Confirmatory Factor Analysis
1 Introduction
Confirmatory Factor Analysis (CFA) is used to verify the factor structure of a set of observed variables.
Unlike its exploratory counterpart EFA, CFA is a theory-driven approach where researchers begin with a strong theoretical foundation (belief) about the relationships between variables and latent constructs in their dataset.
In EFA we ‘discover’ the latent structure. In CFA, we ‘test’ a latent structure we suspect to be present.
While EFA is used to discover the underlying structure of a large set of variables without imposing any preconceived structure, CFA requires us to specify the number of factors and the pattern of indicator-factor loadings in advance. This specification is usually based on prior research, theory, or both.
In a sense, EFA is like unsupervised learning, and CFA is supervised learning.
1.1 Differences between EFA and CFA
Key differences between CFA and EFA include:
- Purpose: CFA tests specific hypotheses about structure, while EFA explores possible structures;
- Theory: CFA is theory-driven and confirmatory; EFA is data-driven and exploratory;
- Factor rotation: CFA doesn’t require rotation methods, unlike EFA;
- Model specification: In CFA, researchers must specify which items load on which factors; in EFA, all items are allowed to load on all factors.
CFA is often used in scale development, construct validation, and testing measurement invariance across groups. It provides a more rigorous evaluation of construct validity than EFA and is often used in advanced research contexts where strong theoretical foundations exist that we may wish to test.
2 Model Specification in CFA
2.1 Introduction
As we’ve noted, CFA is used to test hypotheses about the relationships between observed variables and their underlying latent constructs.
Unlike Exploratory Factor Analysis, which is primarily data-driven, CFA requires us to specify a model based on prior theoretical knowledge or empirical evidence.
This involves defining:
- the number of latent factors;
- the observed variables that load onto each factor; and
- any relationships between the factors.
Our hypothesised model is then tested to determine how well it fits the observed data. This gives us a rigorous framework for evaluating the validity of our hypothesised constructs.
Basically, we’re asking whether our assumptions about the underlying factors are ‘true’ or not.
2.2 Why does model specification matter?
Model specification in CFA is critical because it directly influences the validity and interpretability of the results. A well-specified model will align with both theoretical expectations and the observed data, minimising potential errors such as overfitting or misspecification.
Decisions about model structure, such as the inclusion of error covariances or the choice of estimation method, require us to balance theoretical rigor with statistical fit.
2.3 Defining the measurement model
CFA begins with defining a measurement model, which represents how latent variables (unobserved constructs) are linked to observable indicators.
This process requires a clear hypothesis about the number of factors and their relationships with the indicators. Unlike exploratory factor analysis, which searches for patterns in the data, CFA tests a predefined structure.
The theoretical basis for the model is really important. We can use existing literature, theoretical frameworks, or previous studies to specify which indicators belong to which constructs. This ensures the model reflects the underlying theory, rather than arbitrary patterns in the data.
Measurement models also incorporate assumptions about error. Each indicator includes a portion of error variance, representing aspects unrelated to the latent variable. Accounting for these errors is essential for accurately interpreting factor relationships.
2.4 The role of indicators
Indicators are observable variables used to measure latent constructs. Selecting reliable and valid indicators is key to successful CFA. Each indicator should be closely related to the construct it represents, with minimal overlap with other constructs.
Cross-loadings, where an indicator is associated with multiple constructs, can weaken model validity. For example, if one survey question measures two unrelated traits, it introduces ambiguity into the results. Carefully reviewing indicators ensures they align with the latent variable they are intended to measure.
Measurement error is also important. Errors reduce the precision of the model, making it essential to minimise them through reliable measurement tools and appropriate model specification. Indicators with high error variance are less useful in CFA.
2.5 Constructing path diagrams
Path diagrams provide a visual representation of the measurement model, showing the relationships between latent variables, indicators, and error terms. They help researchers communicate the structure of the model clearly and concisely.
Latent variables are typically represented as circles, while indicators are shown as rectangles. Arrows indicate relationships, with one-headed arrows denoting causal paths and two-headed arrows representing correlations. Including error terms for each indicator emphasizes the model’s complexity.
