Campbell and Fiske (1959: pp. 82-83) proposed four rules for the evaluation of MTMM matrices. The correlations in the monotrait-heteromethod block have to be statistically significant and strong. The correlations in the heterotrait-heteromethod block should be weak or should equal the correlations in the heterotrait-monomethod block. The correlations in the heterotrait-monomethod block should be smaller than the correlations in the monotrait-heteromethod block. The pattern of the trait intercorrelations in all heterotrait triangles should be similar for the mono- as well as the heteromethod blocks. This property also refers to the discriminant validity of the measures (summarized by Koch et al., 2018 ). For details refer also to Eid (2010) . Finally, Trochim (2002) in line with Campbell and Fiske (1959) also suggests the following guidelines for the interpretation of an MTMM matrix: 1) Reliability coefficients must be the highest values in the matrix because the correlations between a trait and itself are expected to be the strongest than with any other construct (also Brown, 2015 ). 2) The coefficients presented in the validity diagonal must be statistically different from zero and preferably moderately strong. 3) Every coefficient in the validity diagonal must be the greatest column and row value of the method block it belongs. Recall that all off-diagonal elements must be the smallest to evidence discriminant validity. 4) Each validity coefficient must be greater than all the off-diagonal elements in the heterotrait monomethod triangles. If this is not true, then a method effect may be present. 5) The relationships of the different traits must be coherent in all triangles, i.e. relationships between every trait dyad must be cohesive, without areas of dramatic changes (Trochim, 2002) . In spite of the ease of application of these rules, in practice, the application of MTMM method was criticized as having several limitations (Marsh, 1988, 1989; Schmitt & Stults, 1986; Eid, 2010) .

Despite the fact that the Campbell and Fiske (1959) technique was a significant progress for examining construct validity, the MTMM approach was not widely accepted the years immediately following its inception (Brown, 2015) . Additionally, several limitations were noted first by Jackson (1969) . Jackson’s criticisms as summarized by Franzen (2002) , mainly focused on the fact that the MTMM method juxtaposes distinct patterns without taking into account the overall structure of the relationships (i.e. between-network; Cronbach & Meehl, 1955 ; Martin & Marsh, 2006 ). This argument is based on the idea that the correlation pattern may be affected by the way variance is distributed when measuring the target traits. To alleviate the MTMM disadvantages, Jackson (1969) suggested to carry out an EFA (PCA actually) in the monomethod matrix. Specifically, the correlation matrix is orthogonalized and then submitted to Principal Components Analysis, with a varimax rotation. The extracted factors equal the number of matrix traits. However, Jackson’s suggestion while offering some advantages, it assumes no relationships exist among the traits and this is rarely the case in psychology. Additionally, the influence of different measurement methods cannot be taken into account when the monomethod matrix is used (Franzen, 2002) . Finally, EFA failed to obtain meaningful solutions of MTMM data due to the inherent restrictions of EFA to specify correlated errors (Brown, 2015) .

Later, Franzen (2002) continues Cole, Howard, and Maxwell (1981) compared the MTMM results with constructs mono-operationalization suggesting that as a rule, the validity coefficients were misleadingly deflated. Therefore, the MTMM method and factor analysis may be used in tandem (Franzen, 2002) . Finally, Cole (1987) repeated to a certain extent Jackson’s criticisms, i.e. absence of concrete guidelines regarding the size of the zero-order correlation coefficients. Additionally, Cole argued that the cost of the MTMM application in clinical settings deems the method inapplicable (Franzen, 2002) . Crucially, Cole noted that the MTMM is prone to correlated errors due to factors like the time of day, or the instrument administrator. Cole suggested the use of CFA instead to examine discriminant and convergent validity (Franzen, 2002) . For a detailed description of MTMM limitations see also Marsh (1988, 1989) and Schmitt & Stults (1986) .

At the same time, non-factorial oriented attempts to find an appropriate method of statistical analysis for the data in the matrix were numerous (Eid et al., 2008; Eid, 2010) . These methods were mainly based on analysis of variance models or nonparametric alternatives and the reason for the lack of success was attributed to conflicting data in the diagonals in relation to the data in the triangles (Sawilowsky, 2007) . For example, Sawilowsky (2002, 2007) proposed a statistical test for analysis MTMM data where the null hypothesis is that the coefficients in the matrix are unordered from high to low. This is tested against the alternative hypothesis of an increasing trend from the lowest level (heterotrait heteromethod) to the highest level i.e. reliability coefficients (Sawilowsky, 2007: p. 180) . See Sawilowsky (2002, 2007) for more details and an applied example.

