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Multidimensionality and Higher-Order Factor Models

MKT 8543 Quantitative Marketing Seminar. Multidimensionality and Higher-Order Factor Models. April 7, 2009. Nicole Ponder. Mississippi State University. Gerbing and Anderson (1984).

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Multidimensionality and Higher-Order Factor Models

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  1. MKT 8543 Quantitative Marketing Seminar Multidimensionality and Higher-Order Factor Models April 7, 2009 Nicole Ponder Mississippi State University

  2. Gerbing and Anderson (1984) • “Adding correlated measurement errors to structural equation models will nearly always improve fit; the important question is whether their addition improves the substantive interpretation of the model.” (p. 572) • Allowing correlated errors to improve fit without providing an explanation could mask a “true” underlying structure • Perhaps the construct may be better represented as a second-order factor • Much better for researchers to think about the possible causes of correlated measurement errors and specify the model accordingly

  3. Gerbing and Anderson (1984)Error Terms in Classical Test Theory • y = t + e • Error is purely random • The indicator y is a combination of two and only two things: the true score and random error • BUT, obtaining a true score is not possible with a survey instrument; some level of measurement error is present

  4. Gerbing and Anderson (1984)Error Terms in First-Order Factor Models • y =  •  is an error of measurement; represents the error that is unique to each indicator • This error is composed of both completely random error as well as error that is “unique” or specific to that indicator • As  represents error unique to the indicator, in theory it should not be correlated with other ’s (or ’s, for that matter!), b/c the factor analytic measurement model follows classical test theory

  5. Gerbing and Anderson (1984)Error Terms in First-Order Factor Models • 1 represents the uniqueness of x1, or whatever variance is left after partitioning out the communality between x1, x2, x3, and x4. • The communality is the latent construct. If measured properly (in a truly reflective manner), what is left over is UNIQUE or specific to x1 and should not be correlated with any other error terms. 1 x1 1 2 x2 x3 3 4 x4  2 5 x5 6 x6

  6. Gerbing and Anderson (1984)Error Terms in First-Order Factor Models • Suppose the previous model had poor fit, and the large MIs for  suggested the addition of these paths – allowing the error terms for x1 and x2 to correlate; and the error terms for x3 and x4 to correlate. • The covariance of the error terms is due to at least one unknown common source 1 x1 1 2 x2 x3 3 4 x4  2 5 x5 6 x6

  7. Gerbing and Anderson (1984)Error Terms in First-Order Factor Models • If MI’s suggest the presence of error term correlations, the unwanted covariation might be due to one of two things: • a method component is present in the data. The correlated error terms may be corrected by adding a “method” construct to the model • the construct of interest may be better represented as a higher-order factor (also called a second-order factor) model

  8. Gerbing and Anderson (1984)The Second-Order Confirmatory Factor Model • The second-order construct (1 ) is the component common to all of the first-order constructs. Its variation is explained outside of the model (i.e., it has no manifest indicators) • Each first-order construct is measured with two indicators. • Models such as this are better representations than Fig A II (p. 575); provides a better explanation of what is really going on (we’ve modeled in “group specificity” with zetas). 1 ζ2 ζ1 ζ3 1 2 3 Y1 Y2 Y3 Y4 Y5 Y6 1 2 3 4 5 6

  9. Law and Wong (1999) • Multidimensional construct: a construct involving more than one dimension (p. 144) (duh!) • Each dimension represents some portion of the overall latent construct • Example: Job satisfaction consists of different dimensions: satisfaction with work, co-workers, supervisor, pay, and promotion • The term “facet” refers to one dimension of a multi-dimensional construct; facet=dimension

  10. Law and Wong (1999)Two Views of Multidimensional Constructs • The Factor View: under this model, the facets are seen as different manifestations of the multidimensional construct • See diagram on p. 145, Fig 1a 1 1 2 3 Y1 Y2 Y3 Y4 Y5 Y6 1 2 3 4 5 6

  11. Law and Wong (1999)Two Views of Multidimensional Constructs • The Composite View: under this model, the multidimensional construct is defined as the outcome of its facets • See diagram on p. 145, Fig 1b 1 1 1 1 X1 X2 X3 X4 X5 X6 1 2 3 4 5 6

  12. Law and Wong (1999) • Differences between the Factor View and the Composite View (p. 146) • The structural relationship b/w indicators and the multidimensional construct • The definition of the error variances • The causal indicators of the construct (reflective, or factor view) versus causes of the construct (formative, or composite view)

  13. Law and Wong (1999)Differences between the Factor View and the Composite View • The structural relationship between indicators and the construct • Factor view: multidimensional construct has structural paths pointing to its facets • Composite view: facets have structural paths pointing to the multidimensional construct • This is an important theoretical difference!! Relates back to the discussion of formative versus reflective indicators, or cause and effect indicators (Bollen and Lennox 1991)…so hopefully this sounded familiar!

