Using Factor Analysis to Study the Effecting Factor on Traffic Accidents

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1 Using Factor Analysis to Study the Effecting Factor on Traffic Accidents Abstract Layla A. Ahmed Deartment of Mathematics, College of Education, University of Garmian, Kurdistan Region Iraq This aer is concerned with the study of the factors that have significant effects on contracting traffic accidents. Another aim is to know whether the influence of each variable is indeendent or has a relation to that of other once. The samle of the research included (150) traffic accidents, the samle was collected from (Directorate of Traffic / Garmian ) in the eriod ( ). Measures of nine variables have been taken. And rincial comonent method was used on the studied variables data to secify the imortance of these variables. As well as, Varimax method was used to rotate the axis to get an easier and more secific result. The results have showed that the following variables have clear influences but their imortance is different in terms of influencing on traffic accidents. We find that the variables (driving license) and (tye of accident) have comes in first rank, while the other factors comes in later rank. Keywords: Traffic accidents, Factor analysis, Princial comonent method, Rotation axes 1.1. Introduction Traffic accidents are considered the most imortant tyes of accidents occurring in the country, which has become necessary to work to find solutions and suggestions to them. The statistical analysis of factor that influences the traffic accidents was carried out by using factor analysis method. Factor analysis is a branch of multivariate analysis rocedure that attemts to identify any und relying factors that are resonsible for co variation among grou indeendent variables. The goals of a factor analysis are tyically to reduce the number of variables used to exlain a relationshi or to determine which variables show a relationshi [9]. Factor analysis originated in sychological theory. Based on the work under taken by Pearson (1901) in which he roosed a method of rincial axes, Searman (1904) began research on the general and secific factors of intelligence [14]. The term factor analysis was first introduced by Thurston (1931)[9]. Lewbel (1991) and Donal (1997) used the rank of a matrix to test for the number of factors, but these theories assume either N or T (the cross-section dimension and the time dimension, resectively) is fixed. Forni, Hallin, Lii and Reichlin (000) suggested a multivariate variant of the Akaike information criterion (AIC) but neither the theoretical nor the emirical roerties of the criterion are known [ 3]. This study aims at determining the factors that have significant effects on traffic accidents. Another aim is to know whether the influence of each variable is indeendent or has a relation to that of other once. 15

2 1.. Factor Analysis The factor analysis model exresses each variable as a linear combination of underlying common factors f 1, f,, f m, with an a comanying error term to account for that art of the variable that is unique, the model is as follows [][1][13]: X n = μ 1 + A m F m 1 + U 1.. (1) Where: m: The number of common factors (m<). A: Loading of the jth variable on the factor. F: Common factors. U: Secific factors. µ: Mean of variables. In factor analysis we begin with a set of variables x 1, x,, x k. These variables are usually standardized so that their variances are each equal to one and their covariance are correlation coefficients [8]. Assume that each x i is a standardized variable, x i = (x i x i ) S i.. () E(x) = μ = 0, V(x) = I Model (1) can be written: X = AF + U The random vectors F and U are unobservable and uncorrelated. (3) E ( F E(F F) E(F U) ) ( F U ) = [ U E( U F) E( U U ] = [Φ m m 0 m ]. (4) 0 m Ψ Where: Φ : Symmetric matrix of factor variance and covariance. Ψ : Diagonal matrix of unique factor variances. Thus the covariance of x can be written as: E(XX) = Σ P P (5) Where Σ is a P P oulation covariance matrix. Σ = E(AF + U)(AF + U) 16

3 Σ = AE(FF )A + AE(FU ) + E(U F )A + E(U U ) (6) Since E(FF ) = Φ E(FU ) = E(U F ) = 0 E(U U )= Ψ Therefore Σ = AΦA + Ψ (7) 1.3. Basic Assumtions of Factor Analysis In factor analysis, we grou variables by their correlations, such that variables in a grou (factor) have high correlations with each other. Thus, for the uroses of factor analysis, it is imortant to understand how much of a variables variance is shared with other variables in that factor versus what cannot be shared. The total variance of any variable can be artitioned in to three tyes of variance [4]: a. Common variance: Is defined as that variance in a variable that is shared with all other variables in the analysis, denoted by h j. h j = a j1 + a j + a j a jm. (8) b. Secific variance (also known as unique variance) is that the variance associated with only a secific variable. This variance cannot be exlained by the correlations to the other variables but is still associated uniquely with a single variable. u j = b j + e j (9) Where: u j : Secific variance. b j : Secial variance to variable j. e j : Error variance. c. Error variance is also variance that cannot be exlained by correlations with other variables, but it is due to unreliability in data gathering rocess, measurement error, or a random comonent in the measured henomenon, denoted by e j. e j = 1 (h j + b j ) (10) 1.4. Commonalties Is the roortion of the variance of an item that is accounted for by the common factors in a factor analysis, denoted by h j. 17

