Factor Analysis: Exploratory and Confirmatory

If your thesis uses a questionnaire with several multi-item scales, factor analysis is almost certainly in your future. It is how researchers show that 20 or 30 survey items really group into the handful of constructs the theory describes. There are two families: exploratory factor analysis (EFA), which discovers structure, and confirmatory factor analysis (CFA), which tests a structure you specified in advance.

The Big Idea

Factor analysis assumes that observed item responses are driven by a smaller number of unobserved, or latent, factors. If items about "job satisfaction" all correlate strongly with each other but weakly with items about "turnover intention", the pattern suggests two underlying factors. Each item's relationship with a factor is its loading, which behaves much like a correlation between the item and the factor.

The method grew from Charles Spearman's work on intelligence in 1904 and was developed further by Louis Thurstone in the 1930s, who introduced multiple factors and rotation. CFA was formalised by Karl Jöreskog in the late 1960s and 1970s.

Exploratory Factor Analysis (EFA)

Use EFA when you have developed new items, adapted a scale to a new culture or language (for example, translating an English scale into Urdu), or have no firm theory about how items group.

Step 1: Check Factorability

• Sample size: at least 5 to 10 respondents per item, and preferably 200 or more in total.
• Kaiser–Meyer–Olkin (KMO): measures sampling adequacy. Values above .60 are acceptable and above .80 are good.
• Bartlett's test of sphericity: should be significant (p < .05), showing that the correlation matrix is not an identity matrix.

Step 2: Extract Factors

Common methods are principal axis factoring and maximum likelihood. Principal component analysis (PCA) is often used too, though strictly it is a data-reduction method rather than true factor analysis.

Step 3: Decide How Many Factors

• Kaiser criterion: keep factors with eigenvalues greater than 1.
• Scree plot: keep factors above the "elbow".
• Parallel analysis: keep factors whose eigenvalues exceed those from random data; this is the most accurate of the three.

Step 4: Rotate

Rotation makes loadings easier to interpret. Varimax (orthogonal) assumes factors are uncorrelated; promax or oblimin (oblique) allow correlations, which is usually more realistic in social science.

Step 5: Interpret and Clean

Retain items with loadings of about .40 or higher on their main factor, and consider removing items that cross-load (load above .30 on two factors).

Worked Example 1: Eigenvalues and Loadings From a Correlation Matrix

Three items about online-learning satisfaction from a survey have this correlation matrix:

$$ R = \begin{pmatrix} 1 & 0.6 & 0.5 \\ 0.6 & 1 & 0.4 \\ 0.5 & 0.4 & 1 \end{pmatrix} $$

Step 1: Total variance. In a correlation matrix each standardised item contributes a variance of 1, so total variance is 3. The eigenvalues must also add up to 3.

Step 2: Find the eigenvalues. Solving \( \det(R - \lambda I) = 0 \) gives a cubic equation. Its roots, which software finds instantly, are

$$ \lambda_1 = 2.004, \quad \lambda_2 = 0.613, \quad \lambda_3 = 0.382 $$

Check: \( 2.004 + 0.613 + 0.382 = 2.999 \approx 3 \).

Step 3: Number of factors. Only \( \lambda_1 \) exceeds 1, so the Kaiser criterion retains one factor.

Step 4: Variance explained. The first factor explains

$$ \frac{2.004}{3} \times 100 = 66.8\% $$

of the total variance.

Step 5: Loadings. The eigenvector for \( \lambda_1 \) is approximately (0.613, 0.580, 0.536). In principal components, a loading equals the eigenvector element multiplied by \( \sqrt{\lambda_1} = \sqrt{2.004} = 1.416 \):

• Item 1: \( 0.613 \times 1.416 = 0.87 \)
• Item 2: \( 0.580 \times 1.416 = 0.82 \)
• Item 3: \( 0.536 \times 1.416 = 0.76 \)

All three loadings are well above .40, so the items form a single coherent factor. As a check, the squared loadings sum to the eigenvalue: \( 0.754 + 0.674 + 0.576 = 2.004 \).

APA report: A principal component analysis of the three satisfaction items yielded a single component with an eigenvalue of 2.00, explaining 66.8% of the variance. Loadings ranged from .76 to .87.

