswithRegressionDiagnostics Unit12PostHole: 6&)"@(:11 ';#:(( &:&&&$ Unit12TechnicalMemoandSchoolBoardMemo: ::& : @#$ Unit12Review: 78$ Unit12Reading: @$633($

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Unit12:TechnicalMemoandSchoolBoardMemo

WorkProducts(PartIofII): I. TechnicalMemo:Haveonesectionperanalysis.Foreachsection, followthisoutline. A. Introduction i. Stateatheory(orperhapshunch)fortherelationship—thinkcausally,becreative.(1Sentence) ii. Statearesearchquestionforeachtheory(orhunch)—thinkcorrelationally,beformal.Nowthatyouknowthestatistical machinerythatjustifiesaninferencefromasampletoapopulation,begineachresearchquestion,“Inthepopulation,…” (1 Sentence)

• iii. Listyourvariables,andlabelthem“outcome” and“predictor,” respectively.
• iv. Includeyourtheoreticalmodel.

B. Univariate Statistics.Describeyourvariables,usingdescriptivestatistics.Whatdotheyrepresentormeasure? i. Describethedataset.(1Sentence) ii. Describeyourvariables.(1ParagraphEach) a. Definethevariable(parentheticallynotingthemeanands.d.asdescriptivestatistics). b. Interpretthemeanandstandarddeviationinsuchawaythatyouraudiencebeginstoformapictureofthewaythe worldis.Neverlosesightofthesubstantivemeaningofthenumbers. c. Polishofftheinterpretationbydiscussingwhetherthemeanand standarddeviationcanbemisleading,referencing themedian,outliersand/orskewasappropriate. d. Notevaliditythreatsduetomeasurementerror. C. Correlations.Provideanoverviewoftherelationshipsbetweenyourvariablesusingdescriptivestatistics.Focusfirstonthe relationshipbetweenyouroutcomeandquestionpredictor,second(tiedontherelationshipsbetweenyouroutcomeandcontrol predictors,second(tiedontherelationshipsbetweenyourquestionpredictorandcontrolpredictors,andfourthonthe relationship(s)betweenyourcontrolvariables. a. Includeyourownsimple/partialcorrelationmatrixwithawell(writtencaption. b. Interpretyoursimplecorrelationmatrix.Notewhatthesimplecorrelationmatrixforeshadowsforyourpartialcorrelation matrix;“cheat” herebypeekingatyourpartialcorrelationandthinkingbackwards.Sometimes,yoursimplecorrelation

• matrixrevealspossibilitiesinyourpartialcorrelationmatrix. Othertimes,yoursimplecorrelationmatrixprovidesforegone

conclusions.Youcanstareatacorrelationmatrixallday,solimityourselftotwoinsights. c. Interpretyourpartialcorrelationmatrixcontrollingforonevariable.Notewhatthepartialcorrelationmatrixforeshadows forapartialcorrelationmatrixthatcontrolsfortwovariables.Limityourselftotwoinsights.

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Unit12:TechnicalMemoandSchoolBoardMemo

WorkProducts(PartIIofII): I. TechnicalMemo(continued) D. RegressionAnalysis.Answeryourresearchquestionusinginferentialstatistics.Weaveyourstrategyintoacoherentstory. i. Includeyourfittedmodel. ii. UsetheR2 statistictoconveythegoodnessoffitforthemodel(i.e.,strength).

• iii. Todeterminestatisticalsignificance,testeachnullhypothesis thatthemagnitudeinthepopulationiszero,reject(ornot)

thenullhypothesis,anddrawaconclusion(ornot)fromthesampletothepopulation.

• iv. Create,displayanddiscussatablewithataxonomyoffittedregressionmodels.

v. Usespreadsheetsoftwaretographtherelationship(s),andincludeawell(writtencaption.

• vi. Describethedirectionandmagnitudeoftherelationship(s)inyoursample,preferablywithillustrativeexamples.Drawout

thesubstanceofyourfindingsthroughyournarrative.

• vii. Useconfidenceintervalstodescribetheprecisionofyourmagnitudeestimatessothatyoucandiscussthemagnitudeinthe

population.

• viii. Ifregressiondiagnosticsrevealaproblem,describetheproblem andtheimplicationsforyouranalysisand,ifpossible,

correcttheproblem. i. Primarily,checkyourresidual(versus(fitted(RVF)plot.(GlanceattheresidualhistogramandP(Pplot.) ii. Checkyourresidual(versus(predictorplots.

• iii. Checkforinfluentialoutliersusingleverage,residualandinfluencestatistics.
• iv. Checkyourmaineffectsassumptionsbycheckingforinteractions beforeyoufinalizeyourmodel.