4 h j = a j1 + a j + a j a jm h m j = 1,,., j = i=1 a ij, { i = 1,,.., m 0 h j 1 (11) Where a i reresent the weight factor for variable j Eigen value The standardized variance associated with a articular factor. The sum of the eigen values cannot exceed the number of items in the analysis, since each item contributes 1 to the sum of variances [1]. An eigen vector of the matrix A as a vector u that satisfies the following equation [6]: Au = λu When rewritten, the equation becomes: (A λi)u = 0 Where λ a scalar is called the eigen value associated to the eigenvector. (1) (13) 1.6. Princial Comonent Method Princial comonent is considered the most imortant stages in the factor analysis method, and working to transform the variables associated to the new variables uncorrelated with each other. The comonents are linear combinations weighted sums of the original variables [13]. Z i = PC i = a 1i X 1 + a i X + + a i X (14) PC i = j=1 a ji X j, i, j = 1,,.., (15) S = CDC (16) Where S is a samle covariance matrix and C is an orthogonal matrix constructed with normalized eigenvectors (c i c i = 1) of S as columns and D is a diagonal matrix with the eigenvalues λ 1, λ,, λ of S on the diagonal: λ 1 0 D = ( ) 0 λ (17) D = D 1 D 1 S = CDC = CD 1 D 1 C = (CD 1 )(C D 1 ) (18) This is of the form S = AÁ, but we do not define A to be CD 1 because CD 1 is P P, and we are seeking a A that is m with m <.We therefore define D 1 = diag(λ 1, λ,., λ m ) with the m largest eigenvalues (λ 1 > λ >. > λ m ) and C 1 = (c 1, c,., c m ) containing the corresonding eigenvectors. We then estimate A by the first m columns of CD 1, 18

5 1 A = C 1 D 1 = ( λ 1 c 1, λ c,., λ m c m ) (19) 1 Where A is m, C 1 is m, and D 1 is m m. The i th diagonal element of AÁ is the sum of squares of the i th row of A, or a i a i = m m j=1 a ij. Hence to comlete the aroximation of S in (16), we define ψ i = s ii j=1 a ij (0) And write S = AA + ψ (1) h i = m j=1 a ij () Which is the sum of squares of the i th row of A. the sum of squares of the j th column of A is the i th eigenvalue of S: i=1 a ij = i=1 ( λ j c ij ) = λ j i=1 c ij = λ j (3) Since the normalized eigenvectors (columns of C) have length 1. By equations (0) and (), the variance of the i th variable is artitioned into a art due to the factors and a art due uniquely to the variable: s ii = h i + ψ i = a j1 + a j + a j a jm + ψ i (4) Thus the jth factor contributes a ij to s ii. The contribution of the jth factor to the total samle variance, tr(s) = s 11 + s + + s, is, therefore, Variance due to jth factor = i=1 a ij = a 1j + a j +.. +a j.. (5) Therefore a ij i=1 tr(s) = λ j tr(s) (6) We can use standardized variables and work with the correlation matrix R. a ij i=1 tr(r) = λ j (7) Where is the number of variables Rotation Axes Simle Structure 19

6 Most of the rationale for rotating comes from Thurston (1947) and Cattell (1978) who defended its use because this rocedure simlifies the factor structure and therefore makes interretation easier and more reliable easier to relicate with different data samles [1],[11].Thurston (1947) first roosed and argued for five criteria that needed to be met for simle structure to be achieved[5]: a. Each variable should roduce at least one zero loading on some factors. b. Each factor should have at least as many zero loadings as there are factors. c. Each air of factors should have variables with significant loadings on one and zero loadings on the other. d. Each air of factors should have a large roortion of zero loadings on both factors. e. Each air of factors should have only a few comlex variables Orthogonal Rotation and Oblique Rotation An orthogonal rotation is secified by a rotation matrix denoted R, where the rows stand for the original factors and the columns for the new (rotated) factors. There are several methods for orthogonal rotation such as the varimax, Quartimax, Equimax and Orthomax [1],[11]. In oblique rotations the new axes are free to take any osition in the factor sace, but the degree of correlation allowed among factors is, in general, small because two highly correlated factors are better interreted as only one factor. There are several methods for orthogonal rotation such as the Quartimin, Promax, Procrustes The Kaiser -Varimax Method A oular scheme for rotation was suggested by Henry Kaiser in (1958). He roduced a method for orthogonal rotation of factors, called the varimax rotation [7], achieved by maximizing the sum of the variances of the squared factor loadings within each factor [] Number of Factors The Kaiser method roosed by Kaiser (1960) is erhas the best know and most utilized in ractice. According to this method, only the factors that have eigen values greater than one are retained for interretation [10].. Data Analysis and Results.1. Data Descrition The data that were used in this research is data from the statistical reort of traffic accidents recorded from (Directorate of Traffic / Garmian) in the eriod ( ). The extraction results of analyzes using the statistical rogram (SPSS V.) includes a set of data variables: X 1 : Age (< 30 years = 1, 30 years = ) X : Tye of comosite (car) (small car (taxi) =1, Bus =, Lorry=3) 130