  14. Law and Wong (1999)Differences between the Factor View and the Composite View • Error variances are different under the two views • Factor view: common variance IS the construct; variance unique to the indicator and random variance are modeled in the error term (p. 147) • Composite view: the composite formed by the facets IS the construct; variance unique to the indicator is a part of the construct, therefore. Only random variance is modeled in the error term. • Importance: Error variances of the facets may be overestimated if the factor view is used, but in reality the composite view should be used

  15. Law and Wong (1999)Differences between the Factor View and the Composite View • Causal indicators versus causes of the construct: how do you know whether to model your construct under the factor view or the composite view? • Consider the definition of the construct (the definitional essay comes in handy here!) • As you develop items for your construct…if the items are based on the components of your definition, then you should use the composite view • If the items come from the overall definition (summary), then it is appropriate to use the factor view • Law and Wong (1999) encourage the use of existing literature to justify the appropriate viewpoint

  16. So Should You Use the Factor View or the Composite View?? • The conceptualization of your constructs (either factor view or composite view) should be theory-driven! • Job perception and job satisfaction have been conceptualized under the composite view, but empirically tested using the factor view • Law and Wong (1999) illustrate that one gets conflicting results when modeling the construct of interest under the factor view versus the composite view • Some problems/unique considerations arise when using causal indicators (rather than effect indicators) in SEM…model is under-identified, unless you also use some effects indicators (p. 157)

  17. Review: Measurement Scales versus Indices • Measurement scale: consists of “effect indicators” whose values are caused by an underlying construct (Bollen 1989) • The reflective model (Bollen and Lennox 1991): var(x1) = 2 var(1) + var(1) The only thing that the 4 indicators should have in common is the latent construct! 1 x1 x2 x3 x4 1 2 3 4

  18. Review: Measurement Scales versus Indices • Index: consists of indicators that, taken together, cause the underlying construct (Diamantopoulos and Winklhofer 2001) • The formative model (Bollen and Lennox 1991): 1= 1x1 + 2x2 + 3x3 + 4 x4 1 Often, researchers mistakenly use this model to run with SEM…and problems occur! x1 x2 x3 x4 1 2 3 4

  19. Bagozzi and Edwards (1998) • In any empirical study, it is essential to be specific as to the depth and dimensionality of constructs and their measures if meaningful results are to be obtained • Interested in studying the different levels of abstraction of a multidimensional construct • Used SEM to test four different levels of aggregation of the Work Aspect Preference Scale (WAPS) • Concluded that work values can only be studied at its lowest levels of abstraction; i.e., overall “work values” cannot be properly analyzed

  20. Levels of abstraction in the Work Aspect Preference Scale (WAPS) Work Values Non-work Orientation Human/ personal concern Freedom Detachment Money Life style Creativity Independence Self-develop Co-workers Security

  21. B&E’s Aggregation Method • Total disaggregation model: tests the subcomponents of work values at their lowest level of abstraction n=437 • the x parameter estimates are all statistically significant, the fit indices show strong model fit, and the modification indices for x and are all nonsignificant • Thus, at this lowest level of abstraction, each indicator can be seen as truly reflective of the construct it is supposed to represent. Det Mon Lif Cre Ind Se Dev Cow Sec 1 2 3 4 5 6 7 8 a b c d e f g h i j k l m n o p q r s t u v w x

  22. B&E’s Aggregation Method • Partial disaggregation model • Use of parcels: reduces number of parameters to be estimated • Overall fit does improve over the total disaggregation model Det Mon Lif Cre Ind Se Dev Cow Sec 1+2 3+4 5+6 7+8 a+b c+d e+f g+h i+j k+l m+n o+p q+r s+t u+v w+x

  23. Quick notes on the use of parcels…from Little, Todd D., William A. Cunningham, Golan Shahar, and Keith F. Widaman (2002), “To Parcel or Not to Parcel: Weighing the Merits,” Structural Equation Modeling, 9 (2), 151-173. • Pros: • Individual items are statistically less reliable than aggregate scores • Overall levels of specific and random error are reduced • Overall fit statistics provide evidence of a better fit of the model to the data • Parsimony! • Cons: • What happens if the construct is multidimensional? And esp. if the dimensions are not related? • Parcels can mask true “problems” that exist with the measurement model • Researchers could “play” with parcels to get the best model fit

  24. B&E’s Aggregation Method • Partial aggregation model Human/ personal concern Nonwork orientation Freedom det sde mon lif cre ind cow sec 2(17, n=437) = 150.52, p = 0.00; CFI = .75 B&E concluded that one must reject the partial aggregation model based on the goodness of fit indices.

