Vahedi, Bevrani, Farrokhi Original Article A Confirmatory Factor Analysis of the Structure of Statistics Anxiety Measure: An examination of four alternative models Shahram vahedi, PhD1 Farahman Farrokhi, PhD2 Hossein Bevrani, PhD3 1 Department of Educational Psychology, Faculty of Education and Psychology, University of Objective: The aim of this study is to explore the confirmatory factor Tabriz, Tabriz, Iran analysis results of the Persian adaptation of Statistics Anxiety Measure 2 English Departments, University (SAM), proposed by Earp. of Tabriz Method: The validity and reliability assessments of the scale were 3 Department of Statistics, performed on 298 college students chosen randomly from Tabriz Faculty of mathematical sciences, University in Iran. Confirmatory factor analysis (CFA) was carried out to university of Tabriz determine the factor structures of the Persian adaptation of SAM. Results: As expected, the second order model provided a better fit to the Corresponding author: data than the three alternative models. Shahram Vahedi, PhD Conclusions: Hence, SAM provides an equally valid measure for use Assistant Professor Department among college students. The study both expands and adds support to the of Psychology, Department of existing body of math anxiety literature. Psychology, 29 bahman, The Tabriz University, Iran. Keywords: Anxiety, Psychometrics, Questionnaire, Statistical factor analysis Tel: 98 411-3392090 Fax: 4113356009 Iran J Psychiatry 2011; 6:92-98 Email: vahedi117@ yahoo.com F or many years, psychologists have been interested oreover, statistics anxiety is situation-specific, inasmuch as the symptoms only emerge at a particular in finding variables that can predict academic time and in a particular situation–specifically, when performance (AP). In recent years, research on the learning or applying statistics in a formal setting (4 and relationships between personality and AP has not only 2). analyzed the general relationships between the two Research indicates that statistics anxiety is a variables but has also focused on the relationships multidimensional construct (5, 6 and 2). Using factor between anxiety and performance in specific academic analysis, Earp (7) identified five components of domains. As a result, several authors have investigated statistics anxiety, namely: (a) anxiety, (b) performance, the predictive power of personality on performance in (c) attitude towards class, (d) attitude towards math, statistics courses. and (e) fearful behavior. It has been estimated that as many as 80% of graduate A growing body of research has documented a students experience uncomfortable levels of statistics consistent negative relationship between statistics anxiety, and statistics examinations are more anxiety- anxiety and course performance (8). In fact, statistics inducing than other types of examinations (1). anxiety has been found to be the best predictor of Statistics anxiety may even hinder a student from achievement in research methods courses (9) and completing a degree or deter a talented student from statistics courses (10). Moreover, a causal link between thinking about a career as a professor (2). Identifying statistics anxiety and course achievement has been individuals suffering from statistics anxiety and established. In particular, Onwuegbuzie and Seaman gaining a better understanding of the domains that (11) found that graduate students with high levels of contribute to such anxiety is a start to addressing the statistics test anxiety who were randomly assigned to a problem of statistical illiteracy today. statistics examination which was administered under Statistics anxiety has been defined as anxiety that timed conditions tended to have lower levels of occurs because of encountering statistics in any form performance than did their low anxious counterparts and at any level, involving a complex array of who took the same test under untimed conditions. emotional reactions (apprehension, fear, nervousness, Earp (7) established an instrument named ‘SAM’ to panic, and worry) that hinder the learning process (3). measure Statistics Anxiety in a community college. 'SAM' had high internal consistency reliability Iranian J Psychiatry 6:3, Summer 2011 92 Published by "Tehran University of Medical Sciences"(www.tums.ac.ir) Factor Analysis of the Statistics Anxiety Measure (Cronbach’s coefficient alpha = 0.82-0.95) and provided with a debrief sheet following completion. construct validity. SAM needs to be more adequately All participants took part on a voluntary basis and were validated because counselors have used it extensively. not remunerated for participation. Based on exploratory factor analysis (EFA) in Earp (7), in this study, we tested four models. Our research Data analysis questions were as follows: The analyses addressed two main questions. First, Do statistics anxiety items generated to reflect the five