应用统计学课件含练习题2013第八章因子分析

2021/7/171第八章因子分析因子分析的目的与类型探索性因子分析的模型因子模型的求解因子旋转因子得分因子分析的SPSS实现实例分析确认性因子分析Origins

of

Factor

AnalysisCharles

Spearman1863-1945In

conjunction

with

his

famous

two-factor

theory

of

intelligence一、因子分析的目的与类型因子分析的目的：用少数几个不可观测的隐变量来解释原始变量间的协方差关系Origins

of

Factor

AnalysisWanted

to

estimate

intelligence

of

24

children

in

avillage

school.Realized

way

of

measuring

intelligence

was

imperfectand

that

the

correlation

between

any

two

variables

(say,one’s

score

on

a

mathematics

exam

and

on

a

classicsexam)

would

be

underestimated.Noticed

that

the

observed

correlations

between

thevariables

he

was

interested

in

were

all

positive

andfollowed

a

pattern.Spearman

wanted

to

develop

a

model

that

wouldreflect

the

pattern

he

saw.What

did

Spearman

notice?ClassicsFrenchEnglishMathDiscrMusicClassics1.83.78.70.66.63French.831.67.67.65.57English.78.671.64.54.51Math.70.67.641.45.51Discr.66.65.54.451.40Music.63.57.51.51.401Correlations

Between

Examination

ScoresNotice

the

trend

across

each

row

on

the

upper

diagonal认为存在着“generalintelligence”，影响着个体在所有智力活动中的表现(解释各变量间的高度相关），而个体在不同智力活动中表现的差异则是由另一些“specificfactors”决定的（解释相关程度差异）。区分这两类因素可以更准确地预测出某个人在某项工作中的表现。two-factor

theory

of

intelligenceCould

model

each

test

score

ashaving

two

types

of

components:

onecommon

to

all

the

scores

and

onespecific

to

the

particular

testsame

degree

for

all

intellectual

actsh:

varies

in

strength

from

one

act

toanotherIf

one

knows

how

a

person

performs

onone

task

that

is

highly

saturated

with

“f",one

can

safely

predict

a

similar

level

ofperformance

for

a

another

highly

“f"saturated

task.the

most

important

information

to

haveabout

a

person's

intellectual

ability

is

anestimate

of

their

“f"classicsf:

available

to

the

same

individual

to

the

frenchenglishmathdiscrmusic=

a11f

+

h1=

a21f

+

h2=

a31f

+

h3=

a41f

+

h4=

a51f

+

h5=

a61f

+

h6Schematicallyclassicsfrenchenglishmathdiscrmusich1f因子：不可观测f可观测h2

h3

h4

h5

h6特殊因子：不可观测，难以估计构成：测量误差+个性因素Goals

of

Factor

Analysismodel

correlation

patterns

in

useful

way通过对多个变量的相关系数矩阵的研究，找出同时影响或支配多个变量的共性因素。allow

for

contextual

interpretation

of

thenew

variablesevaluate

the

original

data

in

light

of

thenew

variables注意：因子分析是一种用来分析隐藏在表象背后的潜在因子作用的统计模型，这些共同因素通常是不可直接观测的基本思想：认为存在一些潜在共性因素影响着事物在多方面的表现实例1考查人体的五项生理指标：收缩压、舒张压、心跳间隔、呼吸间隔和舌下温度。从生理学知识可知，这五项指标是受植物神经支配的，植物神经又分为交感神经和负交感神经，因此这五项指标至少受到两个公共因子的影响，也可用因子模型去处理。五项指标均可观测，而两个公共因子是不可直接观测的：通过指标与公共因子的关系诊病。舒张压心跳间隔呼吸间隔舌下温度收缩压交感神经负交感神经实例2林登根据他收集的来自139名运动员的比赛数据，对第二次世界大战以来奥林匹克十项全能比赛的得分作了因子分析研究。这十个全能项目为：100米跑x1、跳远x2、铅球x3、跳高x4、400米跑x5、110米跨栏x6、铁饼x7、撑杆跳x8、标枪x9、1500米跑x10对10个变量标准化后的因子分析表明，十项得分基本上可归结于他们的短跑速度、爆发性臂力、爆发性腿力和耐力这四个方面，每一方面都称为一个公共因子。因子分析的类型探索性因子分析exploratory

Factor

Analysis根据变量间相关关系探索因子结构实例2确认性因子分析Confirmatory

Factor

Analysis检验对因子结构的先验认识是否合理，评估因子模型的拟合程度实例1二、探索性因子分析模型正交因子模型重要假设因子载荷阵的统计意义1.

