Application of Factor Analysis on Academic Performance of Pupils

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1 Aerican Journal of Applied Matheatics and Statistics, 07, Vol. 5, No. 5, Available online at Science and Education Publishing DOI:0.69/aas-5-5- Application of Factor Analysis on Acadeic Perforance of Pupils Okeke Evelyn Nkiruka *, Okeke Joseph Uchenna, Yawe Ayuba, Anule Josaphat, Bello Abdulrasheed Departent of Matheatics and Statistics, Federal University Wukari, Nigeria *Corresponding author: Abstract Over the years, efforts have been ade by researchers in studying the effects of certain factors on acadeic perforance of students, though with little concentration on the priary section of education. This research therefore worked on the acadeic perforance of pupils in priary school. Variation in the analyzed perforance and the factor(s) causing the variations were studied using factor analysis. The data used is a secondary data, collected fro Federal University Wukari Staff School and it s on the terinal exaination scores of the pupils in seven selected subects over one selected acadeic session (05/06). Fro both the un-rotated and rotated factor analysis results, we observed a fair relationship between the atheatical and less atheatical subects, though they present the aor variation in the pupils perforance to be in the less atheatical subects like English Language, Verbal Aptitude, Social Studies, Creative Art, and Religious Studies. Also, the analysis presented three factors (gender, age, and environent) to be the cause of the variation between the pupils perforance in priary school. Keywords: factor analysis, factor loading, acadeic perforance, quartiax, equaax, veriax, orthoax rotation Cite This Article: Okeke Evelyn Nkiruka, Okeke Joseph Uchenna, Yawe Ayuba, Anule Josaphat, and Bello Abdulrasheed, Application of Factor Analysis on Acadeic Perforance of Pupils. Aerican Journal of Applied Matheatics and Statistics, vol. 5, no. 5 (07): doi: 0.69/aas Introduction Factor analysis is a ultivariate ethod that studies the variability aong observed correlated variables in ters of a potentially lower nuber of unobserved factors that are supposed to contain essential inforation in a larger set of observed variables. The factors are constructed in a way that reduces the overall coplexity of the data by taking advantage of inherent interdependencies. As a result, a few nubers of factors will usually account for approxiately the sae aount of inforation as do the uch larger set of original observations. The broad purpose of factor analysis is to suarize data so that relationships and patterns can be easily interpreted and understood. It is norally used to regroup variables into a liited set of clusters based on shared variance. Hence it helps to isolate constructs and concepts Yong and Pearce []. In factor analysis observed variables are being represented as the linear cobination of the unobserved factors. Richard and Dean [] declared that the essential purpose of factor analysis is to describe, if possible, the covariance relationship aong any variables in ters of a few underlying but unobservable rando quantities called factors. There are two ain types of factor analysis; the exploratory factor analysis (EFA) and confiratory factor analysis (CFA) [3]. According to hi exploratory factor analysis (EFA) is data driven, such that the collected data deterines the resulting factors while confiratory factor analysis (CFA) is used to test factors that have been developed a priori. Ballester et al [4] copared the perforance of principal coponent analysis and factor analysis and stated that principal coponent analysis is recoended for diensional reduction, whereas, factor analysis is adopted for the study of structures in data. Principal coponent analysis is a linear cobination of variables while factor analysis is a easureent ode of a latent variable Karen [3]. This research work is aied at evaluating pupils perforance with respect to the subects that deterine their perforance using factor odel.. Factor Analysis.. Type of Factor Analysis We have two types of factor analysis and this include Exploratory Factor Analysis Confiratory Factor Analysis Exploratory factor analysis (EFA) is data driven; here the data collected deterines the resulting factors. This analysis tries to uncover coplex patterns by exploring the dataset and testing prediction [5]. Confiratory factor analysis (CFA) is used to test factors that have been developed a priori [3]. It s seen as a process for testing what you already think you know. CFA

