MultinomialLogisticRegression_BasicRelationships

7/23/2019 MultinomialLogisticRegression_BasicRelationships

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SW388R7

Data Analysis &

Computers II

Slide 1

Multinomial Logistic Regressionasic Relations!ips

Multinomial Logistic Regression

Descri"ing Relations!ips

Classi#ication Accuracy

Sample $ro"lems


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ters II

Slide 2Multinomial logistic regression

Multinomial logistic regression is used to analy%e relations!ips"eteen a non'metric dependent (aria"le and metric or

dic!otomous independent (aria"les)

Multinomial logistic regression compares multiple groups

t!roug! a com"ination o# "inary logistic regressions)

*!e group comparisons are e+ui(alent to t!e comparisons #or a

dummy'coded dependent (aria"le, it! t!e group it! t!e

!ig!est numeric score used as t!e re#erence group)

-or e.ample, i# e anted to study di##erences in SW, MSW,

and $!D students using multinomial logistic regression, t!e

analysis ould compare SW students to $!D students and MSW

students to $!D students) -or eac! independent (aria"le, t!ere

ould "e to comparisons)


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Slide 3W!at multinomial logistic regression predicts

Multinomial logistic regression pro(ides a set o# coe##icients #or

eac! o# t!e to comparisons) *!e coe##icients #or t!e

re#erence group are all %eros, similar to t!e coe##icients #or t!e

re#erence group #or a dummy'coded (aria"le)

*!us, t!ere are t!ree e+uations, one #or eac! o# t!e groups

de#ined "y t!e dependent (aria"le)

*!e t!ree e+uations can "e used to compute t!e pro"a"ility

t!at a su"/ect is a mem"er o# eac! o# t!e t!ree groups) A caseis predicted to "elong to t!e group associated it! t!e !ig!est

pro"a"ility)

$redicted group mem"ers!ip can "e compared to actual group

mem"ers!ip to o"tain a measure o# classi#ication accuracy)


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ters II

Slide 4Le(el o# measurement re+uirements

Multinomial logistic regression analysis re+uires t!at t!e

dependent (aria"le "e non'metric) Dic!otomous, nominal, and

ordinal (aria"les satis#y t!e le(el o# measurement re+uirement)

Multinomial logistic regression analysis re+uires t!at t!eindependent (aria"les "e metric or dic!otomous) Since S$SS ill

automatically dummy'code nominal le(el (aria"les, t!ey can "e

included since t!ey ill "e dic!otomi%ed in t!e analysis)

In S$SS, non'metric independent (aria"les are included as0#actors) S$SS ill dummy'code non'metric I2s)

In S$SS, metric independent (aria"les are included as

0co(ariates) I# an independent (aria"le is ordinal, e ill

attac! t!e usual caution)


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ters II

Slide 5Assumptions and outliers

Multinomial logistic regression does not mae any assumptions

o# normality, linearity, and !omogeneity o# (ariance #or t!e

independent (aria"les)

ecause it does not impose t!ese re+uirements, it is pre#erredto discriminant analysis !en t!e data does not satis#y t!ese

assumptions)

S$SS does not compute any diagnostic statistics #or outliers) *o

e(aluate outliers, t!e ad(ice is to run multiple "inary logisticregressions and use t!ose results to test t!e e.clusion o#

outliers or in#luential cases)


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Slide 6Sample si%e re+uirements

*!e minimum num"er o# cases per independent (aria"le is 14,

using a guideline pro(ided "y 5osmer and Lemes!o, aut!ors o#

Applied Logistic Regression, one o# t!e main resources #or

Logistic Regression)

-or pre#erred case'to'(aria"le ratios, e ill use 64 to 1)


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ters II

Slide 7Met!ods #or including (aria"les

*!e only met!od #or selecting independent (aria"les in S$SS is

simultaneous or direct entry)


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ters II

Slide 8(erall test o# relations!ip ' 1

*!e o(erall test o# relations!ip among t!e independent

(aria"les and groups de#ined "y t!e dependent is "ased on t!e

reduction in t!e lieli!ood (alues #or a model !ic! does not

contain any independent (aria"les and t!e model t!at contains

t!e independent (aria"les)

*!is di##erence in lieli!ood #ollos a c!i's+uare distri"ution,

and is re#erred to as t!e model c!i's+uare)

*!e signi#icance test #or t!e #inal model c!i's+uare a#ter t!eindependent (aria"les !a(e "een added9 is our statistical

e(idence o# t!e presence o# a relations!ip "eteen t!e

dependent (aria"le and t!e com"ination o# t!e independent

(aria"les)


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ters II

Slide 9(erall test o# relations!ip ' 6

Model Fitting Information

284.429

265.972 18.457 6 .005

Model

Intercept Only

Final

-2 Log

Likeliood !i-"#$are d% "ig.

The presence of a relationship between the dependentvariable and combination of independent variables isbased on the statistical significance of the final modelchi-square in the SPSS table titled "Model FittingInformation".

In this analsis! the probabilit of the model chi-square#$.%&'( was ).))&! less than or equal to the level ofsignificance of ).)&. The null hpothesis that there wasno difference between the model without independentvariables and the model with independent variables wasre*ected. The e+istence of a relationship between theindependent variables and the dependent variable wassupported.


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Slide

!

Strengt! o# multinomial logistic regressionrelations!ip

W!ile multinomial logistic regression does compute correlationmeasures to estimate t!e strengt! o# t!e relations!ip pseudo Rs+uare measures, suc! as :agelere;s R


11/73

Slide

=(aluating use#ulness #or logistic models

*!e "enc!mar t!at e ill use to c!aracteri%e a multinomiallogistic regression model as use#ul is a 6>? impro(ement o(ert!e rate o# accuracy ac!ie(a"le "y c!ance alone)

=(en i# t!e independent (aria"les !ad no relations!ip to t!egroups de#ined "y t!e dependent (aria"le, e ould stille.pect to "e correct in our predictions o# group mem"ers!ipsome percentage o# t!e time) *!is is re#erred to as "y c!anceaccuracy)

*!e estimate o# "y c!ance accuracy t!at e ill use is t!eproportional "y c!ance accuracy rate, computed "y summingt!e s+uared percentage o# cases in eac! group) *!e onlydi##erence "eteen "y c!ance accuracy #or "inary logisticmodels and "y c!ance accuracy #or multinomial logistic modelsis t!e num"er o# groups de#ined "y t!e dependent (aria"le)


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Slide

2Computing "y c!ance accuracy

*!e percentage o# cases in eac! group de#ined "y t!e dependent(aria"le is #ound in t!e @Case $rocessing Summary ta"le)

Case Processing Summary

62 &7.1'

9& 55.7'12 7.2'

167 100.0'

10&

270

15&a

1

2&

(I)(*+,"

+ /I)"

alid

Mi33ing

otal

"$pop$lation

Marginal

ercentage

e dependent ariale a3 only one al$e o3ered

in 146 95.4' 3$pop$lation3.

a.

The proportional b chance accurac rate wascomputed b calculating the proportion of cases foreach group based on the number of cases in eachgroup in the ,ase Processing Summar,! and thensquaring and summing the proportion of cases in eachgroup ).'#/ 0 ).&&'/ 0 ).)'1/ 2 ).%&(.

The proportional b chance accurac criteria is &3.34#.1& + %&.4 2 &3.34(.


13/73

Slide

3Comparing accuracy rates

*o c!aracteri%e our model as use#ul, e compare t!e o(erall

percentage accuracy rate produced "y S$SS at t!e last step in !ic!

(aria"les are entered to 6>? more t!an t!e proportional "y c!ance

accuracy) :oteB S$SS does not compute a cross'(alidated accuracy

rate #or multinomial logistic regression )9

Classification

15 47 0 24.2'

7 86 0 92.5'

5 7 0 .0'

16.2' 8&.8' .0' 60.5'

O3ered

1

2

&

Oerall ercentage

1 2 &

ercent

!orrect

redicted

The classification accurac rate was 3).&4which was greater than or equal to theproportional b chance accurac criteria of&3.34 #.1& + %&.4 2 &3.34(.

The criteria for classification accurac is

satisfied in this e+ample.


