Chapter15: Statistics
143
 SEMEANThestandarderrorofthemean.
 SUMThesumofthevalues.
 MINTheminimumvalue.
 MAXThemaximumvalue.
 RANGEThedifferencebetweenthemaximumandminimumvalues.
 VARIANCEThevariance.
 FIRSTThefirstvalueinthecategory.
 LASTThelastvalueinthecategory.
 SKEWTheskewness.
 SESKEWThestandarderroroftheskewness.
 KURTThekurtosis
 SEKURTThestandarderrorofthekurtosis.
 HARMONICTheharmonicmean.
 GEOMETRICThegeometricmean.
Inaddition,threespecialkeywordsarerecognized:
 DEFAULTThisisthesameasMEANCOUNTSTDDEV.
 ALLAlloftheabovestatisticswillbecalculated.
 NONENostatisticswillbecalculated(onlyasummarywillbeshown).
Morethanonetable canbespecifiedinasinglecommand. Eachtableisseparatedby
a‘/’.Forexample
MEANS TABLES S =
c d d e e BY x
/a b BY x x y
/f BY y BY z.
hasthreetables(the‘TABLE=’isoptional).Thefirsttablehasthreedependentvariables
c,dandeandasinglecategoricalvariablex.Thesecondtablehastwodependentvariables
aandb,andtwo categoricalvariables x x andy. . The e thirdtablehas a singledependent
variablesf andacategoricalvariableformedbythecombinationofy y andz.
By default values are omittedfrom m the analysis s onlyif missingvalues (either system
missingorusermissing)forany ofthevariables directly involvedintheir calculationare
encountered. This s behaviour r can n be modified with the e /MISSING subcommand. . Three
optionsarepossible: TABLE,INCLUDEandDEPENDENT.
/MISSING=TABLE causes s cases tobe e dropped if any variable e is s missing inthe table
specificationcurrentlybeingprocessed,regardlessofwhetheritisneededtocalculatethe
statistic.
/MISSING=INCLUDEsaysthatusermissingvalues,eitherinthedependentvariablesor
inthecategoricalvariablesshouldbetakenattheirfacevalue,andnotexcluded.
/MISSING=DEPENDENTsaysthatusermissingvalues,inthedependentvariablesshould
betakenattheirfacevalue,howevercaseswhichhaveusermissingvaluesforthecategorical
variablesshouldbeomittedfromthecalculation.
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Chapter15: Statistics
144
15.11 NPARTESTS
NPARTESTS
nonparametrictestsubcommands
.
.
.
[/STATISTICS={DESCRIPTIVES}]
[/MISSING={ANALYSIS,LISTWISE}{INCLUDE,EXCLUDE}]
[/METHOD=EXACT[TIMER[(n)]]]
NPARTESTSperformsnonparametrictests.Nonparametrictestsmakeveryfewassump-
tions about the distribution n of the data. . One e or r more e tests s may y be specified d by y using
thecorrespondingsubcommand. Ifthe e /STATISTICS subcommandis alsospecified,then
summarystatisticsareproducesforeachvariablethatisthesubjectofanytest.
Certaintestsmaytakealongtimetoexecute,ifanexactfigureisrequired.Therefore,
bydefaultasymptoticapproximationsareusedunlessthesubcommand/METHOD=EXACTis
specified. Exacttestsgivemoreaccurateresults,butmaytakeanunacceptablylongtime
toperform. IftheTIMERkeywordisused,itsetsamaximumtime,afterwhichthetestwill
beabandoned,andawarningmessageprinted. Thetime,inminutes,shouldbespecified
inparenthesesaftertheTIMERkeyword. IftheTIMERkeywordisgivenwithoutthisfigure,
thenadefaultvalueof5minutesisused.
15.11.1 Binomialtest
[/BINOMIAL[(p)]=var
list[(value1[,value2)]]]
The/BINOMIALsubcommandcomparestheobserveddistributionofadichotomousvari-
able withthatof abinomialdistribution. . Thevariable e pspecifies the test proportionof
thebinomialdistribution. Thedefaultvalueof0.5isassumedifpisomitted.
Ifasinglevalueappearsafterthevariablelist,thenthatvalueisusedasthethreshold
topartitiontheobservedvalues. Valueslessthanorequaltothethresholdvalueformthe
firstcategory.Valuesgreaterthanthethresholdformthesecondcategory.
