1. Factor Analysis
Factor analysis is a statistical procedure utilized for data reduction (Bartholomew et al., 2008). This reduction is reflected in lowering a large number of variables that have similar characteristics, into smaller factors (dimensions). Specifically, based on data reduction, factor analysis is conducted to identify the unobservable and underlying structure of a set of variables which are correlated with each other. These variables are coherent but independent of one another. By coherent it implies that the variables are very consistent and relate with each other harmoniously while independent denotes that the outcomes or the results of each variable are not under control of another variable. Therefore, coherent but independent variables are those variables that have a good relationship with each, but their harmony doesn’t lead to the control of each other. These unobservable latent variables affect the result of observed or manifested variables while factor analysis helps identifying such variables (Bartholomew et al., 2011). Latent variables are those that are not well expressed or have not been developed in a research project but they do exist. By identifying the underlying factors, we can simply understand and describe complex interrelated measures (variables) where the words “dimensions and measures” tend to elucidate the variables in a research study.
Exploratory Factor Analysis (EFA)
EFA is used to discover the underlying structure and relationship between the variables of large sample size and to scrutinize the internal validity of the variables (Norris & Lecavalier, 2009). By internal validity it entails the extent at which errors have been minimized in the research study and if errors have been totally minimized, the conclusions can be warranted. EFA process has three steps that are: deciding the number of factors, choosing an extraction technique and finally choosing a rotation method. On the other hand Confirmatory Factor Analysis is used to test data for fitness regarding the constructed hypothesis (Kline, 2010). CFA does not actually investigate the hypothesis but rather it examines if the factor is acceptable to the sample size or not (Moulding, Anglim, Nedeljkovic, Doron, Kyrios, & Ayalon, 2011). Apart from investigating hypothesis CFA also inspects the correlation/ covariance of factors for heterogeneity. Correlation is a relationship where two of more factors are complementing one another mutually and in the relationship, one factor is dependent or is caused by another. In covariance, two variables change in conjunction with each other. Correlation/ covariance for fact or heterogeneity suggest that the researcher intends to get a wide variance to conclude that there is heterogeneity which is diversity.
Considering the sample size of this study which stands around 600 and nature of study being explanatory, EFA procedure has been used which will help reducing large variables of same characteristics into small factors then it will help in identifying the structure and relationship between large sample sized variables.
2. Purpose of using factor analysis
This current study used factor analysis to reduce the large number of items of the questionnaire. They were about 49 items without the demographic items. Specifically, the researcher intended to summarize the long variables (items) into a smaller factor (dimensions). There are various types of variables and mostly they are based on integers whereby there is a wide interval within comparable variables. After reducing the large numbers of correlated variables, new unrelated factors appeared and represent these large variables. The factors are independent of each other and intended to reflect on the correlation between them. Anglim (2007) points out that using factor analysis is suitable when asking people of their opinions and perceptions about other people and/or things on a scale i.e. five point liker scale. For example, asking employees about their perceptions of their workplace. Influenced by the notified theoretical acuity fulfilled in this study, factor analysis has been used. As students are inquired about the factors which affect their English writing skills and in such a way the study tries to capture their opinion justifying the basic principles of applying factor analysis. Factor analysis is a series of tests and steps that form a complete predefined procedure for conducting a research study and, it is explained further in the next section.
3. Factor Analysis Procedure
Factor analysis was performed for two purposes: firstly, to reduce the data (from a large variables to a smaller factors); secondly, to identify the underlying factors that could illuminate the associations among the variables representing the difficulties faced by Saudi secondary school students when writing in English, and the contributing factors to these difficulties. Four stages were taken before making a final decision about the number of latent factors that ought to be retained. They are correlation matrix, extraction, rotation, and interpretation & summation (Wegener & Fabrigar, 2011).