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Slide

4:umerical pro"lems

*!e ma.imum lieli!ood met!od used to calculate multinomiallogistic regression is an iterati(e #itting process t!at attemptsto cycle t!roug! repetitions to #ind an anser)

Sometimes, t!e met!od ill "rea don and not "e a"le tocon(erge or #ind an anser)

Sometimes t!e met!od ill produce ildly impro"a"le results,reporting t!at a one'unit c!ange in an independent (aria"leincreases t!e odds o# t!e modeled e(ent "y !undreds o#t!ousands or millions) *!ese implausi"le results can "eproduced "y multicollinearity, categories o# predictors !a(ing

no cases or %ero cells, and complete separation !ere"y t!eto groups are per#ectly separated "y t!e scores on one ormore independent (aria"les)

*!e clue t!at e !a(e numerical pro"lems and s!ould notinterpret t!e results are standard errors #or some independent(aria"les t!at are larger t!an 6)4)


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Slide

5

Relations!ip o# indi(idual independent(aria"les and t!e dependent (aria"le

*!ere are to types o# tests #or indi(idual independent

(aria"lesB

*!e lieli!ood ratio test e(aluates t!e o(erall relations!ip

"eteen an independent (aria"le and t!e dependent

(aria"le

*!e Wald test e(aluates !et!er or not t!e independent

(aria"le is statistically signi#icant in di##erentiating "eteen

t!e to groups in eac! o# t!e em"edded "inary logistic

comparisons)

I# an independent (aria"le !as an o(erall relations!ip to t!e

dependent (aria"le, it mig!t or mig!t not "e statistically

signi#icant in di##erentiating "eteen pairs o# groups de#ined "y

t!e dependent (aria"le)


16/73

Slide

6


*!e interpretation #or an independent (aria"le #ocuses on itsa"ility to distinguis! "eteen pairs o# groups and t!e

contri"ution !ic! it maes to c!anging t!e odds o# "eing in

one dependent (aria"le group rat!er t!an t!e ot!er)

We s!ould not interpret t!e signi#icance o# an independent(aria"les role in distinguis!ing "eteen pairs o# groups unless

t!e independent (aria"le also !as an o(erall relations!ip to t!e

dependent (aria"le in t!e lieli!ood ratio test)

*!e interpretation o# an independent (aria"les role indi##erentiating dependent (aria"le groups is t!e same as e

used in "inary logistic regression) *!e di##erence in

multinomial logistic regression is t!at e can !a(e multiple

interpretations #or an independent (aria"le in relation to

di##erent pairs o# groups)


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Slide

7


Parameter Estimates

&.240 2.478 1.709 1 .191

.019 .020 .906 1 .&41 1.019

.071 .108 .427 1 .514 1.07&

-1.&7& .620 4.91& 1 .027 .25&

&.6&9 2.456 2.195 1 .1&8

.00& .020 .017 1 .897 1.00&

.172 .110 2.46& 1 .117 1.188

-1.657 .61& 7.298 1 .007 .191

Intercept

+)

:!

!OL)I"

Intercept

+)

:!

!OL)I"

(I)(*+,"

+ /I)"

a

OO LIL

+/O: I)(

/ "td. rror *ald d% "ig. ;p/

e re%erence category i3= OO M:!(.a.

Parameter Estimates

&.240 2.478 1.709 1 .191

.019 .020 .906 1 .&41 1.019

.071 .108 .427 1 .514 1.07&

-1.&7& .620 4.91& 1 .027 .25&

&.6&9 2.456 2.195 1 .1&8.00& .020 .017 1 .897 1.00&

.172 .110 2.46& 1 .117 1.188

-1.657 .61& 7.298 1 .007 .191

Intercept

+)

:!

!OL)I"

Intercept

+)

:!

!OL)I"

(I)(*+,"

+ /I)"a

1

2


e re%erence category i3= &.a.

SPSS identifies the comparisons it ma5es forgroups defined b the dependent variable inthe table of 6Parameter 7stimates!8 using eitherthe value codes or the value labels! dependingon the options settings for pivot table labeling.

The reference categor is identified in the

footnote to the table.

In this analsis! two comparisons will bemade9

:the T;; T group coded 1 !shaded orange(( will be compared to theT;; M=> group coded ! shaded

purple(.

The reference categor plas the same role inmultinomial logistic regression that it plas inthe dumm-coding of a nominal variable9 it isthe categor that would be coded with Cerosfor all of the dumm-coded variables that allother categories are interpreted against.


18/73

Slide

8


Likelihood Ratio Tests

268.&2& 2.&50 2 .&09

268.625 2.652 2 .265

270.&95 4.42& 2 .110

275.194 9.221 2 .010

%%ect

Intercept

+)

:!

!OL)I"

-2 Log

Likeliood o%

ed$ced

Model !i-"#$are d% "ig.

e ci-3#$are 3tat i3tic i3 te di%%erence in -2 log-likeliood3

etodel and a red$ced >odel. e red$ced >odel i3

%or>ed y o>itting an e%%ect %ro> te %inal >odel. e n$ll ypote3i3

i3 tat all para>eter3 o% tat e%%ect are 0.

Parameter Estimates

&.240 2.478 1.709 1 .191

.019 .020 .906 1 .&41 1.019

.071 .108 .427 1 .514 1.07&-1.&7& .620 4.91& 1 .027 .25&

&.6&9 2.456 2.195 1 .1&8

.00& .020 .017 1 .897 1.00&

.172 .110 2.46& 1 .117 1.188

-1.657 .61& 7.298 1 .007 .191

Intercept

+)

:!!OL)I"

Intercept

+)

:!

!OL)I"

(I)(*+,"

+ /I)"a

1

2



In this e+ample! there is astatisticall significantrelationship between theindependent variable;D


19/73

Slide

9

Interpreting relations!ip o# indi(idual independent(aria"les to t!e dependent (aria"le


268.&2& 2.&50 2 .&09

268.625 2.652 2 .265

270.&95 4.42& 2 .110

275.194 9.221 2 .010

%%ect

Intercept

+)

:!

!OL)I"

-2 Log

Likeliood o%

ed$ced


e ci-3#$are 3tat i3tic i3 te di%%erence in -2 log-likeliood3




Parameter Estimates

&.240 2.478 1.709 1 .191

.019 .020 .906 1 .&41 1.019

.071 .108 .427 1 .514 1.07&-1.&7& .620 4.91& 1 .027 .25&

&.6&9 2.456 2.195 1 .1&8

.00& .020 .017 1 .897 1.00&

.172 .110 2.46& 1 .117 1.188

-1.657 .61& 7.298 1 .007 .191

Intercept

+)

:!!OL)I"

Intercept

+)

:!

!OL)I"

(I)(*+,"

+ /I)"a

1

2



Surve respondents who had less confidence in congress highervalues correspond to lower confidence( were less li5el to be in thegroup of surve respondents who thought we spend too little moneon highwas and bridges G categor #(! rather than the group ofsurve respondents who thought we spend too much mone onhighwas and bridges G categor (.

For each unit increase in confidence in ongress! the odds of being

in the group of surve respondents who thought we spend too littlemone on highwas and bridges decreased b '%.'4. ).1& H #.)2 -).'%'(


20/73

Slide

2!

Interpreting relations!ip o# indi(idual independent(aria"les to t!e dependent (aria"le


268.&2& 2.&50 2 .&09

268.625 2.652 2 .265

270.&95 4.42& 2 .110

275.194 9.221 2 .010

%%ect

Intercept

+)

:!

!OL)I"

-2 Log

Likeliood o%

ed$ced


e ci-3#$are 3tat i3tic i3 te di%%erence in -2 log-likeliood3etodel and a red$ced >odel. e red$ced >odel i3



Parameter Estimates

&.240 2.478 1.709 1 .191

.019 .020 .906 1 .&41 1.019

.071 .108 .427 1 .514 1.07&-1.&7& .620 4.91& 1 .027 .25&

&.6&9 2.456 2.195 1 .1&8

.00& .020 .017 1 .897 1.00&

.172 .110 2.46& 1 .117 1.188

-1.657 .61& 7.298 1 .007 .191

Intercept

+)

:!!OL)I"

Intercept

+)

:!

!OL)I"

(I)(*+,"

+ /I)"a

1

2



Surve respondents who had less confidence in congress highervalues correspond to lower confidence( were less li5el to be in thegroup of surve respondents who thought we spend about the rightamount of mone on highwas and bridges G categor 1(! ratherthan the group of surve respondents who thought we spend toomuch mone on highwas and bridges G ategor (.

For each unit increase in confidence in ongress! the odds of beingin the group of surve respondents who thought we spend about theright amount of mone on highwas and bridges decreased b$).4. ).## H #.) 2 ).$)(


21/73

Slide

2



&27.46&a .000 0 .

&&&.440 5.976 2 .050

&29.606 2.14& 2 .&4&

&&4.6&6 7.17& 2 .028

&&8.985 11.521 2 .00&

%%ect

Intercept

+)

:!

OLI*"

"?

-2 LogLikeliood o%

ed$ced


e ci-3#$are 3tati3 tic i3 te di%%erence in -2 log-likeliood3

etodel and a red$ced >odel. e red$ced >odel

i3 %or>ed y o>itting an e%%ect %ro> te %inal >odel. e n$ll

ypote3i3 i3 tat all para>eter3 o% tat e%%ect are 0.

i3 red$ced >odel i3 e#$ialent to te %inal >odel eca$3e

o>itting te e%%ect doe3 not increa3e te degree3 o% %reedo>.

a.

Parameter Estimates

8.4&4 2.2&& 14.261 1 .000

-.02& .017 1.756 1 .185 .977

-.066 .102 .414 1 .520 .9&6

-.575 .251 5.2&4 1 .022 .56&

-2.167 .805 7.242 1 .007 .115

05 . . 0 . .

4.485 2.255 &.955 1 .047

-.001 .018 .00& 1 .955 .999

.011 .104 .011 1 .916 1.011

-.&97 .257 2.&75 1 .12& .67&

-1.606 .824 &.800 1 .051 .201

05 . . 0 . .

Intercept

+)

:!

OLI*"

@"?A1B

@"?A2B

Intercept

+)

:!

OLI*"

@"?A1B

@"?A2B

+!(La

OO LIL

+/O: I)(


e re%erence cate or i3= OO M:!(.a.

In this e+ample! there isa statisticall significantrelationship between S7Jand the dependentvariable! spending onchildcare assistance.

?s well! S7J plas astatisticall significant rolein differentiating the T;;


22/73

Slide

22

Interpreting relations!ip o# indi(idual independent(aria"les and t!e dependent (aria"le


&27.46&a .000 0 .