正交因子模型mij

=1x

-

mi

=

a

ij

f

j

+hiobserved

variablescommon

factorsspecific

factorsfactorloadings因子载荷设：可观测随机变量xi，E(xi)=μi，i=1,2,

…p,不可观测正交随机变量fj，j=1,2,…m,

E(fj)=0，s(fj)=1，22

22d

21

1x

=

a

f

+a反映了各变量与a1m

fm

+h1

m<pa

2m

fm

+h2TA

p·m则：xd

=

FA

+

ηη

=

(h1,h2

,h

p

),xpd

=

a

p1

f1

+a

p

2

f2

++a

pm

fm

+hp设：xd

=

(x1d

,

x2d

,

xpd

),

F

=

(

f1,

f2

,

fm

),因子载荷阵因子分析：求出各因子载荷量aij，并在此基础上计算各样本的因子得分，据此评价样本，预测。与线性回归模型的区别？因子载荷量一般因子模型：公共因子的关系中心化变量f

++(

j

=1,2,m)x1d

=

a11

f1

+a12

f2

++

f

j

=

b

j1

x1s

+

b

j

2

x2s

++

b

jp

xps因子模型x1x2x3…xixph1f1f2fmf因子：不可观测，可估计可观测h2

h3

hi

hP特殊因子：不可观测，难以估计构成：测量误差+个性因素十项全能例因子模型100米跑=a11短跑速度+a12爆发性臂力+a13爆发性腿力+a14耐力+h1跳远

=

a21短跑速度+

a22爆发性臂力+

a23爆发性腿力+

a24耐力+h2铅球

=

a31短跑速度+

a32爆发性臂力+

a33爆发性腿力+

a34耐力+h31500米=a10，1短跑速度+a10，2爆发性臂力+a10，3爆发性腿力+a10，4耐力+h10因子得分计算公式耐力短跑速度=

b41

x1s

+

b42

x2s

++

b4，10

x10s爆发性臂力=b21

x1s

+b22

x2s

++b2，10

x10s爆发性腿力=b31

x1s

+b32

x2s

++b3，10

x10

s=

b11

x1s

+

b12

x2s

++

b1，10

x10s2.

Important

Assumptionscov(F,

η)

=

E(FT

η)

=

0E(η)

=

0,

var(η)

=

diag(s

2

,s

2

,s

2

),1

2

pf1,

f2,

…,

fm

are

independent,

with

identicaldistributions

having

a

mean

of0

and

avariance

of

1h1、h2

、…hpare

independent,

withdistributions

having

a

mean

of

0

andvariances

si2fi

and

hj

are

independent

for

all

i,

jcombinations即：E(F)=0,var(F)=I,Under

the

assumptions

aboveActually,

the

goal

of

“factor

analysis”

is

to

try

to pose

thecovariance

matrix

(or

correlation

matrix

for

standardized

data)

intotwo

parts

each

in

the

form

dictated

above.2

221s

p

s

2svar(x)

=

AAT

+

T

Tvar(xd

)

=

var(FA

+

η)

=

A

var(F)A

+

var(η)主成分分析：var(Z)=UT

RU

=Λ,R

=

UΛUT

=

(UΛ1/2

)(UΛ1/2

)T

=

FFT当m=p时，var(x)=AAT然而只有当m<<p时，因子分析的优势才能显示出来3.

因子载荷(Factor

loadings)的统计意义These

aij

represent

the

covariance(corelation

if

x

isstandardised)

between

the

original

variable

andthe

corresponding

factor，called

factor

loading，aij表示xi对fj的相关程度mk

=1m=

cov(aik

fk

,

f

j

)

+cov(hi

,

f

j

)k

=1=

aij\

Cov(xi

,

f

j

)

=

cov(aik

fk

+hi

,

f

j

)

xid

=

ai1

f1

+ai

2

f2

++aim

fm

+h1部公共因子对xi的方差贡为变量共同度：因子载全献称荷阵第i行元素平方和无法显示该图片。mmik

i

i

iifor

i

„

jk

=1k

=12非对角元素：cov(xi

,x

j

)=aik

a

jk对角元素：var(x

)=a

+s

2

=

h2

+s

2特殊因子的方差var(x

)

