# Understanding and Using Factor Scores: Considerations for the Applied Researcher

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6 Practical Assessment, Research & Evaluation, Vol 14, No 20 Page 6 Table 1: Advantages and Disadvantages of Refined Methods to Compute Factor Scores Factor scores are correlated to the estimated factor. (Validity) uncorrelated to other orthogonal factors. (Univocality) uncorrelated to factor scores from other orthogonal factors. (Correlational Accuracy) unbiased estimates of factor score parameters. Regression Maximal No No No scores Bartlett factor High Yes No Yes scores Anderson & Rubin factor scores Acceptable No Yes No A second, and paramount, consideration when creating factor scores using refined methods is the problem of indeterminacy of the scores (see Gorsuch, 1983 and Grice, 2001 for detailed explanations). Indeterminacy arises from the fact that, under the common factor model, the parameters are not uniquely defined, due to the researcher s choice of the communality estimate. This means that there is not a unique solution for the factor analysis results and, theoretically, an infinite number of solutions could account for the relationships between the items and factor(s). Therefore, it also follows that the factor scores are not uniquely defined (Grice, 2001). As noted by EFA researchers, the problem of indeterminacy arises with most factor extraction techniques found under the common factor model. EFA methods which have a unique solution (i.e., determinant), such as principal component analysis and image common factor analysis, and their resulting factor scores are thought to be unique. Researchers interested in using factor scores need to be aware of the problem of indeterminacy because it could impact not only the factor scores but also the validity of decisions that rely upon these scores (Grice, 2001). For example, under some conditions, rankings of cases in a data set may vary widely based on different methods to compute factor scores, leaving a researcher unsure as to which ranking to trust. Grice (2001) suggests researchers examine the degree of indeterminacy in their factor solutions using three different measures: (1) validity evidence of correlational relationships between factor scores and factor(s); (2) univocality- the extent to which factor scores are adequately or insufficiently correlated with other factors in the same analysis; and (3) correlational accuracy which reports the extent to which correlations among the estimated factor scores match the correlations among the factors themselves. Table 1 includes these three measures to illustrate differences among the refined methods. As noted by Grice, a high degree of indeterminacy may suggest that a researcher re-examine the number of factors needed or to disregard (or at least, use cautiously), scores from factors which illustrate questionable results. Interested readers should refer to Grice (2001) for an in-depth discussion of indeterminacy, illustrations of how to evaluate if it is present, and a link to SAS programs to examine solutions for indeterminacy. While popular software programs do not yet routinely provide all of these tests, a test for validity using the multiple correlation value is routinely available under the regression method of SPSS when orthogonal factors are requested and with SAS for both orthogonal and oblique solutions. Higher values of the multiple correlation suggest greater validity evidence, meaning greater determinacy of the factor scores. This information is very important for researchers to consider because EFA is an internally driven procedure, and thus, results may be sample specific (Costello & Osborne, 2005). Given the problems with EFA, researchers creating factor scores are urged to replicate the factor structure to ensure that the solution is stable before creating and using factor scores. Factor scores may also be examined for indeterminacy using the procedures described above. A third issue deals with data quality. Once factor scores are obtained, this set of data requires screening and examination to ensure that distribution(s) of factor scores meet assumptions required by the statistical methodology to be used for follow-up testing. While recommendations for data screening and checking assumptions are common before beginning statistical

9 Practical Assessment, Research & Evaluation, Vol 14, No 20 Page 9 Appendix 1 (continued): Refined Methods to Construct Factor Scores Method Procedure Advantages Considerations Regression Scores Multiple regression used to estimate (predict) factor scores. Default procedure to compute factor scores in SAS and SPSS packages; also available in R. Factor scores are standard scores with a Mean =0, Variance = squared multiple correlation (SMC) between items and factor. Procedure maximizes validity of estimates. Factor scores are neither univocal nor unbiased. The scores may be correlated even when factors are orthogonal. Bartlett Method of producing factor scores is similar to regression method, but produces estimates that are most likely to represent the true factor scores. Can be computed using SPSS or R statistical packages. Anderson-Rubin Method of producing factor scores is similar to Bartlett, but allows factor scores to be uncorrelated when factors are orthogonal. Can be computed using SPSS. Factor scores are standard scores (Mean =0, Variance = SMC) Produces unbiased estimates. In an orthogonal solution, factor scores are not correlated with other factors (univocality). Procedure produces high validity estimates between the factor scores and factor. Factor scores have a mean of 0, have a standard deviation of 1. When the factors are orthogonal, factor scores are uncorrelated as well (correlational accuracy). Factor scores have reasonably high correlations with their estimated factor (validity). The scores may be correlated even when factors are orthogonal. Factor scores may be correlated with the other orthogonal factors (i.e.,. not univocal). Factor scores are not unbiased.

10 Practical Assessment, Research & Evaluation, Vol 14, No 20 Page 10 Appendix 2: Computation Procedures for Refined Factor Scores Factor Scores Formulae Where Regression Scores n number of observed variables m number of factors Orthogonal Factors: Oblique Factors: Bartlett F B B 1 xm F 1 xm = = = Z R R 1 xn 1 1 B A A Φ mxm = Z1 xnu A( A' mxn U A) the row vector of m estimated factor scores Z the row vector of n standardized observed variables B the matrix of regression of weights for the m factors on the n observed variables R -1 the inverse of the matrix of correlations between the n observed variables A and A pattern matrices of loadings of n observed variables on m factors or components Ф the correlation matrix of the m factors U -2 the inverse of a diagonal matrix of the variances of the n unique factor scores F Anderson-Rubin F G 1xm = Z = X 1xn U Λ 2 D A X G 1 / 2 X and X matrices of n x n eigenvectors Л D the n x n matrix of eigenvalues G - the matrix product of eigenvalues and eigenvectors G 1/2 the inverse of the symmetric square root of G Formulas for refined factor methods were taken from Hershberger, S. L. (2005).

11 Practical Assessment, Research & Evaluation, Vol 14, No 20 Page 11 Citation DiStefano, Christine, Zhu, Min & Mîndrilă, Diana (2009). Understanding and Using Factor Scores: Considerations for the Applied Researcher. Practical Assessment, Research & Evaluation, 14 (20). Available online: Corresponding Author Christine DiStefano, Associate Professor Department of Educational Studies 137 Wardlaw Hall Columbia, SC distefan [at] mailbox.sc.edu

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