&&&.440 5.976 2 .050

&29.606 2.14& 2 .&4&

&&4.6&6 7.17& 2 .028

&&8.985 11.521 2 .00&

%%ect

Intercept

+)

:!

OLI*"

"?

-2 LogLikeliood o%

ed$ced


e ci-3#$are 3tati3 tic i3 te di%%erence in -2 log-likeliood3






a.

Parameter Estimates

8.4&4 2.2&& 14.261 1 .000

-.02& .017 1.756 1 .185 .977

-.066 .102 .414 1 .520 .9&6

-.575 .251 5.2&4 1 .022 .56&

-2.167 .805 7.242 1 .007 .115

05 . . 0 . .

4.485 2.255 &.955 1 .047

-.001 .018 .00& 1 .955 .999

.011 .104 .011 1 .916 1.011

-.&97 .257 2.&75 1 .12& .67&

-1.606 .824 &.800 1 .051 .201

05 . . 0 . .

Intercept

+)

:!

OLI*"

@"?A1B

@"?A2B

Intercept

+)

:!

OLI*"

@"?A1B

@"?A2B

+!(La

OO LIL

+/O: I)(


e re%erence cate or i3= OO M:!(.a.

Surve respondents who were male code # for se+( were less li5elto be in the group of surve respondents who thought we spend toolittle mone on childcare assistance G categor #(! rather than thegroup of surve respondents who thought we spend too muchmone on childcare assistance G categor (.

Surve respondents who were male were $$.&4 less li5el ).##& H#.) 2 -).$$&( to be in the group of surve respondents who thoughtwe spend too little mone on childcare assistance.


23/73

Slide

23

Interpreting relations!ips #or independent(aria"le in pro"lems

In t!e multinomial logistic regression pro"lems, t!e pro"lem

statement ill as a"out only one o# t!e independent (aria"les)

*!e anser ill "e true or #alse "ased on only t!e relations!ip

"eteen t!e speci#ied independent (aria"le and t!e dependent

(aria"le) *!e indi(idual relations!ips "eteen ot!erindependent (aria"les are t!e dependent (aria"le are not used

in determining !et!er or not t!e anser is true or #alse)


24/73

Slide

24$ro"lem 1

11) In t!e dataset SS6444, is t!e #olloing statement true, #alse, or an incorrect application o#a statisticE Assume t!at t!ere is no pro"lem it! missing data, outliers, or in#luential cases, andt!at t!e (alidation analysis ill con#irm t!e generali%a"ility o# t!e results) Fse a le(el o#signi#icance o# 4)4> #or e(aluating t!e statistical relations!ips)

*!e (aria"les GageG Hage, G!ig!est year o# sc!ool completedG Heduc and Gcon#idence in CongressGHconlegis ere use#ul predictors #or distinguis!ing "eteen groups "ased on responses toGopinion a"out spending on !ig!ays and "ridgesG Hnatroad) *!ese predictors di##erentiatesur(ey respondents !o t!oug!t e spend too little money on !ig!ays and "ridges #rom sur(eyrespondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges and sur(eyrespondents !o t!oug!t e spend a"out t!e rig!t amount o# money on !ig!ays and "ridges#rom sur(ey respondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges)

Among t!is set o# predictors, con#idence in Congress as !elp#ul in distinguis!ing among t!egroups de#ined "y responses to opinion a"out spending on !ig!ays and "ridges) Sur(eyrespondents !o !ad less con#idence in congress ere less liely to "e in t!e group o# sur(eyrespondents !o t!oug!t e spend too little money on !ig!ays and "ridges, rat!er t!an t!egroup o# sur(ey respondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges)-or eac! unit increase in con#idence in Congress, t!e odds o# "eing in t!e group o# sur(eyrespondents !o t!oug!t e spend too little money on !ig!ays and "ridges decreased "y

7J)7?) Sur(ey respondents !o !ad less con#idence in congress ere less liely to "e in t!egroup o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on!ig!ays and "ridges, rat!er t!an t!e group o# sur(ey respondents !o t!oug!t e spend toomuc! money on !ig!ays and "ridges) -or eac! unit increase in con#idence in Congress, t!e oddso# "eing in t!e group o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o#money on !ig!ays and "ridges decreased "y 84)K?)

1) *rue

6) *rue it! caution

3) -alse

J) Inappropriate application o# a statistic


25/73

Slide

25Dissecting pro"lem 1 ' 1

11) In t!e dataset SS6444, is t!e #olloing statement true, #alse, or an incorrect application o# astatisticE Assume t!at t!ere is no pro"lem it! missing data, outliers, or in#luential cases, andt!at t!e (alidation analysis ill con#irm t!e generali%a"ility o# t!e results) Fse a le(el o#signi#icance o# 4)4> #or e(aluating t!e statistical relations!ips)

*!e (aria"les GageG Hage, G!ig!est year o# sc!ool completedG Heduc and Gcon#idence in CongressGHconlegis ere use#ul predictors #or distinguis!ing "eteen groups "ased on responses to Gopiniona"out spending on !ig!ays and "ridgesG Hnatroad) *!ese predictors di##erentiate sur(eyrespondents !o t!oug!t e spend too little money on !ig!ays and "ridges #rom sur(eyrespondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges and sur(eyrespondents !o t!oug!t e spend a"out t!e rig!t amount o# money on !ig!ays and "ridges#rom sur(ey respondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges)

Among t!is set o# predictors, con#idence in Congress as !elp#ul in distinguis!ing among t!egroups de#ined "y responses to opinion a"out spending on !ig!ays and "ridges) Sur(eyrespondents !o !ad less con#idence in congress ere less liely to "e in t!e group o# sur(eyrespondents !o t!oug!t e spend too little money on !ig!ays and "ridges, rat!er t!an t!egroup o# sur(ey respondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges) -oreac! unit increase in con#idence in Congress, t!e odds o# "eing in t!e group o# sur(ey respondents!o t!oug!t e spend too little money on !ig!ays and "ridges decreased "y 7J)7?) Sur(eyrespondents !o !ad less con#idence in congress ere less liely to "e in t!e group o# sur(eyrespondents !o t!oug!t e spend a"out t!e rig!t amount o# money on !ig!ays and "ridges,rat!er t!an t!e group o# sur(ey respondents !o t!oug!t e spend too muc! money on !ig!aysand "ridges) -or eac! unit increase in con#idence in Congress, t!e odds o# "eing in t!e group o#sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on !ig!ays and"ridges decreased "y 84)K?)

1) *rue

6) *rue it! caution

3) -alse


For these problems! we willassume that there is no problem

with missing data! outliers! orinfluential cases! and that thevalidation analsis will confirmthe generaliCabilit of theresults

In this problem! we are told touse ).)& as alpha for themultinomial logistic regression.


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*!e (aria"les "age" [age], "highest year of school completed" [educ] and "confidence in

Congress" [conlegis]ere use#ul predictors #or distinguis!ing "eteen groups "ased onresponses to "opinion about spending on highways and bridges" [natroad].*!ese predictorsdi##erentiate sur(ey respondents !o t!oug!t e spend too little money on !ig!ays and"ridges #rom sur(ey respondents !o t!oug!t e spend too muc! money on !ig!ays and"ridges and sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on!ig!ays and "ridges #rom sur(ey respondents !o t!oug!t e spend too muc! money on!ig!ays and "ridges)

Among t!is set o# predictors, con#idence in Congress as !elp#ul in distinguis!ing among t!egroups de#ined "y responses to opinion a"out spending on !ig!ays and "ridges) Sur(ey

respondents !o !ad less con#idence in congress ere less liely to "e in t!e group o# sur(eyrespondents !o t!oug!t e spend too little money on !ig!ays and "ridges, rat!er t!an t!egroup o# sur(ey respondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges)-or eac! unit increase in con#idence in Congress, t!e odds o# "eing in t!e group o# sur(eyrespondents !o t!oug!t e spend too little money on !ig!ays and "ridges decreased "y7J)7?) Sur(ey respondents !o !ad less con#idence in congress ere less liely to "e in t!egroup o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on!ig!ays and "ridges, rat!er t!an t!e group o# sur(ey respondents !o t!oug!t e spend toomuc! money on !ig!ays and "ridges) -or eac! unit increase in con#idence in Congress, t!e odds

o# "eing in t!e group o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o#money on !ig!ays and "ridges decreased "y 84)K?)

SPSS onl supports direct orsimultaneous entr of independentvariables in multinomial logisticregression! so we have no choice ofmethod for entering variables.

The variables listed first in the problemstatement are the independent variablesIGs(9 "age" Lage! "highest ear of schoolcompleted" Leduc and "confidence inongress" Lconlegis.

The variable used to definegroups is the dependentvariable G(9 "opinion aboutspending on highwas and

bridges" Lnatroad.