=

AAT

+

diag(s

2

,s

2

,s

2

)d

1

2

p变量共同度（communalities）If

Data

are

Standardizedif i

„

jif i

=

jmik

jkmik

jka

ak

=1corr(xi,

xj)

=

k

=1a

a

+s

2

=

h2

+s

2

=1i

i

i三、因子模型的求解方法因子模型求解：估计公共因子个数m、载荷阵A和特殊因子方差已知p个相关变量的n次观测值x

=

(x1

,

x2

,

xp

),2221p

ss

2svar(x)

=

AAT

+

T

Tvar(xd

)

=

var(FA

+

η)

=

A

var(F)A

+

var(η)主成分法i

ii

2

2ijj=12p21'mi=1i

i

i'

p p

'2 2

1 1

p

p2

21

1'p

i

i

ii=1=

r

-

a

00

R

»ss特殊因子方差的估计为：s因子载荷阵的估计为：A

=(对选定的公共因子数m(m

<p),则R

=l1

μ1

,l

μ

μ

+l

μ)

l

μl

μ'l

μ

μ

=l2

μ

2

,

lm

μ

m

),m(

l

μ

,

l

μ

,,

l

μμ1，μ

2，μ

p为相应的单位特征向量。设：l1

>l2

>

>lp是样本相关阵R的特征根，主成分分析：var(Z)=UT

RU

=Λ,R

=

UΛUT

=

UΛ1/2

(UΛ1/2

)Tvar(x

)

=

AAT

+

diag(s

2

,s

2

,s

2

)d

1

2

p主成分解主轴因素法principal

axis

factoring*

2pp12211p121(h

)

=

AA'

r (h*

)2

r(h*

)2

r

r约相关阵R*

=(h*

)2

=1-

(s

*

)2

,i

i2a

itˆ

2m

m1

1(i

=

1,,

p)mt

=1令：

s

i

=

1

-l*

μ*A

=

l*

μ*

,设：R

=AA'+D,D

=diag(s

2

,s

2

)1

pR

-D

=AA'=R*

(约相关阵)i如果我们已知特殊因素方差的初始估计(sˆ*

)2，则初始共同度为*求R的前m个特征值和特征向量，得到:xi对其他所有变量线性回归的R2，然后叠代求解。i实际应用中特殊因素方差未知，可以将初始共同度h2取为三、因子模型的求解方法（续）主轴因素法principal

axis

factoring这是用于因子分析的主成分法，是一种叠代方法极大似然法maximum

likelihood：见书不加权最小二乘unweighted

least

squares)使观测的和再生的相关阵(之差的平方和最小广义最小二乘generalized

least

squaresa因素提取法alpha

factoring映像因子提取法image

factoring)ˆ

ˆˆ

ˆ1

ps

,,s'AA

+

diag(例：消费者对止痛药的感觉消费者对止痛药的调查要求消费者从6个方面给不同牌子的止痛药打分：不伤胃：nstomach没有副作用：nsideeff止痛:stoppain见效快:wksquick保持清醒:kpawake部分止痛:limrelie以主成分法和主轴因素法进行因子分析，说明的总方差提取方法：主成分分析。先用主成分法确定共因子数123456成分?0.00.52.0成分初始特征值提取平方和载入合计方差的%累积%合计方差的%累积%12.43140.51240.5122.43140.51240.51222.07034.49875.0102.07034.49875.0103.4397.31982.3294.3876.45088.7785.3806.34095.1182.56.2934.882100.0001.5特征值1.0主轴因素法主成分法公因子方差初始提取nstomach1.000.720nsideeff1.000.761stoppain1.000.760wksquick1.000.758kpawake1.000.757limrelie1.000.743提取方法：主成分分析。成分矩阵a成分12nstomach.673-.517nsideeff.594-.639stoppain.707.510wksquick.597.634kpawake-.548.676limrelie-.685-.524提取方法:主成分分析法。a.已提取了2个成分。公因子方差初始提取nstomach.453.569nsideeff.509.652stoppain.517.652wksquick.499.634kpawake.487.637limrelie.479.609提取方法：主轴因子分解。因子矩阵a因子12nstomach.596-.462nsideeff.546-.596stoppain.665.457wksquick.559.568kpawake-.499.623limrelie-.631-.459提取方法:主轴因子分解。a.已提取了2个因子。需要7次再生相关性nstomach