which existing factor structure (one, four and five identified domains (Anxiety, Performance, Attitude factor structures) provides an acceptable measurement Towards Class, Attitude Towards Math, and Fearful model for the 44-item SAM? To address this question, Behavior factor) fit appropriately into the five CFA was used to impose each of the three factor domains? structures on two data sets to evaluate each model’s Answering this question required the evaluation of the goodness-of-fit. Second, is there measurement fit of measurement models to SAM data. invariance with respect to gender? To address this Does the Statistics Anxiety Measure provide adequate question, multigroup CFA was used to test hypotheses evidence for reliability and validity? Estimation of about the invariance of the 41-item SAM across males reliability was performed under the framework of and females. One-way analysis of variance (ANOVA) confirmatory factor analysis (CFA). was also used to compare gender differences on the subscale of SAM. Data was analyzed using PASW Materials and Method Statistic18 and AMOS 16 (13 and 14). PASW was used to analyze descriptive statistics and the reliability Participants of the SAM. AMOS was used to perform the CFAs of The SAM was administered to 300 undergraduate the SAM analyzing the fit of models and its respective students (133 males and 165 females) chosen from parameter estimates in two distinct stages. different disciplines of human sciences at Tabriz In stage 1, the four models were subjected to a University in Iran who enrolled in entry-level statistic maximum-likelihood CFA using AMOS 16. Model 1 courses and voluntarily participated in the study. The specified a single factor model, the factor being sample consisted of 133 males and 165 females. Statistics Anxiety. We eliminated 9 items with non- College research examination Board approved the significant factor loadings. Model 2 specified a research protocol. correlated four-factor model with six items loading on the Performance factor (items y29-y34), eleven items Assessment Measures loading on the Anxiety factor (items y1-y11), eight Statistics Anxiety Measure (7): The 43 items of this items loading on the Attitude towards math factor scale are rated on a 5-point scale ranging from 1= (items y21 to y28), and seven items loading on the strongly disagree to 7= strongly agree (higher scores Attitude towards class factor (items y12 to y20). reflect greater statistics anxiety; see Appendix A for Model 3 specified a correlated five-factor model with the list of items). SAM comprises of five discrete seven items loading on the Anxiety factor (items y1- subscales: Anxiety, Performance, Attitude towards y7), six items loading on the Performance factor (items class, Attitude towards math, and Fearful behavior. The y29- y34), four items loading on the Fearful behavior English versions of the scale show a multidimensional factor (items y8 to y11), nine items loading on the structure for students, and have good construct, and attitude towards class factor (items y12 to y20), and discriminate validity (7). The internal consistent eight items loading on the attitude towards math factor reliability of the overall scale (α = 0.93) as well as sub- (items y21 to y28). Model 4 specified the same factor scales generally ranged from high to excellent (α = structure as Model 3 but included a second-order factor .82– .95). labeled Statistics Anxiety. This model was used to The Persian version of the SAM was developed using determine the existence, or robustness, of the five first- the standard back-translation technique (12).The first order factors in the presence of a general factor. The author initially translated the SAM into Persian, and an models are shown in Figure 1 and 2. 1. independent translator unaffiliated with the study then translated this version back into English. Minor differences that emerged during this process were Results resolved between the translators. Between-Group Differences In order to examine possible between-group Procedure differences in responses to the SAM, we ran a one-way All participants were recruited opportunistically using analysis of variance (ANOVA) with the subscales of a cluster-sampling technique initiated by three data SAM (Anxiety factor, Performance factor, Fearful collectors. All participants completed paper-and-pencil behavior factor, the attitude towards class factor, versions of the questionnaire anonymously, and attitude towards math factor) as the dependent variable returned the questionnaires to their contact person. All and participants' sex as independent variable. data were treated confidentially, and participants were Iranian J Psychiatry 6:3, Summer 2011 93 Published by "Tehran University of Medical Sciences"(www.tums.ac.ir) Vahedi, Bevrani, Farrokhi Figure 1. Alternative factor models of Statistics Anxiety Measure: Model 1, Model 2 Iranian J Psychiatry 