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11) In t!e dataset SS6444, is t!e #olloing statement true, #alse, or an incorrect application o# astatisticE Assume t!at t!ere is no pro"lem it! missing data, outliers, or in#luential cases, and t!att!e (alidation analysis ill con#irm t!e generali%a"ility o# t!e results) Fse a le(el o# signi#icance o#4)4> #or e(aluating t!e statistical relations!ips)

*!e (aria"les GageG Hage, G!ig!est year o# sc!ool completedG Heduc and Gcon#idence in CongressG

Hconlegis ere use#ul predictors #or distinguis!ing "eteen groups "ased on responses to Gopiniona"out spending on !ig!ays and "ridgesG Hnatroad) *!ese predictors di##erentiate surveyrespondents who thought we spend too little money on highways and bridges from surveyrespondents who thought we spend too much money on highways and bridgesand surveyrespondents who thought we spend about the right amount of money on highways and bridgesfrom survey respondents who thought we spend too much money on highways and bridges)

Among t!is set o# predictors, con#idence in Congress as !elp#ul in distinguis!ing among t!e groupsde#ined "y responses to opinion a"out spending on !ig!ays and "ridges) Sur(ey respondents !o !adless con#idence in congress ere less liely to "e in t!e group o# sur(ey respondents !o t!oug!t e

spend too little money on !ig!ays and "ridges, rat!er t!an t!e group o# sur(ey respondents !ot!oug!t e spend too muc! money on !ig!ays and "ridges) -or eac! unit increase in con#idence inCongress, t!e odds o# "eing in t!e group o# sur(ey respondents !o t!oug!t e spend too little moneyon !ig!ays and "ridges decreased "y 7J)7?) Sur(ey respondents !o !ad less con#idence in congressere less liely to "e in t!e group o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!tamount o# money on !ig!ays and "ridges, rat!er t!an t!e group o# sur(ey respondents !o t!oug!te spend too muc! money on !ig!ays and "ridges) -or eac! unit increase in con#idence in Congress,t!e odds o# "eing in t!e group o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o#money on !ig!ays and "ridges decreased "y 84)K?)

SPSS multinomial logistic regression models the relationship bcomparing each of the groups defined b the dependent variable to thegroup with the highest code value.

The responses to opinion about spending on highwas and bridges were9

#2 Too little! 1 2 ?bout right! and 2 Too much.

The analsis will result in two comparisons9: surve respondents who thought we spend too little mone

versus surve respondents who thought we spend too muchmone on highwas and bridges

: surve respondents who thought we spend about the rightamount of mone versus surve respondents who thought wespend too much mone on highwas and bridges.


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28Dissecting pro"lem 1 ' J

*!e (aria"les GageG Hage, G!ig!est year o# sc!ool completedG Heduc and Gcon#idence inCongressG Hconlegis ere use#ul predictors #or distinguis!ing "eteen groups "ased onresponses to Gopinion a"out spending on !ig!ays and "ridgesG Hnatroad) *!ese predictors

di##erentiate sur(ey respondents !o t!oug!t e spend too little money on !ig!ays and"ridges #rom sur(ey respondents !o t!oug!t e spend too muc! money on !ig!ays and"ridges and sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on!ig!ays and "ridges #rom sur(ey respondents !o t!oug!t e spend too muc! money on!ig!ays and "ridges)

Among t!is set o# predictors, con#idence in Congress as !elp#ul in distinguis!ing among t!egroups de#ined "y responses to opinion a"out spending on !ig!ays and "ridges) Surveyrespondents who had less confidence in congress were less likely to be in the group ofsurvey respondents who thought we spend too little money on highways and bridges, rather

than the group of survey respondents who thought we spend too much money on highwaysand bridges. For each unit increase in confidence in Congress, the odds of being in thegroup of survey respondents who thought we spend too little money on highways andbridges decreased by !..Survey respondents who had less confidence in congress wereless likely to be in the group of survey respondents who thought we spend about the rightamount of money on highways and bridges, rather than the group of survey respondentswho thought we spend too much money on highways and bridges. For each unit increase inconfidence in Congress, the odds of being in the group of survey respondents who thoughtwe spend about the right amount of money on highways and bridges decreased by #$.%.

7ach problem includes a statement about the relationship betweenone independent variable and the dependent variable. The answerto the problem is based on the stated relationship! ignoring therelationships between the other independent variables and thedependent variable.

This problem identifies a difference for both of the comparisonsamong groups modeled b the multinomial logistic regression.


29/73

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29Dissecting pro"lem 1 ' >

11) In t!e dataset SS6444, is t!e #olloing statement true, #alse, or an incorrect application o# astatisticE Assume t!at t!ere is no pro"lem it! missing data, outliers, or in#luential cases, and t!at t!e(alidation analysis ill con#irm t!e generali%a"ility o# t!e results) Fse a le(el o# signi#icance o# 4)4> #ore(aluating t!e statistical relations!ips)

*!e (aria"les GageG Hage, G!ig!est year o# sc!ool completedG Heduc and Gcon#idence in CongressGHconlegis ere use#ul predictors #or distinguis!ing "eteen groups "ased on responses to Gopinion a"outspending on !ig!ays and "ridgesG Hnatroad) *!ese predictors di##erentiate sur(ey respondents !ot!oug!t e spend too little money on !ig!ays and "ridges #rom sur(ey respondents !o t!oug!t espend too muc! money on !ig!ays and "ridges and sur(ey respondents !o t!oug!t e spend a"out t!erig!t amount o# money on !ig!ays and "ridges #rom sur(ey respondents !o t!oug!t e spend too muc!money on !ig!ays and "ridges)

Among t!is set o# predictors, con#idence in Congress as !elp#ul in distinguis!ing among t!e groupsde#ined "y responses to opinion a"out spending on !ig!ays and "ridges) Sur(ey respondents !o !ad lesscon#idence in congress ere less liely to "e in t!e group o# sur(ey respondents !o t!oug!t e spend toolittle money on !ig!ays and "ridges, rat!er t!an t!e group o# sur(ey respondents !o t!oug!t e spendtoo muc! money on !ig!ays and "ridges) -or eac! unit increase in con#idence in Congress, t!e odds o#"eing in t!e group o# sur(ey respondents !o t!oug!t e spend too little money on !ig!ays and "ridgesdecreased "y 7J)7?) Sur(ey respondents !o !ad less con#idence in congress ere less liely to "e in t!e

group o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on !ig!ays and"ridges, rat!er t!an t!e group o# sur(ey respondents !o t!oug!t e spend too muc! money on !ig!aysand "ridges) -or eac! unit increase in con#idence in Congress, t!e odds o# "eing in t!e group o# sur(eyrespondents !o t!oug!t e spend a"out t!e rig!t amount o# money on !ig!ays and "ridges decreased"y 84)K?)

In order for the multinomial logistic regressionquestion to be true! the overall relationship mustbe statisticall significant! there must be noevidence of numerical problems! the classificationaccurac rate must be substantiall better thancould be obtained b chance alone! and thestated individual relationship must be statisticallsignificant and interpreted correctl.


30/73

Slide

3!Re+uest multinomial logistic regression

Select theRegression |Multinomial Logisticcommand from the

Analyze menu.


31/73

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3Selecting t!e dependent (aria"le

Second! clic5 on the rightarrow button to move thedependent variable to theDependentte+t bo+.

First! highlight thedependent variablenatroadin the listof variables.


32/73

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32Selecting metric independent (aria"les

Move the metricindependent variables!age, educand conlegistothe Covariate(s)list bo+.

Metric independent variables are specified as covariatesin multinomial logistic regression. Metric variables canbe either interval or! b convention! ordinal.

In this analsis! there are no non-metric independent variables. Don-metric independent variables would be

moved to the actor(s)list bo+.


33/73

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33Speci#ying statistics to include in t!e output

Nhile we will accept most ofthe SPSS defaults for theanalsis! we need to specificallrequest the classification table.

lic5 on the !tatisticsO buttonto ma5e a request.


34/73

Slide

34Re+uesting t!e classi#ication ta"le

First! 5eep the SPSSdefaults for !ummarystatistics! Li"eli#oodratio test! and$arameter estimates.

Second! mar5 thechec5bo+ for the

Classi%ication ta&le.

Third! clic5on theContinuebutton tocomplete therequest.


35/73

Slide

35

Completing t!e multinomiallogistic regression re+uest

lic5 on the '

button to requestthe output for themultinomial logisticregression.

The multinomial logistic procedure supportsadditional commands to specif the modelcomputed for the relationships we will use thedefault main effects model(! additionalspecifications for computing the regression!and saving classification results. Ne will notma5e use of these options.


36/73

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36L=2=L - M=ASFR=M=:* ' 1


*!e (aria"les GageG Hage, G!ig!est year o# sc!ool completedG Heduc and Gcon#idence in CongressGHconlegis ere use#ul predictors #or distinguis!ing "eteen groups "ased on responses to"opinion about spending on highways and bridges" [natroad]) *!ese predictors di##erentiatesur(ey respondents !o t!oug!t e spend too little money on !ig!ays and "ridges #rom sur(eyrespondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges and sur(ey

respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on !ig!ays and "ridges#rom sur(ey respondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges)

Among t!is set o# predictors, con#idence in Congress as !elp#ul in distinguis!ing among t!egroups de#ined "y responses to opinion a"out spending on !ig!ays and "ridges) Sur(eyrespondents !o !ad less con#idence in congress ere less liely to "e in t!e group o# sur(eyrespondents !o t!oug!t e spend too little money on !ig!ays and "ridges, rat!er t!an t!egroup o# sur(ey respondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges)-or eac! unit increase in con#idence in Congress, t!e odds o# "eing in t!e group o# sur(eyrespondents !o t!oug!t e spend too little money on !ig!ays and "ridges decreased "y

7J)7?) Sur(ey respondents !o !ad less con#idence in congress ere less liely to "e in t!egroup o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on!ig!ays and "ridges, rat!er t!an t!e group o# sur(ey respondents !o t!oug!t e spend toomuc! money on !ig!ays and "ridges) -or eac! unit increase in con#idence in Congress, t!e oddso# "eing in t!e group o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o#money on !ig!ays and "ridges decreased "y 84)K?)