nsideeff

stoppain

wksquick

kpawake

limrelie再生的相关性

nstomach

.720b

.730

.212

.074 -.719 -.190nsideeff

.730

.761b

.094 -.051 -.757 -.072stoppain

.212

.094

.760b

.746 -.043 -.752wksquick

.074 -.051

.746

.758b

.101 -.741kpawake

-.719 -.757 -.043

.101

.757b

.021limrelie

-.190 -.072 -.752 -.741

.021

.743b残差a

nstomach

-.134 -.050

.015

.126

.020nsideeff

-.134

.037 -.016

.116

.007stoppain

-.050

.037 -.113

.016

.120wksquick

.015 -.016 -.113 -.045

.130kpawake

.126

.116

.016 -.045

.008limrelie

.020

.007

.120

.130

.008提取方法：主成分分析。a.将计算观察到的相关性和重新生成的相关性之间的残差。有6(40.0%)个绝对值大于0.05的非冗余残差。b.重新生成的公因子方差用残差评估因子模型方法：检验原始相关矩阵减再生相关矩阵得到的残差阵中，绝对值大于0.05的元素个数及百分比。残差绝对值大于0.05的个数太多，表明该模型不理想再生相关性nstomachnsideeffstoppainwksquickkpawakelimrelie再生的相关性nstomach.569b.600.185.071-.585-.164nsideeff.600.652b.091-.033-.643-.071stoppain.185.091.652b.631-.047-.630wksquick.071-.033.631.634b.075-.613kpawake-.585-.643-.047.075.637b.029limrelie-.164-.071-.630-.613.029.609b残差anstomach-.004-.023.018-.007-.006nsideeff-.004.040-.034.002.006stoppain-.023.040.001.020-.002wksquick.018-.034.001-.018.003kpawake-.007.002.020-.018.000limrelie-.006.006-.002.003.000提取方法：主轴因子分解。将计算观察到的相关性和重新生成的相关性之间的残差。有0

(.0%)个绝对值大于0.05的非冗余残差。重新生成的公因子方差四、因子旋转——因子的解释观察止痛药因子模型：两个因子与各变量的相关程度都差不多，这使得我们难以解释潜在因子的含义。能否通过某种变换使因子的含义更为清晰？因子矩阵a因子12nstomach.596-.462nsideeff.546-.596stoppain.665.457wksquick.559.568kpawake-.499.623limrelie-.631-.459提取方法:主轴因子分解。a.

已提取了2个因子。需要7次迭四、因子的解释——因子旋转\因子旋转不改变变量共同度和特殊因子方差这说明因子分析的解是不唯一的。这一性质给我们提供了寻找“理想”共因子结构的思路：通过因子旋转使每个变量的载荷都尽可能集中在某个因子上，而在其他因子上的载荷尽可能小，以使公因子易于解释。22221*

*p2

2

2

Td

1

2

pT+

diag(s

,s

,s

)

=

s

,s

,s

)AA

+

diag(var(x

)

=

A

A设：T为任一正交阵，如果A为载荷阵，则:'

*

*xd

=

FA'+η

=

FTTA'+η

=

F

A

'+η其中F*

=FT,A*

=ATA*=AT仍是一个因子载荷阵，因子F*与F有相同的统计特性：E(F*

)

=

E(F)T

=

0,

var(F*

)

=

var(FT)

=

T

var(F)T'

=

IAf1f2RotationsFactor

LoadingsB“Importance”The

orthogonal

rotation

does

notchange

the

overall

covariancematrix,

the

specific

variances

northe

communalities.RotationsAf1Factor

Loadingsf2B“Importance”RotationsAf1Factor

Loadingsf2B“Importance”因子旋转方法正交旋转:保持因素间互不相关方差最大旋转Varimax：使每个因子上具有高载荷的变量数最少——简化对因子的解释四分旋转quartmax：使每个变量中需要解释的因子数最少——简化变量的解释平均正交旋转equamax：前两种方法的结合斜交旋转：允许因素间相关直接斜交旋转direct

oblilminPromax：比直接斜交旋转快旋转因子矩阵a因子12nstomach.141.741nsideeff.014.808stoppain.801.098wksquick.794-.055kpawake.039-.797limrelie-.777-.074提取方法:主轴因子分解。旋转法:具有Kaiser标准化的正交旋转aFactor

