Comparative Fit Index (CFI) Calculator

The Comparative Fit Index (CFI) is an essential tool in structural equation modeling to assess how well a proposed model fits the data compared to a baseline model.

Historical Background

CFI was introduced in the 1980s as a response to the need for a robust fit index in covariance structure modeling. It became widely used due to its ability to adjust for model complexity, providing a more reliable fit measurement than some earlier indices, especially in the field of psychology and social sciences research.

Calculation Formula

The CFI formula is as follows:

\[ \text{CFI} = 1 - \frac{\chi^2{\text{Model}} - \text{df}{\text{Model}}}{\chi^2{\text{Baseline}} - \text{df}{\text{Baseline}}} \]

Where:

• \(\chi^2_{\text{Model}}\) is the chi-square value of the model.
• \(\text{df}_{\text{Model}}\) is the degrees of freedom of the model.
• \(\chi^2_{\text{Baseline}}\) is the chi-square value of the baseline model.
• \(\text{df}_{\text{Baseline}}\) is the degrees of freedom of the baseline model.

Example Calculation

Suppose:

• \(\chi^2_{\text{Model}} = 120\)
• \(\text{df}_{\text{Model}} = 50\)
• \(\chi^2_{\text{Baseline}} = 500\)
• \(\text{df}_{\text{Baseline}} = 200\)

The CFI calculation would be:

\[ \text{CFI} = 1 - \frac{120 - 50}{500 - 200} = 1 - \frac{70}{300} = 1 - 0.2333 = 0.7667 \]

Importance and Usage Scenarios

CFI is widely used in structural equation modeling to compare the fit of a model to a baseline, typically an independent model where all variables are uncorrelated. It is particularly useful in psychological and social science research to evaluate complex theoretical models. A CFI value close to 1 indicates a good fit, while values below 0.90 suggest a poor fit.

Common FAQs

1. What is a good CFI value?

   • A CFI value above 0.90 generally indicates an acceptable fit, while values above 0.95 are considered to represent a very good fit.

2. Why is CFI important?

   • CFI accounts for model complexity and provides a standardized way to assess model fit, allowing researchers to compare models and ensure that their proposed structures are supported by the data.

3. Can CFI be negative?

   • In some cases, especially with poorly fitting models, the CFI value can become negative. However, it is conventionally reported as 0 in such instances, indicating a very poor fit.

4. How is CFI different from other fit indices?

   • Unlike other indices, CFI adjusts for model complexity, offering a more nuanced evaluation of fit, which is less sensitive to sample size and model degrees of freedom.