1) *rue

6) *rue it! caution

Multinomial logistic regression requires that thedependent variable be non-metric and theindependent variables be metric or dichotomous.

";pinion about spending on highwas andbridges" Lnatroad is ordinal! satisfing the non-metric level of measurement requirement for thedependent variable.

It contains three categories9 surve respondentswho thought we spend too little mone! aboutthe right amount of mone! and too much moneon highwas and bridges.


37/73

Slide

37L=2=L - M=ASFR=M=:* ' 6


*!e (aria"les "age" [age], "highest year of school completed" [educ]and "confidence in

Congress" [conlegis]ere use#ul predictors #or distinguis!ing "eteen groups "ased on responsesto Gopinion a"out spending on !ig!ays and "ridgesG Hnatroad) *!ese predictors di##erentiatesur(ey respondents !o t!oug!t e spend too little money on !ig!ays and "ridges #rom sur(eyrespondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges and sur(eyrespondents !o t!oug!t e spend a"out t!e rig!t amount o# money on !ig!ays and "ridges #romsur(ey respondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges)

Among t!is set o# predictors, con#idence in Congress as !elp#ul in distinguis!ing among t!e groupsde#ined "y responses to opinion a"out spending on !ig!ays and "ridges) Sur(ey respondents !o!ad less con#idence in congress ere less liely to "e in t!e group o# sur(ey respondents !ot!oug!t e spend too little money on !ig!ays and "ridges, rat!er t!an t!e group o# sur(eyrespondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges) -or eac! unitincrease in con#idence in Congress, t!e odds o# "eing in t!e group o# sur(ey respondents !ot!oug!t e spend too little money on !ig!ays and "ridges decreased "y 7J)7?) Sur(ey respondents!o !ad less con#idence in congress ere less liely to "e in t!e group o# sur(ey respondents !ot!oug!t e spend a"out t!e rig!t amount o# money on !ig!ays and "ridges, rat!er t!an t!e groupo# sur(ey respondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges) -or eac!unit increase in con#idence in Congress, t!e odds o# "eing in t!e group o# sur(ey respondents !ot!oug!t e spend a"out t!e rig!t amount o# money on !ig!ays and "ridges decreased "y 84)K?)

"?ge" Lage and "highest ear ofschool completed" Leduc are interval!satisfing the metric or dichotomouslevel of measurement requirement forindependent variables.

"onfidence in ongress" Lconlegis is ordinal!satisfing the metric or dichotomous level ofmeasurement requirement for independentvariables. If we follow the convention of treatingordinal level variables as metric variables! the level

of measurement requirement for the analsis issatisfied. Since some data analsts do not agreewith this convention! a note of caution should beincluded in our interpretation.


38/73

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38Sample si%e ratio o# cases to (aria"les


62 &7.1'

9& 55.7'

12 7.2'

167 100.0'

10&

270

15&a

1

2

&

(I)(*+,"

+ /I)"

alid

Mi33ing

otal

"$pop$lation

Marginal

ercentage



a.

Multinomial logistic regression requires that the minimum ratioof valid cases to independent variables be at least #) to #. Theratio of valid cases #3'( to number of independent variables( was &&.' to #! which was equal to or greater than the

minimum ratio. The requirement for a minimum ratio of casesto independent variables was satisfied.

The preferred ratio of valid cases to independent variables is1) to #. The ratio of &&.' to # was equal to or greater than thepreferred ratio. The preferred ratio of cases to independentvariables was satisfied.


39/73

Slide

39

2=RALL R=LA*I:S5I$ =*W==:I:D=$=:D=:* A:D D=$=:D=:* 2ARIAL=S


284.429

265.972 18.457 6 .005

Model

Intercept Only

Final

-2 Log



In this analsis! the probabilit of the model chi-square

#$.%&'( was ).))&! less than or equal to the level ofsignificance of ).)&. The null hpothesis that there wasno difference between the model without independentvariables and the model with independent variables wasre*ected. The e+istence of a relationship between theindependent variables and the dependent variable wassupported.


40/73

Slide

4!

Parameter Estimates

&.240 2.478 1.709 1 .191

.019 .020 .906 1 .&41 1.019

.071 .108 .427 1 .514 1.07&

-1.&7& .620 4.91& 1 .027 .25&

&.6&9 2.456 2.195 1 .1&8

.00& .020 .017 1 .897 1.00&

.172 .110 2.46& 1 .117 1.188

-1.657 .61& 7.298 1 .007 .191

Intercept

+)

:!

!OL)I"

Intercept

+)

:!

!OL)I"

(I)(*+,"

+ /I)"a

1

2



:FM=RICAL $RL=MS

Multicollinearit in the multinomiallogistic regression solution isdetected b e+amining the standarderrors for the b coefficients. ?standard error larger than 1.)indicates numerical problems! such

as multicollinearit among theindependent variables! Cero cells fora dumm-coded independentvariable because all of the sub*ectshave the same value for thevariable! and ,complete separation,whereb the two groups in thedependent event variable can beperfectl separated b scores onone of the independent variables.?nalses that indicate numericalproblems should not be interpreted.

Done of the independent variables inthis analsis had a standard errorlarger than 1.). Ne are notinterested in the standard errorsassociated with the intercept.(


41/73

Slide

4


268.&2& 2.&50 2 .&09

268.625 2.652 2 .265

270.&95 4.42& 2 .110

275.194 9.221 2 .010

%%ect

Intercept

+)

:!

!OL)I"

-2 Log

Likeliood o%

ed$ced


e ci-3#$are 3tati3tic i3 te di%%erence in -2 log-likeliood3




R=LA*I:S5I$ - I:DI2IDFAL I:D=$=:D=:*2ARIAL=S * D=$=:D=:* 2ARIAL= ' 1

The statistical significance of the relationship betweenconfidence in ongress and opinion about spending onhighwas and bridges is based on the statistical significance ofthe chi-square statistic in the SPSS table titled "


42/73

Slide

42

Parameter Estimates

&.240 2.478 1.709 1 .191

.019 .020 .906 1 .&41 1.019

.071 .108 .427 1 .514 1.07&

-1.&7& .620 4.91& 1 .027 .25&&.6&9 2.456 2.195 1 .1&8

.00& .020 .017 1 .897 1.00&

.172 .110 2.46& 1 .117 1.188

-1.657 .61& 7.298 1 .007 .191

Intercept

+)

:!

!OL)I"Intercept

+)

:!

!OL)I"

(I)(*+,"

+ /I)"a

1

2




In the comparison of surve respondents who thought we spend

too little mone on highwas and bridges to surve respondentswho thought we spend too much mone on highwas andbridges! the probabilit of the Nald statistic %.#( for thevariable confidence in ongress Lconlegis was ).)1'. Since theprobabilit was less than or equal to the level of significance of).)&! the null hpothesis that the b coefficient for confidence inongress was equal to Cero for this comparison was re*ected.


43/73

Slide

43

Parameter Estimates

&.240 2.478 1.709 1 .191

.019 .020 .906 1 .&41 1.019

.071 .108 .427 1 .514 1.07&

-1.&7& .620 4.91& 1 .027 .25&

&.6&9 2.456 2.195 1 .1&8

.00& .020 .017 1 .897 1.00&

.172 .110 2.46& 1 .117 1.188

-1.657 .61& 7.298 1 .007 .191

Intercept

+)

:!

!OL)I"

Intercept+)

:!

!OL)I"

(I)(*+,"

+ /I)"a

1

2




The value of 7+p@( was ).1& which implies that for each unitincrease in confidence in ongress the odds decreased b '%.'4).1& - #.) 2 -).'%'(.

The relationship stated in the problem is supported. Surverespondents who had less confidence in congress were less li5elto be in the group of surve respondents who thought we spendtoo little mone on highwas and bridges! rather than the group ofsurve respondents who thought we spend too much mone onhighwas and bridges. For each unit increase in confidence inongress! the odds of being in the group of surve respondentswho thought we spend too little mone on highwas and bridgesdecreased b '%.'4.


44/73

Slide

44

Parameter Estimates

&.240 2.478 1.709 1 .191

.019 .020 .906 1 .&41 1.019

.071 .108 .427 1 .514 1.07&

-1.&7& .620 4.91& 1 .027 .25&&.6&9 2.456 2.195 1 .1&8

.00& .020 .017 1 .897 1.00&

.172 .110 2.46& 1 .117 1.188

-1.657 .61& 7.298 1 .007 .191

Intercept

+)

:!

!OL)I"Intercept

+)

:!

!OL)I"

(I)(*+,"

+ /I)"a

1

2



R=LA*I:S5I$ - I:DI2IDFAL I:D=$=:D=:*2ARIAL=S * D=$=:D=:* 2ARIAL= ' J

In the comparison of surve respondents who thought we spendabout the right amount of mone on highwas and bridges tosurve respondents who thought we spend too much mone onhighwas and bridges! the probabilit of the Nald statistic'.1$( for the variable confidence in ongress Lconlegis was).))'. Since the probabilit was less than or equal to the levelof significance of ).)&! the null hpothesis that the b coefficientfor confidence in ongress was equal to Cero for this comparisonwas re*ected.