Plot1.0.50.0-.5-1.0Factor

21.0.50.0-.5-1.0limreliekpawakewksquickstoppainnstomachnsideeff止痛药因子模型旋转结果Factor

11.0.50.0-.5-1.0Factor

2Factor

Plot

in

Rotated

FactorSpace1.0.50.0-.5-1.0limreliekpawak

estoppainwksquicknsideeffnstomach因子矩阵a因子12nstomach.596-.462nsideeff.546-.596stoppain.665.457wksquick.559.568kpawake-.499.623limrelie-.631-.459提取方法:主轴因子分解。旋转因子矩阵a因子12nstomach.141.741nsideeff.014.808stoppain.801.098wksquick.794-.055kpawake.039-.797limrelie-.777-.074提取方法:主轴因子分解。旋转法:具有Kaiser标准化的正交旋转aFactor1:有效性Factor2:和缓性止痛药潜在因素分析五、因子得分因子模型建立之后，样本评即某个样本在这些公共因子方公共因子得分的计算模式如价需要计算因子得分面的表现下：无法显示该图片。f

j

=

b

j1

x1s

+

bj

2

x2

s

++

b

jp

xps(

j

=1,2,m)classics=

a11f+h1french=

a21f+h2english=

a31f+h3math=

a41f

+h4discr=

a51f+h5music=

a61f

+h6在智力的双因子模型中，每个样本在公共因子f上的得分表示了该样本的一般智力水平f

=

b1classics

+

b2frech

+

b3english

+

b4math

+

b5discr

+

b6musicBartlett

法(加权最小二乘法)x1

-

m1

=

a11

f1

+a12

f2

++a1m

fm

+h1x2

-

m1

=

a

21

f1

+a

22

f2

++a

2m

fm

+h2xp

-

m1

=

a

p1

f1

+a

p

2

f2

++a

pm

fm

+hp其中var(h

)=s

2i

i因子模型：22ˆipim

mi

i

i1

1

i

2[(x

-

m

)-

(a

fˆ

+

afˆ

++a

f

)]/s

达到最小我们可以采用与求解线性回归模型相似的方法来得到f1,f2,…fm的近似解。由于p个特殊方差可以不全相等，因此应采用加权的最小二乘估计法，即寻求一组估计值，使得加权的“残差”平方和2(

j

=1,2,m)fˆj

=

b

j1

x1s

+

b

j

2

x2

s

++

b

jp

xpsi=1这样求得的解就是因子得分因子得分系数的求解方法：止痛药例KE

-

0.009LIMRELIESIDEEFF

+

0.014SSTOPPAIN+

0.355WKQUICK

+

0.043KPAWAfac

_

2(和缓性)=0.286NSTMACH

+0.392N-

0.048WKQUICK

-

0.373KPAWAfac

_1(有效性)=0.34NSTMACH

-0.24NSIDEEFF

+0.372SSTOPPAINKE

-

0.322LIMRELIE-2.00000 -1.00000

0.00000

1.00000

2.00000REGR

factor

score 1

for

analysis

1-2.000000.000002.00000REGR

factor

score2

for

analysis1

3912717112016823245

252627

333437

3824204143485051

5245235760616364656386691

7209

114751

59752676674757

754975484667962808153598283824

431098586

31878288

8991089195289332

4773949597

98

3754

996696123111000

因子得分系数矩阵因子12nstomach.034.286nsideeff-.024.392stoppain.372.014wksquick.355-.048kpawake.043-.373limrelie-.322-.009提取方法:主轴因子分解。旋转法:具有Kaiser标准化的正交旋因子得分方法:回归。SPSS的因子分析过程分析→数据降维→因子分析显示因子分析主对话框六、因子分析的SPSS实现六、因子分析的SPSS实现“描述统计”对话框单变量统计量：均值、标准差初始结果：变量共同度初始估计生成变量相关阵相关阵行列式因子分析的适宜度检验相关阵的逆矩阵反映象相关阵再生相关阵：因子分析后的R及残差“抽取”对话框因子提取方法显示未旋转解特征根图抽取因子个数“旋转”对话框因子旋转方法显示旋转解因子载荷散点图：给出以两两因子为坐标的各变量载荷“因子得分”对话框因子得分作为变量保存回归法：因子得分均值为0“选项”对话框缺失值处理方法有缺失值的样本被剔除载荷系数的显示格式按数值大小排列不显示绝对值小于指定值的载荷系数因子分析的基本步骤因子分析的核心问题有两个：一是如何构造因子变量；二是如何对因子变量进行命名解释。因此，因子分析的基本步骤和解决思路就是围绕这两个核心问题展开的。因子分析常常有以下四个基本步骤：确认待分析的原有变量是否适合作因子分析。构造因子变量（确定公共因子个数）。利用旋转方法使因子变量更具有可解释性。计算因子变量得分。多种算法比较以判断因子模型的稳定性1.巴特利特球度检验（Bartlett

