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Appendix
A
Categorical Variable Coding
Schemes
In many SPSSprocedures,you can requestautomaticreplacementof acategorical
independent variablewith asetofcontrast variables,which willthenbeentered or
removedfromanequationasablock. Youcanspecifyhowthesetofcontrastvariables
isto becoded,usuallyon the
CONTRAST
subcommand. Thisappendix explainsand
illustrateshow differentcontrasttypesrequested on
CONTRAST
actually work.
Deviation
Deviationfromthegrandmean.
In matrix terms,thesecontrasts havetheform:
mean
(1/k
1/k
...
1/k
1/k)
df(1)
(1–1/ k
–1/ k
...
–1/ k
–1/ k)
df(2)
(–1/ k
1–1/ k
...
–1/ k
–1/ k)
.
.
.
.
df(k–1)
(–1/ k
–1/ k
...
1–1/ k
–1/ k)
wherekisthenumberofcategoriesfortheindependent variableandthelastcategory
isomittedbydefault. Forexample,thedeviationcontrastsforanindependentvariable
with threecategoriesareas follows:
(1/3
1/3
1/3)
(2/3
–1/3
–1/3)
(–1/3
2/3
–1/3)
51
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52
Appendix A
To omit acategoryotherthan thelast,specify thenumberoftheomittedcategory in
parenthesesafterthe
DEVIATION
keyword. Forexample, thefollowing subcommand
obtainsthedeviationsforthefirst andthirdcategoriesandomitsthesecond:
/CONTRAST(FACTOR)=DEVIATION(2)
Supposethatfactor hasthreecategories. Theresulting contrast matrix willbe
(1/3
1/3
1/3)
(2/3
–1/3
–1/3)
(–1/3
–1/3
2/3)
Simple
Simplecontrasts.
Compares each level ofafactorto thelast. The general matrix
form is
mean
(1/k
1/k
...
1/k
1/k)
df(1)
(1
0
...
0
–1)
df(2)
(0
1
...
0
–1)
.
.
.
.
df(k–1)
(0
0
...
1
–1)
wherekisthenumber ofcategoriesfortheindependentvariable. For example,the
simplecontrastsforanindependent variablewithfourcategoriesareasfollows:
(1/4
1/4
1/4
1/4)
(1
0
0
–1)
(0
1
0
–1)
(0
0
1
–1)
To useanother category instead ofthe last as areferencecategory, specify in
parenthesesafterthe
SIMPLE
keywordthesequencenumberofthereferencecategory,
which isnotnecessarilythevalueassociatedwith that category. For example,the
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53
Categorical Va
riableCodingSchemes
following
CONTRAST
subcommand obtainsacontrast matrix thatomitsthesecond
category:
/CONTRAST(FACTOR)=SIMPLE(2)
Supposethatfactor hasfourcategories. Theresultingcontrastmatrixwill be
(1/4
1/4
1/4
1/4)
(1
–1
0
0)
(0
–1
1
0)
(0
–1
0
1)
Helmert
Helmertcontrasts.
Comparescategoriesofanindependentvariablewiththemean of
thesubsequent categories. Thegeneral matrix form is
mean
(1/k
1/k
...
1/k
1/k)
df(1)
(1
–1/( k–1)
...
–1/( k–1)
–1/( k–1))
df(2)
(0
1
...
–1/( k–2)
–1/( k–2))
.
.
.
.
df(k–2)
(0
0
1
–1/2
–1/2
df(k–1)
(0
0
...
1
–1)
wherekis thenumberof categoriesoftheindependent variable. Forexample,an
independent variablewith four categories hasaHelmert contrastmatrix of the
following form:
(1/4
1/4
1/4
1/4)
(1
–1/3
–1/3
–1/3)
(0
1
–1/2
–1/2)
(0
0
1
–1)
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54
Appendix A
Difference
DifferenceorreverseHelmertcontrasts.
Compares categoriesof an independent
variablewiththemean ofthepreviouscategories ofthevariable. Thegeneral matrix
form is
mean
(1/k
1/k
1/k
...
1/k)
df(1)
(–1
1
0
...
0)
df(2)
(–1/2
–1/2
1
...
0)
.
.
.
.
df(k–1)
(–1/( k–1)
–1/( k–1)
–1/( k–1)
...
1)
wherekisthenumber ofcategoriesfortheindependentvariable. For example,the
differencecontrastsfor anindependentvariablewithfourcategoriesareasfollows:
(1/4
1/4
1/4
1/4)
(–1
1
0
0)
(–1/2
–1/2
1
0)
(–1/3
–1/3
–1/3
1)
Polynomial
Orthogonalpolynomialcontrasts.
Thefirst degreeof freedom containsthelineareffect
acrossallcategories; thesecond degreeoffreedom,thequadraticeffect; thethird
degreeof freedom, thecubic;andsoon,forthehigher-order effects.