45/73

Slide

45

Parameter Estimates

&.240 2.478 1.709 1 .191

.019 .020 .906 1 .&41 1.019

.071 .108 .427 1 .514 1.07&

-1.&7& .620 4.91& 1 .027 .25&

&.6&9 2.456 2.195 1 .1&8

.00& .020 .017 1 .897 1.00&

.172 .110 2.46& 1 .117 1.188

-1.657 .61& 7.298 1 .007 .191

Intercept

+)

:!

!OL)I"

Intercept

+)

:!

!OL)I"

(I)(*+,"

+ /I)"a

1

2



R=LA*I:S5I$ - I:DI2IDFAL I:D=$=:D=:*2ARIAL=S * D=$=:D=:* 2ARIAL= ' >

The value of 7+p@( was ).## which implies that for each unit increase in

confidence in ongress the odds decreased b $).4 ).##-#.)2-).$)(.The relationship stated in the problem is supported. Surve respondentswho had less confidence in congress were less li5el to be in the group ofsurve respondents who thought we spend about the right amount ofmone on highwas and bridges! rather than the group of surverespondents who thought we spend too much mone on highwas andbridges. For each unit increase in confidence in ongress! the odds ofbeing in the group of surve respondents who thought we spend about theright amount of mone on highwas and bridges decreased b $).4.


46/73

Slide

46


62 &7.1'

9& 55.7'

12 7.2'

167 100.0'

10&

270

15&a

1

2

&

(I)(*+,"

+ /I)"

alid

Mi33ing

otal

"$pop$lation

Marginal

ercentage



a.

CLASSI-ICA*I: FSI: *5= MFL*I:MIAL LIS*ICR=R=SSI: MD=LB C5A:C= ACCFRAC RA*=

The proportional b chance accurac rate was computed bcalculating the proportion of cases for each group based onthe number of cases in each group in the ,ase ProcessingSummar,! and then squaring and summing the proportion ofcases in each group ).'#/ 0 ).&&'/ 0 ).)'1/ 2 ).%&(.

The independent variables could be characteriCed as usefulpredictors distinguishing surve respondents who thought wespend too little mone on highwas and bridges! surverespondents who thought we spend about the right amountof mone on highwas and bridges and surve respondentswho thought we spend too much mone on highwas andbridges if the classification accurac rate was substantiallhigher than the accurac attainable b chance alone.;perationall! the classification accurac rate should be 1&4or more higher than the proportional b chance accuracrate.


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47

Classification

15 47 0 24.2'

7 86 0 92.5'

5 7 0 .0'

16.2' 8&.8' .0' 60.5'

O3ered

1

2

&

Oerall ercentage

1 2 &

ercent

!orrect

redicted

CLASSI-ICA*I: FSI: *5= MFL*I:MIAL LIS*ICR=R=SSI: MD=LB CLASSI-ICA*I: ACCFRAC

The classification accurac rate was 3).&4which was greater than or equal to the

proportional b chance accurac criteria of&3.34 #.1& + %&.4 2 &3.34(.

The criteria for classification accurac issatisfied.


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48Ansering t!e +uestion in pro"lem 1 ' 1


*!e (aria"les GageG Hage, G!ig!est year o# sc!ool completedG Heduc and Gcon#idence in CongressGHconlegis ere use#ul predictors #or distinguis!ing "eteen groups "ased on responses to Gopiniona"out spending on !ig!ays and "ridgesG Hnatroad) *!ese predictors di##erentiate sur(ey respondents!o t!oug!t e spend too little money on !ig!ays and "ridges #rom sur(ey respondents !ot!oug!t e spend too muc! money on !ig!ays and "ridges and sur(ey respondents !o t!oug!t e

spend a"out t!e rig!t amount o# money on !ig!ays and "ridges #rom sur(ey respondents !ot!oug!t e spend too muc! money on !ig!ays and "ridges)

Among t!is set o# predictors, con#idence in Congress as !elp#ul in distinguis!ing among t!e groupsde#ined "y responses to opinion a"out spending on !ig!ays and "ridges) Sur(ey respondents !o !adless con#idence in congress ere less liely to "e in t!e group o# sur(ey respondents !o t!oug!t espend too little money on !ig!ays and "ridges, rat!er t!an t!e group o# sur(ey respondents !ot!oug!t e spend too muc! money on !ig!ays and "ridges) -or eac! unit increase in con#idence inCongress, t!e odds o# "eing in t!e group o# sur(ey respondents !o t!oug!t e spend too littlemoney on !ig!ays and "ridges decreased "y 7J)7?) Sur(ey respondents !o !ad less con#idence incongress ere less liely to "e in t!e group o# sur(ey respondents !o t!oug!t e spend a"out t!erig!t amount o# money on !ig!ays and "ridges, rat!er t!an t!e group o# sur(ey respondents !ot!oug!t e spend too muc! money on !ig!ays and "ridges) -or eac! unit increase in con#idence inCongress, t!e odds o# "eing in t!e group o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!tamount o# money on !ig!ays and "ridges decreased "y 84)K?)

1) *rue

6) *rue it! caution

3) -alse


Ne found a statisticall significant overallrelationship between the combination ofindependent variables and the dependentvariable.

There was no evidence of numerical problems inthe solution.

Moreover! the classification accurac surpassedthe proportional b chance accurac criteria!supporting the utilit of the model.


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49Ansering t!e +uestion in pro"lem 1 ' 6

*!e (aria"les GageG Hage, G!ig!est year o# sc!ool completedG Heduc and Gcon#idence inCongressG Hconlegis ere use#ul predictors #or distinguis!ing "eteen groups "ased onresponses to Gopinion a"out spending on !ig!ays and "ridgesG Hnatroad) *!ese predictorsdi##erentiate sur(ey respondents !o t!oug!t e spend too little money on !ig!ays and"ridges #rom sur(ey respondents !o t!oug!t e spend too muc! money on !ig!ays and"ridges and sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on!ig!ays and "ridges #rom sur(ey respondents !o t!oug!t e spend too muc! money on!ig!ays and "ridges)

Among t!is set o# predictors, con#idence in Congress as !elp#ul in distinguis!ing among t!egroups de#ined "y responses to opinion a"out spending on !ig!ays and "ridges) Sur(eyrespondents !o !ad less con#idence in congress ere less liely to "e in t!e group o# sur(eyrespondents !o t!oug!t e spend too little money on !ig!ays and "ridges, rat!er t!an t!egroup o# sur(ey respondents !o t!oug!t e spend too muc! money on !ig!ays and "ridges)-or eac! unit increase in con#idence in Congress, t!e odds o# "eing in t!e group o# sur(eyrespondents !o t!oug!t e spend too little money on !ig!ays and "ridges decreased "y7J)7?) Sur(ey respondents !o !ad less con#idence in congress ere less liely to "e in t!egroup o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on

!ig!ays and "ridges, rat!er t!an t!e group o# sur(ey respondents !o t!oug!t e spend toomuc! money on !ig!ays and "ridges) -or eac! unit increase in con#idence in Congress, t!eodds o# "eing in t!e group o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amounto# money on !ig!ays and "ridges decreased "y 84)K?)

1) *rue

6) *rue it! caution

3) -alse


Ne verified that each statement about the relationshipbetween an independent variable and the dependentvariable was correct in both direction of the relationshipand the change in li5elihood associated with a one-unitchange of the independent variable! for both of thecomparisons between groups stated in the problem.

The answer to the question is truewith caution.

? caution is added because of theinclusion of ordinal level variables.


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5!$ro"lem 6

1) In t!e dataset SS6444, is t!e #olloing statement true, #alse, or an incorrect application o#a statisticE Assume t!at t!ere is no pro"lem it! missing data, outliers, or in#luential cases,and t!at t!e (alidation analysis ill con#irm t!e generali%a"ility o# t!e results) Fse a le(el o#signi#icance o# 4)4> #or e(aluating t!e statistical relations!ips)

*!e (aria"les G!ig!est year o# sc!ool completedG Heduc, Gse.G Hse. and Gtotal #amily incomeGHincomeK8 ere use#ul predictors #or distinguis!ing "eteen groups "ased on responses toGopinion a"out spending on space e.plorationG Hnatspac) *!ese predictors di##erentiate sur(eyrespondents !o t!oug!t e spend too little money on space e.ploration #rom sur(ey

respondents !o t!oug!t e spend too muc! money on space e.ploration and sur(eyrespondents !o t!oug!t e spend a"out t!e rig!t amount o# money on space e.ploration #romsur(ey respondents !o t!oug!t e spend too muc! money on space e.ploration)

Among t!is set o# predictors, total #amily income as !elp#ul in distinguis!ing among t!egroups de#ined "y responses to opinion a"out spending on space e.ploration) Sur(eyrespondents !o !ad !ig!er total #amily incomes ere more liely to "e in t!e group o# sur(eyrespondents !o t!oug!t e spend a"out t!e rig!t amount o# money on space e.ploration,rat!er t!an t!e group o# sur(ey respondents !o t!oug!t e spend too muc! money on space

e.ploration) -or eac! unit increase in total #amily income, t!e odds o# "eing in t!e group o#sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on spacee.ploration increased "y N)4?)

1) *rue

6) *rue it! caution

3) -alse



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&. 'n the dataset (SS)$$$, is the following statement true, false, or an incorrect

application of a statistic* +ssume that there is no problem with missing data, outliers, orinfluential cases, and that the validation analysis will confirm the generaliability of theresults. -se a level of significance of $.$ for evaluating the statistical relationships.

