10 Exploratory Factor Analysis versus Confirmatory Factor Analysis

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1 128 Teachers Learning, Curriculum Innovations and Knowledge Applications 10 Exploratory Factor Analysis versus Confirmatory Factor Analysis Siti Aishah Hassan and Kaseh Abu Bakar INTRODUCTION Nowadays, most empirical researchers read, hear, and use the various available multivariate data analysis techniques, such as Multiple Regression Analysis (MRA), Multiple Discriminant Analysis (MDA), Multiple Analysis of Variance (MANOVA), Exploratory Factor Analysis/ Principle Component Analysis (PCA), Confirmatory Factor Analysis (CFA), and Structural Equation Modelling (SEM). Some of these data analyses are frequently used by education researchers. However, other data analyses still remain novel to the researchers. Particularly, exploratory factor analysis (EFA) is widely taught in statistics at the level of postgraduate studies but confirmatory factor analysis (CFA) still remains exclusive to advanced researchers. Nevertheless, with the advancement in computer technology, postgraduate students and researchers are expected not to rely on EFA only. More and more user friendly software are available in the market to handle the needs of the more advanced data analysis techniques. CFA, is predominantly beyond the capacity provided by Statistical Package for Social Science (SPSS). Therefore, a number of software programs such as AMOS, SAS, EQS, LISREL, SEPATH, CALIS along with others are available to serve the needs of CFA. Yet, before researchers embark on the more advanced technique of data analysis, they need to conceptually understand the relationships between the two techniques. They also need to compare the similarities and the differences of these two data analyses. Accordingly, they will understand the link between the two techniques and able to use the techniques appropriately. Therefore, this chapter seeks to answer four germane questions: What is factor analysis? Why conduct factor analysis? What are the similarities and differences between exploratory and confirmatory factor analysis? What kind of research questions may employ exploratory or confirmatory factor analysis? Finally, this chapter provides examples of research report writing that employs both exploratory and confirmatory factor analyses.

2 Exploratory Factor Analysis versus Confirmatory Factor Analysis 129 DEFINITION OF TERMS In order to better comprehend this chapter, definitions of relevant terms are provided below. Factor Loadings are multiple correlations between factors and variables (items). High loading indicates the factor significantly explained the variables. A cutting point of.40 for loading is recommended. Factor matrix is the table that displays the factor loadings of all variables on each factor of orthogonal rotation. Factor pattern matrix is the table that displays the factor loadings of all variables on each factor of oblique rotation. Orthogonal rotation is an extraction method of rotation for uncorrelated or very low correlation among the factors. Oblique rotations is an extraction method of rotation for highly correlated factors. Squared multiple correlations (SMC) is the proportion variance explained of the item by the factor. High SMC indicates a high reliability of the item. Eigenvalue is the value that indicates the amount of variance accounted for by each factor. For eigenvalues greater than 1, the total variance explained by the emerging factors is at the optimum point. It is a value that normally represents a sharp elbow or break point in the scree plot. Scree plot is a graphical representation of the relationships between the number of extracted factors and eigenvalues. Bartlett test of sphericity is a test for overall significant correlation matrix. A significant result indicates that the variables are highly correlated and hence suitable for factor analysis study. In other words, some degree of multicollinearity (intercorrelation) is desired. Manifest variable is a directly measured variable. It is the item of the instruments. Latent construct is the unobserved variable. It is the extraction of the hypothesized factor. It is also known as dimension, factor, component, or underlying structure. Exogenous variable is a variable that may roughly be equated with an independent or predictor variable, an arrow pointing outward. Endogenous variable is a variable which may roughly be equated with a dependent or criterion variable, an arrow pointing towards it. WHAT IS FACTOR ANALYSIS? Factor analysis is a generic name given to a group of multivariate statistical techniques whose primary purpose is to define the underlying dimension in a data matrix. It is a technique used to analyze a large number of highly correlated variables. This large number of variables is hypothesized to be explained by or belong to certain common factors. With factor analysis, the researcher can first