Despite the several alternatives proposed―like Sawilowsky’s (2002, 2007) ―the analysis of MTMM data within the CFA framework (a.k.a. CFA-MTMM models) has been a widely accepted methodological strategy (Eid et al., 2008; Byrne, 2012; Koch et al., 2018) .

This method was among the first significant developments in the field of CFA-MTMM models (Koch et al., 2018) and was developed by Keith Widaman (1985) proposing a nested model comparison.

The model comparison presented by Widaman (1985) had (t) traits and (m) methods where (t) is the number of trait factors and (m) is the number of method factors, abiding by five key specifications rules (Brown, 2015) : 1) for model identification, at least three traits (t) and three methods (m) are required; 2) (t) × (m) indicators are specified to define (t) + (m) factors; 3) each indicator loads on two factors, i.e. a trait factor and a method factor and all other cross-loadings are set to zero; 4) trait factors inter-correlations and method factors inter-correlations are freely estimated; 5) trait-method factors correlations are typically constrained to zero; and 6) indicator uniquenesses are estimated freely but cannot be specified to correlate with uniquenesses of other indicators. Each indicator in this specification is a function of a trait, method, and unique factors, Brown (2015) explains.

The specified MTMM model becomes a baseline model, compared against a set of nested models in which parameters are either removed or added. Model comparison criteria used are the χ² (Δχ²) and other fit indicators (see Byrne, 2012 ), in a procedure similar to the multi-group CFA. Based on this comparison, evidence of convergent and discriminant validity emerges, both at the matrix level and at the individual parameter level (Byrne, 2012) . Figure 2 is a path diagram for an MTMM matrix of 3 traits and 3 methods to describe the steps of the correlated methods. Testing for evidence of convergent and discriminant validity involves comparisons between the model in Figure 2 and three alternative nested MTMM models in Figures 3-5 (Byrne, 2012; Brown, 2015) .

Byrne (2012) describes the steps for comparing the nested models as follows.

Step 1―The CTCM Model: Test the first model (see path diagram in Figure 2) is the hypothesized baseline model against which the three alternative CFA models are compared. A common problem in this step of the process is a non-positive matrix error (see Kenny & Kashy, 1992 ; Marsh, 1989 ). Usually, a negative variance is associated, either with a residual or a factor and this problem is generally attributed to overparameterization of the MTMM models, due to their inherent complexity ( Byrne, 2012 ; quoting also Wothke, 1993 ). Possible

solutions are provided by Marsh et al. (1992) and Byrne (2012) . Hopefully, upon model re-specification, the Correlated traits/Correlated Methods model (CTCM) results in a solution and the fit of this CTCM model must be considered (Byrne, 2012) .

Step 2―The NTCM Model: The next hypothesized model of correlated methods specified is a no traits/correlated methods (NTCM) model (see path diagram in Figure 3). Its specification is identical to that of the CTCM model, except that there are no trait factors, as shown in Figure 3 (Widaman, 1985; Byrne,

2012) . Because these two models are nested, a comparison of this model to the CTCM provides a statistical evaluation of whether effects associated with the different traits emerge (Brown, 2015) . First, the goodness-of-fit for this NTCM model is examined in relation to the CTCM model.

Step 3―The PCTCM Model: The specification of a hypothesized Perfectly Correlated Traits/Freely Correlated Methods (PCTCM) MTMM model follows. The path diagram of this model is presented in Figure 4. As with the hypothesized CTCM model (Figure 2), each observed variable loads on a trait and a method factor. Nevertheless, in this MTMM model, the traits are perfectly correlated (i.e., equal to 1.0). The method factors are freely estimated in this model, in line with both the CTCM model and NTCM model. The goodness-of-fit of this model is examined and it is compared to the fit of the previous models (in Figure 2 and Figure 3 respectively).

Step 4―The CTUM Model: The last correlated methods MTMM model is specified which is a Freely Correlated Traits/Uncorrelated Methods (CTUM) see Figure 5. It is different from Model 1 only in the absence of correlations in the method factors. The goodness of-fit for this model is evaluated (see also Byrne, 2012 ).

Step 5―Evaluation of Construct Validity at the Matrix Level: After the evaluation of goodness-of-fit of the four MTMM models and with the assumption that the fit is acceptable (within the acceptability criteria used for typical CFA models, e.g. Hu & Bentler, 1999 ; Brown, 2015 ), evidence of construct and discriminant validity is examined. This information at the matrix level derives through the comparison of particular pairs of models. Note that typically, a summary of goodness-of-fit statistics of all four MTMM models is presented in a table, and a summary of model comparisons in a second table, see an example of both in Table 3 adapted by Byrne (2012) .

