Transcript Unit 5： 因素分析

Unit 5：因素分析
Factor analysis . . . is an
interdependence technique whose
primary purpose is to define the
underlying structure among the
variables in the analysis.
5.1 範例說明
1.行銷研究：生活型態因素之萃取

在行銷方面，消費品之行銷研究人員經常以
生活型態因素為區隔變數，進行市場區隔。

生活型態係一抽象的構念，必須以不同的構
面加以衡量。Plummer(1974)將生活型態分
為三個構面，即活動(Activities) 、興趣
(Interests) 、意見(Opinions)等，通稱為AIO
量表。

AIO量表

生活型態又可分為一般化生活型態即特殊化生活
型態兩類。

一般化生活型態：
指的是與產品無關的活動，如工作活動、休閒活
動、宗教信仰、政治偏好、個人興趣…等。

特殊化生活型態：
指與產品相關的活動，有使用頻率、產品偏好、
產品評價…等。

人文特質分析
2.教育研究

在教育方面，因素分析可應用於學習能力因素之
萃取，進而構建智商指標。

智慧商數之構建
3.財務研究

在財務方面，因素分析可應用於財務因素之萃取，
以重點掌握公司的財務狀況。

放款策略與經營策略
5.2 方法說明
（1）基本概念
FA’s primary purpose is to define the underlying structure among the
variables in the analysis.
Examines the interrelationships among a large
number of
variables and then attempts to explain
them in terms of
their common underlying
dimensions.
 These common underlying dimensions are referred to as
factors.
 Is a summarization and data reduction technique
that does
not have independent and dependent
variables, but an
interdependence technique in
which all variables are
considered simultaneously.
因素分析是一種互依分析技術，也是一類技術的統稱，包含許多縮減空
間（或構面）的技術，其主要目的在以較少的維數（number of
dimensions，即構面的數目）來表示原先的資料結構，而又能保存住
原有資料結構所提供的大部份資訊。

（2）因素分析的用途：








解開多變量資料中各變數間複雜的組合型式。
進行探索性的研究，以找出潛在的特徵，供未來實驗之用。
發展變數間的實證類型（empirical typologies）。
減少多變量資料的維數。
發展一種資料庫單維指數（data-based unidimensional index），
俾便將受測者作差異最大化的區隔。
檢定某些變數間的假設關係。
將預測變數加以轉換，使其結構單純化後，再應用某些技術
（如複迴歸或典型相關）來加以處理。
將知覺與偏好資料尺度化（scaling），並展現在一空間中。
(3) Factor analysis’s outcomes

Data summarization = derives underlying
dimensions that, when interpreted and
understood, describe the data in a much
smaller number of concepts than the
original individual variables.

Data reduction = extends the process of
data summarization by deriving an
empirical value (factor score) for each
dimension (factor) and then substituting
this value for the original values.
（4）因素分析假定樣本單位在某一變數上的反應
（即觀察值或分數）是由二個部份所組成：

各變數共有的部份，稱為共同因素（common
factor）。

各變數所獨有的部份，稱為獨特因素（unique
factor），獨特因素與共同因素無關聯，與其他變
數的獨特因素亦無關聯。
Extraction Method Determines the
Types of Variance Carried into the Factor Matrix
Diagonal Value
Variance
Unity (1)
Communality
Total Variance
Common
Variance extracted
Variance not used
Specific and Error
（4）設Zji為第i個樣本單位在第j個變數的分數，Fei
為第i個樣本單位在第e個共同因素之分數，Uji為第
i個樣本單位在第j個變數的獨特因素之分數。
令Z、F、U均為標準化之分數（即平均數為0，標
準差為1），則第i個樣本單位在變數j的分數可用
下式表示：
Zji＝aj1F1i＋aj2F2i＋…＋ajkFki＋djUji
ajp是因素權重（factor weight），用以表示第p個共
同因素對第j個變數之變異數的貢獻。
dj是指第j個變數之獨特因素的權重。
（5）σj2 ＝ 1 ＝ hj2＋dj2
其中，σj2：變異數（variance）
hj2：共同性（communality）
dj2：獨特性（uniqueness或specificity）
（6）主成分分析（Principal Component Analysis）假
定各共同因素間彼此均無關聯，即相關係數為零，
而且也不考慮變數分數中的獨特因素，故其變數
分數（Zji）及共同性（hj2）分別為：
Zji ＝ aj1F1i＋aj2F2i＋…＋ajkFki
hj2 ＝ aj12＋aj22＋…＋ajk2 ＝ 1
（7）負荷量

