Tips for Conducting a Factor Analysis - 5 Stage Process - an Example
This hub offers an explanation of the steps involved in performing a factor analysis. Factor analysis is a quantitative statistical analytical tool to help researchers boil research data down to common characteristics of phenomena in social science and/or behavioral science research. For instance, this hub attempts to reduce research data to common characteristics of a leadership style called servant leadership. Participants from two countries which I have labeled happyland and dreamland were asked to rate their bosses (on a scale of 1 to 5 - 5 being best) on how well their bosses exemplify information presented in 18 statements. The gathered scores were then entered into a software program called SPSS (or PASW 18.0) to be analyzed for common characteristics of servant leadership. The information in this hub is rather technical in nature and the novice may need to read through it a few or more times to get everything out of it.
A Factor Analysis of Servant Leadership Related Composite Variables
This hub presents an example of a comparative study of two cultures as related to three component variables of a servant leadership theoretical construct including love, humility, and vision involvement. The purpose of the article is to describe the process of a factor analysis as prescribed by Hair, Black, Babin, Anderson, and Tatham in their book Multivariate Data Analysis (2006). This how-to guide for factor analysis is presented in the context of a scenario wherein the object was to determine if the dimensions of the leadership behavior measures are the same for two countries as well as to compare the levels of each dimension in each country.
The method of data analysis for the study was a factor analysis of an 18-item instrument (survey) administered to participants from two fictitious countries named Happyland and Dreamland. In all, data was collected from a sample size of 157 participants comprising 60 participants from Happyland and 97 from the DREAMLAND. Three data analyses were run in the following order: (a) data collected from Happyland participants; (b) data from DREAMLAND participants; and (c) data collected from both countries together. Moreover the three analyses (a) followed the first five stages of Hair's model plus the Chronbach reliability coefficient noted in stage seven of Hair et al.’s suggested process for factor analysis (2006) and (b) were completed with the assistance of SPSS 18.0 data processing software. The five stages outlined by Hair et al. include:
1. Stage 1 – Objectives of Factor Analysis
2. Stage 2 – Designing a Factor Analysis
3. Stage 3 – Assumptions in Factor Analysis
4. Stage 4 – Deriving Factors and Assessing Overall Fit
5. Stage 5 – Interpreting the Factors
After determining the structure of the composite variables, the composite variables were tested for reliability employing the Chronbach alpha reliability coefficient. Finally, it should be noted that the three analyses were run in succession of each other, but the results are reported side by side in the interest of a more parsimonious presentation.
Stage 1 – Objectives of Factor Analysis
According to Hair et al., the starting point in factor analysis is the research problem and the underlying variables related to the research problem to determine whether the research problem and variables lend themselves to factor analysis. First, Hair et al. observed “The general purpose of factor analytic techniques is to find a way to summarize the information contained in a number of original detailed variables into a smaller set of new, more generalized composite dimensions” (p. 107). Hair et al. also suggest perusing the original variables or items involved in the study to determine whether they are related to the suggested study. One of the chief objectives of this study was to summarize 18 variables into a smaller set of more general composite variables related to leadership behaviors. Moreover, the 18-items in the survey seemed related to three leadership behaviors coded by data collectors as Service (items 1-6), Humility (items 7-12), and Vision Involvement (items 13-18). These conditions suggested that factor analysis would be a fitting technique for data analysis of this study.
Stage 2 – Designing a Factor Analysis
Hair et al. contend that the design of a factor analysis involves (a) considering the type of variables involved in the study and (b) how many variables should be included (pp. 111, 112). The type of variables best suited for factor analysis are metric rather than nonmetric variables due to that metric variables are more easily measured by several types of correlations. As to how many variables should be included in a factor analysis study, Hair et al. suggest that the number of variables depends on the sample size. They suggest that the factor analysis technique is most effective with a sample size of 100 or larger (p. 112). Moreover, they suggest that as a general rule a minimum number of five observations per variable with a more acceptable number being ten observations per variable. This study included 18 metric variables and 157 participants, a ratio of over 8:1 both acceptable conditions for factor analysis.
