How do I interpret lavaan output?

Question

I am attempting confirmatory factor analysis (CFA) using lavaan. I am having a hard time interpreting the output produced by lavaan.

I have a simple model - 4 factors each supported by items from collected survey data. The factors are in line with what is measured by the items, to the extent that it appears to be likely that they could serve as a valid measurement.

Please help me understand the following output produced by lavaan's cfa():

 Number of observations                          1730

  Estimator                                         ML
  Minimum Function Test Statistic              196.634
  Degrees of freedom                                21
  P-value (Chi-square)                           0.000

Model test baseline model:

  Minimum Function Test Statistic             3957.231
  Degrees of freedom                                36
  P-value                                        0.000

User model versus baseline model:

  Comparative Fit Index (CFI)                    0.955
  Tucker-Lewis Index (TLI)                       0.923

I have these questions:

1. How is the baseline model defined?
2. Given that for the specified degrees of freedom, the calculated Chi-Sq statistic is larger than what would be expected, is there any interpretation for the p-value that is equal to 0.000?
3. Based on CFI and TLI, it appears that I almost have a reasonable model. Is this a fair interpretation?

Answer 1

1) The baseline is a null model, typically in which all of your observed variables are constrained to covary with no other variables (put another way, the covariances are fixed to 0)--just individual variances are estimated. This is what is often taken as a 'reasonable' worst-possible fitting model, against which your fitted model is compared in order to calculate relative indexes of model fit (e.g., CFI/TLI).

2) The chi-square statistic (labeled as the minimum function test statistic) is used to perform a test of perfect model fit, both for your specified and null/baseline models. It essentially is a measure of deviance between your model-implied variance/covariance matrix, and your observed variance/covariance matrix. In both cases the null of perfect fit is rejected (p < .001), though this is by design in the case of the baseline/null model. Some statisticians (e.g., Klein, 2010) argue that the chi-square test of model fit is useful in evaluating the quality of a model, but most others discourage putting a lot stock in its interpretation, both for conceptual (i.e., the null of perfect fit is unreasonable) and practical (i.e, chi-square test is sensitive to sample size) reasons (see Brown, 2015; Little, 2013, for examples). It is, however, useful for calculating a number of other, more informative, indexes of model fit.

3) Standards for what level of model fit is considered "acceptable" may differ from discipline to discipline, but at least according to Hu & Bentler (1999), you are within the realm of what is considered "acceptable". A CFI of .955 is often considered "good". Keep in mind, however, that both TLI and CFI are relative indexes of model fit--they compare the fit of your model to the fit of your (worst fitting) null model. Hu & Bentler (1999) suggested that you interpret/report both a relative and an absolute index of model fit. Absolute indexes of model fit compare the fit of your model to a perfect fitting model--RMSEA and SRMR are a couple of good candidates (the former is often calculated along with a confidence interval, which is nice).

References

Brown, T. A. (2015). Confirmatory factor analysis for applied research (2nd Edition). New York, NY: Guilford Press.

Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6, 1-55.

Kline, R. B. (2010). Principles and practice of structural equation modeling (3rd Edition). New York, NY: Guilford Press.

Little, T. D. (2013). Longitudinal structural equation modeling. New York, NY: Guilford Press.