X. ExploratoryDataAnalysis.Exploreyourdatausingoutlierresistantstatistics. i. Foreachvariable,useacoherentnarrativetoconveytheresultsofyourexploratoryunivariate analysisofthedata.Don’t losesightofthesubstantivemeaningofthenumbers.(1ParagraphEach) ii. Foreachrelationshipbetweenyouroutcomeandpredictor,useacoherentnarrativetoconveytheresultsofyour exploratorybivariate analysisofthedata.(1ParagraphEach) 1. Ifarelationshipisnon(linear,transformtheoutcomeand/orpredictortomakeitlinear. 2. Ifarelationshipisheteroskedastic,considerusingrobuststandarderrors. II. SchoolBoardMemo:Concisely,preciselyandplainlyconveyyourkeyfindingstoalayaudience.Notethat,whereasyouarebuildingonthe technicalmemoformostofthesemester,yourschoolboardmemoisfresheachweek.(Max200Words)

• III. MemoMetacognitive

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NELS88.savCodebook

NationalEducationLongitudinalStudy Source:U.S.DepartmentofEducation Summary:HereareselectvariablesfromtheNELS88dataset. Notes:IcreatedtheFREELUNCHvariablebasedonannualfamilyincomeoflessthan$25,000. convertedtheHOMEWORKvariablefromanordinal/categoricalvariabletoacontinuous variable,whichiswhyitisso“binny.” Iremovedfromthedatasetstudentswhoself(identified asotherthanAsian,Black,Latino,orWhite.Ithencreatedasetofindicatorvariablesfrom RACE:ASIAN,BLACKANDLATIONwithWHITEasan(implicit)referencecategory. Sample:Anationallyrepresentativesampleof7,8008th graders. Variables: READING&(&,-, FREELUNCH&(,!:(;&"&, HOMEWORK7)&(,8$ ,-$ FREELUNCH9!&(,!:(;&"&,

• ESL!:&::&(,!" ,)!:

RACE& (,2,"3,4&)-,5 ASIAN, &(,1/2, LATINO, &(,1/", BLACK, &(,1/4&),

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SelectVariablesfromtheNELSDataSet

• IntroductiontoRegressionDiagnostics

SearchHI(N(LOfor AssumptionViolations AHeteroskedasticity AIndependence ANormality ALinearity AOutliers

Atleastinsimplelinear regression,diagnostics provideinformationthatwe couldconceivablygleanfrom abivariate scatterplot ofthe

• utome versuspredictor;

neverthelesstheycan provideahelpfullydetailed view.Inmultipleregression, however,diagnosticsprovide informationthatwecould nevergatherbyeye.

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• SettingUpOurQuestion(Part1of3)

88 9 . 7 . 10 92 ˆ READING NG I READ + =

Theaveragestudentscored48pointsasan8th grader.Howmanypointsdoyoupredictthat sheimprovedinthe12th grade?

48 * 9 . 7 . 10 54 + =

Fortheaveragestudent,weexpectan increaseof6readingpointsfromthe8th grade tothe12th grade,from48pointsto54points.

NoticethatIuse“increase.” Thelongitudinaldatawarrantthedevelopmentalconclusion!

Recallthatthey(intercept(i.e.,β0 ,e.g.,10.7)isourpredictionwhenxiszero(e.g.,READING88 = 0).SinceREADING88neverequalszerowithintherangeofourdata,they(interceptismerelya mathematicalabstractioninourfittedmodel.But,wecan makezeromoreinteresting...

*79&& :7&8< :CC@# Forthenextslide, remember54,and6 fortheslideafter.

• SettingUpOurQuestion(Part2of3)

*ExampleSPSSsyntaxforcomputingtransformedvariables. *Thislineartransformationisnotaz(transformationbecauseI didnotdividethedifferencebythestandarddeviation. COMPUTEZEROCENTEREDREADING88=READING88( 48.0155. EXECUTE. *This(goofy)transformationisnon(linearbecauseIdomore thanadd/subtractand/ormultiply/dividebyaconstant.Iuse powersandlogs. COMPUTESean_Is_A_Great_SPSS_Programmer =READING88* 48.0155( 1975/27+FREELUNCH**(1/2)+LN(HOMEWORK+1). EXECUTE. Lookfamiliar?They(interceptnowhasaninterestinginterpretation.Itisourpredictionfortheaveragestudent nowthattheaveragestudenthasanx(valueof0.(Alsonoticethattheslopehas'not changed.)

88 9 . 2 . 54 92 ˆ EDREADING ZEROCENTER NG I READ + =

• SettingUpOurQuestion(Part3of3)

*Ifweareinterestedinchanges,let’scomputeachange scoreanduseitasouroutcome.Thisisnotalinear transformationbecauseIadd/subtractand/ormultiply/divide byavariable,notaconstant. COMPUTEREADINGIMPROVEMENT=READING92( READING88. EXECUTE.

Lookfamiliar?They(interceptnowhasanotherinterestinginterpretation.Itisourpredictedchangefortheaveragestudent nowthattheaveragestudenthasanx(valueof0andouroutcomevariableisachangevariable.Theslopenowtellsusthe differenceinchangeassociatedwithaIpointdifferencein1988readingscore.Ifwetaketwostudentswhodifferedby10 pointsin1988,weexpectthehigherscoringstudenttohaveimprovedherscoreless,byabout1pointless.

READING92hasmeasurementerror,andREADING88hasmeasurement error,whenItaketheirdifference,theirdifferencehasmore measurementerrorthaneither,sincetheyarepositivelycorrelated!Ugh. Wheneverthereisanelementofrandomnessintheoutcome,weexpect regressiontothemean.Measurementerrorisonepossiblesource of randomness,butnottheonlypossiblesourceofrandomness.Ifwe predictadultheightbymother’sheight,wewillgetregressiontothe mean,eventhoughthereisonlytrivialmeasurementerrorwithheight. Why?Thereisgeneticandenvironmentalluckinvolvedinheight. Tobe extremelytall(orshort)requiresluck,butthereisnoguaranteethatthe luckishereditary.Mostextremelytallmomshavenot(as(talldaughters.* *But,ifvarianceintheoutcomeisgreaterthanvariancein thepredictor,therecanbeegressionfrom themean!