Iftwovaluesappearafterthevariablelist,thentheywillbeusedasthevalueswhicha
variablemusttaketobeintherespectivecategory.Casesforwhichavariabletakesavalue
equaltoneitherofthespecifiedvalues,takenopartinthetestforthatvariable.
Ifnovalues appear,thenthe variablemustassumedichotomousvalues. . Ifmorethan
two distinct,non-missing values for avariable under test are encounteredthen an error
occurs.
Ifthetestproportionisequalto0.5,thenatwotailedtestisreported. Foranyothertest
proportion,aonetailedtestisreported. Foronetailedtests,ifthetestproportionisless
thanorequaltothe observedproportion,thenthesignificance ofobservingtheobserved
proportionormoreisreported. Ifthetestproportionismorethantheobservedproportion,
thenthesignificance ofobservingtheobservedproportionor lessis reported. . Thatis s to
say,thetestisalwaysperformedintheobserveddirection.
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Chapter15: Statistics
145
psppusesaverypreciseapproximationtothegammafunctiontocomputethebinomial
significance.Thus,exactresultsarereportedevenforverylargesamplesizes.
15.11.2 ChisquareTest
[/CHISQUARE=var
list[(lo,hi)][/EXPECTED={EQUAL|f1,f2 ...fn}]]
The/CHISQUAREsubcommandproducesachi-squarestatisticforthedifferencesbetween
theexpectedandobservedfrequenciesofthecategoriesofavariable. Optionally,arange
ofvaluesmayappearafterthevariablelist.Ifarangeisgiven,thennonintegervaluesare
truncated,andvaluesoutsidethespecifiedrangeareexcludedfromtheanalysis.
The/EXPECTEDsubcommandspecifiestheexpectedvaluesofeachcategory.Theremust
beexactlyonenon-zeroexpectedvalue,foreachobservedcategory,ortheEQUALkeyword
mustbespecified.Youmayusethenotationn*ftospecifynconsecutiveexpectedcategories
alltakingafrequencyoff. Thefrequenciesgivenareproportions,notabsolutefrequencies.
Thesumof the frequenciesneednotbe1. . Ifno o /EXPECTED subcommandis given, then
thenequalfrequenciesareexpected.
15.11.3 CochranQTest
[/COCHRAN=var
list]
TheCochranQtestisusedtotest fordifferencesbetweenthreeormoregroups. . The
dataforvar
list inallcases must assumeexactly twodistinctvalues(other thanmissing
values).
ThevalueofQwillbedisplayedanditsAsymptoticsignificancebasedonachi-square
distribution.
15.11.4 FriedmanTest
[/FRIEDMAN=var
list]
TheFriedmantestisusedtotestfordifferencesbetweenrepeatedmeasureswhenthere
isnoindicationthatthedistributionsarenormallydistributed.
Alistofvariableswhichcontainthemeasureddatamustbegiven.Theprocedureprints
thesumofranksforeachvariable,theteststatisticanditssignificance.
15.11.5 Kendall’sWTest
[/KENDALL=var
list]
TheKendalltestinvestigateswhetheranarbitrarynumberofrelatedsamplescomefrom
thesamepopulation.ItisidenticaltotheFriedmantestexceptthattheadditionalstatistic
W,Kendall’sCoefficientofConcordanceisprinted.Ithastherange[0,1]—avalueofzero
indicates noagreement betweenthesamples whereas avalue ofunityindicates complete
agreement.
15.11.6 Kolmogorov-SmirnovTest
[/KOLMOGOROV-SMIRNOV({NORMAL[mu,sigma],UNIFORM[min,max],POIS-
SON[lambda],EXPONENTIAL[scale]})=var
list]
The one e sample e Kolmogorov-Smirnov v subcommand d is s usedto test whether r or not t a
dataset is drawn from a a particular r distribution. . Four r distributions are supported, , viz:
Normal,Uniform,PoissonandExponential.