結構負荷量（structure loading）：
各共同因素和變數間之相關係數。

組型負荷量（pattern loading）：
又稱為因素權重，各共同因素之間相關係數為零
時，結構負荷量等於因素上之組型負荷量。

因素負荷量（factor loading）：
組型負荷量或結構負荷量一般均稱為因素負荷量。
5.3 處理流程
P230 圖9-1
5.4 理論探討
（1）共同因素之萃取

主要因素萃取法
探索性因素分析的因素萃取法主要有主軸法，驗
證性因素分析的因素萃取法主要有最大概率法和
多群體分析法。
1.主軸法
 主軸法抽取因素的順序是以能對各變數之共同
性產生最大貢獻之因素優先抽取。

主軸法是目前在探索性因素分析中使用最多的因
素抽取法。應用主軸法進行因素分析的模式很多，
其中最常見的有主成份分析和主要因素法
（Method of Principal Factor）。這兩個模式的主要
差別在於共同性的估計方式不同。主成份分析是
以1置入原相關係數矩陣之對角線上作為共同性之
數值，而不對共同性另作估計。主要因素法不以1
為共同性，而以最高相關係數法、反覆因素抽取
法或複相關係數平方法等估計共同性，再以估計
之共同性置入相關係數矩陣之對角線進行因素分
析。
2.最大概率法
最大概率法不須先估計共同性，而是先假設共同
因素之數目，而後依此假設導出因素和共同性。
3.多群體分析法
多群體分析法適用於驗證性研究。在進行這類因
素分析時，研究人員事前已決定所有變數共含有
多少個共同因素，以及那些變數歸屬於那個因素，
研究人員再進行多群體因素抽取時，通常在一開
始即將所有變數分成幾個群體，每群各歸屬於一
個共同因素，然後按各變數所屬的群體，將整個
相關係數矩陣分割為幾個小矩陣，以每個小矩陣
之橫列和除以該小矩陣總和之平方根，以所得之
商代表各該群體的「重心」（即因素）。

決定共同因素之數目
1.基本原則
抽取的因素愈少愈好，而抽取出之因素能解釋
各變數之變異數則愈大愈好。
2.常見方法
事先決定。參考理論架構及過去有關文獻來決
定抽取共同因素之數目。在驗證有關因素數目
的理論或假設時，可採用此種方法。
 以特徵值（eigenvalue）大於1做為分野。

利用因素的陡階檢定（scree test）來決定。將
每一個因素所能解釋之變異數畫在同一圖形中，
將各點連線，把陡降後曲線走勢趨於平坦之因
素捨棄不用，相對而言，此類因素所能解釋的
變異數較小。
 以變異數的百分比來決定。抽取出之因素所能
解釋的累積變異數已達某一預先設定的百分比
後就中止繼續抽取。
 以各變數的共同性來決定。
 利用平均變數（average variable）的解釋能力來決
定。

Eigenvalue Plot for Scree Test Criterion
（2）因素的轉軸與解釋
為了方便因素的解釋或命名，必須旋轉因素軸，
以使各個因素的意義變得比較清晰明顯。
在因素分析中，研究者最關切的是各變數間的關
係在因素空間中之型態。
旋轉因素座標軸並不會改變各變數間的關係型態，
適當的旋轉因素軸反而能使此種型態更清楚地顯
現出來。
轉軸的準則
1.因素矩陣的每一橫列裡，最少應有一個因素負
荷量為零。
2.如果有k個共同因素，則因素矩陣的每一直行中，
最少應有k個負荷量為零。
3.因素矩陣的任何兩個直行中，應該有幾個變數
在一個因素（即直行）上的負荷量為零，在另
一個因素上的負荷量則不為零。
4.當保留四個或以上的因素時，則在因素矩陣的
任何兩個直行中，大部份變數的負荷量應為零。
5.因素矩陣的任何兩個因素直行中，應該只有少
數幾個變數的負荷量不為零。


