應(yīng)用統(tǒng)計學(xué)課件含練習(xí)題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與類型探索性因子分析的模型因子模型的求解因子旋轉(zhuǎn)因子得分因子分析的SPSS實現(xiàn)實例分析確認(rèn)性因子分析Origins

of

Factor

AnalysisCharles

Spearman1863-1945In

conjunction

with

his

famous

two-factor

theory

of

intelligence一、因子分析的目的與類型因子分析的目的：用少數(shù)幾個不可觀測的隱變量來解釋原始變量間的協(xié)方差關(guān)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認(rèn)為存在著“generalintelligence”，影響著個體在所有智力活動中的表現(xiàn)(解釋各變量間的高度相關(guān)），而個體在不同智力活動中表現(xiàn)的差異則是由另一些“specificfactors”決定的（解釋相關(guān)程度差異）。區(qū)分這兩類因素可以更準(zhǔn)確地預(yù)測出某個人在某項工作中的表現(xiàn)。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：不可觀測f可觀測h2

h3

h4

h5

h6特殊因子：不可觀測，難以估計構(gòu)成：測量誤差+個性因素Goals

of

Factor

Analysismodel

correlation

patterns

in

useful

way通過對多個變量的相關(guān)系數(shù)矩陣的研究，找出同時影響或支配多個變量的共性因素。allow

for

contextual

interpretation

of

thenew

variablesevaluate

the

original

data

in

light

of

thenew

variables注意：因子分析是一種用來分析隱藏在表象背后的潛在因子作用的統(tǒng)計模型，這些共同因素通常是不可直接觀測的基本思想：認(rèn)為存在一些潛在共性因素影響著事物在多方面的表現(xiàn)實例1考查人體的五項生理指標(biāo)：收縮壓、舒張壓、心跳間隔、呼吸間隔和舌下溫度。從生理學(xué)知識可知，這五項指標(biāo)是受植物神經(jīng)支配的，植物神經(jīng)又分為交感神經(jīng)和負(fù)交感神經(jīng)，因此這五項指標(biāo)至少受到兩個公共因子的影響，也可用因子模型去處理。五項指標(biāo)均可觀測，而兩個公共因子是不可直接觀測的：通過指標(biāo)與公共因子的關(guān)系診病。舒張壓心跳間隔呼吸間隔舌下溫度收縮壓交感神經(jīng)負(fù)交感神經(jīng)實例2林登根據(jù)他收集的來自139名運動員的比賽數(shù)據(jù)，對第二次世界大戰(zhàn)以來奧林匹克十項全能比賽的得分作了因子分析研究。這十個全能項目為：100米跑x1、跳遠(yuǎn)x2、鉛球x3、跳高x4、400米跑x5、110米跨欄x6、鐵餅x7、撐桿跳x8、標(biāo)槍x9、1500米跑x10對10個變量標(biāo)準(zhǔn)化后的因子分析表明，十項得分基本上可歸結(jié)于他們的短跑速度、爆發(fā)性臂力、爆發(fā)性腿力和耐力這四個方面，每一方面都稱為一個公共因子。因子分析的類型探索性因子分析exploratory

Factor

Analysis根據(jù)變量間相關(guān)關(guān)系探索因子結(jié)構(gòu)實例2確認(rèn)性因子分析Confirmatory

Factor

Analysis檢驗對因子結(jié)構(gòu)的先驗認(rèn)識是否合理，評估因子模型的擬合程度實例1二、探索性因子分析模型正交因子模型重要假設(shè)因子載荷陣的統(tǒng)計意義1.