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Unit12:ResearchQuestionI C+&D::7 :)&$/7& &7&D& ::(:&( &:&&$ &0+5&::: &779&( ::< *+( ,8#;C!& ": !"$# '(+ %&E1 6::& # .&E :& # @+ ε β β + + = 88

1

EDREADING ZEROCENTER ROVEMENT READINGIMP

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Unit12:ResearchQuestionII

C+::79&:: (77 7:# :: 7(: :$;: 7(:((& (&(&&:$ &0+/(7: ::1&9</ )&7&7&< *+( ,8#;C!& ": !"$# '(+ %&E1 6::& # .&E :& # @+

ε β β + + = 88

1

EDREADING ZEROCENTER ROVEMENT READINGIMP

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ExploratoryGraphs

Aresidual(akaerror)isthedifferencebetweenourobserved

• utcomeandourpredictedoutcome.Iftheresidualisnegativethat

meansweshouldhavepredictedlower(i.e.,weoverpredicted).If theresidualispositive,weshouldhavepredictedhigher(i.e., we underpredicted).Ofcourse,weexpectresidualsbecauseofindividual variation,hiddenvariables,andmeasurementerror.

Observation:(32 Prediction:3 Observed– Predicted=Residual (32– 3 =(35

8

Residuals

Everydatumhasanassociatedresidual, andwecangraphtheresidualswitha histogram:

Whatwouldhappentoourtrendlineifweremovedtheoutlierwitharesidualof(35?Youcanthinkofevery datumaspullingthelinewitharubberband.Whathappenswhenouroutlierletsgoofitsrubberband? Theaverage residualwill bealwaysbe zero.Ifit werenot zero,we wouldneed todrawa bettertrend line.

• PlayingAroundForAFewMinutes

http://www.istics.net/stat/PutPoints/ ExpandingourViewof TheScatterplot

• 1. Scatterplot
• 2. ResidualHistogram
• 3. ResidualVs.FittedPlot

(RVFPlot)

1 2 3

• PlayingAroundForAFewMinutes

ExtremeExample1: Thepart/wholeproblemsolvedbydeletedresiduals. ExtremeExample2: Highleverageisnotnecessarilyhighinfluence.

• PlayingAroundForAFewMinutes

ExtremeExample3: Lowleveragehighresidualsinfluencethey(intercept. ExtremeExample4: Highleveragehighresidualsinfluencetheslope.

• PlayingAroundForAFewMinutes

ExtremeExample5: RVFPlotsblowupnon(linearhorseshoes. ExtremeExample6: RVFPlotsblowupheteroskedastic funnels.

• PlayingAroundForAFewMinutes

ExtremeExample7: Residualhistogramsprovideinsightintonormality. ExtremeExample8: Residualhistogramsdon’tshowconditional normality.

• OutlierDetection:DeletedResiduals(Part1of3)

RegressionWithOutlierRemoved(n=70): OriginalRegression(n=71):

*Identifytheresidual,temporarilyremoveit,andrefit theline. TEMPORARY. SELECTIFNOT(ID=2999973). REGRESSION /MISSINGLISTWISE /STATISTICSCOEFFOUTSCIRANOVA /CRITERIA=PIN(.05)POUT(.10) /NOORIGIN /DEPENDENTREADINGIMPROVEMENT /METHOD=ENTERZEROCENTEREDREADING88. As before, the(raw) residual is(35. The deleted residual is(37.

Noticethattheslopeisnolongerstatsig!

3

OutlierDetection:DeletedResiduals(Part2or3)

*Wedonothavecalculatedeletedresidual“byhand,” wecanhavethecomputerdoitautomaticallyforevery case,and,alongtheway,wecanhavethecomputerdo awholebunchofotherthings. REGRESSION /MISSINGLISTWISE /STATISTICSCOEFFOUTSCIRANOVA /CRITERIA=PIN(.05)POUT(.10) /NOORIGIN /DEPENDENTREADINGIMPROVEMENT /METHOD=ENTERZEROCENTEREDREADING88 /SCATTERPLOT=(*RESID,*PRED) /RESIDUALSHIST(RESID)NORM(RESID) /SAVEPREDRESIDDRESIDLEVERCOOK.

Createaresidualvs.fittedplot(i.e.,aresidualvs.predictedplot). Createahistogramofresidualsandanormalprobabilityplot. Createfivenewvariables: A PRE_#:Apredicted/fittedvalueforeachobservation. A RES_#:Aresidualforeachobservation. A DRE_#:Adeletedresidualforeachobservation. A LEV_#:Aleveragestatisticforeachobservation. A COO_#:Aninfluencestatistic(Cook’sD)foreachobs.

*Onceweproduceourvariables,wecanexaminethem. EXAMINEVARIABLES=DRE_1LEV_1COO_1 /COMPAREGROUP /STATISTICSDESCRIPTIVESEXTREME /CINTERVAL95 /MISSINGLISTWISE /NOTOTAL. GRAPH /HISTOGRAM(NORMAL)=DRE_1. GRAPH /HISTOGRAM=LEV_1. GRAPH /HISTOGRAM=COO_1.

• OutlierDetection:DeletedResiduals(Part3of3)

Adeletedresidualisaresidualbasedonsubtractingthepredictedvaluefromtheobservedvalue,justlikeatypical, rawresidual,exceptthatthepredictedvalueiscalculatedwith theobservationremovedinordertoavoidthe part/wholeprobleminwhichwearelookingforoutliersfromthe trendbuttheoutlierispartofthetrend.

