应用spss进行对数转换教学课件

ComputingTransformationsTransformingvariablesTransformationsfornormalityTransformationsforlinearity1Transformations:

TransformingvariablestosatisfyassumptionsWhenametricvariablefailstosatisfytheassumptionofnormality,homogeneityofvariance,orlinearity,wemaybeabletocorrectthedeficiencybyusingatransformation.Wewillconsiderthreetransformationsfornormality,homogeneityofvariance,andlinearity:thelogarithmictransformationthesquareroottransformation,andtheinversetransformationplusafourththatmaybeusefulforproblemsoflinearity:thesquaretransformation2TransformationschangethemeasurementscaleInthediagramtotheright,thevaluesof5through20areplottedonthedifferentscalesusedinthetransformations.Thesescaleswouldbeusedinplottingthehorizontalaxisofthehistogramdepictingthedistribution.Whencomparingvaluesmeasuredonthedecimalscaletowhichweareaccustomed,weseethateachtransformationchangesthedistancebetweenthebenchmarkmeasurements.Allofthetransformationsincreasethedistancebetweensmallvaluesanddecreasethedistancebetweenlargevalues.Thishastheeffectofmovingthepositivelyskewedvaluestotheleft,reducingtheeffectoftheskewingandproducingadistributionthatmorecloselyresemblesanormaldistribution.3Transformations:

ComputingtransformationsinSPSSInSPSS,transformationsareobtainedbycomputinganewvariable.SPSSfunctionsareavailableforthelogarithmic(LG10)andsquareroot(SQRT)transformations.Theinversetransformationusesaformulawhichdividesonebytheoriginalvalueforeachcase.Foreachofthesecalculations,theremaybedatavalueswhicharenotmathematicallypermissible.Forexample,thelogofzeroisnotdefinedmathematically,divisionbyzeroisnotpermitted,andthesquarerootofanegativenumberresultsinan“imaginary”value.Wewillusuallyadjustthevaluespassedtothefunctiontomakecertainthattheseillegaloperationsdonotoccur.4Transformations:

TwoformsforcomputingtransformationsTherearetwoformsforeachofthetransformationstoinducenormality,dependingonwhetherthedistributionisskewednegativelytotheleftorskewedpositivelytotheright.BothformsusethesameSPSSfunctionsandformulatocalculatethetransformations.Thetwoformsdifferinthevalueorargumentpassedtothefunctionsandformula.Theargumenttothefunctionsisanadjustmenttotheoriginalvalueofthevariabletomakecertainthatallofthecalculationsaremathematicallycorrect.5Transformations:

FunctionsandformulasfortransformationsSymbolically,ifweletxstandfortheargumentpassestothefunctionorformula,thecalculationsforthetransformationsare:Logarithmictransformation:computelog=LG10(x)Squareroottransformation:computesqrt=SQRT(x)Inversetransformation:computeinv=-1/(x)Squaretransformation:computes2=x*xForalltransformations,theargumentmustbegreaterthanzerotoguaranteethatthecalculationsaremathematicallylegitimate.6Transformations:

TransformationofpositivelyskewedvariablesForpositivelyskewedvariables,theargumentisanadjustmenttotheoriginalvaluebasedontheminimumvalueforthevariable.Iftheminimumvalueforavariableiszero,theadjustmentrequiresthatweaddonetoeachvalue,e.g.x+1.Iftheminimumvalueforavariableisanegativenumber(e.g.,–6),theadjustmentrequiresthatweaddtheabsolutevalueoftheminimumvalue(e.g.6)plusone(e.g.x+6+1,whichequalsx+7).7Transformations:

ExampleofpositivelyskewedvariableSupposeourdatasetcontainsthenumberofbooksread(books)for5subjects:1,3,0,5,and2,andthedistributionispositivelyskewed.Theminimumvalueforthevariablebooksis0.Theadjustmentforeachcaseisbooks+1.Thetransformationswouldbecalculatedasfollows:ComputelogBooks=LG10(books+1)ComputesqrBooks=SQRT(books+1)ComputeinvBooks=-1/(books+1)8Transformations:

TransformationofnegativelyskewedvariablesIfthedistributionofavariableisnegativelyskewed,theadjustmentofthevaluesreverses,orreflects,thedistributionsothatitbecomespositivelyskewed.Thetransformationsarethencomputedonthevaluesinthepositivelyskeweddistribution.Reflectioniscomputedbysubtractingallofthevaluesforavariablefromoneplustheabsolutevalueofmaximumvalueforthevariable.Thisresultsinapositivelyskeweddistributionwithallvalueslargerthanzero.Whenananalysisusesatransformationinvolvingreflection,wemustrememberthatthiswillreversethedirectionofalloftherelationshipsinwhichthevariableisinvolved.Ourinterpretationofrelationshipsmustbeadjustedaccordingly.9Transformations:

ExampleofnegativelyskewedvariableSupposeourdatasetcontainsthenumberofbooksread(books)for5subjects:1,3,0,5,and2,andthedistributionisnegativelyskewed.Themaximumvalueforthevariablebooksis5.Theadjustmentforeachcaseis6-books.Thetransformationswouldbecalculatedasfollows:ComputelogBooks=LG10(6-books)ComputesqrBooks=SQRT(6-books)ComputeinvBooks=-1/(6-books)10Transformations:

TheSquareTransformationforLinearityThesquaretransformationiscomputedbymultiplyingthevalueforthevariablebyitself.Itdoesnotmatterwhetherthedistributionispositivelyornegativelyskewed.Itdoesmatterifthevariablehasnegativevalues,sincewewouldnotbeabletodistinguishtheirsquaresfromthesquareofacomparablepositivevalue(e.g.thesquareof-4isequaltothesquareof+4).Ifthevariablehasnegativevalues,weaddtheabsolutevalueoftheminimumvaluetoeachscorebeforesquaringit.11Transformations:

ExampleofthesquaretransformationSupposeourdatasetcontainschangescores(chg)for5subjectsthatindicatethedifferencebetweentestscoresattheendofasemesterandtestscoresatmid-term:-10,0,10,20,and30.Theminimumscoreis-10.Theabsolutevalueoftheminimumscoreis10.Thetransformationwouldbecalculatedasfollows:ComputesquarChg=(chg+10)*(chg+10)12Transformations:

TransformationsfornormalityBoththehistogramandthenormalityplotforTotalTimeSpentontheInternet(netime)indicatethatthevariableisnotnormallydistributed.13Transformations:

DeterminewhetherreflectionisrequiredSkewness,inthetableofDescriptiveStatistics,indicateswhetherornotreflection(reversingthevalues)isrequiredinthetransformation.IfSkewnessispositive,asitisinthisproblem,reflectionisnotrequired.IfSkewnessisnegative,reflectionisrequired.14Transformations:

ComputetheadjustmenttotheargumentInthisproblem,theminimumvalueis0,so1willbeaddedtoeachvalueintheformula,i.e.theargumenttotheSPSSfunctionsandformulafortheinversewillbe:netime+1.15Transformations:

ComputingthelogarithmictransformationTocomputethetransformation,selecttheCompute…commandfromtheTransformmenu.16Transformations:

SpecifyingthetransformvariablenameandfunctionFirst,intheTargetVariabletextbox,typeanameforthelogtransformationvariable,e.g.“lgnetime“.Second,scrolldownthelistoffunctionstofindLG10,whichcalculateslogarithmicvaluesuseabaseof10.(Thelogarithmicvaluesarethepowertowhich10israisedtoproducetheoriginalnumber.)Third,clickontheuparrowbuttontomovethehighlightedfunctiontotheNumericExpressiontextbox.17Transformations:

AddingthevariablenametothefunctionFirst,scrolldownthelistofvariablestolocatethevariablewewanttotransform.Clickonitsnamesothatitishighlighted.Second,clickontherightarrowbutton.SPSSwillreplacethehighlightedtextinthefunction(?)withthenameofthevariable.18Transformations:

AddingtheconstanttothefunctionFollowingtherulesstatedfordeterminingtheconstantthatneedstobeincludedinthefunctioneithertopreventmathematicalerrors,ortodoreflection,weincludetheconstantinthefunctionargument.Inthiscase,weadd1tothenetimevariable.ClickontheOKbuttontocompletethecomputerequest.19Transformations:

ThetransformedvariableThetransformedvariablewhichwerequestedSPSScomputeisshowninthedataeditorinacolumntotherightoftheothervariablesinthedataset.20Transformations:

ComputingthesquareroottransformationTocomputethetransformation,selecttheCompute…commandfromtheTransformmenu.21Transformations:

SpecifyingthetransformvariablenameandfunctionFirst,intheTargetVariabletextbox,typeanameforthesquareroottransformationvariable,e.g.“sqnetime“.Second,scrolldownthelistoffunctionstofindSQRT,whichcalculatesthesquarerootofavariable.Third,clickontheuparrowbuttontomovethehighlightedfunctiontotheNumericExpressiontextbox.22Transformations:

AddingthevariablenametothefunctionSecond,clickontherightarrowbutton.SPSSwillreplacethehighlightedtextinthefunction(?)withthenameofthevariable.First,scrolldownthelistofvariablestolocatethevariablewewanttotransform.Clickonitsnamesothatitishighlighted.23Transformations:

AddingtheconstanttothefunctionFollowingtherulesstatedfordeterminingtheconstantthatneedstobeincludedinthefunctioneithertopreventmathematicalerrors,ortodoreflection,weincludetheconstantinthefunctionargument.Inthiscase,weadd1tothenetimevariable.ClickontheOKbuttontocompletethecomputerequest.24Transformations:

ThetransformedvariableThetransformedvariablewhichwerequestedSPSScomputeisshowninthedataeditorinacolumntotherightoftheothervariablesinthedataset.25Transformations:

ComputingtheinversetransformationTocomputethetransformation,selecttheCompute…commandfromtheTransformmenu.26Transformations:

SpecifyingthetransformvariablenameandformulaFirst,intheTargetVariabletextbox,typeanamefortheinversetransformationvariable,e.g.“innetime“.Second,thereisnotafunctionforcomputingtheinverse,sowetypetheformuladirectlyintotheNumericExpressiontextbox.Third,clickontheOKbuttontocompletethecomputerequest.27Transformations:

ThetransformedvariableThetransformedvariablewhichwerequestedSPSScomputeisshowninthedataeditorinacolumntotherightoftheothervariablesinthedataset.28Transformations:

AdjustmenttotheargumentforthesquaretransformationInthisproblem,theminimumvalueis0,noadjustmentisneededforcomputingthesquare.Iftheminimumwasanumberlessthanzero,wewouldaddtheabsolutevalueoftheminimum(droppingthesign)asanadjustmenttothevariable.Itismathematicallycorrecttosquareavalueofzero,sotheadjustmenttotheargumentforthesquaretransformationisdifferent.Whatweneedtoavoidarenegativenumbers,sincethesquareofanegativenumberproducesthesamevalueasthesquareofapositivenumber.29Transformations:

ComputingthesquaretransformationTocomputethetransformation,selecttheCompute…commandfromtheTransformmenu.30Transformations:

SpecifyingthetransformvariablenameandformulaFirst,intheTargetVariabletextbox,typeanamefortheinversetransformationvariable,e.g.“s2net