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Chapter15: Statistics
146
Ideally youshouldprovidethe parameters of thedistributionagainst which h youwish
totest thedata. . Forexample,withthenormaldistributionthemean(mu)andstandard
deviation(sigma)shouldbegiven;withtheuniformdistribution,theminimum(min)and
maximum(max) valueshouldbeprovided. . However,ifthe e parameters areomittedthey
willbeimputedfromthedata. Imputingtheparametersreducesthepowerofthetestso
shouldbeavoidedifpossible.
Inthefollowingexample,twovariablesscore andagearetestedtoseeiftheyfollowa
normaldistributionwithameanof3.5andastandarddeviationof2.0.
NPAR TESTS
/KOLMOGOROV-SMIRNOV (normal 3.5 2.0) = score age.
If the variablesneedtobetestedagainst different distributions, thenaseparate sub-
command must t be used. . For r example thefollowingsyntax tests s score e against anormal
distributionwithmeanof3.5andstandarddeviationof2.0whilstageis testedagainsta
normaldistributionofmean40andstandarddeviation1.5.
NPAR TESTS
/KOLMOGOROV-SMIRNOV (normal 3.5 2.0) = score
/KOLMOGOROV-SMIRNOV (normal 40 1.5) = = age.
TheabbreviatedsubcommandK-SmaybeusedinplaceofKOLMOGOROV-SMIRNOV.
15.11.7 Kruskal-WallisTest
[/KRUSKAL-WALLIS=var
listBYvar(lower,upper)]
TheKruskal-Wallistestisusedtocomparedatafromanarbitrarynumberofpopulations.
It does not assume normality. . The e data tobe compared arespecified by y var
list. The
categoricalvariabledeterminingthegroupstowhichthedatabelongsisgivenbyvar. The
limitslowerandupper specifythevalidrangeofvar.Anycasesforwhichvarfallsoutside
[lower,upper]willbeignored.
Themeanrankofeachgroupaswellasthechi-squaredvalueandsignificanceofthetest
willbeprinted.TheabbreviatedsubcommandK-WmaybeusedinplaceofKRUSKAL-WALLIS.
15.11.8 Mann-WhitneyUTest
[/MANN-WHITNEY=var
listBYvar(group1,group2)]
TheMann-Whitneysubcommandisusedtotestwhethertwogroupsofdatacomefrom
different populations. . The e variables tobetested shouldbe specified in var
list and d the
grouping variable,that determinestowhichgroupthetestvariables belong,in var. . Var
maybeeitherastringoranalphavariable. Group1 1 andgroup2specifythetwovaluesof
var whichdeterminethegroupsofthetestdata. . Casesforwhichthevar r valueisneither
group1 orgroup2willbeignored.
ThevalueoftheMann-WhitneyUstatistic,theWilcoxonW,andthesignificancewill
beprinted. TheabbreviatedsubcommandM-WmaybeusedinplaceofMANN-WHITNEY.
15.11.9 McNemarTest
[/MCNEMARvar
list[WITHvar
list[(PAIRED)]]]
UseMcNemar’stesttoanalysethesignificanceofthedifferencebetweenpairsofcorre-
latedproportions.
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Chapter15: Statistics
147
IftheWITHkeywordisomitted,thentestsforallcombinationsofthelistedvariablesare
performed.IftheWITHkeywordisgiven,andthe(PAIRED)keywordisalsogiven,thenthe
numberofvariablesprecedingWITHmustbethesameasthenumberfollowingit. Inthis
case,testsforeachrespectivepairofvariablesareperformed. IftheWITHkeywordisgiven,
butthe(PAIRED)keywordisomitted,thentestsforeachcombinationofvariablepreceding
WITHagainstvariablefollowingWITHareperformed.
The data ineachvariable must bedichotomous. . If f there are more thantwo distinct
variablesanerrorwilloccurandthetestwillnotberun.
15.11.10 MedianTest
[/MEDIAN[(value)]=var
listBYvariable(value1,value2)]
The mediantest is usedtotest whether independent samples come from populations
withacommonmedian. Themedianofthepopulationsagainstwhichthesamplesareto
betestedmaybegiveninparenthesesimmediatelyafterthe/MEDIANsubcommand. Ifitis
notgiven,themedianwillbeimputedfromtheunionofallthesamples.
