Is parallel analysis the same as a scree plot?

It often uses a scree display, but adds a simulated or resampled reference comparison.

Does it always select the correct count?

No; results depend on data, assumptions, and implementation choices.

Must I use the 95th percentile?

No; different criteria exist, and the chosen rule should be justified and reported.

AI Glossary / Statistics and probability / Intermediate

Parallel analysis

Definition

Parallel analysis is a method for helping choose how many components or factors to retain by comparing observed eigenvalues with corresponding eigenvalues from random reference data. The reference datasets are constructed to match relevant features of the observed data, especially sample size and number of variables, while removing the substantive association structure being investigated.

What it means

Even unrelated variables can produce apparently large sample eigenvalues. Parallel analysis asks whether the observed dimensions are stronger than the dimensions that a comparable noise model would generate.
The word parallel refers to running comparable analyses on observed and reference data, not necessarily to using multiple computer processors. The method is a decision aid for dimensionality, not proof that the retained dimensions are real psychological traits or useful business concepts.

How it works

Choose whether the goal is principal component analysis or common factor analysis. Components summarize total variance, while common factors model shared covariance. The appropriate eigenvalue calculation and reference procedure must match that goal.
Prepare the data and choose a correlation or covariance approach consistent with the measurement scale. Continuous, ordinal, and binary variables may require different methods and assumptions. Document missing-data handling and any preprocessing.
Generate many reference datasets with the same relevant dimensions. One approach uses independent simulated variables. Another independently permutes each observed column, preserving its marginal values while breaking associations across columns. Permuting every column with the same row order would not remove those associations.
For each reference dataset, calculate and order eigenvalues using the selected analysis. Compare the first observed eigenvalue with the reference distribution of first eigenvalues, the second with second eigenvalues, and so on.
A criterion may use the reference mean or a chosen percentile, such as the 95th percentile. These rules differ, so state the choice. Retain the leading dimensions that exceed the corresponding reference thresholds according to the stated procedure.
Use sufficient simulations to stabilize the threshold estimates and record the random seed. Investigate borderline decisions rather than treating a tiny numerical difference as definitive.
Then evaluate interpretability, model fit, stability, and the research question. Weak dimensions, sample size, correlation estimation, and the factor-analysis communality method can affect the suggested count. Parallel analysis does not replace substantive validation or independent assessment.

Three practical examples

Survey development: analysts investigate how many dimensions may underlie responses to a workplace questionnaire before interpreting factor solutions.
Feature compression: a team compares PCA eigenvalues with noise references to avoid retaining dimensions driven mainly by sampling variation.
AI evaluation research: researchers examine whether several performance measures reflect distinct dimensions, while checking that the sample and variable types suit the analysis.

Think of it this way

Think of looking for ridges in a landscape after first seeing what roughness random terrain produces. A ridge must stand above the appropriate noise background before it is considered a candidate feature. Its statistical prominence still does not tell you what it means.

Worked example

Consider an explicitly hypothetical PCA example with five standardized variables. Suppose the observed correlation-matrix eigenvalues, in descending order, are 2.40, 1.30, 0.70, 0.40, and 0.20. They sum to 5, as required for a five-variable correlation matrix.
Suppose a properly matched reference simulation supplies rank-specific 95th-percentile thresholds of 1.35, 1.15, 1.02, 0.90, and 0.75. These thresholds are stipulated for illustration; they are not claimed to come from a simulation performed for this entry.
The first observed eigenvalue exceeds 1.35, and the second exceeds 1.15. The third, fourth, and fifth fall below their respective thresholds. The stated rule therefore suggests retaining two components.
Those two observed components account for (2.40 + 1.30)/5 = 0.74, or 74% of the standardized total variance. That descriptive percentage is separate from the noise comparison.
The reference percentiles need not sum to 5, because each is calculated separately across its ordered-eigenvalue distribution. The result also does not imply that exactly two common factors exist; the example concerns components under the specified reference procedure.

Illustrative case study

Illustrative fictional case: Larch Research develops a questionnaire about experiences with AI assistants. A first analysis suggests several dimensions, but the team notices that most items use ordered response categories.
It compares a suitable ordinal-data analysis with its initial approach, records reference-generation settings, and examines whether retained dimensions remain interpretable across samples. It keeps weakly supported dimensions provisional rather than naming them as established traits. This fictional case illustrates method selection and validation, not a completed scale-development result.

Common misconceptions and mistakes

Do not compare every observed eigenvalue with the reference's largest eigenvalue. Comparisons are rank-specific. Do not confuse the method with the rule that retains all eigenvalues above one; finite-sample noise thresholds can differ substantially.
Avoid presenting a chosen 95th-percentile threshold as a 95% probability that a factor is real. Do not mix PCA thresholds with a different factor-analysis eigenvalue definition. Correlated rows, inappropriate reference distributions, or sparse categorical data can undermine the analysis.

Business use

Parallel analysis can improve discipline when businesses build survey scales, summarize many indicators, or reduce feature dimensions. A useful report states the analysis type, sample size, variables, correlation method, simulation procedure, threshold criterion, and sensitivity of the retained count. Naming and using dimensions requires additional evidence beyond exceeding a reference line.

Check your understanding

Why does the example retain two components?

Answer: Only the first two eigenvalues exceed their matching reference thresholds.

Why is 74% not the parallel-analysis criterion?

Answer: It summarizes observed variance explained, while retention is based on comparisons with reference eigenvalues.

Related terms

Principal component analysis
Eigenvalue
Dimensionality
Monte Carlo simulation
exploratory factor analysis
scree plot
communality
polychoric correlation

Learn next

Next distinguish principal components from common factors. Then examine how reference generation, ordinal correlations, simulation count, and independent validation affect dimensionality decisions.

Dimensionality

Sources

William Revelle, psych package documentation, fa.parallel: https://personality-project.org/r/psych/help/fa.parallel.html — primary developer explanation of reference analyses, component/factor distinctions, percentile choices, and sensitivity. The accessible page identifies an older package version; no claim is made about current defaults. It cites Horn's 1965 original paper. The eigenvalues and thresholds are explicitly illustrative inputs, with comparison arithmetic independently checked.

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