4.16. Data Analysis
4.16.5. Confirmatory Factor Analysis (CFA)
In the Confirmatory Factor Analysis (CFA) process, an assessment is made as to whether the hypothesised model fits the data, or whether there exists an association between the observed variables and their related fundamental latent or unobserved constructs (Child, 1990 cited in (Holtzman, n.d.). In addition, the CFA is helpful in confirming that all the variables or items are correctly associated with the right aspects of the construct under measurement (Holtzman, n.d:2).
According to (Holtzman, n.d:2, in running a CFA, a number of steps need to be followed. It is important that a model be specified on which basis data needs to be obtained to test the model.
Furthermore, it is important that a minimum of three variables or items are allocated to each factor or dimension for the factor to be strong, but it is regularly tolerable for a model to comprise of, as a maximum, one such dimension or factor (Anderson & Rubin, 1956 cited in Holtzman, n.d:2).
However, “if there are two or more correlated factors, two variables per factor can be sufficient”
(Hancock & Mueller, 2010:106). In addition, the sample for a CFA, there must be a large sample and the rule of thumb is 10 individuals for each variable measuring the construct. (Everitt, 1975 cited in Holtzman, n.d.:2). Normality is also an important requirement and Kline (2005) cited in Harrington (2009) suggests that variables having absolute values of greater than 3 for the skew index and absolute values of greater than 10 for the kurtosis index indicate normality problems.
Finally, the data need to be examined for outliers, no missing data, a multivariate normal distribution and colinearity after which the CFA can be run. It is also important in a CFA analysis to evaluate model fit (Holtzman, n.d.:2).
In conducting a CFA, there are many fit statistics that need to be taken into account indicating whether the model shows an acceptable fit with the data. A chi-square test illustrates the extent of difference between the observed and expected covariance matrices. A chi-square value closer to 0 and an associated p value more than 0.05 is indicative of a small difference between the observed and expected covariance matrices and is an indicator of good model fit, but can be problematical
122 as this test can be sensitive to sample size (Joreskog, 1969 cited in Holtzman, n.d.:2). Hence, other fit statistics are consulted in determining model fit.
Holtzman (n.d:3) provides guidance on the fit statistics for CFA as hereunder. The Root Mean Square Error of Approximation (RMSEA) relates to the model’s residuals. Values for RMSEA have a range of zero to one and a smaller RMSEA value showing a better fit. An RMSEA value of 0.06 or lower indicates good model fit (Hu & Bentler, 1999), although a value of 0.08 or lower is regularly deemed to be adequate (Browne & Cudeck,1993). Comparative Fit Index (CFI) evaluates overall enhancement of a suggested model against an independence model when the observed variables are not correlated (Byrne, 2006). CFI values are found between zero to one and higher values representing improved model fit. An adequate model fit value for CFI is 0.90 or higher (Hu & Bentler, 1999). Other common model fit indicators are The Normed Fit Index (NFI) and Nonnormed fit index (NNFI) (Bentler & Bonett, 1980). For these indicies, a better model fit is brought about with larger values and values higher than 0.90 are are deemed to be tolerable (Holtzmzn, n.d.:3). According to Ong and Van (2007:63) the Goodness-of-Fit Index (GFI) shows the percentage of avaiable variance/co-variance in the dataset explained by the model and cite Bentler and Bonnett (1980) who recommend that the value for GFI should be at minimum 0.9 for acceptable model fit. For Adjusted Goodness-of-Fit Index (AGFI), Kats (2013:103) cites Hooper et al. (2008) who recommends that the value for this index should be greater than or equal to 0.9 for an acceptable model fit. According to Hooper et al. (2008) cited in Fields and Atiku (2015:288) the advice provided is that if at least four indicies are good, a good model fit can be concluded.
A summary of model fit indicies appears in Table 4.2.
123 Table 4.2: Model Fit Index Thresholds
Source: Hu and Bentler (1999) in Statwiki (n.d.)
It is also important to assess the model constructs for convergent and discriminant validity (Statwiki, n.d.). Convergent validity is commonly assessed through the calculation of average variance extracted (AVE). When AVE is less than 0.5, it is considered to be insufficient (Esposito, 2010:696) and therefore no convergent validity exists.
Discriminant validity is described as “the dissimilarity in the measurement tool’s measurement of different constructs” (Esposito, 2010:696) . “A necessary condition for discriminant validity is that the shared variance between the latent variable and its indicators should be larger than the variance shared with other latent variables” (Hulland 1999:199) cited in Esposito (2010:696). Fornell and Larcker (1981:46) cited in Esposito (2010:696) contend that if a latent variables AVE is greater than the shared variances (i.e. “the squared correlations”), of the latent variable (or construct) with other constructs within the model, discriminant validity exists for that construct. Also, when maximum shared variance (MSV) is less than AVE, discriminant validity is confirmed (Gaskin, 2013 cited in Tsiakis, 2015:284). In addition, for discriminant validity to be confirmed, average shared variance (ASV) must be less than AVE (Hair et al., 2009 cited in Ernst, 2015:38).
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