Convergent Validity

Convergent validity assesses the extent to which the items constituting the constructconverge or share a high proportion of variance in common (Straub, Boudreau and Gefen 2004; Hair et al. 2010). In CFA AMOS, the convergence validity of a construct can be assessed using one or a combination of the following measures: Goodness-Of-Fit (GOF) measures; Squared Multiple Correlation (SMC), which is a function of the size of the Standardised Factor Loadings (SFL); Average Variance Extracted (AVE); and Construct Reliability (CR) (Straub, Boudreau and Gefen 2004; Hair et al. 2010). The various measures of convergent validity and the considerations for model re-specification are discussed briefly below.

Goodness-Of-Fit (GOF) Statistics: GOF compares the goodness of fit between

142 the better the theory is said to fit the data. In other words, GOF reflect the model’s ability to represent the data. When the GOF showed a poor fit of the theorised model, the model is required to be re-specified. The CFA approach supports a model to the degree that the fitted population covariance matrix corresponds to the observed sample covariance matrix (Marsh et al. 1988). It statistically tests the entire model simultaneously, to determine its fits with the data (Byrne 2009). Model-fit is a critical concern when conducting SEM. Good model-fit firstly, indicates high correspondence between the data and the relationships represented in the model and secondly, validates the model for the purpose of research (Byrne 2010).

GOF indices used in this study are chi-square, absolute fit indices, incremental fit indices and parsimony fit indices (see Table 6.11). While a number of GOF indices are available, most authors suggest that three to four different types of fit indices can provide adequate support for a model-fit (Hair et al. 2010, p. 672; Kline 2010). For example, Hair et al., (2010) recommend that in addition to the chi-square (2

) value and degrees of freedom, at least one incremental index (CFI or TLI) and at least one absolute index (RMSEA or SRMR) should be reported. Table 6.12 summarises cut-off vaue of the GOF indices for this study.

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Table 6.11. Category of GOF Indices

Category Statistics Definition

Chi-square (2) Chi-square (2) Difference between observed and estimated

covariance matrices

Normed Chi-Square (2/df) Ration of chi-square to degrees of freedom

for a model

Absolute fit indices Root Mean Square Error of

Approximation (RMSEA)

Badness-of-fit index measuring how well a model fits a population taking into account both model complexity and sample size Standardised Root Mean

Square Residual (SRMR)

Standardised value of RMSR Incremental fit

indices

Normed Fit Index (NFI) Assess how well a specific model fits

relative to some alternative baseline model (often a null model that assumes all observed variables are uncorrelated) Comparative Fit Index

(CFI)

Tucker-Lewis Index (TLI) Parsimonious fit

indices

Parsimonious Comparative Fit Index (PCFI)

Evaluates the parsimony ratio of the model compared to the GOF such as CFI and NFI Parsimonious Normed Fit

Index (PNFI)

Source: Hair et al. (2010)

Following the guidelines recommended by Byrne (2010), Kline (2010) and Holmes-Smith (2010), this study evaluates model-fit based on selected fit measured as summarised in Table 6.12.

Table 6.12. A Summary of Selected Fit Measures and Established Criteria

Category Statistics Acceptable

Level Reference Comment Chi-Square (2 ) Chi-square (2) p > 0.05 (Hair et al. 2010, p.666; Holmes- Smith 2010)

This measure is sensitive to large sample size. Normed Chi- square (2 /df) 1.0 ≤ χ2 / df ≤ 5 (Bagozzi et al. 1991; Hair et al. 2010)

The measure is subject to the sample size effects. Absolute fit

indices

Root Mean Square Error of Approximation (RMSEA) ≤ 0.08 (Hair et al. 2006; Hair et al. 2010; Lewis et al. 2005)

Value close to 0 indicates a perfect fit. Standardised Root Mean Square Residual (SRMR) ≤ 0.09 (Hair et al. 2010) Incremental fit indices

Normed Fit Index (NFI)

≥ 0.92 (Hair et al. 2010) Value close to 1 indicates a perfect fit. Comparative Fit Index (CFI) Tucker-Lewis Index (TLI) Parsimonious fit indices Parsimonious Comparative Fit Index (PCFI) ≥ 0.5 (Hair et al. 2010) - Parsimonious

Normed Fit Index (PNFI)

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Squared Multiple Correlations (SMC): Standardised estimates of .5 or above

(preferably .7 and above) and SMC from .3 but preferably .5 and above, suggest construct validity and item reliability (Hair et al. 2010).

AVE and Construct Reliability: With the GOF indices supporting the model’s

fit with the data, the model’s convergent validity is further assessed based on the size of SFL, using the AVE and CR (Hair et al. 2010). The AVE was computed by determining the sum of each individual item’s SFL square and dividing by the total number of items within the factor. CR was computed by squaring the sum of each individual item’s SFL, within the factor and dividing it by the squared sum of each item’s SFL square and sum of each individual item’s error variance within the factor (Hair et al. 2010; Holmes- Smith 2010). Evidence of convergence validity exists if the SFL, AVE and CR values are at least 0.7, 0.5 and 0.6, respectively.

Model Re-specification Considerations: A model is said to be correctly specified when it reproduces the sample covariance matrix well. When instances of specification error are noticed, the critical ratios (t-values), the SMC values, the standardised residuals and the modification indices (MIs) were examined to re-specify the model. Conceptually, all unstandardised estimates should be in the expected direction and statistically different from zero (that is, the critical ratio is larger than ± 1.96 at α = 0.05 significance level) (Byrne 2010; Hair et al. 2010). SMC values should be greater than 0.5. Standardised residual covariance should also be less than the benchmark value of 4 but preferably less than 2.58 (Hair et al. 2010).

A large residual covariance between any two measurement items indicates that the association between these two items is not accounted for sufficiently by the model. This suggests a problem with one or both of the measurement items. A standardised

145 residual value of 2 indicates that a particular covariance is not well reproduced by the hypothesised model (at α = 0.05 significance level) and a standardised residual value of 4 relates to α = 0.001 significance level. When a consistent pattern of large standardised residuals is associated with either a single item or several of the items within the factor, the necessary re-specification had to account for this association between the variables, such as by dropping an item and re-running the measurement model (Hair et al. 2010).

Modification Indices (MIs) also suggest a potential source of model re-

specification. A MI is calculated for each non-free parameter and represents a possible decrease in 2, if the parameter is freely able to be estimated in the re-specified model. A chi-square of 3.84 with one degree of freedom has a p = 0.05 and an MI value greater than 4 suggests that the chi-square could be significantly reduced if the corresponding parameters were estimated. Based on these guidelines, this study examined the measurement items that reveal high MI; that is, above 4 (Byrne 2010; Hair et al. 2010), and made appropriately re-specified the model.