Stage 3 – Testing Assumptions in Factor Analysis
In short Hair et al. specify three rules of thumb when testing the assumptions of factor analysis, those assumptions include
1. A strong conceptual foundation needs to support the assumption that a structure does exist before factor analysis is performed
2. A statistically significant Bartlett’s test of sphericity (significant level < .05) indicates that sufficient correlations exist among the variables to proceed
3. Measure of sampling adequacy values must exceed .50 for both the overall test and each individual variable. (p. 115).
For the purposes of this comparative study, data was examined in four overall steps utilizing data previously coded and entered into SPSS 18.0 analytical software. Those four steps proceeded in the following order:
1. Examination of country data for Happyland participants;
2. Examination of country data for DREAMLAND participants;
3. Examination of country data from both Happyland and DREAMLAND together; and
4. Re-examination of country data for Happyland participants after obtaining results from steps 2 and 3.
In conjunction with the rules of thumb set forth by Hair et al. concerning testing assumptions of factor analysis, data sets for each country separately (Happyland and DREAMLAND) and the two countries together were run through SPSS software which prepared correlation matrices for each data set. Tables 1-3 display the correlation matrices for Happyland, DREAMLAND, and Happyland and DREAMLAND together. Additionally, Bartlett’s test of sphericity and Kaiser, Meyer, and Olkin’s measure of sampling adequacy were run on each of the data sets. Table 4 presents the results of the Bartlett’s and KMO tests.
Tables 1-3 show the correlation matrices for the 18 items measuring for three leadership behaviors of participants from Happyland, DREAMLAND, and Happyland and DREAMLAND together. Table 1 shows the correlation matrix for the data collected from the participants from Happyland. Inspection of the correlation matrix depicted in Table 1 revealed that 132 of 153 correlations (86%) as statistically significant at the .01 level. Moreover, it was noted that of the 21 correlations displayed in Table 1 that were not found to be statistically significant at the .01 level, 16 were significant at the .05 level and five fell outside the acceptable limits of significance. It was also noted that of the 21 correlations not statistically significant at the .01, 20 were linked to the variables 7-12 which data collectors coded as linked to the leadership behavior labeled humility. Table 2 shows the correlation matrix for data collected from the participants from the DREAMLAND. Inspection of the correlation matrix depicted in Table 2 revealed 151 of 153 correlations (97%) significant at the .01 level. Table 3 shows the correlation matrix of DREAMLAND and Happyland together. Inspection of the correlation matrix depicted in Table 3 revealed 153 of 153 correlations (100%) significant at the .000 level. According to Hair et al., these findings of correlation provided an adequate basis for proceeding to an empirical examination of adequacy for factor analysis on both an overall basis and for each variable (p. 141).
After examining the three correlation matrices, the overall significance of those matrices was assessed with the Bartlett’s Test of Sphericity. Additionally, the factorability of the overall set of variables and individual variables were assessed using the Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy. The results of the Bartlett’s and KMO tests as run in SPSS are displayed in Table 4. The Bartlett’s Test found that correlations in all three data sets were significant at the .01 level and KMO tests yielded scores of .878, .927, and .947 respectively well within the acceptable levels claimed by Hair et al. Altogether, the results of the Bartlett’s tests and KMO measures of sampling adequacy indicated that the set of variables was appropriate for factor analysis.
One last note in this stage, measures of sampling area (MSA) aid in assessing partial correlations. Hair et al. suggest that MSA values under .50 should be eliminated from consideration for factor analysis. Checking each of the three correlation matrices revealed that item 7 in the data for Happyland and item 15 in the data for both DREAMLAND and Happyland-DREAMLAND together did not have adequate partial correlations with values over the .50 threshold. Those findings were noted before proceeding to the next stages of analysis. Moreover, in order to confirm results, the Happyland data set was re-run with item 7 was eliminated from the data before running the analysis in SPSS.
Example of Correlation Tables 1-4
Stage 4 – Deriving Factors and Assessing Overall Fit
According to Hair et al., the fourth stage of factor analysis includes decisions to be made concerning (a) the method of extracting factors - i.e. choosing between component analysis (principal component analysis in the SPSS software package) or common factor analysis (principal axis analysis in the SPSS software package); and (b) the number of factors to select to represent the underlying structure. Hair et al. posit that component analysis is used when the objective is to summarize most of the original information (variance) in a minimum number of factors while common factor analysis is used primarily to identify underlying factors or dimensions that reflect what the variables share in common (p. 117). This primary purpose of this study was to summarize the data into a minimum number of composite variables and not necessarily to identify underlying factors or dimensions, therefore principal component analysis was chosen as the method to extract factors.
After selecting the extraction method, this stage also includes identifying the number of factors to select to represent the underlying structure. Without going into the theoretical basis for the decision, this study looked for factors with eigenvalues greater than 1. According to Hair et al., “eigenvalues represent the amount of variance accounted for by a factor” and are employed in accordance with the latent root criterion, which is the most common used technique for determining the number of factors to extract (p. 102). Initial results for the extraction of component factors for each of the three data sets yielded (a) Happyland only – three factors with eigenvalues greater than 1.0; (b) DREAMLAND only – two factors; and (c) Happyland-DREAMLAND together – two factors. Tables 5-7 show the three initial eigenvalue tables.
