Miskolci Egyetem Gazdaságtudományi Kar Üzleti Információgazdálkodási és Módszertani Intézet Factor Analysis

Miskolci Egyetem Gazdaságtudományi Kar Üzleti Információgazdálkodási és Módszertani Intézet

Factor Analysis Factor analysis is a multiple statistical method, which analyzes the correlation relation between data, and it is for data reduction, dimension reduction and to explore the structure.


Aim of the Factor Analysis

Its aim is to group variables into so-called factors in order to an easier interpretation or to avoid the multicollinearity, or to analyze the relation among variables.


Stages of Factor Analysis 1

• General Purpose

2

• Assumptions - DESCRIPTIVES

3

• Factor Method - EXTRACTION

4

• Number of Factors - EXTRACTION

5

• Rotation - ROTATION

6 7

• Validity Tests • Name and Characterization of Factors


General Purpose • if we have too many variable in an analysis, we need to reduce them for an easier interpretation; • to select a group of variables according to their relation to the principle component; • to explore the structure of data, to learn the relation among variables; • to identify groups of cases and/or outliers (in case of type Q factor analysis, see also later);


Assumptions 1. Measure of Variables We need metric variables for factor analysis, however we can apply dummy variables (with outcome 0 and 1), too. Naturally, variables measured on interval or ratio scale can be easier interpreted, because the nominal variables degrade the validity and interpretation.


Assumptions 2. Relations of Variables The analysis of correlation matrix is necessary because without correlation among variables it would not be possible to find variables with similar characteristics and classify them into a single factor. If the correlation coefficient is lower than 0.3, the assumption is violated. Significance level, Partial coefficients


Assumptions 3. Sample Size The higher is the sample size, the more significant is the analysis. However, the ratio of cases to variables is important, too. The number of cases for one variable should be as high as it is possible.


Assumptions 4. General Multivariate Assumptions Besides the correlation coefficients and their significance level, we need to address the assumptions for multiple regression analysis as a pre-assumption for factor analysis: normally distributed variables, homoskedasticity, and linearity are important to not violate the correlation assumption among variables.


Output 1. Correlation Matrix

Correlatio Months since Hire n gender1 Previous Experience (months)

Employment Category Educational Level (years) Beginning Salary Minority Classification

Previous Experienc Employm Educatio Minority ent nal Level Beginning Classificat Months e since Hire gender1 (months) Category (years) Salary ion 1,000 ,066 ,003 ,005 ,047 -,020 ,050

,066

1,000

,165

,378

,356

,457

,076

,003

,165

1,000

,063

-,252

,045

,145

,005

,378

,063

1,000

,514

,755

-,144

,047

,356

-,252

,514

1,000

,633

-,133

-,020

,457

,045

,755

,633

1,000

-,158

,050

,076

,145

-,144

-,133

-,158

1,000


Output 2. KMO and Bartlett's Test Kaiser-Mey er-Olkin Measure of Sampling Adequacy . Bart let t's Test of Sphericity

Approx. Chi-Square df Sig.

,686 887,501 21 ,000


Kaiser-Meyer-Olkin (KMO) Measure


Output 3. Anti-image Matrices

Anti-image Covariance

Anti-image Correlation

Months since Hire gender1 Previous Experience (months) Employment Category Educational Level (years) Beginning Salary Minority Classification Months since Hire gender1 Previous Experience (months) Employment Category Educational Level (years) Beginning Salary

Months since Hire ,985 -,055

Previous Educational Minority Experience Employmen Level Beginning Classificatio gender1 (months) t Category (years) n Salary -,055 -,011 -,015 -,049 ,044 -,035 ,728 -,144 -,023 -,096 -,099 -,120

-,011

-,144

,812

-,042

,245

-,067

-,103

-,015

-,023

-,042

,424

-,039

-,220

,034

-,049

-,096

,245

-,039

,499

-,166

,006

,044

-,099

-,067

-,220

-,166

,322

,055

-,035

-,120

-,103

,034

,006

,055

,928

,356a -,065

-,065 ,799a

-,012 -,187

-,022 -,041

-,070 -,160

,078 -,204

-,037 -,146

-,012

-,187

,348a

-,071

,386

-,132

-,119

-,022

-,041

-,071

,729a

-,084

-,595

,054

-,070

-,160

,386

-,084

,709a

-,416

,009

,078

-,204

-,132

-,595

-,416

,667a

,100


Output 4. Total Variance Explained

Component 1 2 3 4 5 6 7

Total 2,601 1,292 1,028 ,876 ,602 ,385 ,215

Initial Eigenvalues % of Variance Cumulativ e % 37,163 37,163 18,451 55,614 14,688 70,302 12,511 82,812 8,605 91,418 5,507 96,925 3,075 100,000

Extraction Method: Principal Component Analy sis.

Extraction Sums of Squared Loadings Total % of Variance Cumulativ e % 2,601 37,163 37,163 1,292 18,451 55,614 1,028 14,688 70,302

Rotation Sums of Squared Loadings Total % of Variance Cumulativ e % 2,589 36,979 36,979 1,294 18,489 55,468 1,038 14,833 70,302


Number of Factors Retained • • • • •

A prior criterion Kaiser criterion total variance explained method Scree Plot Maximum-likelihood factor analysis


Scree Plot

Outputok 5. 3,0

2,5

Eigenvalue

2,0

1,5

1,0

0,5

0,0 1

2

3

4

5

Component Number

6

7


Output 6.

Goodness-of-fit Test Chi-Square 1,016

df 3

Sig. ,797


Output 7. Rotated Component Matrixa

,909 ,849 ,759 ,663

Component 2 -,044 -,014 -,367 ,350

,063

,841

-,177

Minority Classification -,146 ,573 Months since Hire ,035 -,014 Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.

,350 ,910

Beginning Salary Employment Category Educational Level (years) gender1 Previous Experience (months)

a. Rotation converged in 4 iterations.

1

3 -,092 -,097 ,127 ,152


Readings: Quantitative Information Forming Methods: http://elearning.infotec.hu/ilias.php?baseCla ss=ilSAHSPresentationGUI&ref_id=2774 Naresh K. Malhotra: Marketingkutatás Budapest, 2005. Sajtos-Mitev: SPSS adatelemzési és kutatási kézikönyv


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Miskolci Egyetem Gazdaságtudományi Kar Üzleti Információgazdálkodási és Módszertani Intézet Factor Analysis

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