Exploratory Factor Analysis (EFA)

Validating the construct validity of a measurement scale — a mandatory and decisive transitional step for data reduction and observed variable screening before proceeding to linear regression analysis, Confirmatory Factor Analysis (CFA), or Structural Equation Modeling (SEM).

1 The Scientific Nature of Exploratory Factor Analysis (EFA)

Exploratory Factor Analysis, abbreviated as EFA (Exploratory Factor Analysis), is a multivariate statistical technique widely used to reduce a set of $k$ original observed variables into a set of $F$ factors (where $F < k$) that are more comprehensive and representative in meaning. In quantitative research practice, we typically survey many detailed questions (observed variables) to clarify an abstract concept. These variables often have strong linear correlations with each other.

Practical Simulation Example: Instead of independently analyzing 20 small individual consumer shopping behavior characteristics, EFA will use actual correlations to group them into 4 major factor groups (e.g., Price, Brand, Service Quality, and Convenience). Each of these major factor groups will contain approximately 5 observed variables with strong internal correlations. This helps simplify the theoretical model, saves analysis time, and avoids multicollinearity when running regression.

To better understand EFA's position in the research process, let's compare the fundamental differences between reliability testing using Cronbach's Alpha (performed in the previous section) and scale validity assessment using EFA:

Comparison Criterion Cronbach's Alpha Test Exploratory Factor Analysis (EFA)
Nature of Test Scale reliability (Reliability). Scale validity (Validity).
Scope of Examination Only examines internal linkage between observed variables within the same single group. Examines cross-correlations between all variables across all groups throughout the entire model.
Core Objective Checks whether observed variables are measuring the same concept (internal consistency). Detects variables misassigned to groups, cross-loading variables, or newly altered factor structures.
Processing Outcome Removes isolated questions that do not contribute to the group's overall reliability. Reorganizes, restructures, or removes observed variables that fail convergent or discriminant validity.

Two core values that must be achieved in EFA analysis:

Convergent Validity

Reflects the requirement that observed variables representing the same conceptual factor must have very strong correlations with each other. When displayed on the Rotated Component Matrix, observed variables belonging to the same theoretical group must converge onto a single factor column with factor loadings meeting high statistical standard thresholds.

Discriminant Validity

Reflects the requirement that variables belonging to one conceptual group must be completely distinct and not confused or overlapping with variables from other conceptual groups. When displayed on the Rotated Component Matrix, each group of observed variables must clearly separate into independent columns. If a variable loads strongly onto two different columns with a very small difference in loading coefficients, the discriminant validity of the scale will be violated.

Clear Distinction Between Two Concepts: In quantitative research, a scale with good reliability (high Cronbach's Alpha) does not necessarily achieve practical validity (EFA passing standards). Conversely, if EFA passes standards but internal reliability is poor, the results are equally worthless. Therefore, these two tests always go hand in hand as an inseparable pair.
2 The System of 5 Evaluation Indices in EFA

To confirm that your EFA results meet scientific quality standards and can be used for further research, you must sequentially verify against the following 5 golden indices:

Index 1 — KMO Sampling Adequacy Measure (Kaiser-Meyer-Olkin)

An index used to assess the suitability of the research sample for factor analysis. The $KMO$ value must reach $0.5$ or above ($0.5 \le KMO \le 1.0$) as the minimum prerequisite to proceed with factor analysis.

0.9 ≤ KMO ≤ 1.0 Excellent (Marvelous)
0.8 ≤ KMO < 0.9 Good (Meritorious)
0.5 ≤ KMO < 0.8 Acceptable (Mediocre/Miserable)
KMO < 0.5 Unacceptable

Index 2 — Bartlett's Test of Sphericity

Used to evaluate whether the observed variables have linear correlations with each other in the population. The mandatory condition is that the test must be statistically significant ($Sig. < 0.05$). If $Sig. \ge 0.05$, the observed variables have no cross-linear correlations and the EFA model has no scientific meaning whatsoever.

Statistical Significance Threshold
If the Bartlett Sig. coefficient ≥ 0.05, immediately stop EFA analysis because the variables have no correlation.
Sig. < 0.05

Index 3 — Eigenvalue

Eigenvalue is an index representing the amount of variance (information) explained by each extracted factor in the model. According to Kaiser's commonly used criterion, only factors that accumulate more information than 1 (Eigenvalue > 1) are retained in the final analytical model. Factors with Eigenvalue < 1 are discarded because the explanatory information is even less than a single independent variable.

Factor Retention Rule (Kaiser Criterion)
Factors with an Eigenvalue below 1 will be automatically removed from the extraction list.
Eigenvalue > 1

Index 4 — Total Variance Explained

Represents the level of information condensation of the model after grouping. Total Variance Explained must reach 50% or above ($\ge 50\%$). Treating the total information of the original dataset as 100%, this value shows how much % of the data the newly extracted factors can explain and how much % of the original observed variables is lost.