Step 5a―Convergent Validity at the Matrix Level: Generally, the principles applied when testing for invariance are also applied when comparing models in this step (see Table 3). In line with the procedure proposed by Widaman (1985) , evidence of convergent validity is tested by comparing the CTCM model (Model in Figure 2) to the NTCM model (Model in Figure 3). When the difference in χ² (Δχ²) between these models is highly significant and the fit difference (ΔCFI) is also large evidence of convergent validity is suggested (Byrne, 2012) .

Step 5b―Discriminant Validity at the Matrix Level: Discriminant validity is usually evaluated considering both traits and methods. Evidence of discriminant validity among traits emerges from the comparison of the CTCM Model in which traits are freely correlated (see Model in Figure 2) with the PCTCM model, specified in step 3 in which traits are perfectly correlated (see Model in Figure 4). The larger the difference between the values of χ² and CFI, the stronger the evidence of discriminant validity. The procedure is repeated to show evidence of discriminant validity regarding method effects. Specifically, the CTCM Model in which methods are freely correlated―specified in step 1 and presented as path diagram in Figure 2―, is compared to the CTUM model in which methods are uncorrelated-specified in step 4 and presented as path diagram in Figure 5. A large Δχ² (and/or considerable ΔCFI) suggests common method bias, therefore the lack of discriminant validity across measurement methods (Byrne, 2012) . For an applied example see Byrne (2012) . See Goodness of fit criteria used in Table 3 and the Nested Models Compared in Table 4. Byrne suggests that the difference criteria for the comparison of the nested models to be identical to measurement Invariance criteria (Cheung & Rensvold, 2002) “as an evaluative base upon which to determine evidence of convergent and discriminant validity” (Byrne, 2012: p. 300) . For a more detailed description and an applied example see Byrne (2012) .

Step 6―Evaluation of Construct Validity at the Parameter Level: Trait- and method-related variance can be examined more thoroughly by examining individual parameter estimates, i.e. the factor loadings and factor correlations of

Notes: Based on a table used by Byrne (2012: p. 299) ; Goodness of fit criteria are the same as standard CFA.

Notes: Based on a table used by Byrne (2012: p. 300) ; Byrne suggests difference criteria adopted to be identical to measurement Invariance Criteria (see Cheung and Rensvold, 2002 ; Chen, 2007 for cutoff values).

the CTCM model (the Model of step 1, see path diagram in Figure 2) as Byrne (2012) explains.

Step 6a―Convergent Validity at the Parameter Level: The size of the trait loadings indicates convergent validity. First, all loadings are examined to see if they are statistically significant. Then all factor loadings across traits and factor loadings across methods are compared, to see whether method variance or trait variance is higher over all the available ratings (Byrne, 2012) .

Step 6b―Discriminant Validity at the Parameter Level: Factor correlation matrices contain the evidence of discriminant validity regarding particular traits and methods. Generally, correlations among traits should theoretically be small to show evidence of discriminant validity, but such findings are rare in psychology. Regarding the correlations of the method factors, discriminability of these parameters values suggests the extent to which the methods are dissimilar, testing a major assumption of the MTMM strategy (Campbell & Fiske, 1959; Byrne, 2012) . For an applied example of the correlated method refer to Byrne (2012) and Brown (2015) .

Some simpler alternatives to CTCM models have been proposed, and the correlated-uniqueness (CU) model described next is one of them (Kline, 2016) .

The CU models signify a special case of the general CFA model based on the work of Kenny (1976, 1979) and Marsh (1988, 1989) who proposed this alternative MTMM model parametrization to avoid the common estimation and convergence problems arising when analyzing CTCM models, like the one in Figure 2, ( Brown, 2015 ; Byrne, 2012 ; also quoting Kenny & Kashy, 1992 ; Marsh & Bailey, 1991 ; Tomás, Hontangas, & Oliver, 2000 ).

The CU model identification must abide by the following rules as reported by Brown (2015) . There must be at least two traits (t) and three methods (m). However, a 2 t × 2 m model can be also specified in the factor loadings of indicators that belong to the same trait factor are fixed to be equal. Rules regarding the CU trait model are the same as in the correlated methods approach. Specifically: 1) Each indicator must load on one trait factor with all other cross-loadings fixed to zero; and 2) The trait factor correlations are freely estimated. Consequently, the major difference between the CU method in comparison to the correlated methods is the estimation of method effects. In the CU model, method effects are assessed by adding correlated uniquenesses (errors) among the indicators of the same assessment method instead of specifying method factors (Brown, 2015) . Figure 6 presents the path diagram of the correlated uniqueness CFA specification for an MTMM matrix with 3 traits and 3 methods.