Samples of Eigenvalue Tables
Stages Five – Interpreting the Factors and Validating the Final Results
In Hair et al.’s suggested process of factor analysis, stage five involves interpreting the factors. In order to interpret the factors, Hair et al. recommended a five step process including
1. Examine the Factor Matrix of Loadings
2. Identify the Significant Loading(s) for Each Variable
3. Assess the Communalities of the Variables
4. Respecify the Factor Model If Needed
5. Label the Factors.
Most of the above process was carried out by the SPSS 18.0 software program.
Before beginning the five step process, the respective researcher must select a method of factor rotation and identify the significant factor loadings based on sample size. For the purposes of this study, the VARIMAX factor rotation method was selected. In regards to identifying the significant factor loadings, the sample for Happyland only data was 60 participants requiring factor loadings of .70 or higher; the sample size for DREAMLAND only data was 97 participants requiring factor loadings of .60 or higher; and the overall sample size for Happyland-DREAMLAND together was 157 participants calling for factor loadings of .45 or higher. Beyond examination of factor loadings, Chronbach alpha reliability coefficients were also run to test the reliability of the structure of the revealed composite variables.Tables 10-14 show a progression of Factor Matrices for Happyland only data from inception through elimination according to factor loading requirements; Tables 17-19 show a similar progression of factor matrices for DREAMLAND only data; and Tables 22-24 show a progression of factor matrices for Happyland-DREAMLAND data. Chronbach alpha tests for reliability are reported in the text when appropriate for each data set.
The initial run of the Happyland only data yielded three factors with what seemed to this researcher to be jumbled results. Only 12 of 54 factor loadings were within the limit .70 factor loading limit as recommended by Hair et al. (see Table 8) and only a trace of an underlying structure was revealed. A re-reading of Hair et al.’s chapter on factor analysis led to re-examination of correlation matrix shown above in Table 1 which revealed that item 7 could be eliminated due to no partial correlations of .50 or higher.
Following re-examination of the correlation matrix (Table 8) and elimination of Item 7, the remainder of the Items (1-6; 8-18) were re-run through SPSS and for a second time yielded three factors shown in Table 9. Examination of the factor loadings in Table 9 revealed 12 of 51 factor loadings above the .70 threshold, but seemed to reveal some structure for composite factors 1 and 2. However two items (Items 6 and 13) seemed to be severely cross loaded. Additionally, all factor loadings for Items 8-12 (which seemed connected to the leadership behavior humility) except one (Item 12) fell below the .70 significance threshold. Thus, Items 6, 8-11, and 13 were eliminated from the factor matrix. (It should be noted that other loadings fell below .70 threshold, but when eliminated re-runs of remaining variables through SPSS yielded only one composite variable or as will be discussed later yielded Chronbach alpha reliability coefficients below the industry standard of .70.)
After elimination of those six items, the remaining items were run for a third time with the assistance of SPSS, but yielded only two factors as shown in Table 10. Examination of factor loadings revealed that the last item connected to the leadership behavior humility (Item 12) could be eliminated due to elimination of a corresponding factor. Moreover, a clear structure was revealed for the two remaining factors which were then labeled service and vision involvement, respectively.
Chronbach alpha reliability tests were run on the two composite variables with only the remaining variables included in each test. The tests yielded the following results: Service (5 items, Chronbach α = .93) and Vision Involvement (5 items, α = .84). One final note: two the five items in the vision involvement factor fell below the .70 factor loading threshold and could have been eliminated; however, eliminating them weakened other factors in the structure and or the reliability of the composite variable. For instance, eliminating Item 17 resulted in a Chronbach α of .79 and also caused Item 14 to fall under the .70 threshold, so Item 17 was retained. This determination was in conjunction with Hair et al.’s comment that the good researcher must examine the data both objectively and subjectively and use multiple methods including intuition when attempting to interpret the factors in factor analysis (p.144, 146). Tables 8, 9, and 14 show descriptive statistics, eigenvalues, and final factor matrix for participants from Happyland.
Samples of Descriptive Statistics and Component Matrix Tables
For this study, the sample size for the DREAMLAND was 97 participants, requiring factor loadings of .60 or higher (Hair et al.). The initial run through the SPSS software yielded two factors returning 15 of 36 factor loadings over the .60 threshold (see Table 17). Moreover, the rotated component matrix suggested a fairly well-defined structure with none of the 18 pairs of factor loadings cross-loading at the .60 threshold. It was interesting to note that each of the six items that seemed connected to the leadership behavior humility (Items 7-12) (a) yielded factor loadings above the minimum .60 threshold, (b) did not force a third composite factor of its own, but instead (c) loaded up under the Factor 1 (labeled Service in the analysis of Happyland data). However, since the items that seemed to be underlying variables of the leadership behavior humility (Items 7-12) did not yield their own factor, they were eliminated (perhaps prematurely).