正交因子模型mij

=1x

-

mi

=

a

ij

f

j

+hiobserved

variablescommon

factorsspecific

factorsfactorloadings因子載荷設(shè)：可觀測隨機變量xi，E(xi)=μi，i=1,2,

…p,不可觀測正交隨機變量fj，j=1,2,…m,

E(fj)=0，s(fj)=1，22

22d

21

1x

=

a

f

+a反映了各變量與a1m

fm

+h1

m<pa

2m

fm

+h2TA

p·m則：xd

=

FA

+

ηη

=

(h1,h2

,h

p

),xpd

=

a

p1

f1

+a

p

2

f2

++a

pm

fm

+hp設(shè)：xd

=

(x1d

,

x2d

,

xpd

),

F

=

(

f1,

f2

,

fm

),因子載荷陣因子分析：求出各因子載荷量aij，并在此基礎(chǔ)上計算各樣本的因子得分，據(jù)此評價樣本，預(yù)測。與線性回歸模型的區(qū)別？因子載荷量一般因子模型：公共因子的關(guān)系中心化變量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：不可觀測，可估計可觀測h2

h3

hi

hP特殊因子：不可觀測，難以估計構(gòu)成：測量誤差+個性因素十項全能例因子模型100米跑=a11短跑速度+a12爆發(fā)性臂力+a13爆發(fā)性腿力+a14耐力+h1跳遠(yuǎn)

=

a21短跑速度+

a22爆發(fā)性臂力+

a23爆發(fā)性腿力+

a24耐力+h2鉛球

=

a31短跑速度+

a32爆發(fā)性臂力+

a33爆發(fā)性腿力+

a34耐力+h31500米=a10，1短跑速度+a10，2爆發(fā)性臂力+a10，3爆發(fā)性腿力+a10，4耐力+h10因子得分計算公式耐力短跑速度=

b41

x1s

+

b42

x2s

++

b4，10

x10s爆發(fā)性臂力=b21

x1s

+b22

x2s

++b2，10

x10s爆發(fā)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

、…h(huán)pare

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當(dāng)m=p時，var(x)=AAT然而只有當(dāng)m<<p時，因子分析的優(yōu)勢才能顯示出來3.

因子載荷(Factor

loadings)的統(tǒng)計意義These

aij

represent

the

covariance(corelation

if

x

isstandardised)

between

the

original

variable

andthe

corresponding

factor，called

factor

loading，aij表示xi對fj的相關(guān)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部公共因子對xi的方差貢為變量共同度：因子載全獻(xiàn)稱荷陣第i行元素平方和無法顯示該圖片。mmik

i

i

iifor

i

?

jk

=1k

=12非對角元素：cov(xi

,x

j

)=aik

a

jk對角元素：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變量共同度（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：估計公共因子個數(shù)m、載荷陣A和特殊因子方差已知p個相關(guān)變量的n次觀測值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特殊因子方差的估計為：s因子載荷陣的估計為：A

=(對選定的公共因子數(shù)m(m

<p),則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為相應(yīng)的單位特征向量。設(shè)：l1

>l2

>

>lp是樣本相關(guān)陣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主成分解主軸因素法principal

axis

factoring*

2pp12211p121(h

)

=

AA'

r (h*

)2

r(h*

)2

r

r約相關(guān)陣R*

=(h*

)2

=1-

(s

*

)2

,i

i2a

it?

2m

m1

1(i

=

1,,

p)mt

=1令：

s

i

=

1

-l*

μ*A

=

l*

μ*

,設(shè)：R

=AA'+D,D

=diag(s

2

,s

2

)1

pR

-D

=AA'=R*

(約相關(guān)陣)i如果我們已知特殊因素方差的初始估計(s?*

)2，則初始共同度為*求R的前m個特征值和特征向量，得到:xi對其他所有變量線性回歸的R2，然后疊代求解。i實際應(yīng)用中特殊因素方差未知，可以將初始共同度h2取為三、因子模型的求解方法（續(xù)）主軸因素法principal

axis

factoring這是用于因子分析的主成分法，是一種疊代方法極大似然法maximum

likelihood：見書不加權(quán)最小二乘unweighted

least

squares)使觀測的和再生的相關(guān)陣(之差的平方和最小廣義最小二乘generalized

least

squaresa因素提取法alpha

factoring映像因子提取法image

factoring)?

??