test

of

sphericity）巴特利特球度检验是以变量的相关系数矩阵为出发点。它的零假设是Ho：相关系数矩阵是一个单位阵，即相关系数矩阵对角线上的所有元素都为1，所有非对角线上的元素都为零。巴特利特球度检验的统计量根据相关系数矩阵的行列式计算得到。如果该统计量值比较大，且其对应的相伴概率值小于用户心中的显著性水平，则应拒绝Ho，认为相关系数矩阵不太可能是单位阵，适合作因子分析；相反，如果该统计量值比较小，且其对应的相伴概率值大于用户心中的显著性水平；则不能拒绝Ho，可以认为相关系数矩阵可能是单位阵，不适合作因子分析。因子分析的适宜度检验：变量间相关程度较高才适宜做化简2.

KMO(Kaiser-Meyer-Olkin)KMO统计量是用于比较变量间简单相关系数和偏相关系数的一个指标，计算公式如下：式中：rij是变量和变量之间的简单相关系数，pij是它们之间的偏相关系数。可见，KMO统计量的取值在0和1之间，当所有变量之间的简单相关系数平方和远远大于偏相关系数平方和时，KMO值接近1。KMO值越接近1，则越适合作因子分析，KMO越小，则越不适合作因子分析。Kaiser给出了一个KMO的度量标准：0.9以上非常适合；0.8适合；0.7一般；0.6不太适合；0.5以下不适合。

i„

j

i„

jij

iji„

jijpr

+rKMO

=222止痛药例：因子分析适宜度检验KMO和Bartlett的检验取样足够度的Kaiser-Meyer-Olkin度量。.710Bartlett

的球

近似卡方226.100形度检验

df15Sig..000七、应用（1）研究消费者对速溶麦片的看法：12种品牌速溶麦片的调查(Creating

a

perceptual

map

ofready-to-eat

cereal

brands

in

theAustralian

market,

Roberts

and

Lattin,1991)影响麦片销售的特性有25个每个被调查者从25个方面给三个自己最喜欢的品牌的麦片打分，打分采用5分制共116人作答，得到235个样本研究的目的：有哪些特性影响消费者的购买决策研究过程用主成分法确定因素的个数用不同方法求解因子模型，以确定较好模型因子解释用因子得分对各个品牌做出综合评价。用SPSS中的“aggregate”功能和散点图说明的总方差成分初始特征值提取平方和载入合计方差的%累积%合计方差的%累积%16.50426.01826.0186.50426.01826.01823.82115.28441.3023.82115.28441.30232.50210.00851.3102.50210.00851.31041.6846.73658.0461.6846.73658.04651.0854.34162.3871.0854.34162.3876.9333.73266.1197.8523.40769.5268.7873.14772.6739.7322.92775.60010.6962.78378.38411.6472.58780.97112.5482.19283.16313.5292.11785.27914.4901.95887.23815.4181.67188.90916.3871.54890.45717.3621.45091.90718.3591.43593.34219.3051.21994.56120.2741.09795.65821.2621.05096.70822.242.96997.67723.218.87298.54924.199.79499.34325.164.657100.000提取方法：主成分分析。1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25成分?012567特4征值3保留四个因子还是五个因子？旋转因子矩阵a因子12345natritio.849health.840fibre.816natural.731quality.681regular.657filling.646.488energy.610.341satisfyi.569.387.334sugar.852sweet.696.351salt.689calories.592process.387kids.868family.794economic.409easy.307treat.300.650plain-.638fun.378.538boring-.508crisp.336.458soggy-.454fruit.341.439提取方法:最大似然。旋转法:具有Kaiser标准化的正交旋转法。a.