Twoobviousoutliers. Whoarethey?

>

OutlierDetection:TheLeverageStatistic

Aleveragestatisticisameasureoftheextremityofanobservationbasedonthevalue(s)ofitspredictor(s).Whenwe haveonepredictor,wecaneasilyseewhoisextremeonthatpredictor,butwhenwehave12predictors,itcanbe impossibletoseewhoisgenerally extremeonall predictors. Somehighleverageobservations. Whoarethey?

8

OutlierDetection:TheCook’sDStatistic

Ahighinfluenceobservation. Whoisit?

Aninfluencestatisticcomparesthetrendline(calculatedfromallthedata,includingtheobservation)witha hypotheticaltrendline(calculatedfromallthedataexceptthe observation).Thebiggerthedifferencebetweenthe twotrendlines,thegreatertheinfluence.Cook’sDstatisticistheinfluencestatisticthatwewilluse,butthereare

• thers.

• OutlierDetection:Residuals,Leverage,Influence

Case Number Deleted Residual Leverage Cook’s Distance Result

8 Extreme Minimal Moderate ExtremeinY,NotinX:InfluenceY(Intercept 27 Extreme Extreme Extreme ExtremeinYAndinX:InfluenceSlope 31 Minimal Extreme Minimal NotExtremeinY,ButinX:LittleInfluence 54 Minimal Minimal Minimal NeitherExtremeinYNorinX:LittleInfluence

27 8 31 54

• Non(LinearityDetection:RVFPlot

Aresidualversusfittedplot(RVFplot),alsoknownasaresidualversuspredictedplot,isjustwhatitsaysitis:a scatterplot ofresidualvaluesversusfitted/predictedvalues. Horseshoeshapesindicatenon(linearity.Iftherewereahorseshoeshapeinouroutcomeversus predictorplot,it wouldbemagnifiedintheresidualversusfittedplot,buteverythinglooksokayhere.InUnit13,we’llseeexamplesof non(linearrelationships(andattendanthorseshoes).Ifyouarewondering“what’sthebigdeal?” waituntilwehave7 predictors.Nomatterhowmanypredictors,wewillstillhaveonlyonepredictedvalueandonlyonefittedvaluefor eachobservation,sowecanstilluseanRVFplotforthemultipleregressionmodel,whereaswewouldneednotatwo dimensionalscatterplot oftheoutcomeversuspredictorsbutan8dimensionalscatterplot! GoodOldOutcomeVs.PredictorScatterplot: ShinyNewResidualVs.FittedScatterplot:

• Heteroskedasticity Detection:RVFPlot

Aresidualversusfittedplot(RVFplot),alsoknownasaresidualversuspredictedplot,isjustwhatitsaysitis:a scatterplot ofresidualvaluesversusfitted/predictedvalues. Funnelshapesindicateheterskedasticity.Iftherewereafunnelshapeinouroutcomeversuspredictorplot,itwould bemagnifiedintheresidualversusfittedplot,buteverythinglooksokayhere.InUnit14,we’llseeexamplesof heteroskedastic relationships(andattendantfunnels). GoodOldOutcomeVs.PredictorScatterplot: ShinyNewResidualVs.FittedScatterplot:

3

Non(NormalityDetection:ResidualHistograms

Ahistogramofresidualscangiveanindicationwhetherornotthe residualsarenormallydistributed;however,usewithcaution, becausehistogramsofresidualsshowanunconditionaldistribution (i.e.,theydon’tthinkvertically).Weareultimatelyconcerned withnormality(andhomoskedasticity)conditionalonX. Nevertheless,suchhistogramscanbeuseful,especiallywhen supplementedwithanRVFplotwhichallowsyoutothinkinterms

• fverticalslicesandconsequentlythinkaboutconditional

distributions.

3

Non(NormalityDetection:P(PPlots

Aprobability(probabilityplot(P(Pplot)isanotherwayoflookingat aresidualhistogram,withafocusonnormality.Inanormal distributionweexpect50%oftheobservationstobebelow average,and,becauseit’samathematicalconstruct,weobserve 50%oftheobservationstobebelowaverage.Thissimpletruth formsourbaselineofcomparison(theredlinebelow).Inasample distributionfromapopulationwithanormaldistribution,we expect50%oftheobservationstobebelowaverage,butdueto samplingerror(orperhapsduetoanon(normalpopulation distribution,wemayobservemoreorfewerthan50%of

• bservationstobebelowaverage.

Here,weobserve 50%ofoursample, butweexpecta smidgemorethan 50%. Expect more. Expect less.

Thetake(homemessageforP(Pplots isthatwewantthedottedlinetolie

• ntopofthestraightline,and

wherethedottedlinedeviates,we havenon(normalityinoursample, whichmayindicatenon(normalityin

• urpopulation.

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ReflectingonourUnit12ResearchQuestions

Q2: /(7:: :1&9</)& 7&7&< Q1: 5&:::& 779&( ::<

Toanswerthefirstquestion,wecansortourdatabyresidualsandfindthe largestpositiveresiduals: Toansweroursecondquestion,weseefromourRVFplotthatthe relationshipappearslinear(nohorseshoe)and homoskedastic (nofunnel).