You can specify thespacing between levelsofthetreatmentmeasured bythegiven
categoricalvariable. Equalspacing,whichisthedefaultifyou omit themetric,canbe
specifiedasconsecutiveintegersfrom 1tok,wherekisthenumberofcategories. If
thevariabledrug has threecategories,thesubcommand
/CONTRAST(DRUG)=POLYNOMIAL
isthesameas
/CONTRAST(DRUG)=POLYNOMIAL(1,2,3)
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55
Categorical Va
riableCodingSchemes
Equal spacing is not always necessary, however. For example, suppose that
drugrepresentsdifferentdosagesofadruggiventothreegroups. Ifthedosage
administered to thesecond groupistwicethat given tothefirst group and thedosage
administered to thethird group is three timesthat given to the first group,the
treatmentcategoriesareequally spaced,andan appropriatemetricforthissituation
consistsof consecutiveintegers:
/CONTRAST(DRUG)=POLYNOMIAL(1,2,3)
If,however,thedosageadministered to thesecondgroup isfour times thatgivento
thefirst group, and thedosageadministeredto thethird group isseven timesthat
given to thefirst group,an appropriatemetricis
/CONTRAST(DRUG)=POLYNOMIAL(1,4,7)
In eithercase,theresultofthecontrastspecification isthatthefirstdegreeoffreedom
fordrug contains thelineareffect ofthedosagelevelsandthesecond degreeof
freedom contains thequadraticeffect.
Polynomial contrastsareespecially useful in tests oftrendsand for investigating
thenatureof responsesurfaces. Youcanalso usepolynomial contraststo perform
nonlinearcurvefitting,such ascurvilinear regression.
Repeated
Comparesadjacentlevelsofanindependentvariable.
Thegeneralmatrix form is
mean
(1/k
1/k
1/k
...
1/k
1/k)
df(1)
(1
–1
0
...
0
0)
df(2)
(0
1
–1
...
0
0)
.
.
.
.
df(k–1)
(0
0
0
...
1
–1)
wherekisthenumber ofcategoriesfortheindependentvariable. For example,the
repeated contrastsforan independent variablewithfourcategoriesareasfollows:
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56
Appendix A
(1/4
1/4
1/4
1/4)
(1
–1
0
0)
(0
1
–1
0)
(0
0
1
–1)
Thesecontrasts are useful inprofileanalysisand wherever differencescoresare
needed.
Special
Auser-definedcontrast.
Allowsentry of special contrastsin theform ofsquare
matriceswith as many rows and columns asthereare categoriesof thegiven
independent variable. For
MANOVA
and
LOGLINEAR
,thefirstrowenteredisalways
themean,or constant,effect and representstheset ofweights indicating how to
averageotherindependent variables,if any,overthegivenvariable. Generally,this
contrast isa vectorof ones.
Theremaining rows ofthematrix contain thespecial contrastsindicating the
desired comparisonsbetween categoriesofthevariable. Usually,orthogonal contrasts
are themost useful. Orthogonal contrastsare statistically independent and are
nonredundant. Contrasts areorthogonal if:
ɸ
Foreach row,contrast coefficientssumto 0.
ɸ
The products ofcorresponding coefficients for all pairs ofdisjoint rowsalso
sum to 0.
Forexample,supposethat treatmenthasfourlevelsand thatyou wantto comparethe
variouslevelsoftreatment with eachother. Anappropriatespecialcontrastis
(1
1
1
1)
weights formeancalculation
(3
–1
–1
–1)
compare1
st
with2
nd
through4
th
(0
2
–1
–1)
compare2
nd
with3
rd
and4
th
(0
0
1
–1)
compare3
rd
with4
th
whichyou specify bymeansofthefollowing
CONTRAST
subcommand for
MANOVA
,
LOGISTICREGRESSION
,and
COXREG:
57
Categorical Va
riableCodingSchemes
/CONTRAST(TREATMNT)=SPECIAL( 1
1
1
1
3 -1 -1 -1
0
2 -1 -1
0
0
1 -1 )
For
LOGLINEAR
,you need to specify:
/CONTRAST(TREATMNT)=BASIS SPECIAL( 1
1
1
1
3 -1 -1 -1
0
2 -1 -1
0
0
1 -1 )
Eachrow except themeansrowsumsto 0. Products ofeachpairofdisjointrows
sum to 0 as well:
Rows 2and3:
(3)(0)+(–1)(2)+ (–1)(–1)+ (–1)(–1)=0
Rows 2and4:
(3)(0)+(–1)(0)+ (–1)(1)+ (–1)(–1)=0
Rows 3and4:
(0)(0)+(2)(0)+ (–1)(1)+ (–1)(–1)=0
Thespecial contrasts need not beorthogonal. However,they mustnot belinear
combinationsofeachother. If theyare, theprocedurereportsthelinear dependency
and ceases processing. Helmert, difference,and polynomial contrasts are all
orthogonal contrasts.
Indicator
Indicatorvariablecoding.
Also known asdummy coding,this isnot available in
LOGLINEAR
or
MANOVA
.Thenumber ofnew variablescoded isk–1. Casesin the
referencecategoryarecoded 0for allk–1 variables. A caseinthe ith category is
coded 0for allindicator variables exceptthei
th
,whichis coded 1.
