The measurement model schematically represents the relationship between the
Client Risk-Tolerance latent variables and its corresponding item parcel indicator
variables. The aim of fitting the measurement model was to determine the validity and reliability of the measures used to represent the constructs of interest (Diamantopoulos & Siguaw, 2000).
4.4.1 Confirmatory factor analysis
The substantive research hypothesis was tested by fitting the comprehensive LISREL model. The comprehensive LISREL model encompasses a structural model, an endogenous measurement model and an exogenous measurement model. The endogenous and exogenous measurement models define the nature of the hypothesised relationships between the latent variables and the indicator variables that represent them. Structural model fit indices could only be interpreted unambiguously for or against the fitted structural model if it were proven that the indicator variables used to operationalise the latent variables when fitting the structural model, successfully reflected the latent variables they were tasked to represent (Diamantopoulos & Siguaw, 2000). Therefore, the measurement model fit needed to be evaluated prior to fitting the structural model. As opposed to fitting two separate endogenous and exogenous measurement models, the two models were combined and fitted as a single exogenous measurement model (Swart, 2011).
The covariance matrix was analysed during the fitting of the measurement model. Robust maximum likelihood estimation (RML) was used as the null hypothesis for multivariate normality in the observed data was rejected. LISREL 8.8 (Du Toit & Du Toit, 2001) was used to perform the CFA.
The measurement hypothesis that the measurement model provided a valid description of the process that brought about the observed covariance matrix, was evaluated (Hair et al., 2006). If the measurement hypothesis was taken to mean that the measurement model provided a perfect account of the manner in which the latent
variables manifest themselves in the indicator variables, the measurement hypothesis translated into the following exact fit null hypothesis:
H01a: RMSEA = 0 Ha1a: RMSEA > 0
However, it is somewhat idealistic to assume that the measurement model would provide a perfect account of the manner in which the latent variables manifest themselves in the indicator variables and therefore it was expected to reject the null hypothesis (H01a). If the measurement hypothesis was taken to mean that the
measurement model only provided an approximate description of the process that produced the observed covariance matrix, the measurement hypothesis translated into the following close fit null hypothesis:
H01b: RMSEA ≤ 0.05 Ha1b: RMSEA > 0.05
4.4.2 Interpretation of measurement model fit and parameter estimates
Measurement model fit was interpreted by inspecting the range of indices provided by LISREL (Diamantopoulos & Siguaw, 2000). Further consideration was given to the magnitude and distribution of the standardised residuals, the magnitude of model modification indices calculated for Λ, Θℇ and Θ, the model parameter estimates and the squared multiple correlations (R2) for the indicator variables. Residuals represent a measure of the strength of the difference between elements of the observed and reproduced covariance matrices. If a sample is large enough, the standardised residuals can be interpreted as z-scores, i.e. in terms of standard deviation units from the mean (Diamantopoulos & Siguaw, 2000). Standardised residuals are considered positively or negatively large, i.e. the observed frequency is greater than the reproduced frequency, if they exceed the absolute value of + 2.58 or fall below - 2.58 (Diamantopoulos & Siguaw, 2000). Good model fit is indicated by residuals that are distributed approximately symmetrical around zero. Positive residuals suggest underestimation and imply the need for additional explanatory paths. Negative residuals indicate overestimation and suggest the need to reduce the number of
explanatory paths.
Modification indices indicate the extent to which the value of the chi-square (χ 2) fit statistic will decrease if a currently fixed parameter in the model is freed. Large modification index values indicate the measurement model parameters that, if set free, would improve the fit of the model. A large number of large and significant modification index values comment negatively on the fit of the model and suggests that numerous possibilities exist to improve the fit of the proposed model.
The completely standardised factor loadings reflect the average change, expressed in standard deviation units, in the indicator variables associated with one standard deviation change in the latent variables to which they have been linked, given that the effect of all other variables are held constant (Diamantopoulos & Siguaw, 2000). The factor loading estimates were considered to be satisfactory if the completely standardised factor loading estimates exceeded a stringent cut-off of .71 (Hair et al., 2006). The squared multiple correlations (R2) calculated and interpreted for each of the indicators signify the proportion of the variance in a specific indicator that is explained by its underlying latent variable. High R2 values are preferred.
Operationalisation of the latent variables that encompass the structural model will be considered successful if (a) the measurement model shows close fit, (b) the completely standardised factor loading estimates are statistically significant (p < .05) and exceed the stringent 0.71 cut-off (Hair et al., 2006), (c) the measurement error variances for all items are statistically significant and small, and (d) reasonably large R2 values (R2 ≥ .25) for all items are obtained (Van Heerden, 2013). If at least reasonable fit was obtained for the Client Risk-Tolerance measurement model and if the parameter estimates satisfied the stipulated conditions, the Client Risk-Tolerance structural model could be tested by fitting the reduced structural model with LISREL.
4.4.3 Discriminant validity
The ten latent variables represented in the hypothesised Client Risk-Tolerance structural model were regarded as qualitatively distinct, yet causally related constructs. Due to the causal relations hypothesised in the model, some degree of
correlation was expected as each measure of a construct could be expected to be related to a measure of another construct. However, the ideal would be for the latent variables comprising the study to be measured in such a way that the measurement reflects essentially a single construct, and as such high levels of discriminant validity was sought. That is, the correlations between the latent variables had to be sufficiently low to conclude that the latent variables were successfully operationalised as distinct constructs. Discriminant validity would be indicated if all Φij estimates were smaller than .90. This was investigated by inspecting the Φ matrix.
4.5 Evaluating the Fit of the Client Risk-Tolerance Measurement Model