Minimum Requirement for Explained Information
An extracted variance below 50% indicates insufficient retention of original information.
Var. Expl. ≥ 50%

Index 5 — Factor Loading by Sample Size

Factor loading represents the level of correlation between a single observed variable and the extracted common factor. The higher the factor loading, the stronger the convergence of the variable onto that factor. According to the general standards established by Professor Hair et al. (2010) in Multivariate Data Analysis, the evaluation criteria for factor loading are as follows:

Factor Loading Classification Tiers
• Level ±0.3: Minimum condition for an observed variable to be retained.
• Level ±0.5: Achieves very good statistical significance.
• Level ±0.7: Achieves outstanding statistical significance.
Min ≥ ± 0.3

Importantly, Professor Hair et al. assert that the standard value for the Factor Loading coefficient must necessarily be considered in conjunction with the actual research sample size ($N$). The larger the sample size, the lower the minimum loading requirement, and vice versa. Refer to the detailed standard parameters below:

Sample size N ≥ 350 Factor Loading ≥ 0.30
Sample size N = 250 Factor Loading ≥ 0.35
Sample size N = 150 Factor Loading ≥ 0.45
Sample size N = 120 Factor Loading ≥ 0.50
Sample size N = 85 Factor Loading ≥ 0.60
Sample size N = 50 Factor Loading ≥ 0.75
3 Actual SPSS Output Results & How to Read Them

Below is a collection of extremely detailed simulated EFA results from actual survey data consisting of 15 observed variables belonging to 3 main factor groups: GSA (Graduate Soft Skills), WE (Work Environment), and GE (General Competence).

KMO and Bartlett's Test
Kaiser-Meyer-Olkin Measure of Sampling Adequacy. .696
Bartlett's Test of Sphericity Approx. Chi-Square 380.781
df 105
Sig. .000
KMO Coefficient Achieved
.696
Bartlett Significance (Sig.)
.000
PASSED STATISTICAL STANDARD (Sig. < 0.05)

KMO & Bartlett's Test Interpretation: The KMO coefficient of $0.696 > 0.5$ confirms that the sample dataset is completely compatible and suitable for structural reduction. Additionally, the Bartlett's test achieved absolute statistical significance with $Sig. = 0.000 < 0.05$, rejecting the null hypothesis $H_0$ and confirming that the observed variables have strong linear cross-correlations, establishing a solid foundation to proceed with rotation matrix analysis.

Total Variance Explained Table

Factor Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
1 3.569 23.791 23.791 3.569 23.791 23.791 3.456 23.038 23.038
2 3.419 22.795 46.586 3.419 22.795 46.586 3.263 21.752 44.790
3 2.929 19.529 66.116 2.929 19.529 66.116 3.199 21.326 66.116
4 0.812 5.413 71.529 Factor extraction stopped (Eigenvalue < 1)
5 0.654 4.360 75.889 Factor extraction stopped (Eigenvalue < 1)
Total Factors Extracted
3 Factors
Cumulative Variance Explained (%)
66.116%
MEETS SCIENTIFIC STANDARD (> 50%)

Total Variance Explained Interpretation: The data analysis model successfully extracted 3 factors with cumulative Eigenvalues all greater than 1 (the 3rd factor reached $Eigenvalue = 2.929 > 1$). Furthermore, Total Variance Explained reached 66.116% > 50%. This result demonstrates that the 3 newly formed factors can excellently explain 66.116% of the variance across the original 15 observed variables, with information loss well within an acceptable range.

Rotated Component Matrix

Observed Variable Extracted Component Convergence Quality Check
1 (GSA - Soft Skills) 2 (WE - Environment) 3 (GE - Competence)
GSA4 .876 Convergent Standard Met
GSA5 .832 Convergent Standard Met
GSA2 .826 Convergent Standard Met
GSA3 .803 Convergent Standard Met
GSA1 .792 Convergent Standard Met
WE4 .870 Convergent Standard Met
WE1 .843 Convergent Standard Met
WE3 .788 Convergent Standard Met
WE2 .714 Convergent Standard Met
WE5 .695 Convergent Standard Met
GE4 .843 Convergent Standard Met
GE2 .824 Convergent Standard Met
GE5 .822 Convergent Standard Met
GE1 .778 Convergent Standard Met
GE3 .678 Convergent Standard Met

Perfect Rotation Matrix Analysis

The observed variables grouped perfectly into 3 separate columns. There was absolutely no instance of variables shifting positions, misassigned to theoretical groups, or complex cross-loading phenomena. The factor loadings of all variables are very high, ranging well above the threshold from $0.678$ to $0.876$ (excellently meeting both small-sample and large-sample standards). This fully proves both Convergent Validity and Discriminant Validity of the entire scale.