2 Aerican Journal of Applied Matheatics and Statistics 65 is an integral part of structural equation odeling (SEM) and path analysis. It attepts to confir hypothesis and use path analysis diagras to represent variables and factors. CFA is also a useful tool in checking the reliability of a easureent tool with a new population of subect, or to further refine an instruent which is already in use.. Condition for the Use of Factor Analysis Factor analysis is usually perfored on ordinal or continuous variables, although it can also be perfored on categorical and dichotoous variable. Data that will be used to perfor factor analysis has to be univariate and ultivariate noral [5] and free fro univariate and ultivariate outliers [6] A deterining factor is based on the assuption that there is a linear relationship between the factors and the variable when coputing the correlations [7]. For soething to be called a factor it ust at least has three variables, although this depends on the design of the study [8]. A factor with variables is only considered reliable when the variables are highly correlated with each other (r > 0.70) and or fewer variables should be interpreted with caution. The saple size that should be considered for factor analysis should be at least 30 and the factors should have at least 5 to 0 observations each [9]. It is also recoended that a heterogeneous saple is used rather than a hoogeneous saple as hoogeneous saples lower the variance and factor loadings [0]. Also the correlation, r, existing between the variables ust be 0.30 or greater since anything lower would suggest a really weak relationship between the variables [8]. Generally speaking, cases with issing values are deleted to prevent overestiation [8]. There should be absence of ulticollinearity and singularity with the dataset. Variables that have issue with singularity and ulticollinearity should be reoved fro the dataset..3. Factor Analysis Model Factor analysis is a statistical ethod of investigating whether a nuber of variables of interest, say, XX, XX,, XX pp are linearly related to a nuber of unobservable variable called factors, say, ff, ff,, ff where is relatively less than p, that is, < pp. Given observations on p-variables, XX, XX,, XX pp with ean vector μμ and covariance atrix Ʃ. Noting that the factors ff, ff,, ff are independent of one another and of the error ters εε ii for ii =,, pp, and are such that, EE(ff ii ) = 0 and vvvvvv(ff ii ) =, and holding to the fact that each of the observed variable is a linear function of these factors and error ter, we have the factor analysis odel for a variable X = µ + λf+ λf + + λp fp + ε. () Noting that μμ = 0 when the observations are standardized we have () to be written as X = λk fk + ε () where λλ is the factor loading, such that, λλ is loading of the h variable on the kkkkh factor, ff kk are the underlying variables, the specific or unique factors is denoted by εε, =,,, pp and kk =,,. The factor loading give us an idea about how uch the variable has contributed to the factor; the larger the factor loading the ore the variable has contributed to the factor Haran []. Factor loading are very siilar to weights in ultiple regression analysis, and they represent the strength of the correlation between the variable and the factor [0]. Equation () can be expressed in atrix for as; where X = Λ ƒ + ε (3) ƒ = f, f,, f ; ε = ε, ε,, ε λ λ and Λ=. λp λ p The variance of () we give ( ) λ λ var X = var( f) + var( f) + + λp var( f p ) + var( ε ). p (4) Since it has been stated that the factor ff kk has unit variance, hence (3) can be written as var ( X ) λ λ λ = ψ (5) p var = k + ψ ( X ) λ where λλ is the kth cuunality, and ψ the th specific variance. The covariance of X denoted by Ʃ can be expressed as: Since = EX ( µ )( X µ ). E ( µ ) = 0 = E( X)( X). Recalling that XX = Ʌƒ + εε we have that ( )( ) ( ƒ )( ƒ ) = E X X = E Λ + ε Λ + ε (6) ( ) =ΛE ƒƒ Λ+ E( εε ) ( f ) =Λvar Λ+ var( ε ). Since the vvvvvv(ƒ) =, we have that Where = ΛΛ + Ψ (7) ψ 0 0 Ψ= ψ. 0 0 ψ p

