200
(RMSEA), and the standardized root mean squared residual (SRMR). Incremental fit
measures compare the proposed model to a null model to determine the degree of
improvement over the null model. Common incremental fit measures are the comparative fit
index (CFI), the normed fit index (NFI) and the Tucker-Lewis index (TLI). Parsimonious fit
measures evaluate the fit of the model versus the number of estimated
coefficients needed to
achieve that level of fit. Examples of these measures include the parsimonious normed fit
index (PNFI) and the parsimonious goodness-of-fit index (PGFI) (Tian and Stewart 2007).
Given a lack of consensus on the best measure of fit, SEM scholars recommend the use of
multiple measures rather than rely on a single choice in evaluating the model fit (Bollen 1989,
Hu and Bentler 1999, Schumacker and Lomax 2004, Ho 2006, Byrne 2009, Hair
et al.
2006).
As indicated by Byrne (2009), the chi-square (
χ
2
) is the only statistical test of significance for
testing the fit of a proposed model. A low
χ
2
value relative to the degrees of freedom (
df
),
indicating non-significance (p > 0.05), would point to a good fit. This is because such non-
significance means that there is no statistical difference between the actual and predicted
input matrices (Byrne 2009). Nevertheless, there
are two cases in which the
χ
2
may be
misleading (Hoyle 1995). Firstly, the
χ
2
is very sensitive to the complexity of the model; the
more complex the model, the more likely the results will indicate a poor model fit. Secondly,
it is too sensitive to the sample size; the larger the sample size, the more likely it is that the
model will be rejected even if it is, in reality, a good fit with the data (a
Type II error-
accepting a false null hypothesis).
To recognize these problems, this study complements the
χ
2
measure with other goodness-of-fit measures.
Marsh et al. (1988) propose that the criteria for ideal fit measures are relative independence
of sample size, accuracy and consistency to assess different models, and ease of interpretation
201
aided by a well-defined pre-set range. Based on these stated criteria, Grace and Weaven
(2011)
recommend the use of the GFI, CFI, TLI and RMSEA. Also, in their studies of the
performance of various fit measures in relation to sensitivity to model misspecification, Hu
and Bentler (2010) and Kline (2012) recommend the use of the RMR in tandem with one of
several other indices particularly the TLI and RMSEA. In line with these recommendations,
the present study reports the GFI, CFI, TLI, RMSEA and RMR in combination with reporting
the
χ
2
. The following is a brief discussion of the fit measures and their interpretation taken
into account by the present study when testing model fit.
The GFI is a measure of fit between the hypothesized model and the observed covariance
matrix. Holmes-Smith and Coote (2012) argue that the good model
fit is achieved if GFI is
above 0.90. The CFI estimates the proportion of improvement in the proposed model beyond
the null model, based on the noncentral
χ
2
distribution that allows for unbiased estimation of
small sample size. The CFI has a value ranging from 0 to 1 (1 = perfect fit), with a value of
0.90 or greater representing a good-fitting model. Similarly, the TLI indicates the percentage
improvement in fit over the null model by taking into account
df
. Higher values of the TLI
indicate
a better fitting model, and it is common to apply the 0.90 rule as indicating a good fit
to the data. RMSEA estimates the lack of fit in a model compared to a perfect (saturated)
model. A value of 0.08 or less indicates a good-fitting model, while a value greater than 0.1
represents a poor-fitting model (Steiger 1990, Browne and Cudeck 1993). The RMR is the
square root of the discrepancy between the sample covariance matrix and the model
covariance matrix. The RMR values range from 0 to 1, with value of 0.05 or less indicating a
good model fit (Hooper
et al.
2008).
202
Importantly, the statistical significance of individual parameter estimates for the paths in the
proposed model also needs to be considered for assessing model fit (Schumacker and Lomax
2004). A ratio of the parameter estimate to the estimated standard error can be formed as a
critical value. If the critical ratio exceeds 1.96 at a significant level of less than 0.05, then the
parameter is statistically significant. Furthermore, as suggested by Schumacker and Lomax
(2004), the parameter estimates obtained from the analysis should
be theoretically meaningful,
and should be within an expected range of values (e.g., variances should not have negative
values and correlations should not exceed 1).
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