4.1 Power of the Correlations
In factor analysis, the first step for conducting exploratory study is to check on the power of the correlations and the sample size of the study (Ezekiel, Mordecai & Fox, 1999). Conducting factor analysis requires large number of sample size (Thompson, 2004). As factor analysis is constructed upon correlation matrix, which has all pairs of data sets correlated within a well-illustrated structure or figure while it is stated that correlation can only produce stabilized results if sample size is large. It is advised in this literature that for conducting factor analysis sample size of 50 samples is considered very poor, 100 samples remarks as poor, 200 samples is graded fair while 500 samples is rated very good and over 1000 it is regarded excellent (Tabachnick & Fidell, 2001). So, with large sample size according to the rule of thumb; having at least 10 observations under one factor is preferred to avoid computational difficulties (Comrey and Lee, 1992). The power of the correlations involves the determining how closely related the complementing factors/ variables can be in a research study and in order for that to be determined, there are three specified tests whose results provide foundation for further processing which are appropriateness of factors test, Bartlett test and Kaiser Test.
a. Appropriateness of data for Factorability
Appropriateness, a term that has been used in the determination of the relevance or suitability of some factors and whether they are applicable in a particular research study can be analyzed through the correlation matrix among all factors (Thompson, 2004). Since analyzing each factor independently is next to impossible so, only one value can tell the whole story of appropriateness and that is the determinant. The determinant value is required to be greater than 0. If not, that represents inappropriateness of factors as there lays some computational problem (Fabrigar et al., 1999).
Writing Difficulty with its two factors; sentence level difficulties and paragraph level difficulties has the determinant value of 0.005. Contributing factors to measure difficulties such as Teaching practices with its four factors; teaching paragraphing, motivational strategies in teaching, feedback and teaching techniques have determinant of 0.004. Strategic process with 2 factors; planning, drafting and revising organization, and General strategies have determinant of 0.170, Motivation with only one factor has determinant of 0.529, Anxiety with its only one factor has determinant of 0.436 and finally Arabic writing strategies with 2 factors; general strategies and planning and drafting have determinant of 0.281. All factors have determinants which are elements in a research study that have numerical value greater than 0 which states that there is no computational problem which is a computational problem exists when the elements do not have a numerical value. The problem expected was elements that have no numerical value such as 0.3. Hence, factor analysis can be computed for the above structure of factors, besides the sample size of 600 lies within the category of very good (Tabachnick & Fidell, 2001).
B. Kaiser Meyer Olkin Measure of Sampling Adequacy
KMO measure of sampling adequacy is a test that examines the smallness of a partial correlation. The smallness of partial correlations means the relatively little size and, it implies that adequacy can be achieved, partial correlations are supposed to be relatively smaller against the original correlations for each pair of correlation matrix, for better adequacy of variables the KMO test is required to be close to 1 (Kaiser, 1970). KMO test statistic represent as 0.90 is considered marvelous, 0.80 is meritorious, 0.70 is middling, 0.60 is mediocre, 0.50 is miserable while below 0.50 factors and correlation matric is inadequate and unacceptable.
Writing difficulty, teaching practices, strategic process, motivation, anxiety and Arabic writing strategies have their KMO test values as 0.930, 0.814, 0.784, 0.737, 0.718 and 0.775 respectively which states that the samples are adequate and acceptable as KMO value of all factors is greater than 0.6 (Kaiser et. al., 1974).
c. Bartlett's Test of Sphericity
Bartlett test of spehricity tests the hypothesis. It tests on whether the correlation matrix of factors is an identity matrix or not (Snedecor & Cochran, 1992). If the significant value for this test is rejected, it indicates that no correlation matrix. No correlation matrix is an identity matrix that gives dimensions that are unambiguous, that the values have to be positive and more than zero as it is the case of a correlation matrix. If the values are positive and when multiplied against each other in the identity matrix it gives the positive value or more than it means that all factors have got values exceeding zero. In this study all factors are significant at Bartlett’s Test of sphericity hence null hypothesis is rejected which states that no correlation matrix is an identity matrix.
