PCA, EFA, and CFA

Different tools for different ‘measurement’ questions

Tommaso Feraco

Stop treating PCA, EFA, and CFA as interchangeable.

Why students often confuse them

You can take the same questionnaire and run:

• a PCA,
• an EFA,
• or a CFA.

But you are not asking the same question.

Important

The same items do not imply the same model, the same assumptions, or the same interpretation.

Learning objectives

By the end of this extra, you should be able to:

• distinguish the main goal of PCA, EFA, and CFA;
• recognize what each approach is trying to estimate;
• choose the method that matches a research aim;
• describe the main pros, cons, and common misuses of each approach.

One running example

Imagine you created a 9-item questionnaire on academic self-regulation.

You suspect three facets:

• planning,
• persistence,
• impulse control.

Now ask three different questions:

1. Can I summarize the 9 items with a few composite dimensions?
2. What latent structure seems plausible in these data?
3. Does my hypothesized 3-factor measurement model fit the data?

These questions lead to different methods.

The big idea in one slide

Approach	Main aim	Typical output
PCA	Summarize many observed variables efficiently	Components
EFA	Explore which latent dimensions may underlie the data	Exploratory factors
CFA	Test a theory-driven measurement model	Confirmatory factors + fit

Tip

A useful shorthand: PCA reduces variables; EFA explores structure; CFA tests structure.

Observed summaries vs latent variables

PCA

• works on the variance of the observed variables;
• creates weighted summaries of the variables;
• does not require a reflective latent-variable interpretation.

EFA / CFA

• aim to model common variance among items;
• treat factors as latent dimensions behind the responses;
• separate shared variance from item-specific or residual variance.

Important

If your substantive claim is “I measured a latent construct”, PCA is usually not enough.

PCA: what is it for?

Principal Component Analysis searches for linear combinations of variables that capture as much variance as possible.

The first component explains the largest amount of variance, the second explains the next largest amount, and so on.

Typical goal:

• data reduction,
• simpler composite summaries,
• fewer variables for later analyses.

PCA: how it works, intuitively

• Start from a set of correlated observed variables.
• Build a new variable that captures as much total variance as possible.
• Build a second component that captures additional variance.
• Continue until the remaining components add little value.

Note

A component is a weighted sum of observed variables. It is not automatically a psychological construct.

PCA: strengths and limitations

Strengths

• simple and fast;
• useful for summarizing many variables;
• helpful for descriptive work and preprocessing;
• often easy to communicate.

Limitations

• does not separate common from unique variance;
• components are not necessarily latent traits;
• easy to over-interpret as “factors”;
• less appropriate for theory-based measurement claims.

When PCA is a reasonable choice

Use PCA when your main goal is:

• to compress many correlated measures,
• to create a smaller set of observed summaries,
• to explore patterns descriptively,
• to reduce dimensionality before a later model.

Do not choose PCA just because it is convenient if your real aim is construct measurement.

EFA: what is it for?

Exploratory Factor Analysis asks:

What latent structure seems plausible in these data?

In EFA:

• the factor structure is not fully fixed in advance;
• items can, in principle, relate to multiple factors;
• you are looking for a plausible pattern, not testing a strict theory yet.

EFA: how it works, intuitively

• Estimate a small set of latent factors from the correlations among items.
• Decide how many factors to retain.
• Rotate the solution to improve interpretability.
• Look for a pattern close to simple structure.

A good EFA solution is usually:

• interpretable,
• theoretically sensible,
• and not dominated by messy cross-loadings.

A note on rotation

Rotation does not change the overall information in the solution. It changes the orientation of the factors to make the pattern easier to interpret.

Orthogonal rotation

• factors constrained to be uncorrelated;
• often unrealistic in psychology.

Oblique rotation

• factors allowed to correlate;
• often more realistic for psychological constructs.

EFA: strengths and limitations

Strengths

• useful early in scale development;
• can reveal unexpected structure;
• allows cross-loadings during exploration;
• often a good starting point when theory is still weak.

Limitations

• many researcher decisions affect the result;
• solutions can vary across samples;
• exploratory results are not final proof;
• easy to mistake a neat pattern for confirmation.

Good use of EFA

EFA is most useful when you want to:

• generate a candidate measurement structure,
• refine items,
• identify problematic indicators,
• prepare for a later confirmatory stage.

Tip

A strong workflow is often: theory + item writing → EFA → revise → CFA → validity / invariance / SEM.

CFA: what is it for?

Confirmatory Factor Analysis asks:

Does a theoretically specified measurement model reproduce the observed covariance structure reasonably well?

In CFA:

• you decide in advance which items load on which factor;
• cross-loadings are usually fixed to zero unless theory says otherwise;
• fit is evaluated explicitly.