The variables ofthesamples tobetestedshouldimmediately followthe‘=’sign. . The
keywordBYmustcome next, andthenthegroupingvariable. . Twovaluesinparentheses
shouldfollow. Ifthefirstvalueisgreaterthanthesecond,thena2sampletestisperformed
usingthesetwovalues todeterminethegroups. . Ifhowever,thefirstvariableislessthan
the second, thena k k sample e test is conductedandthe group values s usedare e allvalues
encounteredwhichlieintherange[value1,value2].
15.11.11 RunsTest
[/RUNS({MEAN,MEDIAN,MODE,value}) =var
list]
The/RUNSsubcommandtestswhetheradatasequenceisrandomlyordered.
Itworksbyexaminingthenumberoftimesavariable’svaluecrossesagiventhreshold.
Thedesiredthresholdmustbespecifiedwithinparentheses. Itmayeitherbespecifiedasa
numberorasoneofMEAN,MEDIANorMODE. Followingthethresholdspecificationcomesthe
listofvariableswhosevaluesaretobetested.
Thesubcommandshowsthenumberofruns,theasymptoticsignificancebasedonthe
lengthofthedata.
15.11.12 SignTest
[/SIGNvar
list[WITHvar
list[(PAIRED)]]]
The /SIGN subcommandtests for differences betweenmedians of the variables listed.
Thetestdoesnotmakeanyassumptionsaboutthedistributionofthedata.
IftheWITHkeywordisomitted,thentestsforallcombinationsofthelistedvariablesare
performed.IftheWITHkeywordisgiven,andthe(PAIRED)keywordisalsogiven,thenthe
numberofvariablesprecedingWITHmustbethesameasthenumberfollowingit. Inthis
case,testsforeachrespectivepairofvariablesareperformed. IftheWITHkeywordisgiven,
butthe(PAIRED)keywordisomitted,thentestsforeachcombinationofvariablepreceding
WITHagainstvariablefollowingWITHareperformed.
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Chapter15: Statistics
148
15.11.13 WilcoxonMatchedPairsSignedRanksTest
[/WILCOXONvar
list[WITHvar
list[(PAIRED)]]]
The/WILCOXONsubcommandtestsfordifferencesbetweenmediansofthevariableslisted.
Thetestdoesnotmakeanyassumptionsaboutthevariancesofthesamples.Itdoeshowever
assumethatthedistributionissymmetrical.
IftheWITHkeywordisomitted,thentestsforallcombinationsofthelistedvariablesare
performed.IftheWITHkeywordisgiven,andthe(PAIRED)keywordisalsogiven,thenthe
numberofvariablesprecedingWITHmustbethesameasthenumberfollowingit. Inthis
case,testsforeachrespectivepairofvariablesareperformed. IftheWITHkeywordisgiven,
butthe(PAIRED)keywordisomitted,thentestsforeachcombinationofvariablepreceding
WITHagainstvariablefollowingWITHareperformed.
15.12 T-TEST
T-TEST
/MISSING={ANALYSIS,LISTWISE}{EXCLUDE,INCLUDE}
/CRITERIA=CI(confidence)
(OneSamplemode.)
TESTVAL=test
value
/VARIABLES=var
list
(IndependentSamplesmode.)
GROUPS=var(value1[,value2])
/VARIABLES=var
list
(PairedSamplesmode.)
PAIRS=var
list[WITHvar
list[(PAIRED)]]
TheT-TESTprocedureoutputstablesusedintestinghypothesesaboutmeans.Itoper-
atesinoneofthreemodes:
 OneSamplemode.
 IndependentGroupsmode.
 Pairedmode.
Eachofthesemodesaredescribedinmoredetailbelow. Therearetwooptionalsubcom-
mandswhicharecommontoallmodes.
The/CRITERIAsubcommandtellspspptheconfidenceintervalusedinthetests. The
defaultvalueis0.95.
TheMISSINGsubcommanddeterminesthehandlingofmissingvariables. IfINCLUDEis
set, thenuser-missingvalues s are includedin n the calculations,butsystem-missing values
arenot. IfEXCLUDEisset,whichisthedefault,user-missingvaluesareexcludedaswellas
system-missingvalues. Thisisthedefault.
Chapter15: Statistics
149
IfLISTWISEisset,thentheentirecaseisexcludedfromanalysiswheneveranyvariable
specifiedinthe/VARIABLES,/PAIRSor/GROUPSsubcommandscontainsamissingvalue.If
ANALYSISisset,thenmissingvaluesareexcludedonlyintheanalysisforwhichtheywould
beneeded. Thisisthedefault.