?1

ps

,,s'AA

+

diag(例：消費者對止痛藥的感覺消費者對止痛藥的調(diào)查要求消費者從6個方面給不同牌子的止痛藥打分：不傷胃：nstomach沒有副作用：nsideeff止痛:stoppain見效快:wksquick保持清醒:kpawake部分止痛:limrelie以主成分法和主軸因素法進(jìn)行因子分析，說明的總方差提取方法：主成分分析。先用主成分法確定共因子數(shù)123456成分?0.00.52.0成分初始特征值提取平方和載入合計方差的%累積%合計方差的%累積%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主軸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。成分矩陣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個成分。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：主軸因子分解。因子矩陣a因子12nstomach.596-.462nsideeff.546-.596stoppain.665.457wksquick.559.568kpawake-.499.623limrelie-.631-.459提取方法:主軸因子分解。a.已提取了2個因子。需要7次再生相關(guān)性nstomach

nsideeff

stoppain

wksquick

kpawake

limrelie再生的相關(guān)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殘差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.將計算觀察到的相關(guān)性和重新生成的相關(guān)性之間的殘差。有6(40.0%)個絕對值大于0.05的非冗余殘差。b.重新生成的公因子方差用殘差評估因子模型方法：檢驗原始相關(guān)矩陣減再生相關(guān)矩陣得到的殘差陣中，絕對值大于0.05的元素個數(shù)及百分比。殘差絕對值大于0.05的個數(shù)太多，表明該模型不理想再生相關(guān)性nstomachnsideeffstoppainwksquickkpawakelimrelie再生的相關(guān)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殘差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：主軸因子分解。將計算觀察到的相關(guān)性和重新生成的相關(guān)性之間的殘差。有0

(.0%)個絕對值大于0.05的非冗余殘差。重新生成的公因子方差四、因子旋轉(zhuǎn)——因子的解釋觀察止痛藥因子模型：兩個因子與各變量的相關(guān)程度都差不多，這使得我們難以解釋潛在因子的含義。能否通過某種變換使因子的含義更為清晰？因子矩陣a因子12nstomach.596-.462nsideeff.546-.596stoppain.665.457wksquick.559.568kpawake-.499.623limrelie-.631-.459提取方法:主軸因子分解。a.

已提取了2個因子。需要7次迭四、因子的解釋——因子旋轉(zhuǎn)\因子旋轉(zhuǎn)不改變變量共同度和特殊因子方差這說明因子分析的解是不唯一的。這一性質(zhì)給我們提供了尋找“理想”共因子結(jié)構(gòu)的思路：通過因子旋轉(zhuǎn)使每個變量的載荷都盡可能集中在某個因子上，而在其他因子上的載荷盡可能小，以使公因子易于解釋。22221*

*p2

2

2

Td

1

2

pT+

diag(s

,s

,s

)

=

s

,s

,s

)AA

+

diag(var(x

)

=

A

A設(shè)：T為任一正交陣，如果A為載荷陣，則:'

*

*xd

=

FA'+η

=

FTTA'+η

=

F

A

'+η其中F*

=FT,A*

=ATA*=AT仍是一個因子載荷陣，因子F*與F有相同的統(tǒng)計特性：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”因子旋轉(zhuǎn)方法正交旋轉(zhuǎn):保持因素間互不相關(guān)方差最大旋轉(zhuǎn)Varimax：使每個因子上具有高載荷的變量數(shù)最少——簡化對因子的解釋四分旋轉(zhuǎn)quartmax：使每個變量中需要解釋的因子數(shù)最少——簡化變量的解釋平均正交旋轉(zhuǎn)equamax：前兩種方法的結(jié)合斜交旋轉(zhuǎn)：允許因素間相關(guān)直接斜交旋轉(zhuǎn)direct

oblilminPromax：比直接斜交旋轉(zhuǎn)快旋轉(zhuǎn)因子矩陣a因子12nstomach.141.741nsideeff.014.808stoppain.801.098wksquick.794-.055kpawake.039-.797limrelie-.777-.074提取方法:主軸因子分解。旋轉(zhuǎn)法:具有Kaiser標(biāo)準(zhǔn)化的正交旋轉(zhuǎn)aFactor