轉軸的方法
1.直交轉軸法：
各個因素之間均保持90度的關係，而因素與因素
之間互相獨立。
--四方最大法（Quartimax）
--變異數最大法（Varimax）
2.斜交轉軸法：
--四方最小法（Quartimin）
--共變數最小法（Covarimin）
--雙四方最小法（Biquartimin）
Orthogonal Factor Rotation
Unrotated
Factor II
+1.0
Rotated Factor II
V1
V2
+.50
Unrotated
Factor I
-1.0
-.50
0
+.50
+1.0
V3
V4
-.50
V5
-1.0
Rotated
Factor I
Oblique Factor Rotation
Unrotated
Factor II
+1.0
Orthogonal
Rotation: Factor II
Oblique Rotation:
Factor II
V1
V2
+.50
Unrotated
Factor I
-1.0
-.50
0
+.50
+1.0
V3
V4
Oblique
-.50
V5
Rotation:
Factor I
Orthogonal
Rotation: Factor I
-1.0

結果的解釋
1.因素與變數的關係可用三種方式表示：
--因素權重（因素組型）
--因素和變數之相關係數（因素結構）
--因素和變數的部份相關（part correlation）
2.在直交轉軸後，因各因素間之相關為零，故此三者
皆相同；在斜交轉軸後，三者之數值均不同，在
解釋結果時，通常係以因素結構為主，由因素和
變數間相關係數之大小，可以知道某個因素與那
些變數具有較高的關聯，與那些變數有較小或沒
有關聯，進而可了解該因素之意義，並賦予適當
的名稱。
3.因素分析的用途有二項限制：


因素分析最大限制為它是一高度主觀的分析過程。
在因素分析過程中，並無統計檢定方法可供有規
律的使用。
Rules of Thumb 1





Factor Analysis Design
Factor analysis is performed most often only on metric
variables, although specialized methods exist for the use of
dummy variables. A small number of “dummy variables” can
be included in a set of metric variables that are factor
analyzed.
If a study is being designed to reveal factor structure, strive
to have at least five variables for each proposed factor.
For sample size:
 the sample must have more observations than variables.
 the minimum absolute sample size should be 50
observations.
Maximize the number of observations per variable, with a
minimum of five and hopefully at least ten observations per
variable.
Rules of Thumb 2




Testing Assumptions of Factor Analysis
There must be a strong conceptual foundation to support
the assumption that a structure does exist before the factor
analysis is performed.
A statistically significant Bartlett’s test of sphericity (sig.
> .05) indicates that sufficient correlations exist among the
variables to proceed.
Measure of Sampling Adequacy (MSA) values must
exceed .50 for both the overall test and each individual
variable. Variables with values less than .50 should be
omitted from the factor analysis one at a time, with the
smallest one being omitted each time.
Rules of Thumb 3


Choosing Factor Models and Number of Factors
Although both component and common factor analysis models yield
similar results in common research settings (30 or more variables or
communalities of .60 for most variables):



Any decision on the number of factors to be retained should be based
on several considerations:







the component analysis model is most appropriate when data
reduction is paramount.
the common factor model is best in well-specified theoretical
applications.
use of several stopping criteria to determine the initial number
of factors to retain.
Factors With Eigenvalues greater than 1.0.
A pre-determined number of factors based on research
objectives and/or prior research.
Enough factors to meet a specified percentage of variance
explained, usually 60% or higher.
Factors shown by the scree test to have substantial amounts of
common variance (i.e., factors before inflection point).
More factors when there is heterogeneity among sample
subgroups.
Consideration of several alternative solutions (one more and one less
factor than the initial solution) to ensure the best structure is
identified.
Rules of Thumb 4


