Article summary of Adapting models with confirmatory factor analysis by De Heus - Chapter

What are the different steps of confirmatory factor analysis?

There are several steps in confirmatory factor analysis (CFA).

• The first step is model specification: in which a model is drawn up.
• The next step is model identification: in which it is determined whether the model has a unique solution.
• The third step is to estimate the model: it is temporarily assumed that the model is correct and then the parameters are estimated (such as factor loads and error variances).
• The fourth step is the evaluation of the model: to do this, a new variance-covariance matrix is ​​estimated, which is compared with the original variance-covariance matrix. This makes it possible to test the probability that the differences between the two matrices are based on probability if the model is correct using the chi-squired test. It is also possible to check how well the data fit the model by using goodness-of-fit measurements.
• The fifth step is model respecification: where the model is adjusted if step four shows that this is necessary.

How can we improve the model and what is the difference between trimming and improving the fit?

After the model has been evaluated, there are two ways to improve it. First of all, free parameters can be fixated. This is known as trimming. In addition, fixated parameters can be released. This is known as fit improvement.

How does trimming work?

While a model may have a good fit, it is sometimes desirable to take into account other factors, such as parsimony. This means that, when there are two models that can explain the data equally well, preference is given to the simplest model (with the fewest free parameters). There are two commonly used options in CFA. First, factor loadings can be set to zero. In addition, it is also possible to equate correlations between factors to zero. Using trimming is helpful in simplifying your model, but it will not improve the fit of a model. In addition, it is important to trim in a controlled manner, because it is only acceptable if you can give good reasons for setting certain factor loadings or correlations to zero.

How does fit improvement work?

If a model is rejected based on the chi-squared test and the goodness-of-fit measurements, many researchers try to find a better model for the data (the fit is then improved). There are a few ways in which this can take place in CFA.

• First, it can be allowed for an indicator to also have a charge on a factor other than just the one originally envisioned.
• In addition, it may be allowed that there are correlations between factors or between the errors.
• Finally, more factors can also be added, although this is at the expense of parsimony.

What are modification indices and residuals?

Another important aspect in modifying a model is determining which fixated parameters should be released to improve the model. There are two ways to find out. In the first place, you can look at modification indices. A modification index indicates to what extent releasing a certain parameter leads to a better or worse fit. Because EQS (the computer program used) does not provide useful indexes, a second option is preferred, namely analysis of residuals. A residual is the difference between the actual value and the re-estimated value of an element in the variance-covariance matrix. If a model has a perfect fit, all residuals are equal to zero. The closer a residual is to zero, the better the model can explain the relevant correlation or covariance. To adjust the model, standardized residuals, the difference between the true (Pearson) correlation and the re-estimated correlation, are usually used.

What is residual analysis?

This will be discussed based upon three modification options:

• Correlating factors: if two factors actually have a positive correlation, but the model denies it (the correlation is set to zero), the re-estimated correlations between factors will be lower than the true correlations, leaving largely positive residuals for that combination of variables. In that case it is therefore possible that we can improve the fit of the model if we allow the factors to be correlated. To check if this is actually the case, we need to run the modified model and compare it with the original model.
• Additional factor loading: in reality, if a variable also loads positively on a factor other than their 'own' factor, but the model denies this, the true correlations of that variable with all variables of the other factor will be systematically higher than the re-estimated residuals, leading to positive residuals. In real life situations, this may not be noticeable at first, but only become noticeable if the improved model still has positive residuals.
• Correlating errors: if two variables show a residual that is clearly different from zero, but the residual is not part of a larger pattern, it is possible to look at the errors of those variables. However, this is only allowed if you have a good reason for this correlation. Otherwise you might have a good fit, but not actually be able to understand the model.