Plot1.0.50.0-.5-1.0Factor

21.0.50.0-.5-1.0limreliekpawakewksquickstoppainnstomachnsideeff止痛藥因子模型旋轉(zhuǎn)結(jié)果Factor

11.0.50.0-.5-1.0Factor

2Factor

Plot

in

Rotated

FactorSpace1.0.50.0-.5-1.0limreliekpawak

estoppainwksquicknsideeffnstomach因子矩陣a因子12nstomach.596-.462nsideeff.546-.596stoppain.665.457wksquick.559.568kpawake-.499.623limrelie-.631-.459提取方法:主軸因子分解。旋轉(zhuǎn)因子矩陣a因子12nstomach.141.741nsideeff.014.808stoppain.801.098wksquick.794-.055kpawake.039-.797limrelie-.777-.074提取方法:主軸因子分解。旋轉(zhuǎn)法:具有Kaiser標(biāo)準(zhǔn)化的正交旋轉(zhuǎn)aFactor1:有效性Factor2:和緩性止痛藥潛在因素分析五、因子得分因子模型建立之后，樣本評即某個樣本在這些公共因子方公共因子得分的計算模式如價需要計算因子得分面的表現(xiàn)下：無法顯示該圖片。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在智力的雙因子模型中，每個樣本在公共因子f上的得分表示了該樣本的一般智力水平f

=

b1classics

+

b2frech

+

b3english

+

b4math

+

b5discr

+

b6musicBartlett

法(加權(quán)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：22?ipim

mi

i

i1

1

i

2[(x

-

m

)-

(a

f?

+

af?

++a

f

)]/s

達(dá)到最小我們可以采用與求解線性回歸模型相似的方法來得到f1,f2,…fm的近似解。由于p個特殊方差可以不全相等，因此應(yīng)采用加權(quán)的最小二乘估計法，即尋求一組估計值，使得加權(quán)的“殘差”平方和2(

j

=1,2,m)f?j

=

b

j1

x1s

+

b

j

2

x2

s

++

b

jp

xpsi=1這樣求得的解就是因子得分因子得分系數(shù)的求解方法：止痛藥例KE

-

0.009LIMRELIESIDEEFF

+

0.014SSTOPPAIN+

0.355WKQUICK

+

0.043KPAWAfac

_

2(和緩性)=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

因子得分系數(shù)矩陣因子12nstomach.034.286nsideeff-.024.392stoppain.372.014wksquick.355-.048kpawake.043-.373limrelie-.322-.009提取方法:主軸因子分解。旋轉(zhuǎn)法:具有Kaiser標(biāo)準(zhǔn)化的正交旋因子得分方法:回歸。SPSS的因子分析過程分析→數(shù)據(jù)降維→因子分析顯示因子分析主對話框六、因子分析的SPSS實現(xiàn)六、因子分析的SPSS實現(xiàn)“描述統(tǒng)計”對話框單變量統(tǒng)計量：均值、標(biāo)準(zhǔn)差初始結(jié)果：變量共同度初始估計生成變量相關(guān)陣相關(guān)陣行列式因子分析的適宜度檢驗相關(guān)陣的逆矩陣反映象相關(guān)陣再生相關(guān)陣：因子分析后的R及殘差“抽取”對話框因子提取方法顯示未旋轉(zhuǎn)解特征根圖抽取因子個數(shù)“旋轉(zhuǎn)”對話框因子旋轉(zhuǎn)方法顯示旋轉(zhuǎn)解因子載荷散點圖：給出以兩兩因子為坐標(biāo)的各變量載荷“因子得分”對話框因子得分作為變量保存回歸法：因子得分均值為0“選項”對話框缺失值處理方法有缺失值的樣本被剔除載荷系數(shù)的顯示格式按數(shù)值大小排列不顯示絕對值小于指定值的載荷系數(shù)因子分析的基本步驟因子分析的核心問題有兩個：一是如何構(gòu)造因子變量；二是如何對因子變量進(jìn)行命名解釋。因此，因子分析的基本步驟和解決思路就是圍繞這兩個核心問題展開的。因子分析常常有以下四個基本步驟：確認(rèn)待分析的原有變量是否適合作因子分析。構(gòu)造因子變量（確定公共因子個數(shù)）。利用旋轉(zhuǎn)方法使因子變量更具有可解釋性。計算因子變量得分。多種算法比較以判斷因子模型的穩(wěn)定性1.巴特利特球度檢驗（Bartlett