After the model specification, the model is tested and the fit of the solution to the data is evaluated. Except fit the inspection of standardized residuals, the modification indices must be also evaluated (as proposed by Brown, 2015 ). The parameter estimates Brown (2015) continues are then examined. That is, when loadings of traits factors are substantial and statistically significant, this is an evidence supporting convergent validity. Conversely, when correlations among the trait factors are substantial, this is indicative of low discriminant validity. Method effects are assessed when correlated uniquenesses among indicators of the same method are moderate or large in magnitude (Brown, 2015) .

For example, Brown (2015) explains, when trait factor loadings are large this suggests that the relation of the indicators to their purported latent constructs is strong thus, convergent validity can be supported while adjusting for method effects. When modest correlations emerge among trait factors this is indicative of adequate discriminant validity. The significance of method effects is evaluated by the size of the correlated uniquenesses among indicators. As in the classic MTMM matrix, the method effects in the CFA model must be smaller than those of the indicators. For an applied example of the CU, method sees Byrne (2012)

and Brown (2015) . Saris and Aalberts (2003) also examined different interpretations of CU models.

Some recommend the correlated uniquenesses (CU) method (Kenny, 1976, 1979; Kenny & Kashy, 1992; Marsh, 1989) as superior, whereas others support the general CFA method (Conway, Scullen, Lievens, & Lance, 2004; Lance, Noble, & Scullen, 2002) . Nonetheless, Byrne (2012) comments that the general CFA model is as a rule preferable.

Specifically, an advantage of the CT-CM model is that it permits fragmenting observed variance into a trait, method, and error variance in comparison to the CU model that is more restrictive (Eid et al., 2006; Geiser et al., 2008; Koch et al., 2018) . Kenny and Kashy (1992) argue that this fragmentation of the correlated methods model is the reason why this method corresponds directly to Campbell and Fiske’s (1959) original conceptualization of the MTMM matrix. Moreover, the parameter estimates of the correlated methods models offer direct evidence of construct validity. For instance, substantial trait factor loadings show convergent validity; small and/or non-significant method factor loadings suggest that method effects are absent and modest trait factor correlations suggest good discriminant validity. The addition of method factors permits interpretation of method effects (Brown, 2015) .

Nevertheless, a drawback of the correlated methods model is that it is prone to underidentification errors, or Heywood cases and large standard errors. Because of these problems with the correlated methods model, methodologists suggested (Kenny & Kashy, 1992; Marsh & Grayson, 1995) using the correlated uniqueness model for the analysis of MTMM data. On the other hand, others (Lance, Noble, & Scullen, 2002) argue that given the substantive strengths of the correlated methods model, the CU model should be preferred only as last resort, i.e. when the correlated methods model fails to be identified (Brown, 2015) .

For a summary of CFA-MTMM models see Dumenci (2000) , Eid, Lischetzke, and Nussbeck (2006) , as well as Shrout and Fiske (1995) and Koch et al. (2018) . For applied examples of the above-mentioned procedure both for the Correlated methods and the Correlated Uniquenesses you can refer to Byrne and Goffin (1993) , Byrne and Bazana (1996) as well as to Meyer, Frost, Brown, Steketee, and Tolin (2013) or Koch et al. (2018) . Byrne (2012) refers readers who wish to read more on comparisons of the correlated uniquenesses, composite direct product, and general CFA models to other sources (Bagozzi, 1993; Bagozzi & Yi, 1990; Coenders & Saris, 2000; Hernández & González-Romá, 2002; Marsh & Bailey, 1991; Marsh, Byrne, & Craven, 1992) .

More recent, Koch et al. (2018) comment, MTMM research has gained insight on many aspects of the CTCM minus one (CT-C[M-1]) model coined by Eid, 2000 ; Eid, Lischetzke, Nussbeck, & Trierweiler, 2003 ; see also Geiser, Eid, & Nussbeck, 2008 ; Maydeu-Olivares & Coffman, 2006 ; Pohl & Steyer, 2010 . These comprise the inspection of specifically correlated residuals (Cole, Ciesla, & Steiger, 2007; Saris & Aalberts, 2003) , and extension of the method to longitudinal data (Courvoisier et al. 2008; Grimm, Pianta, & Konold, 2009; LaGrange & Cole, 2008) as well as to multilevel data (Hox & Kleiboer, 2007) .