33

CheckingRegressionAssumptionsWithRegressionDiagnostics

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&HI(N(LO 7&& $

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AnsweringourRoadmapQuestion

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ε β β β β β β β β β β β β β + + + + + + + + + + + + + = LATINO FREELUNCHx BLACK FREELUNCHx ASIAN FREELUNCHx ESLxLATINO ESLxBLACK ESLxASIAN FREELUNCH ESL HOMEWORKP L LATINO BLACK ASIAN READING

12 11 10 9 8 7 6 5 4 3 2 1

1 2

FromtheRVFplot,wedonot appeartohaveaproblemwith meetingthelinearity assumption.However,duetoa ceilingeffect,the homoskedasticity and normalityassumptionsare questionablymet.Otherthan theceilingeffect,the conditionalvariancesappear roughlyequal.Weare concernedthatthehigh(end predictionsarenegatively skewedbecauseoftheceiling effect

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LookingatNormality

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FromahistogramofresidualsandP(Pplot,weseeaslightnegativeskewoftheresidualsthatweattributetothe ceilingeffectofourreadingmeasure.

38

LookingforOutliers

Therearenooutliers

• fconcern,inpart

becausethelarge samplesizeminimizes theinfluenceofany

• nedatum.

Becauseofthelarge samplesize,the histogramsbeloware fairlyuseless,soIwill turnthedistributionof Cook’sDstatisticsinto ascatterplot….

3

ABetterLookatTheInfluentialOutliers

Whenweplotthe Cook’sDstatistics versusanarbitrary x(variable,wesee about10students thatstandoutfrom thepack.Wewill inspectthose10 studentsmore closelytoseeif thereisafurther pattern.

3

LookingForPatternsintheInfluentialOutliers

*HereistheSPSSsyntaxforoutputtingcase(summarytables. *SortthecasesbyCook’sdistance. *Data>SortCases… *SortbytheCook’sD. SORTCASESBYCoo_1(A). *Checkoutthefirsttwentycases, whichhavethehighestCook’sDasperyoursorting. *Analyze>Reports>CaseSummaries… SUMMARIZE /TABLES=IDREADINGHOMEWORKFREELUNCHESLRACE /FORMAT=LISTNOCASENUMTOTALLIMIT=20 /TITLE='CaseSummaries' /MISSING=VARIABLE /CELLS=COUNT.

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Unit12Appendix:KeyConcepts

NoticethatIuse“increase.” Thelongitudinaldatawarrantthedevelopmental conclusion! READING92hasmeasurementerror,andREADING88hasmeasurementerror,whenI taketheirdifference,theirdifferencenecessarilyhasmoremeasurementerrorthan either!Ah,well. Wheneverthereisanelementofrandomnessintheoutcome,weexpectregressionto themean.Measurementerrorisonepossiblesourceofrandomness,butnottheonly possiblesourceofrandomness.Ifwepredictadultheightbymother’sheight,wewill getregressiontothemean,eventhoughthereisonlytrivialmeasurementerrorwith height.

• ? .)

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Unit12Appendix:KeyInterpretations

Fortheaveragestudent,weexpectanincreaseof6readingpointsfromthe8thgradetothe 12thgrade,from48pointsto54points. FromtheRVFplot,wedonotappeartohaveaproblemwithmeetingthelinearity assumption.However,duetoaceilingeffect,thehomoskedasticity andnormality assumptionsarequestionablymet.Otherthantheceilingeffect, theconditionalvariances appearroughlyequal.Weareconcernedthatthehigh(endpredictionsarenegativelyskewed becauseoftheceilingeffect. FromahistogramofresidualsandP(Pplot,weseeaslightnegativeskewoftheresidualsthat weattributetotheceilingeffectofourreadingmeasure. Therearenooutliersofconcern,inpartbecausethelargesamplesizeminimizesthe influenceofanyonedatum. WhenweplottheCook’sDstatisticsversusanarbitraryx(variable,weseeabout10students thatstandoutfromthepack.Wewillinspectthose10studentsmorecloselytoseeifthereis afurtherpattern.

• ? .)

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Unit12Appendix:KeyTerminology