15.12.1 OneSampleMode
TheTESTVALsubcommandinvokestheOneSamplemode.Thismodeisusedtotestapop-
ulationmeanagainstahypothesizedmean.ThevaluegiventotheTESTVALsubcommandis
thevalueagainstwhichyouwishtotest. Inthismode,youmustalsousethe/VARIABLES
subcommandtotellpsppwhichvariablesyouwishtotest.
15.12.2 IndependentSamplesMode
TheGROUPSsubcommandinvokesIndependentSamplesmodeor‘Groups’mode.Thismode
isusedtotestwhethertwogroupsofvalueshavethesamepopulationmean.Inthismode,
youmustalsousethe/VARIABLESsubcommandtotellpsppthedependentvariablesyou
wishtotest.
ThevariablegivenintheGROUPSsubcommandistheindependentvariablewhichdeter-
minestowhichgroupthesamplesbelong.Thevaluesinparenthesesarethespecificvalues
oftheindependentvariable for eachgroup. . Iftheparenthesesareomittedandnovalues
aregiven,thedefaultvaluesof1.0and2.0areassumed.
Iftheindependentvariableisnumeric,itisacceptabletospecifyonlyonevalueinside
theparentheses.Ifyoudothis,caseswheretheindependentvariableisgreaterthanorequal
tothisvaluebelongtothefirstgroup,andcaseslessthanthisvaluebelongtothesecond
group.WhenusingthisformoftheGROUPSsubcommand,missingvaluesintheindependent
variable are excludedona listwise basis,regardless of whether r /MISSING=LISTWISE E was
specified.
15.12.3 PairedSamplesMode
ThePAIRSsubcommandintroducesPairedSamplesmode. Usethis s modewhenrepeated
measureshavebeentakenfromthesame samples. . Ifthe e WITHkeywordisomitted, then
tables forallcombinationsofvariablesgiveninthePAIRSsubcommandaregenerated. . If
theWITH keywordis given,andthe(PAIRED)keywordisalso given,thenthenumber of
variablesprecedingWITHmustbethesameasthenumberfollowingit. Inthiscase,tables
foreachrespectivepairofvariables aregenerated. . IntheeventthattheWITHkeywordis
given,butthe(PAIRED)keywordisomitted,thentablesforeachcombinationofvariable
precedingWITHagainstvariablefollowingWITHaregenerated.
15.13 ONEWAY
ONEWAY
[/VARIABLES=]var
listBYvar
/MISSING={ANALYSIS,LISTWISE}{EXCLUDE,INCLUDE}
/CONTRAST=value1[,value2]...[,valueN]
/STATISTICS={DESCRIPTIVES,HOMOGENEITY}
/POSTHOC={BONFERRONI,GH,LSD,SCHEFFE,SIDAK,TUKEY,AL-
PHA([value])}
Chapter15: Statistics
150
TheONEWAYprocedureperformsaone-wayanalysisofvarianceofvariablesfactoredby
asingleindependentvariable. Itisusedtocomparethemeansofapopulationdividedinto
morethantwogroups.
ThedependentvariablestobeanalysedshouldbegivenintheVARIABLESsubcommand.
ThelistofvariablesmustbefollowedbytheBYkeywordandthenameoftheindependent
(orfactor)variable.
YoucanusetheSTATISTICSsubcommandtotellpspptodisplayancillaryinformation.
Theoptionsacceptedare:
 DESCRIPTIVESDisplaysdescriptivestatistics s aboutthegroups factoredby the in-
dependentvariable.
 HOMOGENEITYDisplaystheLevenetestofHomogeneityofVarianceforthevariables
andtheirgroups.