3 130 Teachers Learning, Curriculum Innovations and Knowledge Applications identify separate dimensions of the structure, which later can be used for two purposes - summarization and data reduction. Summarization is to describe the data using a much smaller number of concepts than the original individual data. Data reduction is to have a composite measure for each dimension and to substitute them in place of individual data. This is particularly important in order to further analyze data by using other statistical techniques such as multiple regressions or path analysis (Abdul Hair et al., 1998). Why Conduct Factor Analysis? The main reason to conduct factor analysis is to condense a large number of observable variables into a smaller set of variables without forfeiting the important information each variable carries. In other words, it is to extract the common factors, structures, components, constructs, underlying dimensions, and latent variables. The factors summarize a large number of variables into a few logical concepts. We label the factors according to logical concepts. We can reduce the data by using the composite measure of the variables under the same dimensions. This type of factor analysis is known as exploratory factor analysis. It is conducted without theoretical constraints imposed upon the solution. Specifically, exploratory factor analysis is conducted to explore how many latent factors underlie a set of variable scores when there are no previous studies conducted on the particular newly adapted or developed instrument. Principle component analysis that is embedded in SPSS is commonly used for exploratory factor analysis. Although the mathematical formula for principle component analysis and exploratory factor analysis are not the same, both methods often yield similar results (Stevens, 1996, p. 362). Hence, principle component analysis is interchangeably used for exploratory factor analysis. However, when there is an established theory or a previous empirical study conducted to identify the common factor for the large number of variables under investigation, then the main reason to conduct factor analysis is to test the measurement model. The theory explains how many factors exist and what the factors are. There is a theoretical constraint that imposes items to load only on a specific factor. Exclusively, confirmatory factor analysis is conducted to test the construct validity of the instrument. Construct validity means how well the construct explains the variables under the construct. An instrument is said to have construct validity when the items are highly correlated with one another within the same construct and the loading or squared multiple correlation of the item is significantly correlated to the assigned construct. In short, as the name implies, exploratory factor analysis is conducted to explore the underlying dimensions of the measured variables. Confirmatory is used to confirm the hypothesized measurement model, i.e. to establish the construct validity of the instrument.

4 Exploratory Factor Analysis versus Confirmatory Factor Analysis 131 What are the Similarities and Differences between Exploratory and Confirmatory Factor Analysis? Since both data analysis techniques are in the same family of data analysis, the relationships between the two are summarized in Table 1. Table 1 Similarities and Differences Between EFA and CFA EFA CFA Similarity To condense a large number of variables into a smaller PURPOSE number of factors Difference To explore the underlying To test the construct construct validity (to confirm the construct) Similarity Application on new population APPLICABILITY Newly developed Has conducted EFA instrument Replication of the Difference Newly adapted instrument construct validity test onto a different sample THEORY ON Similarity Factors conceptually define variables that are highly FACTOR correlated to each other Grounded theory - Theory grounded- test generate theory theory Difference No predetermined number The number of factors of factors is fixed Items free to load on any Items are forced to load factor only on specific factors. INTER FACTOR Similarity A concept or theory explains whether the factors are CORRELATION correlated or uncorrelated Difference Orthogonal rotation for Single order construct uncorrelated factors for uncorrelated /low inter factor correlation Parallel Oblique rotation for Second order construct correlated factors for highly intercorrelated factor SOFTWARE Similarity Both use SPSS Difference SPSS only AMOS embedded in SPSS

5 132 Teachers Learning, Curriculum Innovations and Knowledge Applications What kind of research questions that may employ exploratory or confirmatory factor analysis? Since there are differences between the two techniques of factor analyses, the research questions that they employ can be differentiated accordingly. Research Questions for EFA/PCA 1. What are the underlying factors of...? 2. How many factors explained the concept of...? 3. What do the total proportions variance explain...? Research Questions for CFA 1. Is the hypothesized measurement model of...supported by the observed data? 2. Does the two-factor or three-factor model of...better explain the measured variables? 3. Is there any significant higher order construct that significantly explains the factors of...? CONCLUSION This chapter provides examples from our own research that have been published in journals and the unpublished dissertation. It is worthy to note that due to the limited number of pages that are allowed in journals, the details have been reduced. For dissertation writing however, lengthy report writing is preferred. Hence, we provide both styles of writing. The first example is a Principle Component Analysis (PCA) with orthogonal rotation. The second example is a PCA with an oblique rotation. The third example is a Confirmatory Factor Analysis (CFA) with a single order construct. The fourth example is a CFA with a second order construct. Extensive report writing is provided for each type of analysis. In summary, exploratory factor analysis is conducted based on three focal principles: to determine how many factors are present to label the factors, and to allow the items the freedom to load on any factors. On the other hand, confirmatory factor analysis is conducted to test the construct validity; the researcher s a priori sets the number of factors and forces the items to load only on a specified factor. Both analyses are conducted to condense data and increase reliability and validity of the instruments. Hence, prudent use of both techniques of factor analyses is ensured when their relationships are understood.

