A Practical Example Comparing Principal Component Analysis and Principal Axis Factoring as Methods for the Identification of Latent Constructs

Table of Contents

Executive Summary

List of Abbreviations

List of Figures

List of Tables

1 Introduction/Problem Definition

2 Objectives

3 Methodology

4 Factor Analysis: Historical and Theoretical Basics
4.2 Factor Analysis: Parameters
4.2.1 Factor Score
4.2.2 Factor/Factor Loading
4.2.3 Communality (h2)
4.2.4 Eigenvalue
4.2.5 Rotation
4.3 PCA
4.4 PAF and Oblique Rotation

5 Data Sample and Analysis
5.1 Construct Structure of the Questionnaire
5.2 Sample Structure and Survey Conduction
5.3 Hypothesis/Questions
5.4 Data Analysis
5.5 Results PCA
5.6 Results PAF
5.7 Secondary Factors

6 Conclusion

7 Appendix

8 ITM Checklist

9 Bibliography

Executive Summary

In this paper the historical and theoretical background of the factor analysis is briefly explained. Principal Component Analysis (PCA) and Principal Axis Factoring (PAF) are applied to a data set which has been generated in the scope of the evaluation of the implementation of Company X’s corporate Strategy XX. The results clearly indicate that structural parts of the data collection instrument could be reproduced by the empirical data. The primary factors resulting from an orthogonal respectively oblique rotation are comparable but also show slight differences. Latent constructs like “Trust”, “Job Satisfaction” “Disengagement” and “Pessimism” are indicated by the results. Secondary factors indicate a negative relationship between disengagement and leadership respectively transparency concerning the corporate strategy and job satisfaction. Also aspects of “state negativity” can be identified. This means that a general pessimistic attitude is related to a more pessimistic view on realized customer focus. The application of more elaborated methods would be needed to identify causal relationships.

List of Abbreviations

illustration not visible in this excerpt

List of Figures

Figure 1: Screeplot for PCA Factors.

Figure 2: Screeplot for PAF.

Figure 3: Screeplot for Secondary Factor Extraction.

List of Tables

Table 1: Constructs Operationalized in the Questionnaire.

Table 2: Parameters for the Factor Analysis.

Table 3: PCA: Total Variance Explained by Extracted Factors.

Table 4: Rotated Component Matrix (PCA). Loadings≥ .4 are Highlighted in Colour.

Table 5: PAF: Total Variance Explained by Extracted Factors.

Table 6: Pattern Matrix (PAF). Loadings≥ .4 are Highlighted in Colour.

Table 7: Factor Correlation Matrix.

Table 8: Secondary Factors: Total Variance Explained by Extracted Factors.

Table 9: Communalities for Factor Scores after the 2nd Rotation.

Table 10: Secondary Factors Pattern Matrix. Highest Loadings are Highlighted in Colour.

Table 11: Factor Correlation Matrix for Secondary Factors.

Table 12: Questionnaire.

1 Introduction/Problem Definition

Factor Analysis is a widespread method used for data reduction. Although it is easy to run a factor analysis it is recommended to be cautious concerning the interpretation of results. To improve the empirical validity of the results the data set should meet certain requirements and both the method and the parameters should be chosen carefully.

2 Objectives

The following text should give answers to the following questions concerning the topic of factor analysis:

1. What is the historical and theoretical background?
2. What are the main parameters of a factor analysis?
3. What are the special features of a PCA?
4. How can two different approaches of factor analysis (PCA and PAF) help to structure the results of a complex data set?

3 Methodology

The data sample used for analysis was gathered in 2009 within the scope of an internal survey evaluating the level of intellectual and emotional anchoring of Company X’s corporate strategy “XX” within the national and international workforce.

Both PCA and PAF are calculated with SPSS for Windows 11.5. As a result of the PAF, factor scores are introduced as new variables. Based of the primary factor’s correlation matrix, secondary factors are calculated.

4 Factor Analysis: Historical and Theoretical Basics

Generally the expression “factor analysis” is a collective term for partially very different methods for data reduction. In social sciences factor analysis is used as a heuristic method allowing the generation or validation of hypothesis concerning complex constructs.

The development of the method started over 100 years ago and was mainly driven by psychological research on intelligence and personality. Prominent researches like C. Spearman, H.-J. Eysenck, J.P. Guilford, J. McKeen Cattell and R. Cattell, contributed to the increasing meaning of so called “factor theories”. For more details see for example (Burt, 1966).

Typically a factor analysis is not used when only a few variables are involved. The technique may help if the number of variables is comparably high and the interdependencies can hardly be interpreted (Bortz, 1993, p.474). A factor analysis allows the replacement of many more or less correlated variables by several factors. Based on correlations a factor can be regarded as a “synthetic variable” realizing the highest possible correlation to a certain set of variables of the overall data. The higher the overall level of intercorrelation between the variables of a data set the lower the number of factors which are needed to explain big proportions of the overall variance. Dependent on the rotation method these factors can be independent. This assumption applies for example to the Principal Component Analysis. While the PCA is based on a non-statistical orthogonal linear transformation the PAF is based on a statistical model trying to explain the covariance structure of the data (Noack, 2007, p.32).

4.2 Factor Analysis: Parameters

4.2.1 Factor Score:

the factor score (fmj) of a data set indicates the position of this data set according to the factor j. A factor typically cumulates a certain set of variables. The factor score gives information how strong the characteristics of the cumulated variables are present in a certain data set (e.g. the data of a research subject) (Bortz, 1993, p.480). In the scope of an orthogonal factor model (for example a PCA) there is the underlying assumption that the factors are uncorrelated.

4.2.2 Factor/Factor Loading:

A factor can be regarded as a synthetic variable establishing the basis for all highly intercorrelated variables. The factor loading is defined as the correlation of a variable i and a factor j (aij). According to Guadagnoli & Felicier (Guadagnoli & Velicier, 1988) the following rules are important for the interpretation of factors (Bortz, 1993, p.510):

A factor can be interpreted if at least 4 variables have a factor loading>.6 or 10 variables have a factor loading >.4

If less than 10 variables have a factor loading >.4 the results should only be interpreted if the sample (N) is bigger than 300.

If the prerequisites mentioned above are not met the results should only be interpreted if they can be reproduced.

4.2.3 Communality (h2):

the communality of a variable defines to which extent this variable is explained by the extracted factors. The square of the loading (a2ij) of a variable i and a factor j defines the proportion of shared variance between the variable i and the factor j. Summing up the squared loadings of a variable i over all factors results in a value (h2). h2 defines the proportion of variance of a variable i explained by a certain number of factors. In the scope of a PCA variables are z-standardised. This means that the squared loadings of a variable are ≤1 (Bortz, 1993, ibid.).

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