After Items 7-12 were eliminated, the remaining were re-run employing SPSS software. Item 15 was the only one which did not yield a factor loading of .60 for either factor 1 or 2, so it was eliminated. (Item 15 did not yield factor loadings in the first run either and could have been eliminated then, but the researcher decided to see what impact eliminating Items 7-12 would have before eliminating Item 15.)
After the elimination of Item 15, the remaining eleven variables were run SPSS 18.0 once more and yielded the results presented in Table 19. Just as with the data from Happyland, the resulting two composite factors were tested for reliability with Chronbach alpha test. The tests yielded the following results: Service (6 items, Chronbach α = .94) and Vision Involvement (5 items, α = .84). However, after receiving these results, it seemed to this researcher that eliminating Items 7-12 was pre-mature, for even though they did not yield an anticipated factor they did yield significant factor loadings under Factor 1 (the leadership behavior service). In fact, re-instating them into the model for the DREAMLAND data yielded a stronger Chronbach alpha score suggesting that DREAMLAND participants perceived humility as a dimension of the leadership behavior service. With the dimensions of humility reinstated, Service (12 items, Chronbach α = .96). Tables 15, 16, and 20 show descriptive statistics, eigenvalues, and final factor matrix for participants from DREAMLAND.
More Examples of Descriptive, Eigenvalue, and Component Matrix
Happyland and DREAMLAND
The total sample size of all participants from Happyland and the DREAMLAND together was 157. According to Hair et al., a sample size of 150 or more requires factor loadings of .45 or higher. Given this criteria, the results of the initial factor analysis (aided by SPSS 18.0 and shown in Table 22) were similar to those of the DREAMLAND only data except that four of the six items linked to the leadership behavior humility (Items 8-11) cross loaded at the .45 threshold. Accordingly, those four items were eliminated.
Upon elimination of Items 8-11, the data was run through SPSS a second time which yielded the results presented in Table 23 (see below). While all of the remaining eight items under Factor 1 had more than adequate factor loadings, item 7 was eliminated because (a) eliminating it would not affect the Chronbach reliability score and (b) from the researcher’s perspective, it did not seem to fit the leadership behavior notion of service. This later decision was based on Hair et al.’s suggestion that researchers should use both objective and subjective means including personal intuition in the process of factor analysis. Item 12 was retained because it does seem to the researcher that expressing a demeanor of humility is tied to the behavior of service. Besides item 7, Item 15 was eliminated also because it even though its factor loading of .51 was over the .45 threshold, it was so far below the loadings of other factors that it seemed it should be eliminated. This was in conjunction with Hair et al.’s further suggestion that factor loadings at the low end should meet higher criteria given a factor solution that has a larger number of factors (p.129).
After the elimination of Item 15, the remaining 12 variables were run SPSS 18.0 once more and yielded the results presented in Table 24. The resulting two composite factors were tested for reliability with Chronbach alpha test. The tests yielded the following results: Service (7 items, Chronbach α = .94) and Vision Involvement (5 items, α = .89). Tables 20, 21, and 24 present the descriptive statistics, eigenvalues, and final rotated factor matrix for the Happyland-DREAMLAND participants together.
Final Examples of Descriptive, Eigenvalue, and Component Matrices
The results of this factor analysis from the data gathered from participants from Happyland and DREAMLAND seemed to suggest two factors related to leadership behaviors. Those two leadership behaviors derived from those factors were service and vision involvement. It was anticipated that a third leadership behavior i.e. humility would emerge from the data but it did not.
