Factor analysis

a. The classical common factor model

Based on the correlation or covariance matrix, factor analysis enables the researcher to assess how the variables relate to each other in defining more general, underlying concepts. Such underlying concepts are referred to as latent (unobserved), common factors (e.g., Muthén and Muthén 1998-2007; Kline 2005; Brown 2006), or trait factors (Campbell and Fiske 1959; Kline 2005; Brown 2006).

Based on Spearman’s (1904; 1927) seminal work on factor analysis (Harman 1976, p. 3-4; Fabrigar et al. 1999), the classical Thurstonian common factor analysis model defines the relationships between a set of p observed variables y and m latent continuous factors (with p>m) by multiple linear regression functions (Thurstone 1947 in Mislevy 1986, p.3; Fabrigar et al. 1999, p.275; Brown 2006, p. 13), such as:

2 2 1 21 2 1 1 1 11 1                   m m m m y y    p m pm p p y  ₁₁  

It is assumed that these continuous latent variables (common factors) account for the relationships between the p observed variables, which are also called indicators (Brown 2006, p. 13), manifest or outcome variables (Muthén and Muthén 1998-2007), and that the residuals, or error terms  are uncorrelated (Mislevy 1986, p. 5).

Thus, the two main goals of factor analysis are to find (1) the number of factors underlying the relationship between the manifest (observed) variables and (2) the regression coefficients of the observed variables on the factors so that the discrepancy between the observed and the model correlation/covariance matrices is minimized (Mislevy 1986, p. 3, 7).23

23 _{See Appendix A5, section IV.1.a and b for more details on factor analysis of continuous variables}

Chapter 5: Research design

b. The factor analysis techniques used in the study

Farmers’ environmental attitudes and behaviour were estimated using exploratory and confirmatory factor analysis, exploratory structural equation modeling and structural equation modeling, via the weighted least squares mean- and variance- adjusted (WLSMV) estimator24 (Muthén and Muthén 1998-2007) in the Mplus 5.21 software (Muthén and Muthén 1998-2007).

i. EFA and CFA

Exploratory factor analysis (EFA) and confirmatory factor analysis (CFA) are both based on the common factor model (Brown 2006, p. 14). As they measure the relationship between observed variables and latent factors, they are also referred to as ‘measurement models’ (Kline 2005, p. 71; Brown 2006, p. 1). However, the two techniques serve different purposes. Typically, EFA is used when one wants to explore the relationships between manifest variables without any preconceptions, whereas CFA is used when one wants to test already theoretically formulated models (e.g., Albright 2006, p. 2; Brown 2006, chapter 2). As Brown (2006, p. 42-3) explains: “all indicators in EFA freely load on all factors and the solution is rotated to maximize the magnitude of primary loadings and minimize the magnitude of cross-loadings. […] [R]otation is not necessary in CFA because the simple structure is obtained by specifying indicators to load [usually] on just one factor”. Hence, CFA models are usually more parsimonious than EFA models. Both methods can be used to validate models, in CFA one has access to statistical tools to test the significance of differences between nested models as well as between groups of respondents (Brown 2006, p. 42).

ii. SEM and ESEM

Structural equation modeling (SEM) is an extension of the common factor model as it is composed of (1) a measurement component (CFA), that is, the relationship between observed variables and the latent variables underlying them, as well as (2) a structural component (path analysis), that is, the causation links between

Chapter 5: Research design

latent variables (Kline 2005).25 _{Following the notation in Jöreskog’s paper (1994, p.}

298), the linear structural relation is defined as follows:

  

B  

where  (₁,₂,...,_m) and  (₁,₂,...,_n) are random vectors of latent dependent and independent variables, respectively, B(mm) and (mn) are coefficient matrices and  (1,2,...,m) is a random vector of residuals (errors in equations, random disturbance terms).

Classically, SEM models, via their CFA component, are used to assess the validity of an a priori model (most often theoretically based), where outcome variables are often constrained to depend on only one latent factor (Kline 2005, p. 166-7).26

However, such constrained models rarely render reality. The affect and cognition dimensions in the specific tripartite attitude model can be hard to distinguish, especially by postal surveys (Ajzen 1988, p.21; Morris et al. 2002). Therefore, a more appropriate method than SEM to assess ifaffectandcognitionare indeed two distinct factors would be one that would allow the hypothesized affect andcognition indicators to load freely on two factors – hence, validating or invalidating the presence of an affect and a cognition factor – while still allowing these two factors to predict conation, as postulated by the tripartite model.

Exploratory Structural Equation Modeling (ESEM), recently developed by Asparouhov and Muthen (2008), allows such estimations. As the measurement component of ESEM is based on exploratory factor analysis (as opposed to confirmatory analysis in SEM), it allows observed variables to load freely on all latent factors, therefore resulting in estimations closer to reality. Additionally, SEM structure

25 _{See Appendix A5, section IV.1.b.ii for a discussion on causation in SEM (regression versus}

correlation).

26 _{When variables are constrained to depend on only one latent factor, it is called ‘unidimensional}

measurement’. ‘Multidimensional measurement’ is when an indicator loads on different factors and/or its error term correlates with the error term of another indicator (Kline 2005, p. 166-7).

Chapter 5: Research design

(causal paths) can still be modelled between factors.27 _{By default, ESEM in Mplus uses}

oblique Geomin rotation (where the factors are assumed to be correlated).28