Choosing Factor Rotation Methods
Orthogonal rotation methods:



are the most widely used rotational methods.
are The preferred method when the research goal
is data reduction to either a smaller number of
variables or a set of uncorrelated measures for
subsequent use in other multivariate techniques.
Oblique rotation methods:

best suited to the goal of obtaining several
theoretically
meaningful factors or
constructs because, realistically, very few
constructs in the “real world” are uncorrelated.
Guidelines for Identifying Significant
Factor Loadings Based on Sample Size
Factor Loading
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
Sample Size Needed
for Significance*
350
250
200
150
120
100
85
70
60
50
*Significance is based on a .05 significance level (a), a power level of 80 percent, and
standard errors assumed to be twice those of conventional correlation coefficients.
Rules of Thumb 5



Assessing Factor Loadings
While factor loadings of +.30 to +.40 are minimally
acceptable,
values greater than + .50 are
considered necessary for practical
significance.
To be considered significant:



A smaller loading is needed given either a
larger sample size, or a larger number of
variables being analyzed.
A larger loading is needed given a factor
solution with a larger number of factors,
especially in evaluating the loadings on later
factors.
Statistical tests of significance for factor loadings are
generally
very conservative and should be
considered only as starting points
needed for
including a variable for further consideration.
Rules of Thumb 6





Interpreting The Factors
An optimal structure exists when all variables have high
loadings only on a single factor.
Variables that cross-load (load highly on two or more
factors) are
usually deleted unless theoretically
justified or the objective is strictly data reduction.
Variables should generally have communalities of
greater than .50 to be retained in the analysis.
Respecification of a factor analysis can include options
such as:
 deleting a variable(s),
 changing rotation methods, and/or
 increasing or decreasing the number of factors.
Rules of Thumb 7




Summated Scales
A summated scale is only as good as the items used to
represent the construct. While it may pass all empirical tests,
it is useless without theoretical justification.
Never create a summated scale without first assessing its
unidimensionality with exploratory or confirmatory factor
analysis.
Once a scale is deemed unidimensional, its reliability score, as
easured by Cronbach’s alpha:



should exceed a threshold of .70, although a .60 level can be used
in exploratory research.
the threshold should be raised as the number of items increases,
especially as the number of items approaches 10 or more.
With reliability established, validity should be assessed in
terms of:



convergent validity = scale correlates with other like scales.
discriminant validity = scale is sufficiently different from other
related scales.
nomological validity = scale “predicts” as theoretically suggested.
Rules of Thumb 8


Representing Factor Analysis In Other Analyses
The single surrogate variable:
 Advantages: simple to administer and interpret.
 Disadvantages:



does not represent all “facets” of a factor
prone to measurement error.
Factor scores:
 Advantages:




represents all variables loading on the factor,
best method for complete data reduction.
Are by default orthogonal and can avoid complications
caused by multicollinearity.
Disadvantages:


interpretation more difficult since all variables contribute
through loadings
Difficult to replicate across studies.


Representing Factor Analysis In Other
Analyses
Summated scales:

Advantages:





compromise between the surrogate variable and
factor score options.
reduces measurement error.
represents multiple facets of a concept.
easily replicated across studies.
Disadvantages:



includes only the variables that load highly on the
factor and excludes those having little or marginal
impact.
not necessarily orthogonal.
Require extensive analysis of reliability and validity
issues.
Rotated Component Matrix
“Reduced Set” of HBAT Perceptions Variables
1
Component
2
3
Communality
4
X9 – Complaint Resolution
X18 – Delivery Speed
X16 – Order & Billing
X12 – Salesforce Image
X7 – E-Commerce Activities
X10 – Advertising
X8 – Technical Support
X14 – Warranty & Claims
X6 – Product Quality
X13 – Competitive Pricing
.933
.931
.886
Sum of Squares
Percentage of Trace
2.589
2.216
1.846
1.406
8.057
25.893 22.161 18.457 14.061 80.572
.898
.868
.743
.940
.933
.892
-.730
Extraction Method: Principal Component Analysis.
Rotation Method: Varimax.
.890
.894
.806
.860
.780
.585
.894
.891
.798
.661
Scree Test for HBAT Component Analysis
5.5 實例說明