The procedure for selecting the items for the measurement model is presented in the section below. The measurement model was confirmed by running both PCA and CFA on the items. The assessment of the reliability of the results is based on the results of the CFA.
3.4.1 Validation of the Measurement Model
The validity of the measurement model was evaluated by CFA. The selection criteria for model fit indices are based on SEM methodology research and literature mainly from consumer psychology. Literature distinguishes two kinds of fit indices: those reflecting absolute fit, and those reflecting incremental fit, which refers to the fit of one model relative to another. Absolute indicators of model fit include, for example, chi square and incremental fit statistics include CFI, among others (Iacobucci, 2010).
CFA was done for three different SEM models to validate the measurement constructs and scales. First, the original four-factor BBX model was used to analyze the data, then the model was extended by adding one environmental item in each of the four factors, finally a construct consisting of the four-factor BBX model with an additional fifth factor for eco-friendliness was tested.
The size of the sample determines the significance of the loadings. If the sample size is over 500, a factor loading will be statistically significant if it is greater than or equal to .30. However, Janssens et al. (2008) state that often in practice the factor loading must be at least .50 before a variable may be assigned to a certain factor, and this rule requires a minimum sample size of 100. When examining convergent validity and composite reliability, the factor loadings should be at least .5 and preferably .7 or higher (Bagozzi and Yi, 2012). Factor loadings below .5 or so indicate variables that are not especially aligned with the factors, however, acceptable reliabilities even below .5 may be appear when the CFA model fits satisfactorily (Bagozzi and Yi, 2012).
Opposite signs of factor loadings for different variables in the case of the same factor reflect that the various variables are related with the same factor but in opposite directions (Janssens et al., 2008). Negative values for factor loadings may occur when fitting non-linear functions to data. The negative statements load negatively which is logical as the statements are totally opposite from the positive statements, and the negative and positive statements cannot be valid at the same time. The absolute value of the loadings for the negative statement is used when analyzing the construct validity.
3.4.2 Evaluation of the Model Fit Indices
Based on some of the general recommendations on selecting absolute fit indices, the following model fit indices are used in this research: chi square, the degrees of freedom and its probability (Markland, 2007). In addition, as in most SEM model researches, the Comparative Fit Index (CFI) and Root Mean Square Error of Approximation (RMSEA) are also included (Bentler, 2010). The Tucker-Lewis Index (TLI) is also considered to be a reliable index, and therefore it is referred to in this research as well (Janssens et al., 2008). In some SEM literature, the TLI is also called the non-normed fit index (NNFI) and its use is recommended in addition to RMSEA, CFI (Bagozzi and Yi, 2012).
McIntosh (2007) states the chi square test is the most rigorous global test available in SEM for testing distributional and structural assumptions and thus recommends to start a model evaluation with the chi square test, also Bagozzi and Yi (2012) consider the chi square is one of the most important statistical tests for model, but they remind that it is sensitive to the size of the sample. However, the chi square test is also subjective, just as the approximate fit indices are, and there is no golden rule that can be associated with it, instead it should be demonstrated that the data does not depart considerably from the model (Markland, 2007). Due to the fact that models differ by complexity, parameter
values, sample sizes and data distributions, it is hard to provide golden rules for cut-off values (Markland, 2007). Usually, the larger the size of the sample, the more probable it is that the model will fail to fit according to the chi square fit test (Barrett, 2007).
(Iacobucci, 2010) notes that one should not be too concerned with the chi square, as it does not in most cases fit if the sample size is 50 or more. In accordance with the recommendation of (Mulaik, 2007), the choice for analyzing the model fit is not done between chi square tests and goodness of fit indices, as they are complementary.
It is also important to note, that method effects, such as positively and negatively worded items, might result in some under-parameterization (Markland, 2007). In the data set used in this research, each of the dimensions in the original BBX includes a negatively worded item on each of the four brand experience dimensions. However, it is suitable in these cases to refer to the residual patterning as proof of model fit and also accept the model even if it has a significant chi square value (Markland, 2007).
TLI and CFI are one of the most reliable indices, and they should preferably be greater than .90 (Janssens et al., 2008). In SEM literature, it has also been noted that CFI gets larger as the model and data become more interesting and moves away from a simplistic model of independence, however, CFI is still a very powerful index, and one should not be too critical if the CFI does not quite reach the recommended value (Iacobucci, 2010).
The selection of the cut-off values for the model fit indices is not straight-forward, and it was not unproblematic in this research either. The optimal cutoff criteria for most model fit indices are conditional upon a variety of factors including the estimation method used, sample size, and model complexity. According to Tomarken and Waller (2005), conventional guidelines for the selection and interpretation of fit indices are often incorrect or oversimplified. Due to the complexity of the issues related to interpreting the fit indices, even in the best of cases some subjectivity is included when assessing model fit. Design factors are just as critical as they impact statistical power, and measures of fit are sensitive to poor designs (Tomarken and Waller, 2005). Fabrigar et al. (2010) warn that often researchers interpret model fit indices in a simplistic and dichotomous way, and that dichotomous cut-off values can be arbitrary.
The criterion used for fit is considered to be an abstract concept in the majority of SEM models (Barrett, 2007). Due to the fact that models differ by complexity, parameter values and sample sizes and data distributions it is hard to provide golden rules for cut-off values (Markland, 2007). Generally, the larger the sample size, the more likely a model will fail to fit via using the chi square goodness of fit test (Barrett, 2007).
According to Iacobucci (2010), one should not be too concerned with the chi square, as it does no not in most cases fit if the sample size is 50 or more. Also, one should not be overly critical if the CFI is not quite .95 (Iacobucci, 2010). Iacobucci’s guideline is that a model that fits well often results in a chi square value close to N, which indicates that chi square is sensitive to N. The sample size being 1518 in this study would then mean
that the chi square could around 1518 and even higher for a model to fit still well. Also, the chi square increases as a function of degrees of freedom (Iacobucci, 2010), which can also be seen in the results of this study below.
In SEM literature, different cut-off values area presented for RMSEA. According to Janssens et al. (2008), Hu and Bentler (1999) place the cut-off for RMSEA at .06, whereas Browne and Cudeck (1993) consider that values less than or equal to .05 indicate a good fit and values up to .08 indicate an acceptable fit. The sources of Iacobucci (2010) also indicate that RMSEA is not reliable with all samples, and the fit tends to worsen as the number of variables in the model increase. Table 8 below summarizes the recommendations for the cut-off values for the model fit indices used in this study.
Table 8. Indices for Evaluating Model Fit
Model Fit
Indices Range Recommendation Description
Chi Square sensitive to N
a model that fits well would produce a chi square close to N (Iacobucci, 2010)
A measurement indicating how expectations compare to results.
The data is random, mutually exclusive, drawn from independent variables and from a large enough sample.
- close to 0.95 (Iacobucci, 2010) -greater than 0.90 indicate acceptable fit (Tomarken and Waller, 2005)
- greater than .80 permissible
Comparative Fit Index is an incremental fit index. It takes the fit of one model to the data and compares it to the fit of another model to the same data (Bentler , 2010, Iacobucci, 2010).
4 RESULTS
This chapter concentrates on reporting the findings and presenting the empirical data by starting with the details on the respondents of the survey and then analyzing how well the research questions can be answered and are supported by the data.