test

of

sphericity）巴特利特球度檢驗是以變量的相關(guān)系數(shù)矩陣為出發(fā)點。它的零假設(shè)是Ho：相關(guān)系數(shù)矩陣是一個單位陣，即相關(guān)系數(shù)矩陣對角線上的所有元素都為1，所有非對角線上的元素都為零。巴特利特球度檢驗的統(tǒng)計量根據(jù)相關(guān)系數(shù)矩陣的行列式計算得到。如果該統(tǒng)計量值比較大，且其對應(yīng)的相伴概率值小于用戶心中的顯著性水平，則應(yīng)拒絕Ho，認(rèn)為相關(guān)系數(shù)矩陣不太可能是單位陣，適合作因子分析；相反，如果該統(tǒng)計量值比較小，且其對應(yīng)的相伴概率值大于用戶心中的顯著性水平；則不能拒絕Ho，可以認(rèn)為相關(guān)系數(shù)矩陣可能是單位陣，不適合作因子分析。因子分析的適宜度檢驗：變量間相關(guān)程度較高才適宜做化簡2.

KMO(Kaiser-Meyer-Olkin)KMO統(tǒng)計量是用于比較變量間簡單相關(guān)系數(shù)和偏相關(guān)系數(shù)的一個指標(biāo)，計算公式如下：式中：rij是變量和變量之間的簡單相關(guān)系數(shù)，pij是它們之間的偏相關(guān)系數(shù)?？梢?，KMO統(tǒng)計量的取值在0和1之間，當(dāng)所有變量之間的簡單相關(guān)系數(shù)平方和遠(yuǎn)遠(yuǎn)大于偏相關(guān)系數(shù)平方和時，KMO值接近1。KMO值越接近1，則越適合作因子分析，KMO越小，則越不適合作因子分析。Kaiser給出了一個KMO的度量標(biāo)準(zhǔn)：0.9以上非常適合；0.8適合；0.7一般；0.6不太適合；0.5以下不適合。

i?

j

i?

jij

iji?

jijpr

+rKMO

=222止痛藥例：因子分析適宜度檢驗KMO和Bartlett的檢驗取樣足夠度的Kaiser-Meyer-Olkin度量。.710Bartlett

的球

近似卡方226.100形度檢驗

df15Sig..000七、應(yīng)用（1）研究消費者對速溶麥片的看法：12種品牌速溶麥片的調(diào)查(Creating

a

perceptual

map

ofready-to-eat

cereal

brands

in

theAustralian

market,

Roberts

and

Lattin,1991)影響麥片銷售的特性有25個每個被調(diào)查者從25個方面給三個自己最喜歡的品牌的麥片打分，打分采用5分制共116人作答，得到235個樣本研究的目的：有哪些特性影響消費者的購買決策研究過程用主成分法確定因素的個數(shù)用不同方法求解因子模型，以確定較好模型因子解釋用因子得分對各個品牌做出綜合評價。用SPSS中的“aggregate”功能和散點圖說明的總方差成分初始特征值提取平方和載入合計方差的%累積%合計方差的%累積%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保留四個因子還是五個因子？旋轉(zhuǎn)因子矩陣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。旋轉(zhuǎn)法:具有Kaiser標(biāo)準(zhǔn)化的正交旋轉(zhuǎn)法。a.