旋转在6次迭代后收敛。旋转成分矩阵a成分12345fibre.855natritio.849health.841natural.760regular.715filling.715quality.685energy.684satisfyi.627.481sugar.825salt.774calories.705sweet.702.429process.527-.350kids.860family.824fun.535.400easy.415plain-.701fruit.351.656boring-.338-.537economic.440-.461treat.423.434.381soggy-.799crisp.436.632提取方法:主成分分析法。旋转法:具有Kaiser标准化的正交旋转法。a.旋转在8次迭代后收敛。-2.000000.00000 1.00000

2.00000 3.00000

4.00000GRfactor

score

2

for

analysis

1-2.00000-1.000000.000001.000002.000003.00000REGR

factor

s

core

2

for

analysis

2-4.00000

-3.0-1.000000000

-2.00000

-1.00000

0.00000

1.00000

2.00000

3.00000

REREGR

factor

score

1

for

analysis

1-3.00000-2.00000-1.000000.000001.000002.00000REGR

factor

s

core

1

for

analysis

2-3.00000-2.00000-1.000000.000001.00000

2.00000

3.00000REGR

factor

score

3

foranalysis

1-4.00000-2.000000.000002.000004.00000REGRfactor

s

core

3

for

analysis

2-4.000004.00000-2.00000

0.00000

2.00000REGRfactor

score

4

for

analysis

1-3.00000-2.00000-1.000000.000002.00000REGR

factor

s

core4

for

analysis

2-4.00000

-3.00000

-2.00000

-1.00000

0.00000

1.00000

2.00000

3.00000REGR

factor

score

5

for

analysis

1-4.00000-2.000000.000002.000004.00000REGR

factor

s

core

5

for

analysis

2可能意味着因子数过多-4两.00000

种不同方法得到的同一因子的分散点图越接近45°线1.0000越0

好旋转成分矩阵a成分1

2

3

4natritio

.846fibre

.840health

.835natural

.786filling

.744energy

.703quality

.680regular

.667satisfyi

.651

.441sugar

.822salt

.761sweet

.741

.340calories

.715process

.479kids

.858family

.806economic

-.333

.524easy

.414plain

-.730soggy

-.647treat

.345

.612boring

-.611crisp

.427

.514fun

.456

.504fruit

.416

-.337

.462提取方法:主成分分析法。旋转法:具有Kaiser标准化的正交旋转法。a.

旋转在6次迭代后收敛。旋转因子矩阵a.834.831.815.748.691.663.651.635.613.386.355.822.713.693.613.373.875.798.391.304.300.385.366.331-.650.622.528-.504-.457.444.432natritiohealthfibrenaturalfillingqualityenergyregularsatisfyisugarsweetsaltcaloriesprocesskidsfamilyeconomiceasyplaintreatfunboringsoggyfruitcrisp1234因子提取方法:最大似然。旋转法:具有Kaiser标准化的正交旋转法。a.

旋转在6次迭代后收敛。旋转因子矩阵anatritio.831health.829fibre.821natural.753filling.706energy.660quality.647satisfyi.626.424regular.613sugar.817sweet.702.347salt.686calories.627process.374kids.850family.761economic.415easy.325plain-.657treat.336.602boring-.505soggy-.481fun.417.478fruit.376.443crisp.373.4361234因子提取方法:主轴因子分解。旋转法:具有Kaiser标准化的正交旋转法。a.

旋转在6次迭代后收敛。保留四个因子2.00000-4.00000

-3.00000

-2.00000

-1.00000

0.00000

1.00000REGR

factor

score

1for

analysis

1-4.00-3.00000-2.00000-1.000000.000001.000002.00000REGR

factor

s

core

1

for

analysis

2-2.00000

-1.00000

0.00000 1.00000

2.00000

3.REGR

factor

score

2

foranalysis00000

4.000001-2.00000000-1.000000.000001.000002.000003.000004.00000REGR

factor

s

core

2

for

analysis

2000

3.00000-3.00000

-2.00000

-1.00000

0.00000

1.00000

2.00REGR

factor

score

3

for

analysis

1-4.00000-2.000000.000002.000004.00000REGR

factor

s

core

3

for

analysis

2-4.000004.00000-2.00000

0.00000

2.00000REGR

factor

score

4

for

analysis

1-2.00000-1.000000.000001.000002.00000REGR

factor

s

core

4

for

analysis

2主成分法和极大似然法-4.00000

2.000002-4.00000-3.00000-2.00000-1.000000.000001.000002.00000REGR

factor

s

core

1

for

analysis

300

4.00000-1.000000.000001.000002.000003.000004.00000REGR

factor

s

core

2for

analysis

34.00000-2.00000

-4.000000

-3.00000

-2.00000

-1.00000

0.00000

1.