6:3, Summer 2011 94 Published by "Tehran University of Medical Sciences"(www.tums.ac.ir) Factor Analysis of the Statistics Anxiety Measure Figure 2. Alternative factor models of Statistics Anxiety Measure: Model 3, Model 4 Iranian J Psychiatry 6:3, Summer 2011 95 Published by "Tehran University of Medical Sciences"(www.tums.ac.ir) Vahedi, Bevrani, Farrokhi Results showed participants’ sex was significantly solutions”‘ I do not expect to enjoy this class ’I expect related to the attitude towards class factor, F (1, 298) this class to be boring, “My ability to calculate =4.45, p< .05) such that female student had lower statistics will not affect my chances of getting a job in scores (M=20.52, SD=15.37) than male student (M= my chosen field”, “Taking this class will have little 18.75, SD =13.57). However, participants’ sex was not impact on my life” “ I lack motivation to learn or significantly related to Anxiety, Performance, Fearful continue learning statistics”, “There is no room to be behavior, attitude towards math (p>.05). In other creative in statistics”, and “how satisfied do you think words, female students reported more negative attitude your child has felt about looks and appearance”. On the towards class than male students. basis of the criteria associated with RMSEA and CFI, To evaluate the goodness-of-fit of four alternative IFI, TLI, the four, five -factor and second order models measurement models for the SAM, CFA was first run provide a better fit of the data than the one-factor for a one-factor solution in which all 34 items loaded model. In other words, they account for more variance on to a single general strengths factor (Model 1) and than the one-factor model. subsequently run for other models suggested by Earp We also directly compared the models with the Δχ2 (7). statistics. Both statistics directly compare the fit of the In order to determine which of the four proposed models after adjusting for differences in the degrees of models is the `best' model we can use both statistical freedom. In every case, the Δχ2 was significant at .001. criteria and information from the parameter estimates These results again strongly support the superiority of from each of the models. The results of the CFAs for the second order model over the one factor and four- each model are shown in Table 1. In all the analyses, factor model; thus, this model was considered optimal. the chi-square goodness of fit statistic is large and Therefore, Model 4 is preferred to Model 3 because of significant beyond the 0.001 level, rather than being parsimony; it provides an adequate description of the small and associated with a high probability, which sample data, and provides a better description than the would indicate a close fit between model and data. three alternative models. However, this statistic is sensitive to sample size and Descriptive statistics and factor correlations for second does not provide a realistic test of the fit of models order model are reported in Table 3. (16). In model 1, items y29-y34 did not get hold of a magnitudes of the factor loadings to be equal for male statistical significant loading (.40). Following the and female students, and the other omitting this removal of these factors loadings, indices for this invariance constraint . model improved. The results of the initial estimation of Table 2 presents the fit statistics for the models. the one factor model did not provide a satisfactory Several fit indices were examined to evaluate the result with a chi-square value of 3473.316 (df =495), overall fit of each model: χ2 tests the hypothesis that an which was significant at the P < .001 level. Other fit unconstrained model fits the covariance or correlation indices revealed a low fit (RMSEA =.14; TLI=.53; matrix as well as the given model; ideally values CFI=.56; IFI = .56). To justify a unidimensional should not be significant); Comparative Fit Index (CFI; construct, we compared the four-factor model with a 15; comparison of the hypothesized model with a unidimensional model. model in which all correlations among variables are In accordance with Bentler and Bonett (15), data from zero, and where values around .90 indicate very good modification indices in models of four, five factors and fit; Root-Mean-Square Error of Approximation second order suggested that six paths reflecting (RMSEA; values of .08 or below indicate reasonable fit covariance be added between error terms to improve for the the fit of the model that was obtained. These paths model; Tucker-Lewis index (TLI) and the incremental involved pairs of items that shared variance from fit index (IFI), with values close to .95 being indicative variance accounted for by various factors. These items of good fit; Akaike Information Criterion (AIC), AIC included “Developing conclusions based on close to zero reflects good fit and between two AIC mathematical solutions”, “Solving