Well-constructed path diagrams make it easier to identify errors or ambiguities in the model. They also serve as a useful reference when conducting parameter estimation and interpreting results.
3 Parameter Estimation
3.1 Introduction
Parameter estimation is an important step in CFA, where numerical values are assigned to the model’s relationships based on the data. These parameters include factor loadings, variances, covariances, and residuals, all of which help determine how well the model fits the observed data.
This section covers estimating factor loadings, understanding variances and covariances, and choosing suitable estimation methods.
3.2 Estimating factor loadings
As we noted in the section on EFA, factor loadings represent the strength of the relationship between latent variables and their indicators. Higher loadings indicate that an indicator strongly reflects its associated construct, while lower loadings suggest weaker alignment.
Loadings can be standardised or unstandardised: - Standardised loadings range from -1 to 1 and allow for easy comparison across indicators - Unstandardised loadings depend on the scale of measurement.
Both types are useful, depending on the context of analysis.
Interpreting factor loadings involves setting thresholds for acceptability. Loadings above 0.7 are generally considered strong, though lower values (e.g., 0.5) might be acceptable in exploratory research. Consistently low loadings might indicate poorly chosen indicators.
3.3 Covariances and variances
“Covariances” reflect the relationships between latent constructs, while “variances” describe the spread of data within each construct or indicator. These parameters are critical for understanding how constructs interact and whether they are distinct.
High covariances between constructs can suggest overlap, potentially indicating redundancy in the model. Conversely, low covariances might indicate insufficient conceptual links between constructs, challenging the theoretical basis of the model.
Residual variances represent the portion of variance in an indicator not explained by the latent construct. Large residuals may signal poor model specification or high measurement error, both of which require further investigation.
3.4 Methods for estimation
Maximum Likelihood (ML) estimation is the most commonly used method in CFA. It assumes multivariate normality and maximises the likelihood that the model-generated data matches the observed data. ML is robust and versatile, making it a popular choice.
For non-normal data, alternatives like Weighted Least Squares (WLS) or Robust ML are better suited. WLS is ideal for ordinal or categorical data, while Robust ML handles violations of normality. Each method has specific assumptions and strengths.
Selecting the right estimation method depends on the data type, sample size, and model complexity. Understanding these factors ensures accurate parameter estimation and reliable results.
4 Assessing Model Fit in CFA
4.1 Introduction
Assessing model fit determines how well the specified CFA model aligns with the observed data. Fit indices provide quantitative measures of this alignment, guiding researchers in evaluating model adequacy. This section explores absolute fit indices, incremental fit indices, and overall goodness-of-fit criteria.
4.2 Absolute fit indices
Absolute fit indices, such as the chi-square statistic, compare the observed data with the model’s predicted values. A significant chi-square indicates a poor fit, but it is sensitive to sample size, often misrepresenting model adequacy in large samples.
RMSEA (Root Mean Square Error of Approximation) addresses some chi-square limitations. It measures the model’s error per degree of freedom, with values below 0.06 indicating good fit. SRMR (Standardized Root Mean Square Residual) evaluates the average residual, with values under 0.08 deemed acceptable.
These indices provide a foundation for assessing fit but should be interpreted alongside other measures to ensure a balanced evaluation.
4.3 Incremental fit indices
Incremental fit indices, such as CFI (Comparative Fit Index) and TLI (Tucker-Lewis Index), compare the specified model with a baseline model that assumes no relationships among variables. High values (above 0.90) suggest good fit.
Incremental indices complement absolute indices by highlighting improvements over the baseline. They are less sensitive to sample size and provide a more nuanced view of model adequacy.
By combining incremental and absolute measures, researchers can develop a comprehensive understanding of model performance, ensuring results are both accurate and interpretable.
4.4 Goodness-of-fit indices
Goodness-of-fit criteria involve integrating multiple indices to assess overall model adequacy. Relying on a single index can be misleading, as each has unique limitations.
Sample size and model complexity influence fit indices. Larger samples often produce significant chi-square results, while complex models struggle to achieve high fit values. We must account for these factors in our evaluations.
A good practice is to report multiple indices (e.g., chi-square, RMSEA, CFI) and justify their selection based on the research context. This ensures transparency and reliability in our analysis.