Atleastinsimplelinearregression,diagnosticsprovideinformationthatwecouldconceivablygleanfromabivariate scatterplot oftheoutome versuspredictor;neverthelesstheycanprovideahelpfullydetailedview.Inmultiple regression,however,diagnosticsprovideinformationthatwecouldnevergatherbyeye. Aresidual(akaerror)isthedifferencebetweenourobservedoutcomeandourpredictedoutcome.Iftheresidualis negativethatmeansweshouldhavepredictedlower(i.e.,weoverpredicted).Iftheresidualispositive,weshouldhave predictedhigher(i.e.,weunderpredicted).Ofcourse,weexpectresidualsbecauseofindividualvariation,hidden variables,andmeasurementerror. Everydatumhasanassociatedresidual,andwecangraphtheresidualswithahistogram. Adeletedresidualisaresidualbasedonsubtractingthepredictedvaluefromtheobservedvalue,justlikeatypical,raw residual,exceptthatthepredictedvalueiscalculatedwiththe observationremovedinordertoavoidthepart/whole probleminwhichwearelookingforoutliersfromthetrendbuttheoutlierispartofthetrend. Aleveragestatisticisameasureoftheextremityofanobservationbasedonthevalue(s)ofitspredictor(s).Whenwe haveonepredictor,wecaneasilyseewhoisextremeonthatpredictor,butwhenwehave12predictors,itcanbe impossibletoseewhoisgenerally extremeonall predictors. Aninfluencestatisticcomparesthetrendline(calculatedfromallthedata,includingtheobservation)withahypothetical trendline(calculatedfromallthedataexcepttheobservation).Thebiggerthedifferencebetweenthetwotrendlines, thegreattheinfluence.Cook’sDstatisticistheinfluencestatisticthatwewilluse,butthereareothers. Aresidualversusfittedplot(RVFplot),alsoknownasaresidualversuspredictedplot,isjustwhatitsaysitis:a scatterplot ofresidualvaluesversusfitted/predictedvalues. Ahistogramofresidualscangiveanindicationwhetherornottheresidualsarenormallydistributed;however,usewith caution,becausehistogramsofresidualsshowanunconditionaldistribution(i.e.,theydon’tthinkvertically).Weare ultimatelyconcernedwithnormality(andhomoskedasticity)conditionalonX.Nevertheless,suchhistogramscanbe useful,especiallywhensupplementedwithanRVFplotwhichallowsyoutothinkintermsofverticalslicesand consequentlythinkaboutconditionaldistributions. Aprobability(probabilityplot(P(Pplot)isanotherwayoflookingataresidualhistogram,withafocusonnormality.Ina normaldistributionweexpect50%oftheobservationstobebelowaverageand,becauseit’samathematicalconstruct, weobserve50%oftheobservationstobebelowaverage.Thissimpletruthformsourbaselineofcomparison(inred below).Inasampledistributionfromapopulationwithanormal distribution,weexpect50%oftheobservationstobe belowaverage,butduetosamplingerror,wemayobservemoreor fewerthan50%ofobservationstobebelowaverage.

• ThepopulationvarianceofmeasurementXisdenoted:.

ThepopulationstandarddeviationofmeasurementXisdenoted: . ThepopulationcorrelationbetweenmeasurementsXandYis denoted:.

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Unit12Appendix:Formulas

2 2 2 2

2 2

X Y XY X Y DD X Y X Y X Y X YY X

σ σ ρ ρ ρ σ σ ρ σ σ ρ σ σ

′ ′ ′

+ − = + −

2 X X σ ′

2

Y X

ρ σ

ThereliabilityofmeasurementXisdenoted:ρXX‘ ,'wheretheGreek letterrho,standsforthepopulationcorrelation,thesubscript X standsforoneformofmeasurementX,andthesubscriptX'standsfor aparallelformofthemeasurement. ReliabilityNotation TheReliabilityofaDifference

Y X

σ σ

• 3

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Unit12Appendix:Formulas

2 2 2 2

2 2

X Y XY X Y DD X Y X Y X Y X YY X

σ σ ρ ρ ρ σ σ ρ σ σ ρ σ σ

′ ′ ′

+ − = + −

TheReliabilityofaDifference

1 : y Reliabilit Perfect Of Baseline Small Want Big Want Big Want Small Want Big Want Big Want

'

= = − + = = =

′ ′ Y Y X X DD

ρ ρ ρ

AWewantthereliablevarianceinmeasurementXtobebig. AWewantthereliablevarianceinmeasurementYtobebig. AWewantthecorrelationbetweenmeasurementsXandYtobesmall. Ifthecorrelationisnegative,thenthereliabilityofthedifferencecanactuallyexceedthereliabilityoftheindividualtests! AWhathappenswhenmeasurementXandYareperfectlyreliable? AWhathappenswhenmeasurementXandYareperfectlyunreliable? Notethat,ifmeasurementsXandYareperfectlyunreliable,thentheymustbeperfectlyuncorrelatedaswell. AWhathappenswhenmeasurementXandYareperfectlycorrelated? Notethat,ifmeasurementsXandYareperfectlycorrelated(and theyhavethesamestandarddeviation),theneverybodyhasthe sameexact differencescore.

;)&FBG& &D&#D+

XY XY Y Y X X DD

ρ ρ ρ ρ ρ 2 2 2

'

− − + =

′ ′

• ? .)

!$%:

Unit12Appendix:SPSSSyntax

*ExampleSPSSsyntaxforcomputingtransformedvariables. *Thislineartransformationisnotaz(transformationbecauseIdidnotdividethedifferencebythestandard deviation. COMPUTEZEROCENTEREDREADING88=READING88( 48.0155. EXECUTE. *This(goofy)transformationisnon(linearbecauseIdomorethanadd/subtractand/ormultiply/dividebyaconstant.I usepowersandlogs. COMPUTESean_Is_A_Great_SPSS_Programmer =READING88*48.0155( 1975/27+FREELUNCH**(1/2)+ LN(HOMEWORK+1). EXECUTE. *Ifweareinterestedinchanges,let’scomputeachangescoreanduseitasouroutcome.Thisisnotalinear transformationbecauseIadd/subtractand/ormultiply/dividebyavariable,notaconstant. COMPUTEREADINGIMPROVEMENT=READING92( READING88. EXECUTE. *Identifytheresidual,temporarilyremoveit,andrefittheline. TEMPORARY. SELECTIFNOT(ID=2999973). REGRESSION /MISSINGLISTWISE /STATISTICSCOEFFOUTSCIRANOVA /CRITERIA=PIN(.05)POUT(.10) /NOORIGIN /DEPENDENTREADINGIMPROVEMENT /METHOD=ENTERZEROCENTEREDREADING88.