6 Exploratory Factor Analysis versus Confirmatory Factor Analysis 133 Example 1: Principle Component Analysis with Orthogonal Rotation An Extensive Report Writing for Exploratory Factor Analysis Preliminary Analysis In order to justify the use of principle component analysis, one of the important assumptions to be assessed is the intercorrelations among the items. Barlett Sphericity Test was statistically significant, χ 2 (36) = , p =.001 the variables were higly correlated to one another. Besides that, Kaiser-Meyer- Olkin Measure of Sampling Adequacy was.650. Thus, there was evidence for the overall measurement of sampling adequacy fulfilling the requirement of at least.50. The individual measure of sampling adequacy is listed in Table 1. The individual measure of sampling adequacy revealed that item 10 was low, MSA =.492. Thus, this item was considered as a candidate to be deleted. Next, communality was assessed. It revealed that there were three items needed to be deleted items 12, 13, and 14. As a result, four items were deleted from the original list of maternal involvement. Hence, parental involvement inventory left with ten items. Table Table 1: Preliminary 1 Preliminary analysis analysis and and descriptive descriptive statistics statistics of all of items all items Item Commu Mean SD Note: The diagonal ent ry is MSA The Factor Solutions Beginning with 14 items, a varimax rotation was conducted, resulting in fourfactor solutions with eigenvalues greater than 1. The total variance explained was 56.32%. However, due to the constraint that we adopted for the standard requirement of MSA and communality greater than 0.5, thus this four factorsolutions were not acceptable. Hence, we decided to conduct a second varimax rotation with the 10 items left. This time, the result showed three factors were extracted with 58.17% of

7 134 Teachers Learning, Curriculum Innovations and Knowledge Applications total variance explained. Factor 1 consisted of items 8, 9, 10, and 12. Whereas factor 2 consisted of items 1, 2, and 3 and finally factor 3 consisted of items 4, 5, and 6. Analyzing the items of each factor, we noticed that factor 1 could not be logically labeled, which consisted of item 11. This item is related to Islamic dimension and does not logically related to the existing factor. Thus, we finally conducted the third varimax rotation for the 9 items left after discarding item 11. The results of the analysis showed there were three latent variables emerged. This indicated that the three underlying dimensions accounted for 61.82% of the total variance explained of nine items and found to be higher as compared to the first (56.32%) and the second (58.17%) times varimax rotations were conducted with fourteen and ten items respectively. Thus, the solution of three latent structures of maternal involvement seemed to be the best. The dimensions can be logically labelled as reading encouragement for factor 1, homework monitoring for factor 2, and school relationship for factor 3. The variance of the first dimension, the largest eigenvalue was 2.47, while the other subsequence eigenvalues were 1.76 and 1.34, respectively. Inspection on the scree plot also pointed out the 9-items measured three factors, with the sharp break point after the three factors. All estimated factor loadings were large enough to be of practical significance at p =.05; even the weakest loading extracted was.549. In addition, all of the directions of loadings were positive and free from factorial complexity. The results of the final factor solution are illustrated in Table 2. Table 2 Rotated Component Matrix In summary, the study suggests that there are three underlying dimensions of maternal involvement construct among mothers of a selected Integrated Primary Islamic Schools. The dimensions are maternal reading encouragement, maternal homework monitoring, and maternal school relationship.

8 Exploratory Factor Analysis versus Confirmatory Factor Analysis 135 Example 2: Principle Component Analysis with Oblique Rotation A brief report writing style for exploratory factor analysis Example 3: Confirmatory Factor Analysis - Single Order Measurement Model A brief report writing style for confirmatory factor analysis Figure 3 shows the results of the three-factor measurement model of Maternal Accountability. All fit indices exceeded the recommended threshold values GFI, AGFI, IFI, TLI, CFI >.90, RMSEA<.08, indicating that the model fit the data. The inter-factor correlations were r = -.03,.19 and.35, substantiated the hypothesis that the three factors were distinct. The negative correlations were due to the negatively worded items for factor named Reliable. The loadings range was.49 to.82. Succinctly, construct validity for maternal accountability is supported. Figure 3 Measurement Model for Maternal Accountability