Taken together all the tests provide a minimum standard for maintaining the powers of the correlations and should be passed to further stages of factor analysis. Since both KMO and Bartlett tests are passed, which suggests that minimum standards that is a case of minimum standards is assumptions based on the results of the two tests, KMO and Barlet tests for conducting factor analysis are fulfilled and the study can further proceed to next stage.
4.2 Factor Extraction
There are many approaches to extract factors such as Principal Components Analysis (PCA), Principal Axis Factoring (PAF), Alpha factoring, Maximum likelihood, Image factoring, generalized least squares and unweight least squares. However, in most of the works PCA and PAF are very common. PFA and PEA have common results given that they yield some factors that need explanation because their components are large with the underlying aspects of the study being the variables. Having better sample adequacy implies that the results from the sample can be relied upon as compared to those results from other samples (Thompson & Vidal-Brown, 2001). PFA is an ideal tool for establishing and exploring a hypothesis or assumption of complex variables which can be justified by two reasons that this study is exploratory in nature which explores the fit theoretical model of latent variables which influence the manifest factors (Costello & Osborne, 2005). Secondly all the factors have passed Kaiser Meyer test and have better results sample adequacy and sphericity which suggests that the factors are correlated while in case of correlations among factors, it is required to employ factor analysis axis because factor extraction is required for correlated variables otherwise for uncorrelated factors component factor analysis might be the best choice (Finch & West, 1997).
After examining the power of correlations, the next step is extraction and it is carried out through communalities which are shared characteristics within the factors under study in which certain factors are retained and then their initial and extracted variances which are statement is not an example but a procedure or a step within the factor extraction where higher value of variance is appreciated. Similarly to the communalities, eigenvalues along with their extraction sum of squared holdings can also help in evaluating and determining the factors for extraction and selection (Pett et. al., 2003). On the other hand screen plot which is derived from eigenvalues can also serve the purpose of selecting the factors with high variance and accountability to the research model. In English Writing skills factor out of 12 extracted items, only 7 items had well representation of the factor due to their higher value of variance. In teaching practices out 13 extracted items, 7 were well representing the factor. In strategic method out of 9 retained extracted items, only 2 were having high variance rest 7 were not well representing the factor. In motivation no item had high variance; all 5 items were not well representing the factor. In anxiety 5 items were retained extracted while only one had well representation. In Arabic writing strategies 6 items were extracted while only 3 items had high variances and were well represented. In process of extraction and dropping low variance items, it is essential to take care of such variables with low variance but have strong correlation with any dependent variable. So, factor rotation is conducted after factor extraction for analyzing the correlation between each factor.
4.3 Factor Rotation
Un-rotated outputs result in variance maximization from first to successive factors while rotating factors makes the output more comprehensible (Russell, 2002). Factor Rotations has two type orthogonal (Varimax, Quartimax & Equimax) and oblique (Direct oblimin & Promax). In this study varimax method of factor rotation examines coordinates and minimizes the sum of variance. It is usually preferred when dealing with single variables where the variance is expected to be at minimum levels and have been used because it easily identifies each factor with Single variable (Kieffer, 1999).
a. Eigen Values
Eigenvalues are the variances of selected factors under study and after rotation represent the variance of variables and their correlation to their factors (Gorsuch, 1983). The eigenvalue of first factor in writing difficulty is 48% variance of adjustable first variable of teaching practices explains 30% variance while second explain 14% variance with decreasing variance of successors. First variable of strategic process explains 34% variance while second variance stands at 14%, Motivation has consistent variance with 43%, 17%, 14% and 13% variance at first, second, third and fourth variable. Eigenvalue of Anxiety and Arabic teaching factors cover 44% variance at their first variable which clearly desire need for rotation.
b. Extraction Sum of Squared Loadings
The Extraction Sums of Squared Loadings have the same interpretation and purpose as of eigenvalues. The Factor Matrix gives the loading which tells about the correlations between each variable and each factor (Hayashi & Sen, 1997). In writing difficulty two items are retained with their common variance at 44% and 5% respectively. In teaching practices four variables are retained in loading with common variance as 25%, 12%, 5% and 3% respectively. Strategic process has retained two items having 27% and 8% variance. Motivation and Anxiety have reserved only one variable with 28% & 32% common variance while Arabic teaching has reserved two items having 35% and 7% variance of squared loading. After rotation, rotated factor matrix is derived which provide clusters of items which describe the factors.
