Reconciling factor-based and composite-based approaches to structural equation modeling

Transcription

1 Reconciling factor-based and composite-based approaches to structural equation modeling Edward E. Rigdon Modern Modeling Methods Conference May 20, 2015

2 Thesis: Arguments for factor-based SEM being better than composite-based approaches (particularly PLS path modeling) are invalid. 2

3 Factor-based SEM Common factors A, B, C Observed variables a 1 a j, b 1 b k, c 1 c m Factor / factor: factor / indicator C g1a g2b e a A e, i 1, j i ia ia b B e, i 1, k i ib ib c C e, i 1, m i ic ic 3

4 PLS path modeling Composites A, B, C Observed variables a 1 a j, b 1 b k, c 1 c m Inner model outer model C g A g B e A w a 1 2 B C With routine standardization j k m ai w b bi w c ci i i i 4

5 Mode A, Mode B Mode B : regression-weighted sum of indicators Mode A : correlation-weighted sum of indicators, ignoring collinearity among predictors (Dana & Dawes 2004, Waller & Jones 2010) Regardless of specifics, PLS (like GSCA) uses weighted composites where factor-based SEM uses common factors 5

6 The tidy theory of error laid down for psychology at the start of the century by Spearman and Brown has always seemed just a little too tidy to describe the perverse behavior of real data. Cronbach, Gleser, Nanda & Rajaratnam (1972), p. v 6

7 Measurement Model Conceptual Variable (In)validity Proxy (Un)reliability Mathematical Operations Observed Variables 7

8 Thesis: Arguments for factor-based SEM being better than composite-based approaches (particularly PLS path modeling) are invalid. 8

9 The arguments Bias: Factor-based SEM yields unbiased parameter estimates, while PLS parameter estimates are biased. Measurement error: Factor-based SEM accounts for measurement error, while PLS does not. Latent variables: Factor-based SEM involves latent variables, while PLS does not. Overall fit: Factor-based SEM assesses overall fit, while PLS does not. 9

10 1. Factor-based SEM yields unbiased parameter estimates, while PLS estimates are biased Wold (1982): PLS employs an intentional approximation Recently, Aguirre-Urreta & Marakas (2013), Rönkkö & Evermann (2013) criticized the quality of PLS estimates. The nature of Dijkstra and Henseler s (2015) consistent PLS implies that bias exists. 10

11 Counter: Bias is due to model misspecification, not intrinsic in PLS path modeling Rigdon (2012) speculated that, for a correctly specified population, PLS parameter estimates would be at least consistent. Becker et al. (2013) specified a composite-based population, and found PLS parameter estimates to be consistent, converging on specified values as n increases. BUT for small n, bias can be substantial (PLS also has its own identification requirements) 11

12 Population model Becker et al. (2013, in prep) x 1 w 11 x 2 x 3 x 4 x 5 x 6 x 7 w 12 w 13 w 14 w 25 w 26 w 27 C X1 C X2 p 1 p 2 C Y v 1 v 2 v 3 v 4 y 1 y 2 y 3 y 4 w 28 x 8 12

13 x 1 w 11 x 2 x 3 x 4 x 5 x 6 x 7 w 12 w 13 w 14 w 25 w 26 w 27 C X1 C X2 p 1 p 2 C Y v 1 v 2 v 3 v 4 y 1 y 2 y 3 y 4 w 28 x 8 YY XY XY XX 13

14 Specifying a composite-based population Becker et al. (2013, in prep) YY, XX : Specify within covariances for each set of components W, V: Specify weights, which define composite variances P: Specify structural path weights (and / or covariances) between composites XY : Calculate across covariances consistent with desired path weights, using path tracing rules V PW XY YY XX 14

15 Sample results p 1 p 2 15

16 2. Factor-based SEM accounts for measurement error, while PLS does not Factor-based SEM separates common factor from specific factor or residual, which has been called measurement error (Jöreskog 1983) Bollen (1989) explicitly associated common factor with true score, thus linking residuals with measurement error PLS composites share in any error variance contained in their components, to some extent 16

17 Counter: common factors retain an error component via factor indeterminacy De-meaned factor model with p indicators Y, k common factors F: = F + Z Covariance model, common factors orthogonal to specific factors: = + Consolidating, into : Rank ( ) = p. Rank ( ) = p + k. (Guttman 1955)

18 The indeterminate error component in common factors 1 F ' Y PS PP ' M 1 ' Schönemann and Steiger (1976) 18

19 Factor indeterminacy and error 2 Guttman s (1955) determinacy metric: min 2 FY, 1 where 2 F,Y is R 2 for this common factor predicted by all observed variables in the model Mulaik 2010, p. 380: 2 F,Y calculated from model parameters: 2 1 FY, diag McDonald (1999, p. 89): for a single common factor, 2 F,Y equals McDonald s, i.e., composite reliability BUT ONLY when loadings are equal 19

20 Values of Guttman s min Number of Indicators Loading

21 Factor-based SEM does not eliminate error it repackages error as indeterminacy Neither approach to SEM truly eliminates error, except at the limit 21

22 Indeterminacy does not affect estimation of relations within the factor model Indeterminacy does blur relations with variables outside the model (Steiger 1979) 22