  25. Our Proposed Aggregation Method • Creation of “reflective combinations” to use in the partial aggregation model: Det Mon Lif Cre Ind Se Dev Cow Sec 1 2 3 4 5 6 7 8 a b c d e f g h i j k l m n o p q r s t u v w x Reflective measures may be created for nonwork orientation, freedom, and human/personal concern as follows:NWO1 = 1 + 5 + aNWO2 = 2 + 6 + bNWO3 = 3 + 7 + cNWO4 = 4 + 8 + d FR1 = e + i FR2 = f + j FR3 = g + k FR4 = h + l HPC1 = m + q + u HPC2 = n + r + v HPC3 = o + s + w HPC4 = p + t + x

  26. Our Method • Full variance-covariance matrix provided by B&E • Used a FORTRAN-based multivariate normal data generator to generate 1000 individual observations that will reproduce the given matrix • Able to replicate results that B&E got for their models • Used SEM to re-analyze partial and total aggregation models using our method

  27. Re-analyzed Partial Aggregation Model Freedom Non-work Orientation Human/personal Concern NWO1 NWO2 NWO3 NWO4 FR1 FR2 FR3 FR4 HPC1 HPC2 HPC3 HPC4 2(51, n=1000) = 320.08, p = 0.00; CFI = 0.96; GFI = 0.95; AGFI = 0.93 Here, SMCs are high; MIs are low, no cross-loadings.

  28. B&E’s Aggregation Method • Total aggregation model Work Values Non-work Orientation Human/personal Concern Freedom 2(2, n=437) = 66.78, p = 0.00; CFI = .62 The three factor loadings were constrained to be equal. B&E concluded that the fit of the model was poor; therefore, one can only study work values at its “facet” levels.

  29. Re-analyzed Total Aggregation Model • Can use reflective combinations in the total aggregation model as well: Freedom Non-work Orientation Human/personal Concern NWO1 NWO2 NWO3 NWO4 FR1 FR2 FR3 FR4 HPC1 HPC2 HPC3 HPC4 Reflective measures may be created for work values as follows: WV1 = NWO1 + FR1 + HPC1WV2 = NWO2 + FR2 + HPC2WV3 = NWO3 + FR3 + HPC3WV4 = NWO4 + FR4 + HPC4

  30. Re-analyzed Total Aggregation Model Work Values WV1 WV2 WV3 WV4 2(5, n=1000) = 8.60, p = 0.13; CFI = .99; GFI = .99; AGFI = .99 The three factor loadings were constrained to be equal. If measured properly, work values can be studied at a global level.

  31. Comments on Our Aggregation Results • Much better fit, now measures are properly reflective • In order for reflective combinations to be proper indicators, it is mandatory that the total disaggregation model displays properties of excellent model fit • parameter estimates for x must be large and statistically significant (0.70 or higher if phi is standardized) • SMCs for each indicator must be large (well above 0.50, preferably 0.70 or higher) • modification indices for x must be statistically non-significant (values < 3.84) • modification indices for must be statistically non-significant (values < 3.84)

  32. Comments/Guidelines for Aggregation • Need clean results at the total disaggregation level • Take time to develop proper conceptual definitions of constructs • Pay attention to the assumption of reflective measures • How do you know which indicators to combine? • To create reflective indicators of NWO, the combination of 1, 5, and a is arbitrary • Just ensure each dimension is represented! NWO1 = 1 + 5 + aNWO2 = 2 + 6 + bNWO3 = 3 + 7 + cNWO4 = 4 + 8 + d Money Detach Lifestyle 1 2 3 4 5 6 7 8 a b c d

  33. Conclusions • Better alternative than HOF models • Forces the researcher to place importance on the development of conceptual definitions, and to “get it right” at the total disaggregation level! • Reflective combinations approach may be applied to other multidimensional constructs • Trust: reliability, integrity, and confidence • Communication: informing, answering, listening • Service quality: tangibles, reliability, responsiveness, assurance,empathy • Market orientation: customer orientation, competitor orientation, interfunctional coordination

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