*!e (aria"les G!ig!est year o# sc!ool completedG Heduc, Gse.G Hse. and Gtotal #amily incomeGHincomeK8 ere use#ul predictors #or distinguis!ing "eteen groups "ased on responses toGopinion a"out spending on space e.plorationG Hnatspac) *!ese predictors di##erentiate sur(eyrespondents !o t!oug!t e spend too little money on space e.ploration #rom sur(eyrespondents !o t!oug!t e spend too muc! money on space e.ploration and sur(eyrespondents !o t!oug!t e spend a"out t!e rig!t amount o# money on space e.ploration #romsur(ey respondents !o t!oug!t e spend too muc! money on space e.ploration)

Among t!is set o# predictors, total #amily income as !elp#ul in distinguis!ing among t!egroups de#ined "y responses to opinion a"out spending on space e.ploration) Sur(eyrespondents !o !ad !ig!er total #amily incomes ere more liely to "e in t!e group o# sur(eyrespondents !o t!oug!t e spend a"out t!e rig!t amount o# money on space e.ploration,rat!er t!an t!e group o# sur(ey respondents !o t!oug!t e spend too muc! money on space

e.ploration) -or eac! unit increase in total #amily income, t!e odds o# "eing in t!e group o#sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on spacee.ploration increased "y N)4?)

1) *rue

6) *rue it! caution

3) -alse


For these problems! we willassume that there is no problemwith missing data! outliers! or

influential cases! and that thevalidation analsis will confirmthe generaliCabilit of theresults

In this problem! we are told touse ).)& as alpha for themultinomial logistic regression.


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/he variables "highest year of school completed" [educ], "se0" [se0] and "total family income"[income%#]ere use#ul predictors #or distinguis!ing "eteen groups based on responses to"opinion about spending on space e0ploration" [natspac].*!ese predictors di##erentiate sur(eyrespondents !o t!oug!t e spend too little money on space e.ploration #rom sur(ey respondents!o t!oug!t e spend too muc! money on space e.ploration and sur(ey respondents !o t!oug!t espend a"out t!e rig!t amount o# money on space e.ploration #rom sur(ey respondents !o t!oug!te spend too muc! money on space e.ploration)

Among t!is set o# predictors, total #amily income as !elp#ul in distinguis!ing among t!e groupsde#ined "y responses to opinion a"out spending on space e.ploration) Sur(ey respondents !o !ad!ig!er total #amily incomes ere more liely to "e in t!e group o# sur(ey respondents !o t!oug!t

e spend a"out t!e rig!t amount o# money on space e.ploration, rat!er t!an t!e group o# sur(eyrespondents !o t!oug!t e spend too muc! money on space e.ploration) -or eac! unit increase intotal #amily income, t!e odds o# "eing in t!e group o# sur(ey respondents !o t!oug!t e spenda"out t!e rig!t amount o# money on space e.ploration increased "y N)4?)

1) *rue

6) *rue it! caution

3) -alse


SPSS onl supports direct orsimultaneous entr of independentvariables in multinomial logisticregression! so we have no choice ofmethod for entering variables.

The variables listed first in the problemstatement are the independent variablesIGs(9 "highest ear of school completed"Leduc! "se+" Lse+ and "total familincome" Lincome$.

The variable used to definegroups is the dependentvariable G(9 "opinion aboutspending on space

e+ploration" Lnatspac.


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1) In t!e dataset SS6444, is t!e #olloing statement true, #alse, or an incorrect application o# a statisticEAssume t!at t!ere is no pro"lem it! missing data, outliers, or in#luential cases, and t!at t!e (alidationanalysis ill con#irm t!e generali%a"ility o# t!e results) Fse a le(el o# signi#icance o# 4)4> #or e(aluating t!estatistical relations!ips)

*!e (aria"les G!ig!est year o# sc!ool completedG Heduc, Gse.G Hse. and Gtotal #amily incomeG HincomeK8ere use#ul predictors #or distinguis!ing "eteen groups "ased on responses to Gopinion a"out spending onspace e.plorationG Hnatspac) /hese predictors differentiate survey respondents who thought we spendtoo little money on space e0ploration from survey respondents who thought we spend too much moneyon space e0ploration and survey respondents who thought we spend about the right amount of money onspace e0ploration from survey respondents who thought we spend too much money on spacee0ploration.

Among t!is set o# predictors, total #amily income as !elp#ul in distinguis!ing among t!e groups de#ined "yresponses to opinion a"out spending on space e.ploration) Sur(ey respondents !o !ad !ig!er total #amily

incomes ere more liely to "e in t!e group o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!tamount o# money on space e.ploration, rat!er t!an t!e group o# sur(ey respondents !o t!oug!t e spendtoo muc! money on space e.ploration) -or eac! unit increase in total #amily income, t!e odds o# "eing in t!egroup o# sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on space e.plorationincreased "y N)4?)

1) *rue

6) *rue it! caution

3) -alse

SPSS multinomial logistic regression models the relationshipb comparing each of the groups defined b the dependentvariable to the group with the highest code value.

The responses to opinion about spending on the spaceprogram were9#2 Too little! 1 2 ?bout right! and 2 Too much.

The analsis will result in two comparisons9: surve respondents who thought we spend too little mone

versus surve respondents who thought we spend too muchmone on space e+ploration

: surve respondents who thought we spend about the rightamount of mone versus surve respondents who thought wespend too much mone on space e+ploration.


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54Dissecting pro"lem 6 ' J


+mong this set of predictors, total family income was helpful in distinguishing among thegroups defined by responses to opinion about spending on space e0ploration. Surveyrespondents who had higher total family incomes were more likely to be in the group ofsurvey respondents who thought we spend about the right amount of money on spacee0ploration, rather than the group of survey respondents who thought we spend too muchmoney on space e0ploration. For each unit increase in total family income, the odds ofbeing in the group of survey respondents who thought we spend about the right amount ofmoney on space e0ploration increased by 1.$.

1) *rue

6) *rue it! caution

3) -alse


7ach problem includes a statement about therelationship between one independent variable andthe dependent variable. The answer to theproblem is based on the stated relationship!ignoring the relationships between the otherindependent variables and the dependent variable.

This problem identifies a difference for onl oneof the two comparisons based on the three valuesof the dependent variable.

;ther problems will specif both of the possiblecomparisons.


55/73

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55Dissecting pro"lem 6 ' >


Among t!is set o# predictors, total #amily income as !elp#ul in distinguis!ing among t!egroups de#ined "y responses to opinion a"out spending on space e.ploration) Sur(eyrespondents !o !ad !ig!er total #amily incomes ere more liely to "e in t!e group o# sur(eyrespondents !o t!oug!t e spend a"out t!e rig!t amount o# money on space e.ploration,rat!er t!an t!e group o# sur(ey respondents !o t!oug!t e spend too muc! money on spacee.ploration) -or eac! unit increase in total #amily income, t!e odds o# "eing in t!e group o#sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on spacee.ploration increased "y N)4?)

1) *rue 6) *rue it! caution

3) -alse


In order for the multinomial logistic regressionquestion to be true! the overall relationship mustbe statisticall significant! there must be noevidence of numerical problems! the classificationaccurac rate must be substantiall better thancould be obtained b chance alone! and thestated individual relationship must be statisticallsignificant and interpreted correctl.


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56L=2=L - M=ASFR=M=:* ' 1

1) In t!e dataset SS6444, is t!e #olloing statement true, #alse, or an incorrect application o#a statisticE Assume t!at t!ere is no pro"lem it! missing data, outliers, or in#luential cases,and t!at t!e (alidation analysis ill con#irm t!e generali%a"ility o# t!e results) Fse a le(el o#signi#icance o# 4)4> #or e(aluating t!e statistical relations!ips)

*!e (aria"les G!ig!est year o# sc!ool completedG Heduc, Gse.G Hse. and Gtotal #amily incomeGHincomeK8 ere use#ul predictors #or distinguis!ing "eteen groups based on responses to"opinion about spending on space e0ploration" [natspac]. /hese predictors differentiatesurvey respondents who thought we spend too little money on space e0ploration from

survey respondents who thought we spend too much money on space e0ploration andsurvey respondents who thought we spend about the right amount of money on spacee0ploration from survey respondents who thought we spend too much money on spacee0ploration.

Among t!is set o# predictors, total #amily income as !elp#ul in distinguis!ing among t!egroups de#ined "y responses to opinion a"out spending on space e.ploration) Sur(eyrespondents !o !ad !ig!er total #amily incomes ere more liely to "e in t!e group o# sur(eyrespondents !o t!oug!t e spend a"out t!e rig!t amount o# money on space e.ploration,

rat!er t!an t!e group o# sur(ey respondents !o t!oug!t e spend too muc! money on spacee.ploration) -or eac! unit increase in total #amily income, t!e odds o# "eing in t!e group o#sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on spacee.ploration increased "y N)4?)

1) *rue

6) *rue it! caution

3) -alse


Multinomial logistic regression requires that thedependent variable be non-metric and theindependent variables be metric or dichotomous.

";pinion about spending on space e+ploration"Lnatspac is ordinal! satisfing the non-metriclevel of measurement requirement for thedependent variable.

It contains three categories9 surve respondentswho thought we spend too little mone! aboutthe right amount of mone! and too much moneon space e+ploration.