Confirmatory Factor Analysis (CFA)

Use CFA when you are using an established scale or when your EFA, or theory, has already suggested a structure. You specify in advance which items load on which factor, and the software tests how well that model reproduces the observed covariances. CFA is also the measurement-model stage of structural equation modelling (SEM), and is common in PLS-SEM or covariance-based SEM theses.

Model Fit Indices

Typical cut-offs, based largely on Hu and Bentler (1999), are:

• χ²/df: below 3 (the raw chi-square is nearly always significant with large samples).
• CFI and TLI: .90 or above is acceptable; .95 or above is good.
• RMSEA: .08 or below is acceptable; .06 or below is good.
• SRMR: .08 or below.

Convergent and Discriminant Validity

After acceptable fit, you assess each construct using its standardised loadings \( \lambda_i \):

$$ CR = \frac{\left(\sum \lambda_i\right)^2}{\left(\sum \lambda_i\right)^2 + \sum (1 - \lambda_i^2)}, \qquad AVE = \frac{\sum \lambda_i^2}{k} $$

Composite reliability (CR) should be at least .70 and average variance extracted (AVE) at least .50.

Worked Example 2: Composite Reliability and AVE

A CFA of a four-item "organisational commitment" construct gives standardised loadings of 0.78, 0.72, 0.81 and 0.69.

Step 1: Sum the loadings. \( 0.78 + 0.72 + 0.81 + 0.69 = 3.00 \), so \( (\sum \lambda_i)^2 = 9.00 \).

Step 2: Squared loadings. 0.6084, 0.5184, 0.6561 and 0.4761, which sum to 2.259.

Step 3: Error variances. \( 1 - \lambda_i^2 \) gives 0.3916, 0.4816, 0.3439 and 0.5239, which sum to 1.741.

Step 4: Composite reliability.

$$ CR = \frac{9.00}{9.00 + 1.741} = \frac{9.00}{10.741} = 0.838 $$

Step 5: AVE.

$$ AVE = \frac{2.259}{4} = 0.565 $$

Both thresholds are met: CR = .84 > .70 and AVE = .56 > .50, so convergent validity is supported.

Step 6: Discriminant validity (Fornell–Larcker). \( \sqrt{AVE} = \sqrt{0.565} = 0.75 \). If organisational commitment correlates, say, .52 with job satisfaction, then since .75 > .52 discriminant validity holds. Many reviewers now also expect the HTMT ratio, which should be below .85 or .90.

APA report: The CFA showed acceptable fit, and the organisational commitment construct demonstrated good composite reliability (CR = .84) and convergent validity (AVE = .56).

Software

• EFA: SPSS (Analyze → Dimension Reduction → Factor), R's psych::fa(), Stata's factor, or Python's factor_analyzer.
• CFA: AMOS, R's lavaan, Stata's sem, Mplus, or SmartPLS for PLS-based models.

r library(lavaan) model <- 'commit =~ c1 + c2 + c3 + c4' fit <- cfa(model, data = df) summary(fit, fit.measures = TRUE, standardized = TRUE)

Common Mistakes

• Running EFA and CFA on the same sample to "confirm" what you just explored; split the sample or collect new data.
• Using Varimax by default when constructs are clearly correlated.
• Deleting many items purely to improve fit without theoretical justification.

Did You Know?

• Spearman's single "g" factor of intelligence was the first factor model, published in 1904.
• The word "eigen" is German for "own" or "characteristic", hence eigenvalue.
• Thurstone argued for several "primary mental abilities" rather than one general factor, and factor analysis became the battleground for that debate.

Key Takeaways

• EFA discovers how items group; CFA tests a structure specified in advance.
• Check KMO, Bartlett's test and sample size before EFA; use eigenvalues, scree plots and parallel analysis to choose factors.
• Retain items with loadings around .40 or higher and avoid cross-loadings.
• In CFA, report CFI, TLI, RMSEA and SRMR, plus CR ≥ .70 and AVE ≥ .50 for each construct.

If you need EFA or CFA run and interpreted for your thesis, Amathify's experts and PhD scholars can deliver step-by-step analysis and write-ups; message us on WhatsApp or use the Ask a Question form.