More specifically, the CT-C (M-1) model is analogous to the correlated methods model (presented in Figure 2) but it comprises one method factor less than the traits included (M-1). For example, the model presented in Figure 2 could be transformed into a CT-C (M-1) model by eliminating the Method 1 latent variable. This specification answers to identification problems of the correlated methods model (Brown, 2015) . They can be defined in three steps: 1) Selection of a reference method (standard), on the basis of theory and research questions (Geiser, Eid, & Nussbeck, 2008) . 2) Selection of general trait variables (common or indicator-specific) and M-1 method factors. 3) Define additional constraints as a function of how non-correlationally measurement errors are, how non-correlationally method variables with trait variables of the same TMU are, and what is the homogeneity of the variables. Note, that if parameters of the model are restricted, the model data is equivalent to the common factor TMU model ( Koch et al., 2018 also quoting Geiser et al., 2008 ; Geiser, Eid, West, Lischetzke, & Nussbeck, 2012 ).

Latent difference models, Koch et al. (2018) argue, offer a possibility to contrast methods. Initially, they were developed for longitudinal research (Steyer, Partchev, & Shanahan, 2000) but they can successfully use in multimethod research (see Pohl, Steyer & Kraus, 2008 ). This technique also involves a reference method. Nevertheless, the latent method variables are not specified as residuals regarding the true-score variables of the reference method. It is hypothesized that each indicator taps a trait variable, but method variables are unidimensional. Therefore, a common method factor is specified for all indicators of the same method and the same trait (Koch et al., 2018) .

One more alternative approach is the direct product model (Browne, 1984; Cudeck, 1988; Wothke & Browne, 1990; Verhees & Wansbeek, 1990) . In this approach, the method factors relate to the trait factors in a multiplicatively instead of additively. That is, method effects may increase the correlations of strongly correlated traits, more than they increase the correlations of weakly correlated traits. In other words, the higher the correlation between the traits, the greater the method effects. However, the data must be suitable for the direct product model parameterization. Additionally, the implementation and interpretation of the method are rather complex and prone to improper solutions, but less prone than the correlated methods (Brown, 2015; Lievens & Conway, 2001; Campbell & O’Connell, 1967) .

On the other hand, methodologists disagree on the merits of the multiplicative versus the additive models. Some support the additive ones as more successful (Bagozzi & Yi, 1990) , whereas others either disagreed (Goffin & Jackson, 1992; Coovert et al., 1997) or present mixed evidence (Byrne & Goffin, 1993) . For example, Corten et al. (2002) reported better results in the additive models in 71 out of 79 datasets used (Loehlin & Beaujean, 2017; Loehlin, 2004) .

CFA-MTMM models have received additional criticism because the meaning of the trait and method factors often remains unclear. This happens because CFA-MTMM analysis just assumes that trait and method factors are present. In order to properly interpret trait and method effects, what the latent factors mean must be clear (Koch et al., 2018) . Koch et al., argue that developments in stochastic measurement theory (see Steyer, 1989 ; Zimmerman, 1975 ) proposed a technique to define the MTMM model factors in terms of random variables in a well-defined random experiment. Koch et al. (2018) add that a wide range of models for different purposes and data structures can be defined with this method but not all the CFA MTMM models.

With so many alternative approaches, is difficult for the researcher to choose the most appropriate. One of the most important criterion for selecting the appropriate model is the type of method to use. The researcher must examine: 1) the methods used in the study; 2) the data and the structure of the random experiment implied by the measurement design, and 3) the possible conceptualization of method effects. For a study with interchangeable methods, a classic CFA modelling strategy can be chosen. For a study with structurally different methods, several other modelling alternatives can be used. Each model has distinct features that should be appropriate for the research question of the study. This means, there is no optimal model for all types of CFA MTMM measurement designs. The selection of the model must be driven by theory and hypotheses studied (Koch et al., 2018) . For comparisons of these alternatives, readers can refer to other sources (Eid et al., 2008; Pohl & Steyer, 2010; Saris & Aalberts, 2003) .

Regarding CFA-MTMM research in general, Loehlin and Beaujean (2017) and Loehlin (2004) review that the implementation of CFA MTMM models within and across groups has also been discussed (Marsh & Byrne, 1993) . Marsh and Hocevar, (1988) reported using CFA MTMM with higher-order structures (Loehlin & Beaujean, 2017; Loehlin, 2004) . For applied examples the reader can refer to Bagozzi and Yi (1990) , Coovert, Craiger, and Teachout, (1997) and Lievens and Conway (2001) . Next we will focus on the longitudinal CFA-MTMM research.