4.4 Interpretation
In writing difficulties factor 1 with 7 items and factor 2 with 5 items have considerable correlation in explaining the relationship of underlying variable over manifest. In teaching practices factor 1 with 4 items, factor 2 with 2 items, factor 3 with 2 items and factor 4 with only 1 item have considerably strong relationship. In strategic process, factor 1 with only 2 items and factor 2 with 4 items have strong relation with manifest. Motivation has only one factor with 5 items having high loadings. Anxiety also has one factor with 4 items having high correlation and finally in Arabic writing, factor 1 with 4 items and factor 2 with no item have high correlation of latent variables to the manifest. Out of 53 items, and 12 factors; now we have 40 items with their 12 factors which have strong correlation among each other. Though factor analysis has provided some structure and relationship of variables to their manifest but the question for suitable procedure has certain limitations that contain decision of processing separately because one item may be positive but correlated to other latent variables which collectively have immense impact on manifest but as a result these variables are transformed into cluster of latent variables which can best explain the relationship of latent-manifest variables. The next step after rotation and interpretation of correlated item, the study requires item analysis for examining the internal consistency of the individual items.
a. Item Analysis
Item analysis is a test which is applied to selected items for examining the proportion of their selection as it helps in determining the internal consistency of each item. In classical item analysis testing “reliability analysis” is used while alternatively Cronbach Alpha also provides the same results because reliability of items is actually identified from the consistency of items in the sum scale which in other word is referred to as internal consistency of the factors. In this study Cronbach Alpha test for reliability is used whereby they are used to suggest that the internal consistency can be acceptable when Cronbach Alpha is 0.6≤ α. When Cronbach Alpha test for reliability is used, it signifies that the internal consistence value is next to zero and can, therefore, be valuable in terms of reliability.
The output of Cronbach Alpha test of English writing skill for the first 12 items is .899 acceptable. Alpha value for teaching practices for the next 13 items stands at .779 indicating reliability. Strategic processes with its next 9 items has reliable alpha at .753. The output of motivation factor for its next 5 items shows acceptable alpha at .666. Output of anxiety for 5 items has alpha at .719 representing acceptable. Finally Arabic writing strategies for 6 variables has cronbach alpha of .727 indicating reliable.
4.5 Summing the Score
Summing the factor scores actually provides a numerical value which designates to the standing of latent variables. There are two approaches to sum the scores which are refined and non-refined. This study uses the sum scores by factors under non-refined approach because this study is designed upon exploratory factor analysis for which simple score are well enough to describe the standing of latent variables. The sum score of English writing skills by adding all the rotation holdings on difficulty in English writing is 5.984. Sum rotation loadings score against teaching practices is 6.814. Strategic process factor has the sum rotation loadings score of 3.182. Sum score by rotation loading of motivation is 1.439 while sum score for anxiety factor is 1.617. Finally the sum rotation holdings score of Arabic writing strategies stands at 2.57.
4. Conclusion
In this study factor analysis provided great help in reducing the large variables and in identifying the underlying factors of association for analyzing the difficulties faced by the Saudi Secondary School student. Through four stages of factor analysis procedure finally the factor scores of all the factors are attained. Starting from analysis of Correlation Matrix to factor extraction then factor rotation then examination of reliability and internal consistency led to achieve the final small variables which purely have their underlying association to the problems faced by Saudi Secondary School students.
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