TheCONTRASTsubcommandisusedwhenyouanticipatecertaindifferencesbetweenthe
groups. Thesubcommandmustbefollowedbyalistofnumeralswhicharethecoefficients
ofthegroupstobetested. Thenumber r ofcoefficientsmust correspondtothenumberof
distinctgroups(orvaluesoftheindependentvariable).Ifthetotalsumofthecoefficientsare
notzero,thenpsppwilldisplayawarning,butwillproceedwiththeanalysis.TheCONTRAST
subcommandmaybegivenupto10timesinordertospecifydifferentcontrasttests.The
MISSINGsubcommanddefineshowmissingvaluesarehandled. IfLISTWISEisspecifiedthen
caseswhichhavemissingvaluesfortheindependentvariableoranydependentvariablewill
beignored. IfANALYSISisspecified,thencaseswillbeignorediftheindependentvariable
ismissingorifthedependentvariablecurrentlybeinganalysedismissing. Thedefaultis
ANALYSIS. AsettingofEXCLUDEmeansthatvariableswhosevaluesareuser-missingareto
beexcludedfromtheanalysis. AsettingofINCLUDEmeanstheyaretobeincluded. The
defaultisEXCLUDE.
UsingthePOSTHOCsubcommandyoucanperformmultiplepairwisecomparisonsonthe
data. Thefollowingcomparisonmethodsareavailable:
 LSDLeastSignificantDifference.
 TUKEYTukeyHonestlySignificantDifference.
 BONFERRONIBonferronitest.
 SCHEFFEScheff´e’stest.
 SIDAKSidaktest.
 GHTheGames-Howelltest.
Theoptionalsyntax ALPHA(value) is usedtoindicate that value shouldbe used d as s the
confidencelevelforwhichtheposthoctestswillbeperformed. Thedefaultis0.05.
15.14 QUICKCLUSTER
QUICKCLUSTERvar
list
[/CRITERIA=CLUSTERS(k)[MXITER(max
iter)]CONVERGE(epsilon)[NOINITIAL]]
[/MISSING={EXCLUDE,INCLUDE}{LISTWISE,PAIRWISE}]
[/PRINT={INITIAL}{CLUSTER}]
TheQUICKCLUSTERcommandperformsk-meansclusteringonthedataset.Thisisuseful
whenyouwishtoallocatecasesintoclustersof similarvalues andyoualreadyknowthe
numberofclusters.
Chapter15: Statistics
151
The minimum m specification is ‘QUICKCLUSTER’ followed by the names of f the
variables which contain n the cluster r data. . Normally y you u will l also o want t to o specify
/CRITERIA=CLUSTERS(k)wherek isthenumberofclusters.Ifthisisnotspecified,thenk
defaultsto2.
Ifyouuse/CRITERIA=NOINITIALthenanaivealgorithmtoselecttheinitialclustersis
used. Thiswillprovideforfasterexecutionbutlesswellseparatedinitialclustersandhence
possiblyaninferiorfinalresult.
QUICKCLUSTERusesaniterativealgorithmtoselecttheclusterscenters. Thesubcom-
mand/CRITERIA=MXITER(max_iter)setsthemaximumnumberofiterations.Duringclas-
sification,psppwillcontinueiteratinguntiluntilmax
iter iterationshavebeendoneorthe
convergencecriterion(seebelow)isfulfilled. Thedefaultvalueofmax
iteris2.
Ifhowever,youspecify/CRITERIA=NOUPDATEthenafterselectingtheinitialcenters,no
furtherupdatetotheclustercentersisdone.Inthiscase,max
iter,ifspecified. isignored.
Thesubcommand/CRITERIA=CONVERGE(epsilon)isusedtosettheconvergencecrite-
rion.Thevalueofconvergencecriterionisepsilontimestheminimumdistancebetweenthe
initialclustercenters.Iterationstopswhenthemeanclusterdistancebetweenoneiteration
andthenextislessthantheconvergencecriterion. Thedefaultvalueofepsiloniszero.
TheMISSINGsubcommanddeterminesthehandlingofmissingvariables. IfINCLUDEis
set,thenuser-missingvaluesareconsideredattheir facevalueandnotas missingvalues.
IfEXCLUDEisset,whichisthedefault,user-missingvaluesareexcludedaswellassystem-
missingvalues.
IfLISTWISEisset,thentheentirecaseisexcludedfromtheanalysiswheneveranyofthe
clusteringvariablescontainsamissingvalue. IfPAIRWISEisset,thenacaseisconsidered
missingonlyifalltheclusteringvariablescontainmissingvalues. Otherwiseitisclustered
onthebasisofthenon-missingvalues.ThedefaultisLISTWISE.