Index
asymptoticregression,35
inNonlinearRegression,35
backwardelimination, 6
inLogisticRegression, 6
binarylogisticregression, 1
categoricalcovariates, 7
cellprobabilitiestables,18
inMultinomialLogisticRegression,18
cellswithzeroobservations,19
inMultinomialLogisticRegression,19
classification,13
inMultinomialLogisticRegression,13
classificationtables,18
inMultinomialLogisticRegression,18
confidence intervals,18
inMultinomialLogisticRegression,18
constantterm,10
inLinearRegression,10
constrainedregression
inNonlinearRegression,37
contrasts, 7
inLogisticRegression, 7
convergencecriterion,19
inMultinomialLogisticRegression,19
Cook's D, 9
inLogisticReg
ression, 9
correlationmatrix,18
inMultinomialLogisticRegression,18
covariance matrix,18
inMultinomialLogisticRegression,18
covariates
inLogisticRegression, 7
CoxandSnellR-square,18
inMultinomialLogisticRegression,18
custom models,15
inMultinomialLogisticRegression,15
delta,19
ascorrectionforcellswithzeroobservations,19
densitymodel,35
inNonlinearRegression,35
deviancefunction,21
forestimatingdispersionscalingvalue,21
DfBeta, 9
inLogisticRegression, 9
dispersionscalingvalue,21
inMultinomialLogisticRegression,21
fiducialconfidenceintervals,28
inProbit Analysis,28
forwardselection, 6
inLogisticRegression, 6
fullfactorial models,15
inMultinomialLogisticRegression,15
Gaussmodel,35
inNonlinearRegression,35
Gompertzmodel,35
inNonlinearRegression,35
goodnessoffit,18
inMultinomialLogisticRegression,18
Hosmer-Lemeshowgoodness-of-fitstatistic,10
inLogisticRegression,10
intercept,15
includeorexclude,15
iterationhistory,19
inMultinomialLogisticRegression,19
iterations,10,19,28
inLogisticRegression,10
inMultinomialLogisticRegression,19
inProbit Analy
sis,28
Johnson-Schumachermodel,35
inNonlinearRegression,35
leverage values, 9
inLogisticRegression, 9
likelihoodratio,18,21
forestimatingdispersionscalingvalue,21
goodnessoffit,18
LinearRegression,43,47
Two-StageLeast-SquaresRegression,47
weightestimation,43
logistic regression, 1
binary, 1
LogisticRegression, 3, 6
assumptions, 4
categoricalcovariates, 7
59
60
Index
classificationcutoff,10
coefficients, 3
commandaddi
tionalfeatures,11
constantterm,10
contrasts, 7
dataconsiderations, 4
define selectionrule, 6
dependentvariable, 4
displayoptions,10
example, 3
Hosmer-Lemeshowgoodness-of-fitstatistic,10
influencemeasures, 9
iterations,10
predictedvalues, 9
probabilityforstepwise,10
relatedprocedures, 4
residuals, 9
savingnewvariables, 9
setrule, 6
statistics, 3
statistics andplots,10
stringcovariates, 7
variableselectionmethods, 6
log-likelihood,18,43
inMultinomialLogisticRegression,18
inWeightEstimation,43
log-modifiedmodel,35
inNonlinearRegression,35
main-effectsmodels,15
inMultinomialLogisticRegression,15
McFaddenR-square,18
inMultinomialLogisticRegression,18
Metcherlichlaw
ofdiminishingreturns,35
inNonlinearRegression,35
MichaelisMentenmodel,35
inNonlinearRegression,35
Morgan-Mercer-Florinmodel,35
inNonlinearRegression,35
MultinomialLogistic Regression,13,13,15,18,
19,22,23
assumptions,13
commandadditionalfeatures,23
criteria,19
exportingmodelinformation,22
models,15
referencecategory,17
save,22
statistics,
18
NagelkerkeR-square,18
inMultinomialLogisticRegression,18
nonlinearmodels,35
inNonlinearRegression,35
NonlinearRegression,31,35
assumptions,32
bootstrapestimates,39
commandadditionalfeatures,40
commonnonlinearmodels,35
conditionallogic,33
dataconsiderations,32
derivatives,38
estimationmethods,39
example,31
interpretationofresults,40
Levenberg-Marquardtalgorithm,39
loss function,36
parameterconstraints,37
parameters,34
predictedvalues,38
relatedprocedures,32
residuals,38
savenew variables,38
segmentedmodel,33
sequentialquadraticprogramming,39
startingvalues,32,34
statistics,31
parallelism test,28
inProbit Analysis,28
parameterconst
raints
inNonlinearRegression,37
parameterestimates,18
inMultinomialLogisticRegression,18
Peal-Reedmodel,35
inNonlinearRegression,35
Pearsonchi-square,18,21
forestimatingdispersionscalingvalue,21
goodnessoffit,18
ProbitAnalysis,25
assumptions,25
commandadditionalfeatures,29
criteria,28
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