-2.00000

-1.00000

0.00000

1.00000

2.00000

3.0000000-2.000000.000002.000004.00000REGR

factor

s

core

3

for

analysis

3-4.00000

-2.00000

0.00000

2.00000

-2.00000

-1.00000

0.00000

1.000002.00000REGR

factor

score

1

for

analysis

REGRfactor

score

2

for

analysis

2

REGR

factor

score

3

for

analysis

2

REGRfactor

score

4

for

analysis

2-2.00000-1.000000.000001.000002.00000REGR

factor

s

core

4for

analysis

3主轴因素法和极大似然法公因子方差初始提取filling.633.569natural.636.615fibre.703.703sweet.612.623easy.230.171salt.448.486satisfyi.615.606energy.571.522fun.483.461kids.636.725soggy.314.241economic.372.308health.750.774family.599.594calories.417.420plain.408.461crisp.410.356regular.521.395sugar.664.731fruit.488.444process.297.212quality.614.549treat.561.590boring.334.337natritio.711.728提取方法：主轴因子分解。注意：“easy”的变量共同度非常低，说明

easy83%的变异都是特殊因子解释的，4个因子对这一特性的解释能力非常低，而且与每个因子的相关程度很低（最大0.3)，为什么？因子个数太少？easy这一特性指标选得不好？所有品牌都是速溶麦片，消费者在这一特性上难以作出明确判断公因子方差初始提取filling.633.590natural.636.638fibre.703.728sweet.612.646easy.230.170salt.448.4955个公因子选用主轴因素法旋转因子矩阵a.424.831.829.821.753.706.660.647.626.613.347.817.702.686.627.374.850.761.415.325.336.417.376.373-.657.602-.505-.481.478.443.436natritiohealthfibrenaturalfillingenergyqualitysatisfyiregularsugarsweetsaltcaloriesprocesskidsfamilyeconomiceasyplaintreatboringsoggyfunfruitcrisp1234因子提取方法:主轴因子分解。healthfulartificialNon-adultPopularityinteresting对各品牌的评价调用”数据”→“分类汇总”过程，按照cerealid分类统计各品牌麦片的因素得分-1.00-0.500.000.50nonadu_1-1.00-0.500.000.501_eretni1315161713114921232425-1.000.50-0.50

0.00health_1-0.400.000.400.801_ifitra3131415161719212324125应用（2）303名MBA学生对10个品牌汽车的评价BMW328i,

Ford

Explorer,

Infiniti

J130,

Cherokee,Lexus

ES300,

Chrysler

Town&Country,

MercedesC280,

Saab9000,

Porsche

Boxster,

Volvo

V90每个学生对每种车型就16个方面打分Exciting,

dependable,

luxurious,

ourdoorsy,

powerful,

stylish,comfortable,

rugged,

fun

to

drive,

safe,

high-performance

car,family

car,

versatile,

sporty,

high-status

car,

practical从每个学生对10个品牌汽车的评价中随机抽取一份组成样本，共303个样本。作因子分析：可提取多少个共因子？如何解释这些因子？保存因子得分，计算每个品牌汽车的平均因子得分并作图和解释KMO和Bartlett的检验取样足够度的Kaiser-Meyer-Olkin度量。.880Bartlett

的球

近似卡方3364.722形度检验

df120Sig..000说明的总方差因子初始特征值提取平方和载入合计方差的%累积%合计方差的%累积%15.92737.04237.0425.64435.27535.27523.18719.92056.9612.79117.44652.72132.54215.88772.8492.22213.88566.6064.6484.05276.9015.5893.68480.5856.4772.98383.5687.4062.53986.1078.3692.30788.4149.3372.10790.52110.2961.85292.37311.2771.72994.10212.2421.51095.61313.2051.28296.89514.1851.15498.04915.1691.05499.10316.144.897100.000提取方法：主轴因子分解。-.399.560.384.312.862.859.594stylish.876exciting.864fun.864status.831safe.769comfort.719dependab.641practicaruggedoutdoorsversatil123旋转因子矩阵a因子提取方法:主轴因子分解。旋转法:具有Kaiser标准化的正交旋转法。aperforma.772现代powerful.715sports.695.355luxuriou.644.425family-.637.533实用越野1234

5678910保留四个因子？再生相关阵残差明显下降旋转因子矩阵a因子1234stylish.874exciting.864fun.863status.804performa.752sports.712.348powerful.692luxuriou.588.515safe.770comfort.720dependab.640rugged.895outdoors.863practica.347.694versatil.438.597family-.568.345.576提取方法:主轴因子分解。旋转法:具有Kaiser标准化的正交旋转法。a.

旋转在7次