Before presenting the estimation procedure for the structural equation model, this paragraph first briefly discusses the data requirements for a successful SEM analysis and whether the sample data in this study meets these requirements. First of all, SEM is in some ways more sensitive to sample size than other multivariate approaches. Given the model complexity and the measurement model characteristics in this study, which are elaborated below, the survey sample size73 (N=261) can be considered modest, but sufficient to serve the study aim (see, Hair et al., 2010 for a detailed discussion of the issue about sample size)74. Secondly, since the survey sample data deviates from multivariate normality, the estimation uses the MLM method (maximum likelihood estimation with robust standard errors and a Satorra-Bentler scaled test statistics), which is robust to non-normality and often used in SEM studies to deal with non-normality (Satorra and Bentler, 1988, 1994). Depending on the complexity of study model and the severity of non-normality problem, sample sizes of 200-500 are sufficient for good estimates with the MLM estimation method (Curran et al., 1996; Hu et al., 1992; Rosseel, 2012). Thirdly, like to other multivariate analysis, missing data could be a cause of problem to SEM analysis in estimation or interpretation, and the issue needs remedies if the missing data is substantial and nonrandom. Various procedures can be utilized to remedy missing data; however, each approach has its own advantages and disadvantages, and there is no golden rule in determining which approach is superior to others (see, Hair et al., 2010 for a
72 See Chapter 2 for a literature overview of the studies that have used SEM for data analysis for study
of travelers’ choices of ICT and ICT’s effects on travel behavior.
73A larger sample is believed to be able to reduce effects from deviation from normality and
multicollinearity in model estimation and to produce more stable solutions that are more likely to be replicable (Hair et al., 2010). However, to obtain a larger sample is constrained in this study because of the available budget for the data collection.
74 The suggested sample size according to Hair et al. is as follows:
- Minimum sample size-100: models containing five or fewer constructs, each with more than
three items (observed variables) and with high item communalities (.6 or higher).
- Minimum sample size-150: models with seven constructs or less, modest communalities (.5),
and no under-identified constructs.
- Minimum sample size-300: models with seven or fewer constructs, lower communalities
(below .45), and/or multiple under-identified (fewer than three) constructs.
- Minimum sample size-500: models with large numbers of constructs, some with lower
detailed comparison of these different approaches). In this study, pre-tests75 show that missing data is not a serious issue in the collected data. For the missing data, model-based EM imputation approach was used to remedy the missing data.
In addition to these tests and remedies conducted to meet the data requirements for a successful SEM analysis, the entire estimation process also involves a series of preparation and other types of analysis before reaching the final model that is theoretically meaningful and statistically well fitting (to be presented later in section 3.4.2). The estimation procedure is described as follows.
First, the basic statistics of all the variables are calculated, and the bivariate correlations between the variables are calculated in order to obtain a whole picture of all the variables and relations between these variables.
Second, an explorative factor analysis (EFA) is conducted in the statistical software package SPSS to explore the patterns of the measurement variables – whether the a-priori expectation that is used to define the measurement variables for the latent constructs in the operationalized model is consistent with the patterns concluded from the analysis based on the empirical data76. EFA is a technique particularly suitable for examining the underlying patterns or relationships for a large number of variables and it can be utilized to determine whether large number of variables can be summarized in a smaller set of factors or components (Hair et al., 2010). The explorative factor analysis shows that a majority of the variables demonstrate underlying patterns that are in line with the a-priori conceptualization, except for the variable “frequency of teleworking – how many days per month, on average, do you telework part of the day or the whole day at home” and the variable “Teleworking is an option for me, because of my house situation”, which are loaded on factors different from what was expected. These two variables are hence considered to be excluded in the SEM analysis later.
A confirmatory factor analysis (CFA) is conducted next to estimate the measurement model in software Mplus 6 (Muthén and Muthén, 1998-2012) – that is, to estimate the loadings of all the measured variables for the latent constructs in the operationalized model, and to estimate the mutual correlations, rather than the hypothesized paths, between all the constructs. Achieving a statistically well-fitting measurement model is an essential step to take before conducting a SEM analysis (Bryne, 2001; Jöreskog and Sorbom, 1996). Several tests are
75 The pre-test involves the following processes: a) determine the type of missing data – whether or not
the missing data is ignorable; b) determine the extent of missing data – whether the extent of missing data is low enough to not affect the results; c) diagnose the randomness of the missing data process – are the missing data missing at random or missing completely at random; and d) select the imputation method for missing data. See Hair et al. (2010) for more details of these processes.
76
As SEM should always be attempted with a strong theoretical basis for model specification, it does not require explorative factor analysis to explore the underlying patterns of measurement variables in advance. Rather, confirmatory factor analysis, which will be introduced later, should be used to estimate the measurement model (Bryne, 2001). However, as introduced, the conceptual model in this study is developed based on the anticipation of travelers’ behavior and preference in realism, rather than some underlying theory or established insights in the literature. Hence, the explorative factor analysis is conducted to test statistically whether the underlying patterns of the measurement variables are in line with the a-priori expectation in order to form a reasonable measurement model for SEM analysis.
conducted based on different specifications. The variables with factor loading on construct smaller than 0.5 or insignificant are deleted (Hair et al., 2010) and the model is re-estimated. The results of the measurement model with best model fit are used for the following SEM analysis. The details are elaborated in section 3.4.2.
After obtaining the measurement model with best model fit, the structural model is constructed to include the hypothesized paths between the latent constructs into model estimation. The model test and parameter estimates are based on the covariance matrix and estimated using MLM estimation method in Mplus 6. After estimation of the full model wherein all hypothesized paths are included, insignificant77 paths (but not insignificant correlations), which can be considered irrelevant to the model, are deleted and the model is re- estimated. In addition, modification indices (MI) are used to assess whether the paths that were not theoretically expected should be included into the model. Modification indices indicate the decrease in the chi-square value (i.e., model fit improvement) if an extra path is added for estimation (Bryne, 2001). The estimation results of different model specifications are compared, and the final model is chosen based on the criteria of model fit to the data and theoretical plausibility. Some test results are reported in the appendix of this chapter, and the final model is presented in section 3.4.2.