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DOCUMENT RESUME TM 025 063 ED 395 042 AUTHOR Garbarino, Jennifer J. Alternate Ways of Computing Factor Scores. TITLE PUB DATE 27 Jan 96 24p.; Paper presented at the Annual Meeting of the NOTE Southwest Educational Research Association (19th, New Orleans, LA, January 25-27, 1996). Evaluative/Feasibility (142) PUB TYPE Reports Speeches/Conference Papers (150) MF01/PC01 Plus Postage. EDRS PRICE Algorithms; *Factor Analysis; *Regression DESCRIPTORS (Statistics); *Scores Parametric Analysis; *Variables IDENTIFIERS ABSTRACT All parametric analysis focuses on the "synthetic" variables created by applying weights to "observed" variables, but these synthetic variables are called by different names across methods. This paper explains four ways of computing the synthetic scores in factor analysis: (1) regression scores; (2) M. S. (3) the Anderson-Rubin method (T. W. Bartlett's algorithm (1937); Anderson and H. Rubin, 1956); and (4) standardized, noncentered factor scores. A description and illustration of the derivation and utility of factor scores in multivariate analysis were undertaken. In addition, an attempt was made to explain the concept that factor scores are synthetic variables, or weighted combinations of the observed scores, and that they are similar to those in regression. An appendix contains a Statistical Package for the Social Sciences program. (Contains 6 tables and 11 references.) (Author/SLD) *********************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. *********************************************************************** Factor Scores Ut DEPARTMENT OF EDUCATION "PERMISSION TO REPRODUCE THIS Once a/ &Imams* Reemecn end Improvement MATERIAL HAS BEEN GRANTED BY RESOURCES INFORMATION EDUCA CENTER (ERIC) TExiti/rEe 7-6,6649,e/4de) 60Cumnt nes been risseducgd as reeseved trom th PIKSOA Og aeganitilawn it Ontlifthng IngOS neve bow made to Improve MMOt Roproduaam qUalitt hams ot vow or woman* stated in this docu- TO THE EDUCATIONAL otlicill RESOURCES ment eo nol necestierrty tOresent INFORMATION CENTER OEFII positron or poPcT (ERIC)." Scores Alternate Ways of Computing Factor Jennifer J. Garbarino Texas A&M University BEST COPY AVAIIABLE Educational Research annual meeting of the Southwest Paper presented at the 19th 27, 1996. Association, New Orleans, LA, January ii 2 Factor Scores 2 Abstract "synthetic" rariables created by applying All parametric analysis focus on the synthetic variable scores are called by different weights to "observed" variables. These computing the synthetic The present paper explains four ways of names across methods. algorithm, (c) Anderson- ) regression scores, (b) Bartlett's scores in factor analysis: ( A description and illustration of Rubin, and (d) standardized, noncentered factor scores. in multivariate analysis was undertaken. the derivation and utility of factor scores that factor scores are synthetic variables, Additionally, an attempt to explain the concept yhat in egression was of the observed scores, and are similar to or weighted combinations undertaken as well. Factor Scores 3 understand that all parametric analyses are It is important for researchers to "synthetic" "observed" variables to create the correlational and invoke weights applied to variance-accounted-for effect the analysis and yield variables that are actually the focus of language we use Thompson, 1991). The conventional sizes (Fan, 1992; Knapp, 1978; of those attempting to gain an understanding obscure these realizations and confuses "equations" in call the same systems of weights parametric analyses. For example, we analysis, and "functions" or "rules" in discriminant regression, "factors" in factor analysis, "beta" analysis. We call the weights themselves "functions" in canonical correlation function in factor analysis, and "standardized weights in regression, "pattern coefficients" synthetic canonical correlation analysis. The coefficients" in discriminant function or in factor analysis, "discrimant "yhat" in regression, "factor scores" scores are called in canonical "canonical function (or variate) scores" scores" in discriminant analysis, and correlation analysis. in factor detail the synthetic scores computed The present paper explains in be of computing factor scores will analysis--factor scores. Specifically, four ways (b) Bartlett's algorithm, and (c) the explained. These include: (a) regression scores, 1983). A fourth alternativestandardized, Anderson-Rubin method (Gorsuch, 1993)--will also be explained. noncentered factor scores (Thompson, heuristically and Swineford (1939) will be used to A small data