7 X 3 : Tye of accidents (Cou =1, Collision =, Run over =3) X 4 : Accident time (Day =1, Night =) X 5 : Weather conditions (Rainy =1, Sunny =, Cloudy =3) X 6 : The lace of the accident (Inside the city =1, Outside the city = ) X 7 : Driving license (Yes =1, No =) X 8 : Due to the drinking (Yes =1, No =) X 9 : The cause of traffic accidents (Excess seed =1, Passing wrong =).. Analysis of the Results After analyzing the correlation matrix by rincial comonent method this method is the most widely used for determining a first set of the loadings and seeks values of the loadings that bring the estimate of the total communality as close as ossible to the total of the observed variances, it became clear the existence of five factors reresented by the number of egien values that greater than one. Where the extraction accounted for % of the total variance of the variables as shown in table (1). factors that Table (1) Initial Eigen values and Rotation Sums of Squared Loadings Initial Eigen values Rotation Sums of Squared Loadings % of Cumulative % of Cumulative Comonent Total Variance % Total Variance % Age Tye of comosite Tye of accident Time of accident Weather conditions lace of accident Driving license Due to the drinking Cause of accident The results in table (), refers to the comonents matrix which reresents the results of extraction factors after rotation calculated according the method of Varimax with Kaisers normalization. 131

8 Table () Rotated Comonent Matrix Rotated Comonent Matrix a Comonent Comonent Age Tye of comosite Tye of accident Time of accident Weather conditions lace of accident Driving license Due to the drinking Cause of accident Extraction Method: Princial Comonent Analysis. Rotation Method: Varimax with Kaiser Normalization. Rotation converged in 9 iterations. Conclusions The results have showed that the following factors that accounted for % of the total variance of the variables have clear influences but their imortance is different in terms of influencing on traffic accidents. 1. The First Factor This factor has a great imortance in influencing road accidents where he exlains (14.447) of the total variance, included two variables have the greatest imact on this factor which are driving license, tye of accident with comonents (0.868, 0.596) resectively.. The Second Factor This factor ranked second in terms of imortance in the interretation of the relationshi between the variables, where he exlains (14.018) of the total variance, this factor contains about three variables that includes: due to the drinking, the lace of the accident and age, with comonents (0.609, , 0.357) resectively. 3. The Third Factor This factor ranked third in terms of imortance in influencing road accidents, where he exlains (13.144) of the total variance, this factor contains only one variable tye of comosite, with comonent (0.864). 4. The Fourth Factor 13

9 This factor ranked fourth in terms of imortance in the interretation of the relationshi between the variables, where he exlains (1.371) of the total variance, this factor contains about two variables which contains: accident time and the cause of traffic accidents, with comonents (0.8, 0.538) resectively. 5. The Fifth Factor This factor ranked fifth and last in terms of imortance in the imortance in influencing road accidents, where he exlains (11.998) of the total variance, this factor contains one variable weather conditions, with its comonents (0.869). Recommendations 1. Militancy in giving licenses leadershi and increased attention to verify the fitness standards for drivers and accuracy of the medical examination rocess for those who want to get driving licenses.. Militancy in the activation of the traffic lows and increase the number of seed cameras. 3. Enter reorter traffic culture in different academic levels commensurate with each stage. References [1] Abdi, H. and Williams, l. j.,(010), Princial Comonent Analysis, Wiley Interdiscilinary Reviews, Comutational Statistics, Vol., [] Afifi, A., Clark, V. A. and Susanne M.,(004), Comuter- Aided Multivariate Analysis, 4 th Edition, Chaman & Hall/CRC, London, New York. [3] Bai, J. and Serena, N., (00), Determining the Number of Factor in Aroximate Factor Models, Journal of Econometrica, Vol.70, No.1, [4] Brace, N., Kem, R. and Snelgar, R.,(006), SPSS for Psychologists, Versions 1and 13, 3 rd Edition, Palgrave Macmillan, New York. [5] Brown, J. D., (009), Choosing the Right Tye of Rotation in PCA and EFA, JALT Testing and Evaluation SIG Newsletter, Vol.13, No.3, 0-5. [6] Coombe, G., (006), An Introduction to Princial Comonent Analysis and Online Singular Value Decomosition. [7] Hair, J. F., and others, (006), Multivariate Data Analysis, 6 th Edition, Pearson rentice Hall, Uer Saddle River, Inc. [8] Hardle,W. and Simar, L., (003), Alied Multivariate Statistical Analysis, Method and Technologies, Version 9. [9] Lawley, D. N. and Maxwell, A. E., (196), Factor Analysis as a Statistical Method, Journal of the Royal Statistical Society, Vol.1, No.3,

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