Patterson (2003) developed a theoretical construct of servant leadership that included seven composite variables including agapao love, humility, altruism, trust, vision, empowerment, and service. Dennis and Bocarnea (2005) developed a servant leadership assessment instrument to test the relationships of the seven composite variables in Patterson’s theoretical construct of servant leadership. An examination of the 18-item survey employed in this comparison of Happyland and DREAMLAND leadership behaviors revealed those 18-items came from Dennis and Bocarnea’s SLAI. Thus, it seems reasonable that the operational definitions of the three composite variables examined in this current study came from Patterson’s theoretical construct of servant leadership. In study of servant leadership in Cambodia, Coggins (2010, unpublished) summarized Patterson’s concepts of humility, vision, and service as follows:
Sandage and Wiens (2001) defined “humility” as “the ability to keep one’s accomplishments and talents in perspective.” Essentially this means that those who practice humility maintain a sober view of themselves. They think neither too highly nor too lowly about themselves and practice self-acceptance without being self-centered. Swindoll (1981) agreed with this view of humility and argued that humility is not to be equated with a poor self-esteem, but rather a healthy ego. As such, humility is not groveling in the dirt as one with a low self-worth, but having a right view of one’s self and others. Hunter (2004) explains the paradox of humility in leadership by saying humble leaders realize they came into the world with nothing and will leave with nothing (Waddell, 2006). Furthermore Waddell explains
People mistakenly associate being humble with being overly modest, passive, or self-effacing. To the contrary, humble leaders can be very bold when it comes to their sense of values, morality, and doing the right thing. They view their leadership as an awesome responsibility that affords them a position of trust and stewardship to take care of the people entrusted to them. (p. 3).
As applied to servant leadership, humility guides the servant leader to lead from an authentic desire to help others and search for ways to serve others by staying in touch with subordinates within the organization (Swindoll). Moreover, those leaders who practice humility are (a) willing listeners; (b) accept accountability to those they serve; and (c) openly accept criticism and advice (Patterson, 2003; Harrison, 2002; Blanchard, 2000).
Patterson (2003) acknowledged that vision is normally associated with the organization and not the individual. However, she further observed that
In servant leadership theory, vision refers to the idea that the leader looks forward and sees the person as a viable and worthy person, believes in the future state for each individual, and seeks to assist each one in reaching that state. (p. 18).
Greenleaf (1977); Buchen (1998); Farling, Stone, and Winston (1999); and Harvey (2001) all acknowledged that vision for each individual in the organization was a fundamental aspect of servant leadership (Patterson, 2003).
At the core of servant leadership is the trait that is inherent in the label servanthood or service (Russell & Stone, 2002; Dennis & Bocarnea, 2005; Waddell, 2006). From their respective research, Patterson, 2003, Dimitrova and Bocarnea (2010), and Dennis and Bocarnea found that “the act of serving includes a mission of responsibility to others and that people are accountable to those they serve, whether they serve customers or subordinates (Wis, 2002; Greenleaf, 1996). Furthermore, Winston (2003) noted that the servant leader sees his or her role to the follower as one of providing the follower with what is needed for the follower to accomplish his/her task (Dimitrova & Bocarnea, 2010). After reviewing the literature, Kimura (2007), following Patterson (2003) and Swindoll (1981), observed that service requires the leaders’
1. giving of time, energy, compassion, and one’s belongings;
2. personal involvement in followers’ lives;
3. authenticity in what they do;
4. and, serving as a role model in behavior and style that sets the foundation the servant-led organizational culture (p. 7; Lytle, Horn, & Mokwa, 1998).
Comparison of Happyland and DREAMLAND
The three composite variables presented above were used in this study to compare leadership behaviors between participants from Happyland and the DREAMLAND. The overall result yielded seven dimensions related to the leadership behavior service, five related to vision (or vision involvement as in the survey), but none specifically related to the composite variable of humility. However, a comparison of the respective country data of Happyland and DREAMLAND separately seemed to indicate that participants from the DREAMLAND perceived qualities of humility as part of the leadership behavior of service. In contrast, the data from participants from Happyland did not indicate any aspect of humility as part of the leadership behavior of service and furthermore did not see modeling service in behaviors and attitudes as part of the leadership behavior of service (Item 6) either.
In regards to the leadership behavior of vision (vision involvement), a comparison of the three data sets yielded five dimensions of the construct with four of the five similar including items 14, 16-18. However, the Happyland participants perceived item 15 – i.e. “and I have written a clear and concise vision statement for our company” as the fifth dimension of leadership behavior of visionwhereas the DREAMLAND and Happyland-DREAMLAND data yielded Item 13 – i.e. “has sought my vision regarding the organization's vision” as the fifth dimension. There seems to be a subtle difference between these two statements and their underlying outlooks, suggesting that (a) leaders from Happyland allow subordinates to be involved in the actual writing of the corporate vision whereas (b) those from the DREAMLAND take personal goals of subordinates into consideration when crafting a corporate vision, but may not include subordinates in the actual writing of that vision. This subtle difference between the two cultures in this area could reflect difference orientations in regards to the cultural dimension of collectivist and individualist societies (House et al., 2004). Happyland has been described as a more collectivist society which may lead to joint decisions in crafting corporate visions whereas the DREAMLAND has been described as a more individualistic society which may lead to personal considerations being included in vision crafting (King, MacLaughlin, & Thomas, n.d.; House et al.).
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