• >

? .) !$%:

Unit12Appendix:SPSSSyntax

*Wedonothavecalculatedeletedresidual“byhand,” wecanhavethecomputerdoitautomaticallyforeverycase, and,alongtheway,wecanhavethecomputerdoawholebunchof otherthings. REGRESSION /MISSINGLISTWISE /STATISTICSCOEFFOUTSCIRANOVA /CRITERIA=PIN(.05)POUT(.10) /NOORIGIN /DEPENDENTREADINGIMPROVEMENT /METHOD=ENTERZEROCENTEREDREADING88 /SCATTERPLOT=(*RESID,*PRED) /RESIDUALSHIST(RESID)NORM(RESID) /SAVEPREDRESIDDRESIDLEVERCOOK. *Onceweproduceourvariables,wecanexaminethem. EXAMINEVARIABLES=DRE_1LEV_1COO_1 /COMPAREGROUP /STATISTICSDESCRIPTIVESEXTREME /CINTERVAL95 /MISSINGLISTWISE /NOTOTAL. GRAPH /HISTOGRAM(NORMAL)=DRE_1. GRAPH /HISTOGRAM=LEV_1. GRAPH /HISTOGRAM=COO_1.

• 8

OutputYourNewVariablesandNiftyPlots

Startfrom: Analyze> Regression> Linear

• ExamineYourNewVariables

Lookaroundand checkoutyour

• ptions.

• ? .)

!$%:

PerceivedIntimacyofAdolescentGirls(Intimacy.sav)

A Source:HGSEthesisbyDr.LindaKilner entitledIntimacyinFemale Adolescent'sRelationshipswithParentsandFriends(1991).Kilner collectedtheratingsusingtheAdolescentIntimacyScale. A Sample:64adolescentgirlsinthesophomore,juniorandseniorclasses

• falocalsuburbanpublicschoolsystem.

A Variables:

*&@ @H# C@ @HC# @6:7@ @H6# )'(7@ @H'# .&2&7@ @H.# 6&7@ @H6# *&4 4H# C4 4HC# @6:74 4H6# )'(74 4H'# .&2&74 4H.# 6&74 4H6#

A Overview:Datasetcontainsself(ratingsoftheintimacythat adolescentgirlsperceivethemselvesashavingwith:(a)their motherand(b)theirboyfriend.

• ? .)

!$%:

PerceivedIntimacyofAdolescentGirls(Intimacy.sav)

> ? .) !$%:

PerceivedIntimacyofAdolescentGirls(Intimacy.sav)

> ? .) !$%:

PerceivedIntimacyofAdolescentGirls(Intimacy.sav)

> ? .) !$%:

PerceivedIntimacyofAdolescentGirls(Intimacy.sav)

>3 ? .) !$%:

HighSchoolandBeyond(HSB.sav)

A Source:SubsetofdatagraciouslyprovidedbyValerieLee,Universityof Michigan. A Sample:Thissubsamplehas1044studentsin205schools.Missing data

• ntheoutcometestscoreandfamilySESwereeliminated.Inaddition,

schoolswithfewerthan3studentsincludedinthissubsetofdatawere excluded. A Variables:

'((E

4&)#,4&),% "#,",% 9#,;,@ 4B!#4! .2#=.2 .#=.2 4BC#4&: 446&#4&& ;!6&#;;71&&

'((I&E

.&@#J=.&: =G#=G .&*#J=.&: 4B!H#2:!= .2H#2:.2= .2H#2:.2= 4BCH#2:&= 446&H#2:(&&= ;!6&H#2:71&&=

A Overview:HighSchool&Beyond– Subsetofdata focusedonselectedstudentandschoolcharacteristics aspredictorsofacademicachievement.

>- ? .) !$%:

HighSchoolandBeyond(HSB.sav)

>> ? .) !$%:

HighSchoolandBeyond(HSB.sav)

>8 ? .) !$%:

HighSchoolandBeyond(HSB.sav)

> ? .) !$%:

HighSchoolandBeyond(HSB.sav)

> ? .) !$%:

UnderstandingCausesofIllness(ILLCAUSE.sav)

A Source:PerrinE.C.,Sayer A.G.,andWillettJ.B.(1991). SticksAndStonesMayBreakMyBones:ReasoningAboutIllness CausalityAndBodyFunctioningInChildrenWhoHaveAChronicIllness, Pediatrics,88(3),608(19. A Sample:301children,includingasub(sampleof205whowere describedasasthmatic,diabetic,or healthy.Afterfurtherreductions duetothelist0wise'deletion'ofcaseswithmissingdataononeormore variables,theanalyticsub(sampleusedinclassendsupcontaining:33 diabeticchildren,68asthmaticchildrenand93healthychildren. A Variables:

/""62!# 6I:/6 !# 6I! :&7!$# ..'C# 6I&.(.&'&(C 2!# 6I2:/@ !!2# 6I&:C 6&/# ,2&*(&,= 2&# ,2&,= *(&# ,*(&,=

A Overview:Dataforinvestigatingdifferencesinchildren’s understandingofthecausesofillness,bytheirhealth status.

