Model fit in factor analysis is determined by assessing how well the hypothesized model reproduces the observed covariance or correlation matrix. Several statistical measures and indices are used to evaluate this fit.
Here's a breakdown of the common approaches:
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Chi-Square Test (χ²): This is a fundamental test that assesses the discrepancy between the observed and model-implied covariance matrices. A non-significant p-value (typically p > 0.05) suggests a good fit, meaning the difference between the matrices is small enough to be attributable to sampling error. However, the chi-square test is highly sensitive to sample size; with large samples, even trivial discrepancies can lead to a significant (poor fit) result.
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Degrees of Freedom (df): The degrees of freedom represent the number of independent pieces of information used to estimate the model parameters. A model with more parameters will have fewer degrees of freedom. Generally, a model with more degrees of freedom is preferred, provided it still fits the data adequately.
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Root Mean Square Error of Approximation (RMSEA): RMSEA estimates the amount of error in the model's approximation of the population covariance matrix. Values closer to zero indicate a better fit. Generally:
- RMSEA < 0.05: Close fit
- 0.05 < RMSEA < 0.08: Reasonable fit
- 0.08 < RMSEA < 0.10: Mediocre fit
- RMSEA > 0.10: Poor fit
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Standardized Root Mean Square Residual (SRMR): SRMR represents the average discrepancy between the observed and predicted correlations. Like RMSEA, lower values indicate a better fit. A common guideline is that SRMR < 0.08 suggests acceptable fit.
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Comparative Fit Index (CFI): CFI assesses the fit of the proposed model relative to a null model (a model with no relationships between variables). CFI values range from 0 to 1, with values closer to 1 indicating a better fit. A CFI of 0.90 or higher is generally considered indicative of acceptable model fit, and values above 0.95 are considered excellent. The CFI is particularly useful because it adjusts for the issues of sample size inherent in the chi-squared test of model fit, and the normed fit index.
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Tucker-Lewis Index (TLI): Similar to the CFI, the TLI compares the fit of the proposed model to that of a null model. It also adjusts for model complexity. TLI values range from 0 to 1, with values closer to 1 indicating a better fit. A TLI of 0.90 or higher is generally considered acceptable.
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Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC): AIC and BIC are information criteria used to compare different models. They consider both model fit and model complexity (number of parameters). Lower AIC and BIC values indicate a better balance between fit and parsimony. They are particularly useful for comparing non-nested models (models that cannot be derived from one another by adding or removing parameters).
In summary, model fit in factor analysis is determined by examining a combination of statistical tests and fit indices. No single measure is definitive; researchers typically consider multiple indices to make a comprehensive assessment of model fit. Important considerations are the size of the sample, the complexity of the model, and theoretical justification for the proposed factor structure.