5 Modifying CFA Models
5.1 Introduction
Even the most carefully specified CFA models may require adjustments to improve fit and align with the data.
Modifying models involves diagnosing issues, making changes to paths or parameters, and validating the revised structure. However, modifications must be made cautiously to avoid overfitting or deviating from theoretical underpinnings.
This section focuses on diagnosing model misspecification, introducing changes, and validating the revised model.
5.2 Diagnosing model misspecification
Diagnosing model misspecification begins with our examination of fit indices.
Poor fit indices, such as high RMSEA or low CFI, indicate that the model does not adequately capture the data structure. We need to investigate these issues systematically.
Modification indices (MIs) provide guidance by suggesting changes that could improve model fit. For example, adding a path between indicators with high residual correlations may resolve the issue. However, these indices should not dictate changes without theoretical justification.
Residual analysis offers additional insight. Large residuals between observed and predicted values highlight areas where the model fails to account for data patterns. Identifying these discrepancies helps refine the model while maintaining its theoretical integrity.
5.3 Introducing changes to the model
When making changes to a model, it’s important to find a balance between improving the stats and keeping things meaningful and grounded in theory. Adding or removing paths, changing indicators, or adjusting factor structures should only be done if there’s solid support from existing research or theory.
Simplifying the model can often help it fit better and make it easier to understand. For instance, taking out indicators that don’t work well or dropping unnecessary factors can make things clearer and less complicated. However, it’s crucial not to oversimplify in a way that damages the validity of what you’re trying to measure.
One big risk when tweaking models is overfitting. An overfitted model might look perfect with your current data but could fail to work with new data. To avoid this, it’s a good idea to use cross-validation or split your data into training and testing sets to check how well the model holds up.
5.4 Cross-validation after modifications
Cross-validation ensures that modifications are not solely tailored to one dataset. Researchers can split their data into two subsets, using one for model modification and the other for validation. Consistent fit across both subsets indicates a robust model.
Theoretical justification is critical when presenting modifications. Every change should be clearly explained and linked to the research context, ensuring the revised model remains meaningful and interpretable.
Documenting the entire modification process, including rejected changes, enhances transparency. This allows other researchers to evaluate the model’s development and apply it in their own work.
6 Model Validation
6.1 Introduction
Model validation is the final step in CFA, ensuring that our model is reliable and generalisable across different datasets.
This process involves testing reliability, evaluating validity, and replicating the model to confirm its robustness. Validation strengthens confidence in the model’s findings and supports its application in future research.
6.2 Testing model reliability
Reliability assesses the consistency of a model’s measurements. Composite Reliability (CR) is a common measure, with values above 0.7 indicating acceptable reliability. CR accounts for the combined contributions of all indicators to a construct.
Cronbach’s Alpha is another reliability measure, widely used for internal consistency. While alpha values above 0.7 are generally considered good, very high values (above 0.9) may indicate redundancy among indicators.
Indicator reliability focuses on individual observed variables. An indicator’s reliability can be assessed through its factor loading, with loadings above 0.7 suggesting strong alignment with the construct.
6.3 Evaluating model reliability
Validity ensures that the model measures what it is intended to measure:
Convergent validity assesses whether indicators of a construct are highly correlated. Average Variance Extracted (AVE) is a common metric for this, with values above 0.5 indicating sufficient validity.
Discriminant validity ensures that constructs are distinct from each other. Low correlations between latent variables and high AVE values support discriminant validity. Violations of discriminant validity may signal issues like overlapping constructs or poor indicator selection.
Establishing validity involves rigorous testing and theoretical backing. Researchers must demonstrate that their constructs are both conceptually sound and empirically distinct.
6.4 Replicating the CFA model
Replication is essential for confirming model robustness and generalisability. Applying the model to a new dataset tests whether its structure and fit hold across different samples. Consistent results indicate strong external validity.
External validation involves using independent datasets, ideally from different populations or contexts. This process strengthens the model’s credibility and ensures its applicability beyond the initial study.
Reporting the replication process and results transparently is crucial. Detailed documentation allows others to assess our model’s generalisability and build upon its findings in future research.