> ? .) !$%:

UnderstandingCausesofIllness(ILLCAUSE.sav)

8 ? .) !$%:

UnderstandingCausesofIllness(ILLCAUSE.sav)

8 ? .) !$%:

UnderstandingCausesofIllness(ILLCAUSE.sav)

8 ? .) !$%:

UnderstandingCausesofIllness(ILLCAUSE.sav)

83 ? .) !$%:

UnderstandingCausesofIllness(ILLCAUSE.sav)

8- ? .) !$%:

UnderstandingCausesofIllness(ILLCAUSE.sav)

8> ? .) !$%:

UnderstandingCausesofIllness(ILLCAUSE.sav)

88 ? .) !$%:

UnderstandingCausesofIllness(ILLCAUSE.sav)

8 ? .) !$%:

ChildrenofImmigrants(ChildrenOfImmigrants.sav)

A Source:Portes,Alejandro,&RubenG.Rumbaut (2001).'Legacies:'The'Story'of' the'Immigrant'SecondGeneration.BerkeleyCA:UniversityofCaliforniaPress. A Sample:Randomsampleof880participantsobtainedthroughthewebsite. A Variables:

:#

:2&&

;&# J&7:(&: @#

,@,;

*#

*& =:&#

!#

6!& A Overview:“CILSisalongitudinalstudydesignedtostudythe adaptationprocessoftheimmigrantsecondgenerationwhichis definedbroadlyasU.S.(bornchildrenwithatleastoneforeign(born parentorchildrenbornabroadbutbroughtatanearlyagetothe UnitedStates.Theoriginalsurveywasconductedwithlargesamples

• fsecond(generationchildrenattendingthe8thand9thgradesin

publicandprivateschoolsinthemetropolitanareasofMiami/Ft. LauderdaleinFloridaandSanDiego,California” (fromthewebsite descriptionofthedataset).

8 ? .) !$%:

ChildrenofImmigrants(ChildrenOfImmigrants.sav)

8 ? .) !$%:

ChildrenofImmigrants(ChildrenOfImmigrants.sav)

• ? .)

!$%:

ChildrenofImmigrants(ChildrenOfImmigrants.sav)

• ? .)

!$%:

ChildrenofImmigrants(ChildrenOfImmigrants.sav)

• ? .)

!$%:

HumanDevelopmentinChicagoNeighborhoods(Neighborhoods.sav)

A Source:Sampson,R.J.,Raudenbush,S.W.,&Earls,F.(1997).Neighborhoods andviolentcrime:Amultilevelstudyofcollectiveefficacy.Science,'277,918( 924. A Sample:Thedatadescribedhereconsistofinformationfrom343Neighborhood ClustersinChicagoIllinois.Someofthevariableswereobtainedbyprojectstaff fromthe1990Censusandcityrecords.Othervariableswereobtainedthrough questionnaireinterviewswith8782Chicagoresidentswhowereinterviewedin theirhomes. A Variables: =# =&&$ @># =&> *# 6&*: /H6&#/: (# ( .# . 6!# 6&!&& '&# J55'&'& .&'# J5.&'&

A ThesedatawerecollectedaspartoftheProjecton HumanDevelopmentinChicagoNeighborhoodsin1995.

3 ? .) !$%:

HumanDevelopmentinChicagoNeighborhoods(Neighborhoods.sav)

• ? .)

!$%:

HumanDevelopmentinChicagoNeighborhoods(Neighborhoods.sav)

> ? .) !$%:

HumanDevelopmentinChicagoNeighborhoods(Neighborhoods.sav)

8 ? .) !$%:

HumanDevelopmentinChicagoNeighborhoods(Neighborhoods.sav)

• ? .)

!$%:

HumanDevelopmentinChicagoNeighborhoods(Neighborhoods.sav)

• ? .)

!$%:

HumanDevelopmentinChicagoNeighborhoods(Neighborhoods.sav)

• ? .)

!$%:

HumanDevelopmentinChicagoNeighborhoods(Neighborhoods.sav)

• ? .)

!$%:

HumanDevelopmentinChicagoNeighborhoods(Neighborhoods.sav)

• ? .)

!$%:

HumanDevelopmentinChicagoNeighborhoods(Neighborhoods.sav)

• ? .)

!$%:

4(HStudyofPositiveYouthDevelopment(4H.sav)

A Sample:Thesedataconsistofseventhgraderswhoparticipatedin Wave3ofthe4(HStudyofPositiveYouthDevelopmentatTufts University.Thissubfile isasubstantiallysampled(downversionofthe

• riginalfile,asallthecaseswithanymissingdataontheseselected

variableswereeliminated. A Variables:

9;# ,;,@ @!# B@I!& # 1 *# * 6# ;/# ;I ./& .# . *# , 1>*# ,B 8K*#

A 4(HStudyofPositiveYouthDevelopment A Source:SubsetofdatafromIARYD,TuftsUniversity

2&6# 1.&2&&6& &6# 1.&&6& .6# 1.&.&6& .2# 1.&.&2& 64# 1.&6&4 5# 15

3 ? .) !$%:

4(HStudyofPositiveYouthDevelopment(4H.sav)

• ? .)

!$%:

4(HStudyofPositiveYouthDevelopment(4H.sav)

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4(HStudyofPositiveYouthDevelopment(4H.sav)

8 ? .) !$%:

4(HStudyofPositiveYouthDevelopment(4H.sav)

• ? .)

!$%:

4(HStudyofPositiveYouthDevelopment(4H.sav)

• ? .)

!$%:

4(HStudyofPositiveYouthDevelopment(4H.sav)

• ? .)

!$%:

4(HStudyofPositiveYouthDevelopment(4H.sav)

• ? .)

!$%:

4(HStudyofPositiveYouthDevelopment(4H.sav)