9 136 Teachers Learning, Curriculum Innovations and Knowledge Applications Confirmatory Factor Analysis -Second Order Measurement Model An extensive report writing style for confirmatory factor analysis Model Testing Results for Maternal Piety Maternal Piety (MP) was hypothesized to be a second order latent construct for three underlying dimensions, namely Faith in Allah and the Hereafter (Faith), Call for Virtue (Virtue), and Forbidding F-Vices (F-Vices). The results as illustrated in Figure 4, γ 2 (31) = 38.87, p =.156, suggested that there was no significant difference between the hypothesized model and the observed model. GFI, AGFI, IFI, TLI, CFI >.90 and RMSEA <.08 indicated that the model fits the data. Moreover, there were no offending estimates, suggesting that the hypothesized model of Maternal Piety was admissible. The result above was achieved after taking into consideration the Modification Index (MI). We allowed the residuals for items MSC9 and MSC7 to correlate as suggested by MI. Figure 4.1 Measurement Model for Maternal Piety

10 Exploratory Factor Analysis versus Confirmatory Factor Analysis 137 Precisely, Maternal Piety was statistically significant, as indicated by the first latent construct named Virtue (γ 12 =.69, R 2 =.48, p = 0.001). Faith, the second latent construct, was a priori set to be the reference indicator due to the conceptually sound indicator of Piety. Hence, the p-value was not estimated and presumed that Faith was significant indicator of Piety. This was substantiated when the estimated parameter (γ 12 =.90, R 2 =.80) indicated 80% of total variance for Faith. Finally, the third latent construct F-Vices was statistically significant as well (γ 31 =.74, R 2 =.55, p =. 001). In addition, all of the items (except item MSC 11) were statistically significant at α =.01, with loadings greater than The details are outlined in Table 4. Table 4: 4 Results of the CFA of of Maternal Piety Piety In summary, the hypothesized model of Maternal Piety was a second order construct for tri-dimensional measurement model; comprising of the Call for Virtue (Virtue), Forbidding Vices (F-Vices), and Faith in hereafter (Faith). It was supported by all the estimated fit statistics. Faith was found to be the best indicator for Maternal Piety, followed by F-Vices and finally Virtue. Illustratively, Faith best indicated by item MSC4, mom reminds me about hereafter. F-Vices were best indicated by item MSC7; mom forbids me from watching indecent TV programs. Virtue was equally best indicated by items MSC2 and MSC8, congregational prayer and reading Al-Qur an respectively. REFERENCE Arbuckle, J.L. (2007). Amos TM 16.0 User s Guide. Bethelhem Pike, PA: Amos Development Corporation. Arbuckle, J.L. and Wothke, W. (1999). AMOS 4.0 User s Guide. Chicago, IL: Small Waters Corporation. Armsden, G.C. and Greenberg, M.T. (1987). The inventory of parent and peer attachment: Individual differences and their relationship to psychological wellbeing in adolescence. Journal of Youth and Adolescence, 16,

11 138 Teachers Learning, Curriculum Innovations and Knowledge Applications DeVellis, R. (1991). Scale development: Theory and Applications. Newbury Park, California: Sage Publication. Hair, J.F., Black, W.C., Babin, B.R., Anderson, R.E. and Tantham, R.L. (2006). Multivariate Data Analysis (6 th Edn.). New Jersey: prentice Hall. Kline, P. (1994). An Easy Guide to Factor Analysis. London: Routledge. Lantos, E. and Rezmovic, V. (1981). A confirmatory factor analysis approach to construct validation. Educational and Psychological Measurement, 41, Sidek Mohd.Noah, Jamiah Manap, Azimi Hamzah, Hasnan Kasan, Turiman Suandi, Kairul Anwar Mastor, Rumaya Juhari, Azma Mahmood, Zanariah Mohd Nor and Steven Eric Krauss. (2006). Pembinaan inventori personaliti muslim (InPm) untuk kegunaan belia Malaysia. Jurnal Pekama, 12, Siti Aishah Hassan, Abdullah Seif Abdullah, Noriah Ishak and Hassan Langgulung. (2008). Measuring the unmeasurable: Maternal piety scales. Pertanika Journal of Social Sciences & Humanities, 6(1), Siti Aishah Hassan. (2006). Maternal quality time, children s emotional intelligence and their academic performance: A structural equation modeling analysis. Unpublished Doctoral Dissertation, International Islamic University of Malaysia. Stevens, J. (2002). Applied Multivariate Statistics for The Social Sciences. New Jersey: Lawrence Erlbaum Associates, Publishers. Stevens, J. (1996). Applied Multivariate Statistics for The Social Sciences. New Jersey: Lawrence Erlbaum Associates, Publishers. Thompson, B. (2004). Exploratory and Confirmatory Factor Analysis: Understanding Concepts and Applications. Washington, DC: American Psychological Association. Thorndike, R. (1997). Measurement and Evaluation in Psychology and Education (6 th Edn.). New Jersey: Prentice Hall.