旋轉(zhuǎn)在6次迭代后收斂。旋轉(zhuǎn)成分矩陣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。旋轉(zhuǎn)法:具有Kaiser標(biāo)準(zhǔn)化的正交旋轉(zhuǎn)法。a.旋轉(zhuǎn)在8次迭代后收斂。-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可能意味著因子數(shù)過多-4兩.00000

種不同方法得到的同一因子的分散點圖越接近45°線1.0000越0

好旋轉(zhuǎn)成分矩陣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。旋轉(zhuǎn)法:具有Kaiser標(biāo)準(zhǔn)化的正交旋轉(zhuǎn)法。a.

旋轉(zhuǎn)在6次迭代后收斂。旋轉(zhuǎn)因子矩陣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。旋轉(zhuǎn)法:具有Kaiser標(biāo)準(zhǔn)化的正交旋轉(zhuǎn)法。a.

旋轉(zhuǎn)在6次迭代后收斂。旋轉(zhuǎn)因子矩陣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:主軸因子分解。旋轉(zhuǎn)法:具有Kaiser標(biāo)準(zhǔn)化的正交旋轉(zhuǎn)法。a.

旋轉(zhuǎn)在6次迭代后收斂。保留四個因子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主成分法和極大似然法-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主軸因素法和極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：主軸因子分解。注意：“easy”的變量共同度非常低，說明

easy83%的變異都是特殊因子解釋的，4個因子對這一特性的解釋能力非常低，而且與每個因子的相關(guān)程度很低（最大0.3)，為什么？因子個數(shù)太少？easy這一特性指標(biāo)選得不好？所有品牌都是速溶麥片，消費者在這一特性上難以作出明確判斷公因子方差初始提取filling.633.590natural.636.638fibre.703.728sweet.612.646easy.230.170salt.448.4955個公因子選用主軸因素法旋轉(zhuǎn)因子矩陣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:主軸因子分解。healthfulartificialNon-adultPopularityinteresting對各品牌的評價調(diào)用”數(shù)據(jù)”→“分類匯總”過程，按照cerealid分類統(tǒng)計各品牌麥片的因素得分-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應(yīng)用（2）303名MBA學(xué)生對10個品牌汽車的評價BMW328i,

Ford

Explorer,

Infiniti

J130,

Cherokee,Lexus

ES300,

Chrysler

Town&Country,

MercedesC280,

Saab9000,

Porsche

Boxster,

Volvo

V90每個學(xué)生對每種車型就16個方面打分Exciting,

dependable,

luxurious,

ourdoorsy,

powerful,

stylish,comfortable,

rugged,

fun

to

drive,

safe,

high-performance

car,family

car,

versatile,

sporty,

high-status

car,

practical從每個學(xué)生對10個品牌汽車的評價中隨機抽取一份組成樣本，共303個樣本。作因子分析：可提取多少個共因子？如何解釋這些因子？保存因子得分，計算每個品牌汽車的平均因子得分并作圖和解釋KMO和Bartlett的檢驗取樣足夠度的Kaiser-Meyer-Olkin度量。.880Bartlett

的球

近似卡方3364.722形度檢驗

df120Sig..000說明的總方差因子初始特征值提取平方和載入合計方差的%累積%合計方差的%累積%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：主軸因子分解。-.399.560.384.312.862.859.594stylish.876exciting.864fun.864status.831safe.769comfort.719dependab.641practicaruggedoutdoorsversatil123旋轉(zhuǎn)因子矩陣a因子提取方法:主軸因子分解。旋轉(zhuǎn)法:具有Kaiser標(biāo)準(zhǔn)化的正交旋轉(zhuǎn)法。aperforma.772現(xiàn)代powerful.715sports.695.355luxuriou.644.425family-.637.533實用越野1234

5678910保留四個因子？再生相關(guān)陣殘差明顯下降旋轉(zhuǎn)因子矩陣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:主軸因子分解。旋轉(zhuǎn)法:具有Kaiser標(biāo)準(zhǔn)化的正交旋轉(zhuǎn)法。a.

旋轉(zhuǎn)在7次