Three statistical methods are vital categories in the field of data analysis, especially in social sciences and psychology namely, Principal Component Analysis (PCA), Exploratory Factor Analysis (EFA), and Confirmatory Factor Analysis (CFA). All three methods have specific functions, and research approaches and are applied in different settings. This article will define each of the techniques and explain the distinction between them.

This is a statistical method that is used in forming techniques of dimensionality reduction. It changes a high variance with many variables to a new set of orthogonal variables called the principal components. These components include the maximum amount of variance of the information obtained from the initial dataset.

**Objective:** The main purpose of PCA is to explain as much variability in the data as it is possible after reducing their dimensionality. This is particularly valuable within large dimensional data sets.

**Methodology:** The mechanism of PCA can be explained as follows; calculating the covariance matrix of the data set, finding the eigenvalues and corresponding eigenvectors, and profiling the top components from the eigenvectors in the eigenvalue computation. The components obtained are a linear transformation of the ones formed initially.

**Assumptions:** PCA operationalizes this supposition through the belief that the principal components are uncorrelated or orthogonal and there are linear relationships between the existent variables.

Applications: It is frequently used in data preprocessing and exploratory data analysis techniques and image and signal encoders.

EFA is a technique used to define hidden patterns in measured variables. It is used to search for meanings of hidden concepts that are capable of predicting observed co-variation between two variables.

**Objective:** The main use of EFA is to simply examine a set of variables to see what structure, if any, may be found in the data rather than to force a given structure. It makes a great ease for a researcher to know how these variables cluster.

**Methodology:** EFA makes an effort to calculate the nonlinear relationships within the observed variables and estimate factors that are indicative of resident constructs. However, EFA is concerned with commonality rather than total variance as is PCA.

**Assumptions:** EFA assumes that there are hidden factors, which affect variables and are measurable from the data available.

**Applications:** Used mainly in scale construction, psychological assessment, and measuring variables in social science research for purposes of revealing the factors in the construct such as attitudes or personality traits.

CFA is a procedure for examining whether or not a particular hypothesized factor structure is an adequate fit for a particular data set. It enables researchers to confirm or reject theories that they may set a priori that is Expected theories or Models.

**Objective:** CFA mainly serves the purpose of testing hypotheses about the associations of measured variables with the factors. It tells how well the proposed model has explained the data.

**Methodology:** When doing CFA, researchers determine the number of factors and the variables that should load on a certain factor. The model fitness is assessed utilizing chi-square tests, Root Mean Square Error Approximation (RMSEA), and Comparative Fit Index (CFI).

**Assumptions:** What is on the suppositions in CFA is that there is a particular association between the assessed variables and factors based on theoretical conceptions.

**Applications:** Common in psychometrics in both the multivariate context of instrument validation and the testing of hypothesized structural models.

Feature | PCA | EFA | CFA |

Purpose | Dimensionality reduction | Discovering latent structures | Testing hypothesized models |

Assumptions | No underlying factors assumed | Assumes latent factors influence data | Requires predefined factor structure |

Data Usage | Utilizes total variance | Analyzes covariance among variables | Tests specific models against data |

Output Interpretation | Components are linear combinations | Factors represent underlying constructs | Factors defined by a theoretical framework |

Complexity Level | Generally simpler | More complex due to factor extraction | Complex due to model testing |

- While PCA aims at dimensionality reduction it does not make any assumption on underlying structure; it derives its components from variance only.
- EFA uses correlations without prior assumptions about the stands of variables to them.
- CFA evaluates the degree to which indicators align with the constructs that have been posited to underlie them.

- PCA does not involve model specification; it produces a few components based on variance.
- EFA thus permits the flexibility for the number of factors to be defined, however, no specific model is validated.
- CFA, unlike other methods, does not allow unstandardized coefficients whereby the researcher is forced to define which observed variables should load on which factors before conducting the analysis.

- Apply PCA when data complexity needs to be reduced or, in case of trying to visualize high-dimensional data.
- When conducting new constructs or new measurement scales without the prior knowledge of the factors to be included.
- Advanced users should use CFA when there is an existing theoretical framework or hypothesized framework of variables, against which the empirical data can be tested.

- If a consumer behavior study is conducted on a large number of survey items which has been followed by several indicator variables, the researcher may employ PCA to bring down the various items into various principal components which is easier to analyze.
- If researchers for instance wish to determine whether specific survey items measure a latent construct such as ‘customer satisfaction’, EFA is used to determine the items that are most similar in terms of the measure of inter-item correlation.
- If the researchers have a particular notion about which facets of customer satisfaction the survey should capture (for example, service quality, and product quality) then they would employ CFA to determine whether the cross-sectionary model that has been hypothesized is a good fit to the collected survey sample.

Altogether, the methods of PCA, EFA, and CFA are interrelated but are used for different objectives as components of multivariate analysis required for various investigations. Researchers need to know these differences for them to be able to determine which method to use as to their goals which include but are not limited to reducing dimensionality, examining latent structures, or as a test of theoretical models. While choosing the most suitable technique, the investigators are guaranteed a correct understanding of the results to make valuable additions to the subject matter.