3 Aerican Journal of Applied Matheatics and Statistics Estiation of Factor Score Factor score are the score of each of the variables on each factor. To copute the factor score for a given variable of a given factor, one takes the standardized score on each variable ultiplied by the corresponding factor loading of the variable for the given factor, and sus these products. A nuber of ethods are being used to estiate the factor scores, for this research work, the ethod of ordinary least squares will be eployed. Using factor odel fro (3) above, we have that ε = X Λ f. (8) Squaring both sides of (8) and taking suation gives ε ( ƒ ) k = X Λ k Miniizing the above equation with respect to ƒ kk and setting the derivative to zero gives ( X Λfk) Λ= 0 since 0, then the expression becoes This then gives ( X fk) Λ Λ= 0. ΛXˆ ΛΛ ˆƒ. Dividing the equation above by and setting it in ters of ƒ kk we have ΛΛ ˆ = Λ Xˆ ƒ k Multiplying the equation above by inverse of the coefficient of ƒ kk we have that; ˆ k. k ( ) Xˆ ƒ = ΛΛ Λ. (9) Equation (9) can therefore be referred to as factor scores. The nuber of factors to be retained in the work will be deterined using scree plot and principal coponent analysis ethod..5. Factor Rotation Methods Factors are rotated for better interpretation of results since unrotated factors are abiguous. The goal of rotation is to attain an optial siple structure which attepts to have each variable load on as few factors as possible, but axiizes the nuber of high loading on each variable [8]. Ultiately, the siple structure attepts to have each factor define a distinct cluster of interrelated variables so that interpretation is easier. In this research work we considered four different rotational ethods which include equaax, veriax, quartiax, and orthoax (at gaa 0.00 and.00 values). Aong these are orthogonal and oblique rotations. Orthogonal rotation is when the factors are rotated 90 oo fro each other, and it is. assued that the factors are uncorrelated [,3]. Oblique rotation is when the factors are not rotated 90 oo fro each other, and the factors are considered to be correlated. Oblique rotation is ore coplex than orthogonal, since it can involve one of two coordinate systes: a syste of priary axes or a syste of reference axes [3]. Quartiax rotation - is an orthogonal technique that involves the iniization of the nuber of factors needed to explain each variable [7]. Veriax rotation - is an orthogonal technique that iniizes the nuber of variables that have high loadings on each factor and works to ake sall loadings even saller. Equaax rotation is orthogonal rotation invented with intention of sharpening soe properties of variax. It is self-adusted for the nuber of the being rotated factors. It tends to distribute variables (highly loaded) ore uniforly between factors than veriax does and thus further is less prone to giving general factors. Orthoax is a procedure for siple structure rotation of a atrix, that is, a two-way array. A loading atrix has perfect siple structure if each row has at ost one nonzero eleent. It is shown that there is an orthogonal rotation of an initial loading atrix that has perfect siple structure, and then orthoax rotation with 0 gggggggggg of the initial loading atrix will produce the perfect siple structure [4]. Gaa specifies the eber of the faily to use. Varying Gaa changes axiization of the variances of the loading fro colun (Variax) to rows (Quartiax). 3. Acadeic Perforance of Pupils Factor analysis of pupils perforance is a for of exploratory ultivariate analysis that is used to either reduce the nuber of variables in a odel or to detect relationship aong variables affecting acadeic perforance of pupils. Many studies have been conducted in the area of acadeic perforance and those studies identified soe nuber of factors that affect acadeic perforance of pupils at priary school, students at college and even at university level. According to Epatein et al., [5], Wing et al., [6], Uphoff and Gilore [7], Anene [8], acadeic perforance has been known to be affected by a nuber of factors such as learning style, age, gender, teacher factor, location, econoic status of parents, and any others. Coley [9] found that feales scored higher than ales in reading and writing across different ethnic and age group. Also Alton-Lee and Praat [0] and Mullis et al [] noted that educational statistics have indicated that feales outperfored ales at all levels of the school syste, attaining school and post school qualification and attending university at higher nubers. De Meis and Steas [] and Diwtz and Wilson [3] found no significance relationship between age and acadeic perforance. In contrast, La Paro and Pianta [4] and Crosser [5] presented evidence that older children fare better acadeically than their younger age appropriate peers. According to Kibra and Kikooti [6], the language spoken at the hoes of the people of high class is often the ediu of instruction in school; hence children fro