CFA: how it works, intuitively

A CFA model specifies:

• which items belong to each latent factor,
• the size of the factor loadings,
• residual variances,
• factor covariances,
• and, when justified, selected residual correlations or other constraints.

This makes CFA a measurement model, not just a descriptive summary.

A measurement-first reminder

Important

In CFA, you are not just saying that items “go together”. You are making a stronger claim: the responses are treated as indicators of a latent construct.

That is why CFA belongs naturally in an SEM workflow:

• it supports a measurement-first / two-step mindset;
• it makes assumptions explicit;
• it lets you evaluate fit before moving to structural relations.

CFA: what do we inspect?

Global fit

• \(\chi^2\)
• CFI / TLI
• RMSEA (+ CI)
• SRMR

These tell you whether the whole model reproduces the covariance pattern reasonably well.

Local diagnostics

• standardized residuals,
• modification indices,
• size and plausibility of loadings,
• residual variances and factor correlations.

These tell you where strain or misspecification may be located.

CFA: strengths and limitations

Strengths

• theory-driven;
• explicit and testable;
• handles measurement error;
• integrates naturally with reliability, validity, and invariance work.

Limitations

• requires stronger assumptions;
• can be too rigid if the theory is poor (is this actually a limit?);
• fit does not prove the model is “true”;
• easy to misuse as post-hoc model cleaning.

One important caution about fit

Warning

Good fit is not the same as truth.

A CFA model may fit well because:

• it is substantively reasonable,
• it is flexible enough,
• or it was tuned too closely to one sample.

So in this course we keep the same rule as elsewhere:

• use global fit,
• inspect local diagnostics,
• revise only when changes are theory-justified,
• report decisions transparently.

Head-to-head comparison

Question	PCA	EFA	CFA
Main goal	Reduce observed variables	Explore latent structure	Test latent structure
Theory needed beforehand	Low	Moderate	High
Latent-variable claim	Usually no	Usually yes	Yes
Cross-loadings	Not relevant in the same way	Allowed	Usually fixed unless justified
Measurement error modeled explicitly	No	Partly / yes in factor logic	Yes
Model fit in SEM sense	No	Limited / different traditions	Yes
Best use	Summaries and reduction	Structure discovery	Measurement testing
Common misuse	Calling components “constructs”	Treating exploration as confirmation	Chasing fit without theory

A simple decision tree

Choose the tool that matches your question.

• I want fewer observed summaries → PCA
• I think latent dimensions exist, but I am not yet sure how items organize → EFA
• I have a measurement theory and want to test it explicitly → CFA

Tip

In psychology, the best answer is often not “which one is best?” but “which one fits my current research stage?”

Common mistakes in applied psychology

• Calling PCA a “factor analysis” and then claiming construct validity.
• Treating an EFA solution as if it were already confirmatory evidence.
• Using CFA only because it looks more advanced.
• Ignoring theory and choosing the model that gives cleaner output.
• Thinking that good CFA fit proves the construct is real.
• Forgetting that scale development and model testing are iterative.

Minimal R orientation

# same item set, different tools
items <- dat[, paste0("sr", 1:9)]

# PCA: observed summaries
pca_fit <- psych::principal(items, nfactors = 3, rotate = "none")

# EFA: exploratory latent structure
efa_fit <- psych::fa(items, nfactors = 3, fm = "ml", rotate = "oblimin")

# CFA: confirmatory measurement model
mod_cfa <- '
Planning =~ sr1 + sr2 + sr3
Persistence =~ sr4 + sr5 + sr6
ImpulseCtrl =~ sr7 + sr8 + sr9
'

cfa_fit <- lavaan::cfa(mod_cfa, data = dat)

What you typically inspect in each output

PCA

• explained variance,
• scree pattern,
• component loadings,
• component scores.

EFA

• number of factors,
• loading pattern,
• communalities,
• cross-loadings,
• rotated solution.

CFA

• standardized loadings,
• factor correlations,
• global fit,
• local diagnostics,
• reliability / validity evidence.

Exercises

1. For each research aim below, decide whether PCA, EFA, or CFA is the best starting point.
2. Justify your answer in one sentence.

Research aims:

• “I need a few summary scores from 24 highly redundant items.”
• “I developed a new scale and do not yet know how many dimensions are present.”
• “I hypothesized three correlated factors based on theory and want to test them.”
• “I want to move to SEM, but first I need to check whether my indicators measure the intended latent variables.”

For hands-on follow-up, see the CFA lab: lab04_cfa_reliability_omegas.qmd

Three things to remember

1. PCA, EFA, and CFA answer different questions.
2. PCA is mainly about reducing observed variables, not testing a latent measurement model.
3. CFA is strongest when theory is clear and fit is interpreted with discipline, not as proof of truth.

References / suggested readings

Final message

Do not ask:

Which method is best?

Ask instead:

Which method answers my question at this stage of the research process?