mathematical measures, the lower one reflects the model with the equations”, “Calculating probabilities” and better fit) ( cited in 14) “Developing conclusions based on mathematical Table 1. Goodness-of-fit statistics and their Comparisons for four alternative measurement models SAM Models and χ2 df χ2/df CFI TLI IFI RMSEA AIC χ2 difference Comparisons Model 1 3694.43 527 7.01 .53 .50 .53 .14 3898.431 Model 2 1539.28 515 2.99 .85 .83 .85 .8 1767.282 Model 3 972.727 511 1.90 .93 .93 .93 .055 1208.727 Model 4 982.54 52 1.90 .93 .93 .93 .055 1208.536 M1–M2 2155.15* M2– M3 544.55* M3–M4 9.81* Note: Model 1= One factor; Model 2=four-factor model; Model 3= five-factor model; Model 4= second order * P < 0.001. Iranian J Psychiatry 6:3, Summer 2011 96 Published by "Tehran University of Medical Sciences"(www.tums.ac.ir) Factor Analysis of the Statistics Anxiety Measure Table2. Reliability Estimates and factor correlations Table 3: Factor correlations and Reliability from the second order model of statistic anxiety measure 1 2 3 4 5 6 α Mean Std. Deviation 1) statistic anxiety .90 84.08 14.65 2) performance -.39 .86 22.37 4.02 3) fearful behavior .90 -.35 .85 8 2.67 4) anxiety .48 -.29 .43 .91 15.86 5.33 5) attitude towards class .91 -.35 .81 .44 .89 20.22 5.56 6) attitude towards math .69 -.27 .62 .33 .62 .95 17.63 6.53 All correlations were significant p < 0.05. On the this basis of this, Model 4 should be preferred Discussion to Model 3 based on parsimony, ; it provides an The primary purpose of this study was to use adequate description of the sample data, and is confirmatory factor analytic techniques in a sample of provides a better description than the four alternative young adult college students to explore the fit of the models. We reported the standardized factor loadings five-factor model of the SAM proposed by Earp. The for Models 3 and 4 in Table 2 (estimates for Model 4 in second aim of the present study was to evaluate the parenthesis). For Models 3 and 4 all the factor loadings psychometric properties of a Persian version of the are positive, high and statistically significant. The SAM and to test measurement invariance across sex. factor correlations for Model 4 are reported in Table 2. The present study showed that there was a significant gender difference for attitude towards class factor but Iranian J Psychiatry 6:3, Summer 2011 97 Published by "Tehran University of Medical Sciences"(www.tums.ac.ir) Vahedi, Bevrani, Farrokhi not for other subscales of SAM. This is consistent with 2. Onwuegbuzie, A. J, DaRos D, Ryan, J. M. The the research that reported significant gender components of statistics of statistics anxiety: a differences in women who experienced higher levels of phenomenological study. Focus on Learning Problems in Mathematics 1997; 19: 11–35. statistics anxiety (4 and 11). On the other hand, the 3. Onwuegbuzie AJ, Daley C. 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Academic procrastination statistic anxiety symptoms indicated that the lower and statistics anxiety. Assessment and order factor of Fearful behavior and attitude towards Evaluation in Higher Education 2004; 29: 1-19. class made the largest unique contribution to the 9. Onwuegbuzie AJ, Slate J, Paterson F, prediction of statistic anxiety measure. Watson M, Schwartz R. Factors associated In addition, results showed that the SAM has high with underachievement in educational internal consistency, with Cronbach are a reaching research courses. Research in the Schools 0.90. These data are further supported by the parameter 2000; 7: 53- 65. 10. Fitzgerald SM, Jurs S, Hudson LM. A model estimates of the CFAs, and is generally consistent with predicting statistics achievement among previous works showing that SAM has high internal graduate students. Coll Stud J 1996;30: 361- consistency (e.g., 7). 366. Several limitations of this study are as follows: First, 11. Onwuegbuzie AJ, Seaman M. The effect of the construct validity of the results reported in this time constraints and statistics test anxiety on article is mainly derived from the student sample (the test performance in a statistics course. J Exp University of Tabriz). Further research is necessary to Educ 1995; 63: 115–124. replicate the results scale in other geographical settings 12. Breslin RW. Back-translation for cross-cultural to validate the Persian version of SAM. A second research. J Cross Cult Psychol 1970; 1:185– 216. limitation is the relatively small sample size. Due to 13. Arbuckle JL. Amos™ 18 User’s Guide. limited sample size, structures found in this study may Chicago: Amos Development Corporation; not hold in future administrations given larger sample 2009. sizes. 14. Byrne BM. Structural equation modeling with AMOS: Basic concepts, applications, and Acknowledgements programming, 2th. New York: Routledge; 2010. 15. Bentler PM, Bonett DG. 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