4 Aerican Journal of Applied Matheatics and Statistics 67 this class have advantage over children who speak their other tongue at hoes. 4. Methodology For proper and effective analysis of this research work, the Federal University Wukari Staff School is selected as a case study. The data used is a secondary data collected fro the school; it is on the outcoe of grades scored by pupils of the school through the terinal exaination conducted by the school. It is collected fro priary one to priary five by a rando selection of ters fro the 05/06 acadeic session. The saple of size 9 was randoly selected fro each class aking a total of 45 observations. The inforation on the factors and observed variables entioned above were collected fro a rando selection across the classes of the school. In this study, the pupils scores per 00 percent in selected subects which are English Language, Matheatics, Social Studies, Religious Studies, Verbal Reasoning, Quantitative Reasoning and Creative Arts are the observed correlated variables and their age, gender and location are the factors. 5. Result and Discussion Fro the correlation atrix of the entire data we observed the variables are correlated and that the correlations aong the variables are relatively fair, ranging fro to 0.76 which indicates no singularity proble in the data. Fro the Bartlett s test for the hoogeneity of variance we observed the Kaiser-Meyer-Olkin value of 0.85 which falls within the range of superb ( ) according to Hutcheson and Sofroniou [7], which iplies that the variances across the variable are not equal. In this regard, this calls for the use of factor analysis. Fro the Table of counalities which easures the percentage of variance explained by all the coponents before and after extraction we observed that 75.7% of the variance associated with English language was extracted; 75.8% of the variance associated with Verbal reasoning extracted; 8.0% of the variance associated with Social studies extracted and so on. The extracted counalities is the percentage of variance in a given variable explained by the factors that are extracted, which are norally fewer in nuber than the original variable. Fro the scree plot and Table of Total variance explained, we observed that three out of seven factors accounted for 85.06% of the total variance in the data, hence, it can be seen that three factors are responsible for the variation in the original data used. A factor s eigen value ay be coputed as the su of its square loading for the entire variable. Fro the unrotated factor loadings we observed that Social Studies, verbal attitude, English language and creative art have high loading in the three selected factors. This goes a long way to prove that location and econoic class of a child have influence in his acadeic perforance. It also tells us that art subects are ain subects that ainly deterine the pupils perforance at the end of exaination. When we considered equaax, veriax and orthoax (at gaa =.00) rotations we observed that creative act, religious studies and quantitative reasoning are highlighted by the three selected factors, which goes a long way to indicate that age has a lot to do with the pupils perforance. Fro quartiax rotation and orthoax rotation (at gaa = 0.00) we observed that verbal attitude, English language, social studies, creative art, and religious studies are subects that deterine the position of pupil at the end of exainations. Since at priary school ale pupils are good in creating thing like constructing of kits, toy car, catapults and feale pupils are ore social we can deduced fro this that gender of a pupil has influence in his acadeic perforance. These discoveries are in line with what we have in the literature under acadeic perforance of pupils. References [] Yong, A. G. and Pearce, P. (03). A beginner s guide to factor analysis: Focusing on exploratory factor analysis, Tutorials in quantitative ethods for Psychology, 9(): [] Richard, A. and Dean, W. (99). Applied ultivariate analysis, Pretence Hall Englewood Cliffs, New Jersey 0763 [3] Karen, G. (07). Confiratory Factor analysis, Webiner, [4] Ballester, M.A., Linguraru, M.G., Aguirre, M.R., and Ayache, N. (004). On the adequacy variability, Epidaure Research Proect 004 route deslucioles BP3, Sophia Antipolis, 0690, France. [5] Child, D. (006). The essentials of factor analysis in behavioral and life science, New York, NY: Plenu Press. [6] Field, A. 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