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57L=2=L - M=ASFR=M=:* ' 6

1) In t!e dataset SS6444, is t!e #olloing statement true, #alse, or an incorrect application o# astatisticE Assume t!at t!ere is no pro"lem it! missing data, outliers, or in#luential cases, andt!at t!e (alidation analysis ill con#irm t!e generali%a"ility o# t!e results) Fse a le(el o#signi#icance o# 4)4> #or e(aluating t!e statistical relations!ips)

*!e (aria"les "highest year of school completed" [educ], "se0" [se0] and "total family income"

[income%#]ere use#ul predictors #or distinguis!ing "eteen groups "ased on responses toGopinion a"out spending on space e.plorationG Hnatspac) *!ese predictors di##erentiate sur(eyrespondents !o t!oug!t e spend too little money on space e.ploration #rom sur(eyrespondents !o t!oug!t e spend too muc! money on space e.ploration and sur(eyrespondents !o t!oug!t e spend a"out t!e rig!t amount o# money on space e.ploration #romsur(ey respondents !o t!oug!t e spend too muc! money on space e.ploration)

Among t!is set o# predictors, total #amily income as !elp#ul in distinguis!ing among t!e groupsde#ined "y responses to opinion a"out spending on space e.ploration) Sur(ey respondents !o!ad !ig!er total #amily incomes ere more liely to "e in t!e group o# sur(ey respondents !ot!oug!t e spend a"out t!e rig!t amount o# money on space e.ploration, rat!er t!an t!e groupo# sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on spacee.ploration) -or eac! unit increase in total #amily income, t!e odds o# "eing in t!e group o#sur(ey respondents !o t!oug!t e spend a"out t!e rig!t amount o# money on spacee.ploration increased "y N)4?)

1) *rue

6) *rue it! caution

"Se+" Lse+ is dichotomous!satisfing the metric ordichotomous level of measurementrequirement for independentvariables.

">ighest ear of schoolcompleted" Leduc is interval!satisfing the metric ordichotomous level ofmeasurement requirement forindependent variables.

"Total famil income" Lincome$ is ordinal!satisfing the metric or dichotomous level ofmeasurement requirement for independentvariables. If we follow the convention of treatingordinal level variables as metric variables! the level

of measurement requirement for the analsis issatisfied. Since some data analsts do not agreewith this convention! a note of caution should beincluded in our interpretation.


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58Re+uest multinomial logistic regression

Select theRegression |Multinomial Logisticcommand from the

Analyze menu.


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59Selecting t!e dependent (aria"le

Second! clic5 on the rightarrow button to move thedependent variable to theDependentte+t bo+.

First! highlight thedependent variablenatspacin the listof variables.


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Slide

6!Selecting non'metric independent (aria"les

Move the non-metricindependent variableslisted in the problem tothe actor(s)list bo+.

Select thedichotomousvariable se.

Don-metric independent variables are specified asfactors in multinomial logistic regression. Don-metricvariables can be either dichotomous! nominal! or ordinal.

These variables will be dumm coded as needed andeach value will be listed separatel in the output.


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6Selecting metric independent (aria"les

Move the metricindependent variables!educand income*+! tothe Covariate(s)list bo+.

Metric independent variables are specified as covariatesin multinomial logistic regression. Metric variables canbe either interval or! b convention! ordinal.


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62Speci#ying statistics to include in t!e output

Nhile we will accept most ofthe SPSS defaults for theanalsis! we need to specificallrequest the classification table.

lic5 on the !tatisticsO buttonto ma5e a request.


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63Re+uesting t!e classi#ication ta"le

First! 5eep the SPSSdefaults for !ummarystatistics! Li"eli#ood

ratio test! and$arameter estimates.

Second! mar5 thechec5bo+ for the

Classi%ication ta&le.

Third! clic5on theContinuebutton to

complete therequest.

C l ti g t! lti i l


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64

Completing t!e multinomiallogistic regression re+uest

lic5 on the '

button to requestthe output for themultinomial logisticregression.

The multinomial logistic procedure supportsadditional commands to specif the modelcomputed for the relationships we will use thedefault main effects model(! additionalspecifications for computing the regression!and saving classification results. Ne will notma5e use of these options.

Slid


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65


&& 15.9'

90 4&.&'

85 40.9'

94 45.2'

114 54.8'

208 100.0'

62

270

1&8a

1

2

&

"+! ?LO+IO

O)+M

1

2

"O" "?

alid

Mi33ingotal

"$pop$lation

Marginal

ercentage

e dependent ariale a3 only one al$e o3ered in 112

81.2' 3$pop$lation3.

a.

Sample si%e ratio o# cases to (aria"les

Multinomial logistic regression requires that the minimum ratioof valid cases to independent variables be at least #) to #. Theratio of valid cases 1)$( to number of independentvariables ( was 3. to #! which was equal to or greater than

the minimum ratio. The requirement for a minimum ratio ofcases to independent variables was satisfied.

The preferred ratio of valid cases to independent variables is1) to #. The ratio of 3. to # was equal to or greater than thepreferred ratio. The preferred ratio of cases to independentvariables was satisfied.

Slid 2=RALL R=LA*I:S5I$ =*W==:


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66


&54.268

&&4.967 19.&01 6 .004

Model

Intercept Only

Final

-2 Log


2=RALL R=LA*I:S5I$ =*W==:I:D=$=:D=:* A:D D=$=:D=:* 2ARIAL=S


In this analsis! the probabilit of the model chi-square#.)#( was ).))%! less than or equal to the level ofsignificance of ).)&. The null hpothesis that there wasno difference between the model without independentvariables and the model with independent variables wasre*ected. The e+istence of a relationship between theindependent variables and the dependent variable wassupported.

Slid


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67

Parameter Estimates

-4.1&6 1.157 12.779 1 .000

.101 .089 1.276 1 .259 1.106

.097 .050 &.701 1 .054 1.102

.672 .426 2.488 1 .115 1.959

0 . . 0 . .

-2.487 .840 8.774 1 .00&

.108 .068 2.521 1 .112 1.114

.058 .0&4 2.9&2 1 .087 1.060

.501 .&17 2.492 1 .114 1.650

0 . . 0 . .

Intercept

:!

I!OM98

@"?A1B

@"?A2B

Intercept

:!

I!OM98

@"?A1B

@"?A2B

"+! ?LO+IO

O)+Ma

1

2



i3 para>eter i3 3et to Cero eca$3e it i3 red$ndant..

:FM=RICAL $RL=MS

Multicollinearit in the multinomiallogistic regression solution isdetected b e+amining thestandard errors for the bcoefficients. ? standard errorlarger than 1.) indicates numerical

problems! such as multicollinearitamong the independent variables!Cero cells for a dumm-codedindependent variable because all ofthe sub*ects have the same valuefor the variable! and ,completeseparation, whereb the twogroups in the dependent eventvariable can be perfectl separatedb scores on one of the

independent variables. ?nalsesthat indicate numerical problemsshould not be interpreted.

Done of the independent variablesin this analsis had a standarderror larger than 1.).

Slid R=LA*I:S5I$ - I:DI2IDFAL I:D=$=:D=:*


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68



&&4.967a .000 0 .

&&7.788 2.821 2 .244

&40.154 5.187 2 .075

&&8.511 &.544 2 .170

%%ect

Intercept

:!

I!OM98

"?

-2 Log

Likeliood o%

ed$ced


e ci-3#$are 3tati3tic i3 te di%%erence in -2 log-likeliood3






a.The statistical significance of the relationship betweentotal famil income and opinion about spending on spacee+ploration is based on the statistical significance of thechi-square statistic in the SPSS table titled "


69/73

Slide

69Ansering t!e +uestion in pro"lem 6


*!e (aria"les G!ig!est year o# sc!ool completedG Heduc, Gse.G Hse. and Gtotal #amily incomeGHincomeK8 ere use#ul predictors #or distinguis!ing "eteen groups "ased on responses toGopinion a"out spending on space e.plorationG Hnatspac) *!ese predictors di##erentiate sur(eyrespondents !o t!oug!t e spend too little money on space e.ploration #rom sur(ey

respondents !o t!oug!t e spend too muc! money on space e.ploration and sur(eyrespondents !o t!oug!t e spend a"out t!e rig!t amount o# money on space e.ploration #romsur(ey respondents !o t!oug!t e spend too muc! money on space e.ploration)

Among t!is set o# predictors, total #amily income as !elp#ul in distinguis!ing among t!e groupsde#ined "y responses to opinion a"out spending on space e.ploration) Sur(ey respondents !o!ad !ig!er total #amily incomes ere more liely to "e in t!e group o# sur(ey respondents !ot!oug!t e spend a"out t!e rig!t amount o# money on space e.ploration, rat!er t!an t!e groupo# sur(ey respondents !o t!oug!t e spend too muc! money on space e.ploration) -or eac!

unit increase in total #amily income, t!e odds o# "eing in t!e group o# sur(ey respondents !ot!oug!t e spend a"out t!e rig!t amount o# money on space e.ploration increased "y N)4?)

1) *rue

6) *rue it! caution

3) -alse


Ne found a statisticall significant overallrelationship between the combination ofindependent variables and the dependentvariable.

There was no evidence of numerical problems inthe solution.

>owever! the individual relationship betweentotal famil income a

MultinomialLogisticRegression_BasicRelationships

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