set from Holzinger 9 achievement variables are from a battery of 301 scores on illustrate all calculations. The analyzed were: tests. The nine tests I. Paragraph comprehension test 4 Factor Scores 4 2. Sentence completion test 3. Word meaning test 4. Speeded addition test 5. Speeded counting of dots in shape curved caps 6. Speeded discrimination of straight and 7. Memory of target words 8. Memory of target numbers 9. Memory of object-number association targets the matrix of association by finding the The purpose of factor analysis is to simplify separating these into a smaller set of underlying constructs between the variables and be directly observed or counted or factors. The factors are latent variables that cannot factors, which are interpreted with measured. Interest centers mainly on the common study, factor analysis indicated the reference to the observed variables. In the pt esent these examination of the observed variables indicated that presence of three factors. An being measured by the achievement tests-- fators represent three dimensions or constructs comprehension, speed, and memory. computation of a matrix of association An initial step in factor analysis is the study, a correlation matrix was used, coefficients from the raw data matrix. In the present matrix used. Examination of the correlation however, any matrix of association may be matrix with in Table 1, indicates a symmetric derived from the heuristic data set, presented redundant off-diagonal triangles (11). For any 9 rows and 9 columns, and with this R intercorrelation matrix, there is an inverse of symmetric matrix, such as a variable Factor Scores 5 1,, is called an Identity such that R,, RVXV1 = Ivx (Gorsuch, 1983). matrix called matrix you started with. The I matrix matrix, because any matrix times I equals the same algebra to multiplying by 1. When the symmetric in matrix algebra is equivalent in regular saying the variables are perfectly matrix is R, saying R is an I matrix is the same as the factor correlation matrix is always an uncorrelated. When factors are first extracted, factor (orthogonal = uncorrelated) rotation, the Identity matrix. After an orthogonal correlated) However, in an oblique (oblique = correlation matrix is still an Identity matrix. longer an Identity matrix. rotation, the factor correlation matrix is no Insert Table 1 About Here the variables that are and are Lot The interpretation of a factor is based upon uncorrelated, one matrix summarizes the related to that factor. When the factors are matrix. factor pattern/structure coefficient factor-variable relationships and is called the the beta weights and structure coefficients However, just as in regression, in some cases be interpreted (Thompson, when the two are not equal, both must are the same, but each variable perfectly uncorrelated, the contribution of 1992). When the factors are not several factor matrices are will differ depending upon which of to the interpretation coefficient matrix and the factor pattern examined, thus both the factor structure bivariate factor structure matrix presents the coefficient matrix must be interpreted. The latent variable variable scores of the n people with the correlations between the observed 6 Factor Scores 6 for the people. This is the most commonly used matrix or factor scores of the same n interpretation of the factors. calculate factor scores. The factor pattern Matrix is the weight matrix used to combinations of the observed scores of Factor scores are synthetic variables, or weighted regression. In regression, you use the beta the n people, and are similar to yhat scores in yhat scores. In factor analysis, if the weights, not the structure coefficients, to compute equal, you must use the pattern coefficients to pattern and structure coefficients are not the f the predicted scores of the n individuals on compute factor scores. Factor scores are only tne unique contribution of each factor factors. The factor pattern matrix represents variables removing redundancies among any correlated to each variable, arbitrarily used to relate the factors to other originating from the same factor. Factor scores are each variable. variables. They reflect the impact of the factor on would have the following The best procedure for computing factor scores characteristics (McDonald & Burr, 1967): high correlation with the factors First, the cornmon factor scores should have a factor scores should be they are attempting to measure. Second, common Third, in the case of conditionally unbiased estimates of the true factor scores. from one factor should have zero orthogonal factors, the factor scores estimated