A Highly Dangerous Real-World Error (Cross Loading)

If in actual results you discover that an observed variable loads simultaneously onto 2 different factor columns with coefficients both greater than $0.30$, you must perform a difference comparison: take the largest loading minus the second largest loading. If this difference is less than $0.30$, the observed variable completely fails discriminant validity. You must immediately remove this problematic variable and re-run a new EFA rotation cycle!

4 Standard EFA Procedure in 7 Steps on SPSS

The EFA factor analysis procedure requires strict sequential steps. Follow the 7 scientifically standard steps below to achieve the smoothest results:

1

Open the Factor Analysis Settings Window

On the top toolbar of SPSS, click to navigate via the exact path: Analyze > Dimension Reduction > Factor...

2

Add Observed Variables to the Analysis

Select all observed variables belonging to the independent factor system to be tested from the left list, click the arrow button to move them into the Variables box on the right. (Special note: Only add independent variables to run together in EFA. Dependent variables must be separated and run in EFA independently in a separate analysis round to avoid structural model bias!).

3

Configure Basic Statistical Indices (Descriptives)

Click the Descriptives... button on the right side. In the Correlation Matrix group, check the boxes for KMO and Bartlett's test of sphericity and Initial solution. Then click Continue.

4

Select Factor Extraction Method (Extraction)

Click the Extraction... button. Under Method, select the popular extraction method Principal components. In the Extract area, select Eigenvalues over: 1. Click Continue.

5

Set Up Factor Rotation Method (Rotation)

Click the Rotation... button. To achieve the best discriminant validity in most research, select the orthogonal rotation method Varimax. Alternatively, if the hypothesis suggests factors have strong cross-correlations, you may select the oblique rotation method Promax. Click Continue.

6

Format and Clean Output Display (Options)

Click the Options... button. Select Sorted by size to sort loading coefficients from largest to smallest for easy reading. Check the box for Suppress small coefficients and enter the value Absolute value below: .30 (or .40 / .50 depending on your sample size) so the system automatically hides small loading coefficients, making the rotation matrix display as clean as possible. Click Continue then click OK to output the tables.

7

Analyze Rotation Matrix and Cyclically Filter Poor Variables

Read the rotation matrix results. If any variable fails to meet the minimum loading condition (e.g., $< 0.3$) or violates discriminant validity (cross-loading difference less than $0.3$), proceed to remove only the single most severely violating variable from the list, return to Step 1 to re-run the analysis from the beginning. The cycle repeats until the model is completely clean.

Advanced Expert Tip: When dealing with EFA violations in Step 7, never rush to remove multiple variables all at once. Patiently remove only the single worst variable first, then re-run. Often the cross-correlation structure will automatically shift in a positive direction, helping retain many other previously suspected problematic variables!
5 EFA Decision Algorithm Flowchart

The complete logical process and handling of all error situations arising in EFA analysis has been visually diagrammed below.

Fail

ERROR FLOW — Model violates initial indices (KMO / Bartlett)

Pass

PASS FLOW — Model achieves preliminary convergent & discriminant validity

Check: Is $KMO < 0.5$ or $Sig. Bartlett \ge 0.05$?
True (Serious Violation) False (Preliminary Pass)
Sample Unsuitable: EFA analysis must be stopped immediately. Review questionnaire translation, remove extreme outliers, or expand the actual survey sample size.
Total Variance Explained $< 50\%$ or $Eigenvalue < 1$?
No Error Error Found
Optimal Fix: Remove variables with Communalities $< 0.5$, or set a fixed number of factors to increase extracted variance, then re-run EFA.
Abandon Model: The theoretical model has lost too much original information and has severely disintegrated. Insufficient conditions to continue deeper research.
Check Rotation Matrix: Any poor variables ($Loading < 0.3$) or cross-loading present?
No Yes
EFA Successful: The scale achieves reliability, excellent convergent and discriminant validity. Save the new factor model structure and proceed to CFA/SEM.
Check loading difference: Is the difference between the two largest cross-loadings of that observed variable $\ge 0.3$?
Yes (Pass) No (Error)
Variable Accepted: Cross-loading difference meets discriminant standard. Retain this observed variable in the model without any cleanup needed.
Remove Variable & Re-run: Immediately remove this violating variable from the analysis list, return to Step 1 to perform a new EFA model from scratch.
FILTER VARIABLES & RE-RUN NEW CYCLE

Golden Rule for Protecting the EFA Model

If the total number of removed observed variables exceeds 20% of the original count, you should consider lowering the Factor Loading threshold (from 0.50 down to 0.40 or 0.35) to protect the core theoretical structure of the model.