ThePRINTsubcommandrequestsadditionaloutputtobeprinted.IfINITIALisset,then
theinitialclustermembershipswillbeprinted. IfCLUSTERisset,theclustermemberships
oftheindividualcaseswillbedisplayed(potentiallygeneratinglengthyoutput).
15.15 RANK
RANK
[VARIABLES=]var
list[{A,D}][BYvar
list]
/TIES={MEAN,LOW,HIGH,CONDENSE}
/FRACTION={BLOM,TUKEY,VW,RANKIT}
/PRINT[={YES,NO}
/MISSING={EXCLUDE,INCLUDE}
/RANK[INTOvar
list]
/NTILES(k)[INTOvar
list]
/NORMAL[INTOvar
list]
/PERCENT[INTOvar
list]
/RFRACTION[INTOvar
list]
/PROPORTION[INTOvar
list]
/N[INTOvar
list]
/SAVAGE[INTOvar
list]
Chapter15: Statistics
152
TheRANKcommandranksvariablesandstorestheresultsintonewvariables.
TheVARIABLESsubcommand,whichismandatory,specifiesoneormorevariableswhose
values areto be ranked. . After r eachvariable, , ‘A’or r ‘D’may appear,indicating that the
variableis toberankedinascendingor descendingorder. . Ascendingis s the default. . Ifa
BYkeywordappears,itshouldbefollowedbyalistofvariableswhicharetoserveasgroup
variables. Inthiscase,thecases s aregatheredintogroups,andranks calculatedfor each
group.
The TIES subcommandspecifies howtied values aretobetreated. . The e default is to
takethemeanvalueofallthetiedcases.
TheFRACTIONsubcommandspecifieshowproportionalranksaretobecalculated.This
onlyhasanyeffectifNORMALorPROPORTIONALrankfunctionsarerequested.
ThePRINTsubcommandmaybeusedtospecifythatasummaryoftherankvariables
createdshouldappearintheoutput.
The function subcommands are RANK, , NTILES, NORMAL, , PERCENT, , RFRACTION,
PROPORTION and d SAVAGE. . Any y number of f functionsubcommands may y appear. . If f none
are given, then the default is RANK. The NTILES subcommand must t take e an integer
specifyingthenumberofpartitionsintowhichvaluesshouldberanked.Eachsubcommand
maybefollowedbytheINTOkeywordandalistofvariableswhicharethevariablestobe
createdandreceivetherankscores. Theremaybeasmanyvariablesspecifiedasthereare
variablesnamedontheVARIABLESsubcommand. Iffewer r arespecified,thenthevariable
namesareautomaticallycreated.
The MISSING subcommanddetermines how user missing values are tobe treated. . A
settingofEXCLUDEmeansthatvariableswhosevaluesareuser-missingaretobeexcluded
fromtherankscores. AsettingofINCLUDEmeanstheyaretobeincluded. . Thedefaultis
EXCLUDE.
15.16 REGRESSION
TheREGRESSIONprocedurefitslinearmodelstodatavialeast-squaresestimation. Thepro-
cedureisappropriatefordatawhichsatisfythoseassumptionstypicalinlinearregression:
 The e dataset contains nobservations ofa dependent variable,say Y
1
;:::;Y
n
, andn
observationsofoneormoreexplanatoryvariables. LetX
11
;X
12
,...,X
1n
denotethe
nobservationsofthefirstexplanatoryvariable;X
21
,...,X
2n
denotethenobservations
ofthesecondexplanatoryvariable;X
k1
,...,X
kn
denotethenobservationsofthekth
explanatoryvariable.
 ThedependentvariableY Y hasthefollowingrelationshiptotheexplanatoryvariables:
Y
i
= b
0
+b
1
X
1i
+:::+b
k
X
ki
+Z
i
where b
0
;b
1
;:::;b
k
are unknowncoefficients,and
Z
1
;:::;Z
n
areindependent,normallydistributednoisetermswithmeanzeroandcom-
monvariance.Thenoise,orerrortermsareunobserved.Thisrelationshipiscalledthe
linearmodel.
The REGRESSION procedure e estimates the e coefficients b
0
;:::;b
k
and produces output
relevanttoinferencesforthelinearmodel.
15.16.1 Syntax
REGRESSION
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