23 The conceptual variable is outside the model Conceptual Variable Proxy Observed Variables 23

24 3. Factor-based SEM involves latent variables, while PLS does not Latent variable can mean either: (statistical) common factor, or (substantive) conceptual variable Obviously, factor-based SEM involves factors, while composite-based SEM does not 24

25 Counter: conceptual variables are important to both Either approach can be used to learn about conceptual variables though not all applications of either method are well-suited to that purpose Without conceptual variables and validated proxies, multiple indicator methods may be just fancified data description 25

26 4. Factor-based SEM assesses overall fit, while PLS does not No doubt, factor-based SEM has an overall 2 statistic, and composite-based PLS path modeling does not (Composite-based generalized structured component analysis (GSCA) claims an overall R 2 criterion) But does it matter? 26

27 Measurement Model Conceptual Variable Proxy (In)validity (Un)reliability Does 2 fit tell us anything about the quality of the proxy thus created? Observed Variables Mathematical Operations 27

28 If a value of 2 is obtained, which is large compared to the number of degrees of freedom, this is an indication that more information can be extracted from the data. Jöreskog (1969, p. 201) In itself, the existence of more information in the data says nothing about the validity of a given proxy (though indeterminacy will mean unreliability) 28

29 A P F B P F might be a great proxy for conceptual variable A but not so much for B. 2 fit of the statistical model will be the same, either way y 1 y 2 y 3 y 4 29

30 Overall fit assessment in factor-based SEM does not make factor proxies better than composite proxies. 30

31 None of these arguments offers a valid basis for always preferring a factor proxy over a composite proxy. 31

32 So, what would make one kind of proxy better than another, in a given situation? 32

33 v1 A B C D v2 v3 v4 v1 A* B* C* D* v2 v3 v4 33

34 What does all this mean? Learning about conceptual variables can be done with either factor-based SEM or composite-based SEM, or possibly a combination (via Djkstra and Henseler s consistent PLS ) BUT we must enable meaningful assessment of our proxies Besides using large and relevant samples and properly accounting for data distributions We must take full advantage of prior information, and embrace the discipline of the scientific imagination (Feynman, 1994) 34

35 We also need to think about the value of exploratory multiple indicator analyses, where nothing is known about the conceptual variables supposedly involved. 35

36 If we take the simile of the bridge crossing a river by way of an island, there is a statistical span from the near bank to the island, and a subject-matter span from the island to the far bank. Both are important. Cornfield & Tukey (1956), p

37 Thank You! / Questions 37

38 References Aguirre-Urreta, M.I., Marakas, G.M. (2013). Partial least squares and models with formatively measured endogenous constructs: A cautionary note. Inform Syst Res (25:4), Becker, J.-M., Rai, A., Rigdon, E.E. (2013). Predictive validity and formative measurement in structural equation modeling: Embracing practical relevance. Thirty-Fourth International Conference on Information Systems. Becker, J.-M., Rai, A., Rigdon, E.E. (in prep). Parameter recovery, statistical power and predictive utility in composite-based populations using partial least squares path modeling. Bollen, K.A. (1989). Structural Equations with Latent Variables. New York: John Wiley & Sons. Cornfield, J., Tukey, J.W. (1956). Average values of mean squares in factorials. Ann Math Stat (27:4), Cronbach, L.J., Gleser, G.C., Nanda, H., Rajaratnam, N. (1972). The Dependability of Behavioral Measurements: Theory of Generalizability for Scores and Profiles. New York: Wiley. Dana, J., Dawes, R.M. (2004). The superiority of simple alternatives to regression for social science predictions. J Educ Behav Stat (29:3), Dijkstra, T.K., Henseler, Jörg (2015). Consistent partial least squares path modeling, MIS Quart (39:2), Feynman, R. (1994). The Character of Physical Law. New York: The Modern Library. Guttman, L. (1955). The determinacy of factor score matrices with implications for five other basic problems of common-factor theory. Brit J Statist Psych (8:2), Jöreskog, K. G. (1969). A general approach to maximum likelihood confirmatory factor analysis. Psychometrika (34: 2), Jöreskog, K. G. (1983). Factor analysis as an errors-in-variables model. In Principles of Modern Psychological Measurement: A Festschrift for Frederick M. Lord, H. Wainer and S. Messick (eds.), Hillsdale, NJ: Lawrence Erlbaum, McDonald, R.P. (1999). Test Theory: A Unified Treatment. Mahwah, NJ: LEA. Mulaik, S.A. (2010). Foundations of Factor Analysis (2 nd ed.). Boca Raton, FL: Chapman & Hall / CRC. Rigdon, E.E, (2012). Rethinking partial least squares path modeling: In praise of simple methods. Long Range Plann (45:5-6), Rönkkö, M., Evermann, J. (2013). A critical examination of common beliefs about partial least squares path modeling. Organ Res Methods (16:3), Schönemann, P.H. Steiger, J.H. (1976). Regression component analysis. Brit J Math Statist Psych (29:2), Steiger, J.H. (1996). The relationship between external variables and common factors. Psychometrika (44:1), Waller, N., Jones, J. (2010). Correlation weights in multiple regression. Psychometrika (75:1), Wold, H. (1982). Soft modeling: The basic design, and some extensions. In Systems Under Indirect Observation (Part I), K. G. Jöreskog and H. Wold (eds.). Amsterdam: North-Holland,