factors are orthogonal, the correlations with all other factors. Fourth, if the correlate zero with each other. (Gorsuch, 1983, common factor scores should p. 260) - "wigAlgr,g7wFrwrirgry-7-77-7=-,-. Factor Scores 7 by the the number of variables pattern/structure matrix is The rank of a factor variables by the matrix is the number of The rank of a correlation number of factors, v x f. number of factors correlation matrix is the The rank of a factor number of variables, v x v order to multipy matrices, Gorsuch (1983), in factors, fx f. According to by the number of of rows in the be equal to the number in the left matrix must the number of columns resulting matrix in a "conformability"). The right (called adjacent matrix to the and columns in the leftmost matrix the number of rows multiplication has rows equal to heuristic data In the case of the rightmost matrix. of columns in the equal to the number (n) synthetic scores of 301 matrix will indicate the study, the factor score set used in this Table 2. factors, as noted in individuals on thc, 3 (f) Here Insert Table 2 About (a) the in the present study: factor scores are explored Four ways to compute BTF score. The Anderson-Rubin and (d) the (b) Bartlett, (c) Regression Algorithm, Appendix A. A factor commands presented in with the SPSS analysis was performed each of the variables combination of the scores on variable, a weighted score is a new Z- factor scores based on Algorithm derives the The Regression (McMurray, 1987). is the is the Zscore matrix, P = F), where Z (Z the matrix formula scores and uses is the factor coefficient matrix, and F matrix, P is the pattern inverse of the correlation and the regression approaches, familiarity with multiple Due to widespread score matrix. Factor Scores 8 popular factor scores (Gorsuch, 1983, p. availability of programs, regression estimates are 262). minimize the sum of Bartlett (1937) suggested a Ieast-squares procedure to procedure is factors over the range of the variables. Bartlett's squares of the unique their place" so that they are used only to intended to keep the noncommon factors "in and those reproduced from the explain the discrepancies between the observed scores between the Bartlett's procedure leads to high correlations common factors (p. 264). estimated. factor scores and the factors that are being Bartlett except they Anderson and Rubin (1956) proceeded in the same manner as required to be orthogonal, resulting in a added the condition that the factor scores were factor The Anderson-Rubin equation produces than Bartlett's. more complex equation matrix (p. 265). estimates whose correlations form an identity Anderson-Rubin) algorithms yield However, these three (Regression, Bartlett, of factor scores has a mean of zero and a factor scores that are in Zscore form (each set allow comparison of the mean factor standard deviation of one). The result does not data set. The BTF factor with the mean on other factors for the same score on any given noncentered factor score, Thompson (1983), and yields a standardized, score is based on illustrated in Table 3. which allows comparisons of Fscore means, as Insert Table 3 About Here Factor Scores 9 original variable means are added The variables are converted to Zscore form, then the information is retrieved, then multiplied by back onto the Zscores, so that central tendency matrix as in the original Regression the inverse of the correlation matrix and by the pattern transformed P = F). Thus, in this procedure, the raw data are algorithm ((Z+mean) deviation of the variable, with the only by the division of each raw score by the standard of one, but a non-zero mean. This allows result that each variable has a standard deviation factors to make judgments regarding the the comparison of the mean factor scores across factor scores for Factor I are relative importance of given factors. All four types of factor score relationships for Factor 1. shown in Table 4, allowing a comparison of the Insert Table 4 About Here equal, when the Although the means of the four types of factor scores are not correlate perfectly with correlations among them are examined, all of the factor scores other factor scores (see Table 5). This themselves, and are perfectly uncorrelated with all McDonald and Burr (1967) specifications. indicates that the factor scores conform to the Insert Table 5 About Here matrix is the bivariate correlation As previously indicated, the factor structure latent variables. Thus, a structure coefl: sient between the n observed variables with the n observed variables (n=301) and the indicates the correlation between the scores on the 1 0