# Confirmatory Factor Analysis (CFA): A Detailed Overview

### Introduction to Confirmatory Factor Analysis

Confirmatory Factor Analysis (CFA) is a sophisticated statistical technique used to verify the factor structure of a set of observed variables. It allows researchers to test the hypothesis that a relationship between observed variables and their underlying latent constructs exists. CFA is distinct from Exploratory Factor Analysis (EFA), where the structure of the data is not predefined and is instead determined through the analysis.

### Purpose and Procedure of CFA

The primary goal of CFA is to confirm whether the data fits a hypothesized measurement model based on theory or prior research. This involves several critical steps:

1. Defining Constructs: The process begins by clearly defining the theoretical constructs. This stage often involves a pretest to evaluate the construct’s items and ensure they are well-defined and represent the concept accurately.

2. Developing the Measurement Model: In CFA, it is essential to establish the concept of unidimensionality, where each factor or construct is represented by multiple observed variables that are presumed to measure only that specific construct. Typically, a good practice involves having at least three items per construct.

3. Specifying the Model: Researchers must specify the number of factors and the pattern of loadings (which variables load on which factors). This specification is based on theoretical expectations or results from previous studies.

4. Assessing Model Fit: The validity of the measurement model is assessed by comparing the theoretical model with the actual data. This includes examining factor loadings (with a standard threshold of 0.7 or higher for adequate loadings), and fit indices such as Chi-square, Root Mean Square Error of Approximation (RMSEA), Goodness of Fit Index (GFI), and Comparative Fit Index (CFI).

Can the proposed five factors in a 20-question instrument be identified and validated through the specific items designed to measure them?

Do four specific survey questions reliably measure a single underlying factor?

### Assumptions in CFA

Multivariate Normality: The data should follow a multivariate normal distribution.

Sample Size: Adequate sample size is crucial, generally n > 200, to ensure reliable results.

Model Specification: The model should be correctly specified a priori based on theoretical or empirical justification.

Random Sampling: Data must be collected from a random sample to generalize findings.

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### Key Terms and Concepts in CFA

• Theory and Model: A theory is a systematic set of causal relationships explaining a phenomenon, while a model is a specified set of dependent relationships within that theory used for testing.
• Path Analysis and Diagram: Path analysis is utilized to test structural equation models, with path diagrams visually representing the cause-effect relationships.
• Endogenous and Exogenous Variables: Endogenous variables are outcomes within the model, influenced by other variables, while exogenous variables are predictors not influenced by other variables within the model.
• Confirmatory Analysis and Cronbachâ€™s Alpha: Confirmatory analysis tests pre-specified relationships, and Cronbachâ€™s Alpha assesses the reliability of construct indicators.
• Identification: This refers to the ability of the data to provide sufficient information to estimate the model. Models can be under-identified, exactly identified, or over-identified.
• Goodness of Fit: This measures how well the model fits the observed data. Fit indices help in evaluating whether the model is acceptable.

### Conclusion

CFA is an essential tool in the toolkit of researchers aiming to validate the structure of their measurement instruments. It provides a rigorous method to ensure that the data aligns with expected theoretical constructs, enhancing the reliability and validity of subsequent analyses based on these measurements.

Confirmatory factor analysis (CFA) and statistical software:

Usually, statistical software like Intellectus Statistics, AMOS, LISREL, and SAS are used for confirmatory factor analysis.  In AMOS, visual paths are manually drawn on the graphic window and analysis is performed. In LISREL, confirmatory factor analysis can be performed graphically as well as from the menu.  In SAS, confirmatory factor analysis can be performed by using the programming languages.

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